https://eeecon.uibk.ac.at/~zeileis/
Achim Zeileis
2018-10-08T03:44:05+02:00
Research homepage of Achim Zeileis, Universität Innsbruck. <br/>Department of Statistics, Faculty of Economics and Statistics. <br/>Universitätsstr. 15, 6020 Innsbruck, Austria<br/>Tel: +43/512/507-70403
Achim Zeileis
Achim.Zeileis@R-project.org
https://eeecon.uibk.ac.at/~zeileis/
Jekyll
https://eeecon.uibk.ac.at/~zeileis/news/palmtree/
Partially additive (generalized) linear model trees
2018-10-08T00:00:00+02:00
Achim Zeileis
Achim.Zeileis@R-project.org
https://eeecon.uibk.ac.at/~zeileis/
The PALM tree algorithm for partially additive (generalized) linear model trees is introduced along with the R package palmtree. One potential application is modeling of treatment-subgroup interactions while adjusting for global additive effects.
<p>The PALM tree algorithm for partially additive (generalized) linear model trees is introduced along with the R package palmtree. One potential application is modeling of treatment-subgroup interactions while adjusting for global additive effects.</p> <h2 id="citation">Citation</h2> <p>Heidi Seibold, Torsten Hothorn, Achim Zeileis (2018). “Generalised Linear Model Trees with Global Additive Effects.” <em>Advances in Data Analysis and Classification</em>. Forthcoming. <a href="https://doi.org/10.1007/s11634-018-0342-1">doi:10.1007/s11634-018-0342-1</a> <a href="http://arxiv.org/abs/1612.07498">arXiv</a></p> <h2 id="abstract">Abstract</h2> <p>Model-based trees are used to find subgroups in data which differ with respect to model parameters. In some applications it is natural to keep some parameters fixed globally for all observations while asking if and how other parameters vary across subgroups. Existing implementations of model-based trees can only deal with the scenario where all parameters depend on the subgroups. We propose partially additive linear model trees (PALM trees) as an extension of (generalised) linear model trees (LM and GLM trees, respectively), in which the model parameters are specified a priori to be estimated either globally from all observations or locally from the observations within the subgroups determined by the tree. Simulations show that the method has high power for detecting subgroups in the presence of global effects and reliably recovers the true parameters. Furthermore, treatment-subgroup differences are detected in an empirical application of the method to data from a mathematics exam: the PALM tree is able to detect a small subgroup of students that had a disadvantage in an exam with two versions while adjusting for overall ability effects.</p> <h2 id="software">Software</h2> <p><a href="https://CRAN.R-project.org/package=palmtree">https://CRAN.R-project.org/package=palmtree</a></p> <h2 id="illustration-treatment-differences-in-mathematics-exam">Illustration: Treatment differences in mathematics exam</h2> <p>PALM trees are employed to investigate treatment differences in a mathematics 101 exam (for first-year business and economics students) at Universität Innsbruck. Due to limited availability of seats in the exam room, students could self-select into one of two exam tracks that were conducted back to back with slightly different questions on the same topics. The question is whether this “treatment” of splitting the students into two tracks was fair in the sense that it is on average equally difficult for the two groups. To investigate the question the data are loaded from the <a href="https://CRAN.R-project.org/package=psychotools">psychotools</a> package, points are scaled to achieved percent in [0, 100], and the subset of variables for the analysis are selected:</p> <pre><code class="language-{r}">data("MathExam14W", package = "psychotools") MathExam14W$tests <- 100 * MathExam14W$tests/26 MathExam14W$pcorrect <- 100 * MathExam14W$nsolved/13 MathExam <- MathExam14W[ , c("pcorrect", "group", "tests", "study", "attempt", "semester", "gender")] </code></pre> <p>A naive check could be whether the percentage of correct points (<code class="highlighter-rouge">pcorrect</code>) differs between the two <code class="highlighter-rouge">group</code>s:</p> <pre><code class="language-{r}">ci <- function(object) cbind("Coefficient" = coef(object), confint(object)) ci(lm(pcorrect ~ group, data = MathExam)) ## Coefficient 2.5 % 97.5 % ## (Intercept) 57.60 55.1 60.08 ## group2 -2.33 -5.7 1.03 </code></pre> <p>This shows that the second group achieved on average 2.33 percentage points less than the first group. But the corresponding confidence interval conveys that this difference is not significant.</p> <p>However, it is conceivable that stronger (or weaker) students selected themselves more into one of the two groups. And if the assignment had been random, then the “treatment effect” might have been larger or even smaller. Luckily, an independent measure of the students’ ability is available, namely the percentage of points achieved in the online <code class="highlighter-rouge">tests</code> conducted during the semester prior to the exam. Adjusting for that increases the treatment effect to a decrease of 4.37 percentage points which is still non-significant, though. This is due to weaker students self-selecting into the second group. Moreover, the <code class="highlighter-rouge">tests</code> coefficient signals that 1 more percentage point from the online tests lead on average to 0.855 more percentage points in the written exam.</p> <pre><code class="language-{r}">ci(lm(pcorrect ~ group + tests, data = MathExam)) ## Coefficient 2.5 % 97.5 % ## (Intercept) -5.846 -13.521 1.828 ## group2 -4.366 -7.231 -1.502 ## tests 0.855 0.756 0.955 </code></pre> <p>Finally, PALM trees are used to assess whether there are subgroups of differential <code class="highlighter-rouge">group</code> treatment effects when adjusting for a global additive <code class="highlighter-rouge">tests</code> effect. Potential subgroups can be formed from the covariates <code class="highlighter-rouge">tests</code>, type of <code class="highlighter-rouge">study</code> (three-year bachelor vs. four-year diploma), the number of times the students <code class="highlighter-rouge">attempt</code>ed the exam, number of <code class="highlighter-rouge">semester</code>s, and <code class="highlighter-rouge">gender</code>. Using <a href="https://CRAN.R-project.org/package=palmtree">palmtree</a> this can be easily carried out:</p> <pre><code class="language-{r}">library("palmtree") palmtree_math <- palmtree(pcorrect ~ group | tests | tests + study + attempt + semester + gender, data = MathExam) print(palmtree_math) ## Partially additive linear model tree ## ## Model formula: ## pcorrect ~ group | tests + study + attempt + semester + gender ## ## Fitted party: ## [1] root ## | [2] attempt <= 1 ## | | [3] tests <= 92.3: n = 352 ## | | (Intercept) group2 ## | | -7.09 -3.00 ## | | [4] tests > 92.3: n = 79 ## | | (Intercept) group2 ## | | 14.0 -14.5 ## | [5] attempt > 1: n = 298 ## | (Intercept) group2 ## | 2.33 -1.70 ## ## Number of inner nodes: 2 ## Number of terminal nodes: 3 ## Number of parameters per node: 2 ## Objective function (residual sum of squares): 253218 ## ## Linear fixed effects (from palm model): ## tests ## 0.787 </code></pre> <p>A somewhat enhanced version of <code class="highlighter-rouge">plot(palmtree_math)</code> is shown below:</p> <p><a href="https://eeecon.uibk.ac.at/~zeileis/assets/posts/2018-10-08-palmtree/palmtree-math.png"><img src="https://eeecon.uibk.ac.at/~zeileis/assets/posts/2018-10-08-palmtree/palmtree-math.png" alt="PALM tree for mathematics exam data" /></a></p> <p>This indicates that for most students the <code class="highlighter-rouge">group</code> treatment effect is indeed negligible. However, for the subgroup of “good” students (with high percentage correct in the online tests) in the first attempt, the exam in the second group was indeed more difficult. On average the students in the second group obtained -14.5 percentage points less than in the first group.</p> <pre><code class="language-{r}">ci(palmtree_math$palm) ## Coefficient 2.5 % 97.5 % ## (Intercept) -7.088 -16.148 1.971 ## .tree4 21.069 13.348 28.791 ## .tree5 9.421 5.168 13.673 ## tests 0.787 0.671 0.903 ## .tree3:group2 -2.997 -6.971 0.976 ## .tree4:group2 -14.494 -22.921 -6.068 ## .tree5:group2 -1.704 -5.965 2.557 </code></pre> <p>The absolute size of this group difference is still moderate, though, corresponding to about half an exercise out of 13.</p> <h2 id="simulation-study">Simulation study</h2> <p>In addition to the empirical case study the manuscript also provides an extensive simulation study comparing the performance of PALM trees in treatment-subgroup scenarios to standard linear model (LM) trees, optimal treatment regime (OTR) trees (following Zhang et al. 2012), and the STIMA algorithm (simultaneous threshold interaction modeling algorithm). The study evaluates the methods with respect to (1) finding the correct subgroups, (2) not splitting when there are no subgroups, (3) finding the optimal treatment regime, and (4) correctly estimating the treatment effect.</p> <p>Here we just briefly highlight the results for question (1): Are the correct subgroups found? The figure below shows the mean number of subgroups (over 150 simulated data sets and mean adjusted rand index (ARI) for increasing treatment effect differences Δ<sub>β</sub> and number of observations n.</p> <p><a href="https://eeecon.uibk.ac.at/~zeileis/assets/posts/2018-10-08-palmtree/palmtree-sim.png"><img src="https://eeecon.uibk.ac.at/~zeileis/assets/posts/2018-10-08-palmtree/palmtree-sim.png" alt="Simulation results" /></a></p> <p>This shows that PALM trees perform increasingly well and somewhat better with respect to these metrics than the competitors. More details on the different scenarios and corresponding evaluations can be found in the manuscript. More replication materials are provided along with the manuscript on the publisher’s web page.</p>
2018-10-08T00:00:00+02:00
https://eeecon.uibk.ac.at/~zeileis/news/thunderstorm_forecasting/
Thunderstorm forecasting with GAMs
2018-09-16T00:00:00+02:00
Achim Zeileis
Achim.Zeileis@R-project.org
https://eeecon.uibk.ac.at/~zeileis/
Boosted binary generalized additive models (GAMs) with stability selection and corresponding MCMC-based credibility intervals are discussed in a new MWR paper as a probabilistic forecasting method for the occurrence of thunderstorms.
<p>Boosted binary generalized additive models (GAMs) with stability selection and corresponding MCMC-based credibility intervals are discussed in a new MWR paper as a probabilistic forecasting method for the occurrence of thunderstorms.</p> <h2 id="citation">Citation</h2> <p>Thorsten Simon, Peter Fabsic, Georg J. Mayr, Nikolaus Umlauf, Achim Zeileis (2018). “Probabilistic Forecasting of Thunderstorms in the Eastern Alps.” <em>Monthly Weather Review</em>. <strong>146</strong>(9), 2999-3009. <a href="https://dx.doi.org/10.1175/MWR-D-17-0366.1">doi:10.1175/MWR-D-17-0366.1</a></p> <h2 id="abstract">Abstract</h2> <p>A probabilistic forecasting method to predict thunderstorms in the European eastern Alps is developed. A statistical model links lightning occurrence from the ground-based Austrian Lightning Detection and Information System (ALDIS) detection network to a large set of direct and derived variables from a numerical weather prediction (NWP) system. The NWP system is the high-resolution run (HRES) of the European Centre for Medium-Range Weather Forecasts (ECMWF) with a grid spacing of 16 km. The statistical model is a generalized additive model (GAM) framework, which is estimated by Markov chain Monte Carlo (MCMC) simulation. Gradient boosting with stability selection serves as a tool for selecting a stable set of potentially nonlinear terms. Three grids from 64 x 64 to 16 x 16 km<sup>2</sup> and five forecast horizons from 5 days to 1 day ahead are investigated to predict thunderstorms during afternoons (1200–1800 UTC). Frequently selected covariates for the nonlinear terms are variants of convective precipitation, convective potential available energy, relative humidity, and temperature in the midlayers of the troposphere, among others. All models, even for a lead time of 5 days, outperform a forecast based on climatology in an out-of-sample comparison. An example case illustrates that coarse spatial patterns are already successfully forecast 5 days ahead.</p> <h2 id="software">Software</h2> <p><a href="https://CRAN.R-project.org/package=bamlss">https://CRAN.R-project.org/package=bamlss</a></p> <h2 id="case-study">Case study</h2> <p>Predicting thunderstorms in complex terrain (like the Austrian Alps) is a challenging task since one of the main forecasting tools, NWP systems, cannot fully resolve convective processes or circulations and exchange processes over complex topography. However, using a boosted binary GAM based on a broad range of NWP outputs useful forecasts can be obtained up to 5 days ahead. As an illustration, lightning activity for the afternoon of 2015-07-22 is shown in the top-left panel below, indicating thunderstorms in many areas in the west but not the east. While the corresponding baseline climatology (top middle) has a low probability of thunderstorms for the entire region, the NWP-based probabilistic forecasts (bottom row) highlight increased probabilities already 5 days ahead, becoming much more clear cut when moving to 3 days and 1 day ahead.</p> <p><a href="https://eeecon.uibk.ac.at/~zeileis/assets/posts/2018-09-16-thunderstorm_forecasting/map.png"><img src="https://eeecon.uibk.ac.at/~zeileis/assets/posts/2018-09-16-thunderstorm_forecasting/map.png" alt="observed and forecasted occurrence of thunderstorms on 2015-07-22" /></a></p> <p>More precisely, the probability of thunderstorms is predicted based on a binary logit GAM that allows for potentially nonlinear smooth effects in all NWP variables considered. It selects the relevant variables by gradient boosting coupled with stability selection. Effects and 95% credible intervals of the model for day 1 are estimated via MCMC sampling and shown below (on the logit scale). The number in the bottom-right corner of each panel indicates the absolute range of the effect. The x-axes are cropped at the 1% and 99% quantiles of the respective covariate to enhance graphical representation.</p> <p><a href="https://eeecon.uibk.ac.at/~zeileis/assets/posts/2018-09-16-thunderstorm_forecasting/effects.png"><img src="https://eeecon.uibk.ac.at/~zeileis/assets/posts/2018-09-16-thunderstorm_forecasting/effects.png" alt="stability-selected effects of the boosted binary logit GAM" /></a></p> <p><em>(Note: As the data cannot be shared freely, the customary replication materials unfortunately cannot be provided.)</em></p>
2018-09-16T00:00:00+02:00
https://eeecon.uibk.ac.at/~zeileis/news/mpttree/
MPT trees published in BRM
2018-08-27T00:00:00+02:00
Achim Zeileis
Achim.Zeileis@R-project.org
https://eeecon.uibk.ac.at/~zeileis/
Multinomial processing trees are recursively partitioned to capture heterogeneity in latent cognitive processing steps. Accompanied by the R function mpttree in the psychotree package, combining partykit::mob and psychotools::mpt.
<p>Multinomial processing trees are recursively partitioned to capture heterogeneity in latent cognitive processing steps. Accompanied by the R function mpttree in the psychotree package, combining partykit::mob and psychotools::mpt.</p> <h2 id="citation">Citation</h2> <p>Florian Wickelmaier, Achim Zeileis (2018). “Using Recursive Partitioning to Account for Parameter Heterogeneity in Multinomial Processing Tree Models.” <em>Behavior Research Methods</em>, <strong>50</strong>(3), 1217-1233. <a href="https://doi.org/10.3758/s13428-017-0937-z">doi:10.3758/s13428-017-0937-z</a></p> <h2 id="abstract">Abstract</h2> <p>In multinomial processing tree (MPT) models, individual differences between the participants in a study can lead to heterogeneity of the model parameters. While subject covariates may explain these differences, it is often unknown in advance how the parameters depend on the available covariates, that is, which variables play a role at all, interact, or have a nonlinear influence, etc. Therefore, a new approach for capturing parameter heterogeneity in MPT models is proposed based on the machine learning method MOB for model-based recursive partitioning. This procedure recursively partitions the covariate space, leading to an MPT tree with subgroups that are directly interpretable in terms of effects and interactions of the covariates. The pros and cons of MPT trees as a means of analyzing the effects of covariates in MPT model parameters are discussed based on simulation experiments as well as on two empirical applications from memory research. Software that implements MPT trees is provided via the <code class="highlighter-rouge">mpttree</code> function in the <em>psychotree</em> package in R.</p> <h2 id="software">Software</h2> <p><a href="https://CRAN.R-project.org/package=psychotree">https://CRAN.R-project.org/package=psychotree</a></p> <h2 id="illustration-source-monitoring">Illustration: Source monitoring</h2> <p>To highlight how MPT trees can capture the influence of covariates on the parameters in MPT models, data from a source monitoring experiment are analyzed, that was conducted at the Department of Psychology, University of Tübingen.</p> <p><strong>Study:</strong> Participants were presented with items from two different sources (labeled A vs. B) and afterwards, in a memory test, were shown old and new items intermixed and asked to classify them as either A, B, or new (N). In the experiment the two sources were controlled such that half of the respondents had to read the presented items either quietly (A = think) or aloud (B = say). The other half wrote them down (A = write) or read them aloud (B = say). Items were presented on a computer screen at a self-paced rate. In the final memory test, the studied items and distractor items had to be classified as either A, B, or new (N) by pressing a button on the screen.</p> <p><strong>Model:</strong> To infer the cognitive processes a well-known MPT model is employed that was established by the late Bill Batchelder (who passed away earlier this month) and David Riefer for the source monitoring paradigm:</p> <p><a href="https://eeecon.uibk.ac.at/~zeileis/assets/posts/2018-08-27-mpttree/sourcemonitoring-mpt.png"><img src="https://eeecon.uibk.ac.at/~zeileis/assets/posts/2018-08-27-mpttree/sourcemonitoring-mpt.png" alt="MPT graph" /></a></p> <p><strong>Explanation:</strong> Consider the paths from the root to an A response for a Source A item (left). With probability D1, a respondent detects an item as old. If, in a second step, he/she is able to discriminate the item from a Source B item (d1), then the response will correctly be A; else, if discrimination fails (1 - d1), a correct A response can only be guessed with probability a. If the item was not detected as old in the first place (1 - D1), the response will be A only if there are both a response bias for “old” (b) and a guess for the item being Source A (g). The remaining paths in the left tree lead to classification errors (B, N). The trees for Source B and new items work analogously. Moreover, a = g is assumed for identifiability and discriminability is assumed to be equal for both sources (d1 = d2) as in a similar example in Batchelder and Riefer (1990).</p> <p><strong>Question:</strong> Do these probabilities in the source monitoring (D1, D2, d, b, g) depend on the source condition (think-say vs. write-say), or gender or age of the participants?</p> <p><strong>Answer:</strong> The MPT-based model tree (MOB) finds a highly significant difference between the think-say and write-say source condition. Furthermore, there is an age difference in the think-say condition that is significant at a Bonferroni-corrected 5% level. Gender is not found to play a significant role.</p> <p><a href="https://eeecon.uibk.ac.at/~zeileis/assets/posts/2018-08-27-mpttree/sourcemonitoring-mpttree.png"><img src="https://eeecon.uibk.ac.at/~zeileis/assets/posts/2018-08-27-mpttree/sourcemonitoring-mpttree.png" alt="MPT tree" /></a></p> <p><strong>Probabilities:</strong> For the think-say sources (Nodes 3 and 4), probability D2 exceeds D1 indicating an advantage of say items over think items with respect to detectability. For the write-say sources (Node 5), D2 and D1 are about the same indicating that for these sources no such advantage exists. The think-say subgroup is further split by age with the older participants having lower values on D1 and d, which suggests lower detectability of think items and lower discriminability as compared to the younger participants. This age effect seems to depend on the type of sources as there is no such effect for the write-say sources. In addition, there are only small effects for the bias parameters b and g, which are psychologically less interesting. Some of the differences in the probabilities across groups/nodes can be brought out even more clearly by parameter estimates and corresponding 95% Wald confidence intervals:</p> <p><a href="https://eeecon.uibk.ac.at/~zeileis/assets/posts/2018-08-27-mpttree/sourcemonitoring-coef.png"><img src="https://eeecon.uibk.ac.at/~zeileis/assets/posts/2018-08-27-mpttree/sourcemonitoring-coef.png" alt="MPT coefficients" /></a></p>
2018-08-27T00:00:00+02:00
https://eeecon.uibk.ac.at/~zeileis/news/sandwich250/
Clustered Covariances in sandwich 2.5-0
2018-08-20T00:00:00+02:00
Achim Zeileis
Achim.Zeileis@R-project.org
https://eeecon.uibk.ac.at/~zeileis/
Version 2.5-0 of the R package 'sandwich' is available from CRAN now with enhanced object-oriented clustered covariances (for lm, glm, survreg, polr, hurdle, zeroinfl, betareg, ...). The software and corresponding vignette have been improved considerably based on helpful and constructive reviewer feedback as well as various bug reports.
<p>Version 2.5-0 of the R package 'sandwich' is available from CRAN now with enhanced object-oriented clustered covariances (for lm, glm, survreg, polr, hurdle, zeroinfl, betareg, ...). The software and corresponding vignette have been improved considerably based on helpful and constructive reviewer feedback as well as various bug reports.</p> <h2 id="enhancements-in-version-25-0">Enhancements in version 2.5-0</h2> <p>Most of the improvements and new features pertain to clustered covariances which had been introduced to the <a href="https://CRAN.R-project.org/package=sandwich">sandwich</a> package last year in version 2.4-0. For this my PhD student Susanne Berger and myself (= <a href="https://eeecon.uibk.ac.at/~zeileis/">Achim Zeileis</a>) teamed up with <a href="https://sites.google.com/site/npgraham1/">Nathaniel Graham</a>, the maintainer of the <a href="https://CRAN.R-project.org/package=multiwayvcov">multiwayvcov</a> package. With the new version 2.5-0 almost all features from <em>multiwayvcov</em> have been ported to <em>sandwich</em>, mostly implemented from scratch along with generalizations, extensions, speed-ups, etc.</p> <p>The full list of changes can be seen in the <a href="https://CRAN.R-project.org/web/packages/sandwich/NEWS">NEWS</a> file. The most important changes are:</p> <ul> <li> <p>The manuscript <code class="highlighter-rouge">vignette("sandwich-CL", package = "sandwich")</code> has been significantly improved based on very helpful and constructive reviewer feedback. See also below.</p> </li> <li> <p>The <code class="highlighter-rouge">cluster</code> argument for the <code class="highlighter-rouge">vcov*()</code> functions can now be a formula, simplifying its usage (see below). <code class="highlighter-rouge">NA</code> handling has been added as well.</p> </li> <li> <p>Clustered bootstrap covariances have been reimplemented and extended in <code class="highlighter-rouge">vcovBS()</code>. A dedicated method for <code class="highlighter-rouge">lm</code> objects is considerably faster now and also includes various wild bootstraps.</p> </li> <li> <p>Convenient parallelization for bootstrap covariances is now available.</p> </li> <li> <p>Bugs reported by James Pustejovsky and Brian Tsay, respectively, have been fixed.</p> </li> </ul> <h2 id="manuscriptvignette">Manuscript/vignette</h2> <p>Susanne Berger, Nathaniel Graham, Achim Zeileis: <a href="https://CRAN.R-project.org/web/packages/sandwich/vignettes/sandwich-CL.pdf">Various Versatile Variances: An Object-Oriented Implementation of Clustered Covariances in R</a></p> <p>Clustered covariances or clustered standard errors are very widely used to account for correlated or clustered data, especially in economics, political sciences, or other social sciences. They are employed to adjust the inference following estimation of a standard least-squares regression or generalized linear model estimated by maximum likelihood. Although many publications just refer to “the” clustered standard errors, there is a surprisingly wide variety of clustered covariances, particularly due to different flavors of bias corrections. Furthermore, while the linear regression model is certainly the most important application case, the same strategies can be employed in more general models (e.g. for zero-inflated, censored, or limited responses).</p> <p>In R, functions for covariances in clustered or panel models have been somewhat scattered or available only for certain modeling functions, notably the (generalized) linear regression model. In contrast, an object-oriented approach to “robust” covariance matrix estimation - applicable beyond <code class="highlighter-rouge">lm()</code> and <code class="highlighter-rouge">glm()</code> - is available in the sandwich package but has been limited to the case of cross-section or time series data. Now, this shortcoming has been corrected in <em>sandwich</em> (starting from version 2.4.0): Based on methods for two generic functions (<code class="highlighter-rouge">estfun()</code> and <code class="highlighter-rouge">bread()</code>), clustered and panel covariances are now provided in <code class="highlighter-rouge">vcovCL()</code>, <code class="highlighter-rouge">vcovPL()</code>, and <code class="highlighter-rouge">vcovPC()</code>. Moreover, clustered bootstrap covariances, based on <code class="highlighter-rouge">update()</code> for models on bootstrap samples of the data, are provided in <code class="highlighter-rouge">vcovBS()</code>. These are directly applicable to models from many packages, e.g., including <em>MASS</em>, <em>pscl</em>, <em>countreg</em>, <em>betareg</em>, among others. Some empirical illustrations are provided as well as an assessment of the methods’ performance in a simulation study.</p> <h2 id="illustrations">Illustrations</h2> <p>To show how easily the clustered covariances from <code class="highlighter-rouge">sandwich</code> can be applied in practice, two short illustrations from the manuscript/vignette are used. In addition to the <code class="highlighter-rouge">sandwich</code> package the <code class="highlighter-rouge">lmtest</code> package is employed to easily obtain Wald tests of all coefficients:</p> <pre><code class="language-{r}">library("sandwich") library("lmtest") options(digits = 4) </code></pre> <p>First, a Poisson model with clustered standard errors from <a href="https://doi.org/10.1257/aer.103.1.277">Aghion <em>et al.</em> (2013, <em>American Economic Review</em>)</a> is replicated. To investigate the effect of institutional ownership on innovation (as captured by citation-weighted patent counts) they employ a (pseudo-)Poisson model with industry/year fixed effects and standard errors clustered by company, see their Table I(3):</p> <pre><code class="language-{r}">data("InstInnovation", package = "sandwich") ii <- glm(cites ~ institutions + log(capital/employment) + log(sales) + industry + year, data = InstInnovation, family = poisson) coeftest(ii, vcov = vcovCL, cluster = ~ company)[2:4, ] ## Estimate Std. Error z value Pr(>|z|) ## institutions 0.009687 0.002406 4.026 5.682e-05 ## log(capital/employment) 0.482884 0.135953 3.552 3.826e-04 ## log(sales) 0.820318 0.041523 19.756 7.187e-87 </code></pre> <p>Second, a simple linear regression model with double-clustered standard errors is replicated using the well-known <a href="https://www.kellogg.northwestern.edu/faculty/petersen/htm/papers/se/test_data.htm">Petersen data</a> from Petersen (2009, <em>Review of Financial Studies</em>):</p> <pre><code class="language-{r}">data("PetersenCL", package = "sandwich") p <- lm(y ~ x, data = PetersenCL) coeftest(p, vcov = vcovCL, cluster = ~ firm + year) ## t test of coefficients: ## ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 0.0297 0.0651 0.46 0.65 ## x 1.0348 0.0536 19.32 <2e-16 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 </code></pre> <h2 id="simulation">Simulation</h2> <p>In addition to the description of the methods and the software, the manuscript/vignette also contains a simulation study that investigates the properties of clustered covariances. In particular, this assesses how well the methods perfom in models beyond linear regression but also compares different types of bias adjustments (HC0-HC3) and alternative estimation techniques (generalized estimating equations, mixed effects).</p> <p>The detailed results are presented in the manuscript - here we just show the results from one of the simulation experiments: The empirical coverage of 95% Wald confidence intervals is depicted for a beta regression, zero-inflated Poisson, and zero-truncated Poisson model. With increasing correlation within the clusters the conventional “standard” errors and “basic” robust sandwich standard errors become too small thus leading to a drop in empirical coverage. However, both clustered HC0 standard errors (CL-0) and clustered bootstrap standard errors (BS) perform reasonably well, leading to empirical coverages close to the nominal 0.95.</p> <p><a href="https://eeecon.uibk.ac.at/~zeileis/assets/posts/2018-08-20-sandwich250/sim.png"><img src="https://eeecon.uibk.ac.at/~zeileis/assets/posts/2018-08-20-sandwich250/sim.png" alt="Simulation: Experiment III" /></a></p> <p>Details: Data sets were simulated with 100 clusters of 5 observations each. The cluster correlation (on the x-axis) was generated with a Gaussian copula. The only regressor had a correlation of 0.25 with the clustering variable. Empirical coverages were computed from 10,000 replications.</p>
2018-08-20T00:00:00+02:00
https://eeecon.uibk.ac.at/~zeileis/news/fifa2018eval/
Evaluation of the 2018 FIFA World Cup Forecast
2018-07-23T00:00:00+02:00
Achim Zeileis
Achim.Zeileis@R-project.org
https://eeecon.uibk.ac.at/~zeileis/
A look back the 2018 FIFA World Cup in Russia to check whether our tournament forecast based on the bookmaker consensus model was any good...
<p>A look back the 2018 FIFA World Cup in Russia to check whether our tournament forecast based on the bookmaker consensus model was any good...</p> <h2 id="how-surprising-was-the-tournament">How surprising was the tournament?</h2> <p>Last week France won the 2018 FIFA World Cup in a match against Croatia in Russia, thus delivering an entertaining final to a sportful tournament. Many perceived the course of the tournament as very unexpected and surprising because many of the “usual” favorites like Brazil, Germany, Spain, or Argentina did not even make it to the semi-finals. And in contrast, teams like host Russia and finalist Croatia proceeded further than expected. However, does this really mean that expectations of experts and fans were so wrong? Or, how surprising was the result given pre-tournament predictions?</p> <p>Therefore, we want to take a critical look back at our own <a href="https://eeecon.uibk.ac.at/~zeileis/news/fifa2018/">Probabilistic Forecast for the 2018 FIFA World Cup</a> based on the bookmaker consensus model that aggregated the expert judgments of 26 bookmakers and betting exchanges. A set of presentation slides (in PDF format) with explanations of the model and its evaluation are available to accompany this blog post: <a href="https://eeecon.uibk.ac.at/~zeileis/assets/posts/2018-07-23-fifa2018eval/slides.pdf">slides.pdf</a></p> <h2 id="tldr">TL;DR</h2> <p>Despite some surprises in the tournament, the probabilistic bookmaker consensus forecast fitted reasonably well. Although it is hard to evaluate probabilistic forecasts with only one realization of the tournament but by and large most outcomes do not deviate systematically from the probabilities assigned to them.</p> <p>However, there is one notable exception: Expectations about defending champion Germany were clearly wrong. “Die Mannschaft” was predicted to advance from the group stage to the round of 16 with probability 89.1% - and they not only failed to do so but instead came in last in their group.</p> <p>Other events that were perceived as surprising were not so unlikely to begin, e.g., for Argentina it was more likely to get eliminated before the quarter finals (predicted probability: 51%) than to proceed further. Or they were not unlikely conditional on previous tournament events. Examples for the latter are the pre-tournament prediction for Belgium beating Brazil in a match (40%) or Russia beating Spain (33%). Of course, another outcome of those matches was more likely but compared with these predictions the results were maybe not as surprising as perceived by many. Finally, the pre-tournament prediction of Croatia making it to the final was only 6% but conditional on the events from the round of 16 (especially with Spain being eliminated) this increased to 27% (only surpassed by England with 36%).</p> <h2 id="tournament-animation">Tournament animation</h2> <p>The animated GIF below shows the pre-tournament predictions for each team winning the 2018 FIFA world cup. In the animation the teams that “survived” over the course of the tournament are highlighted. This clearly shows that the elimination of Germany (winning probability: 15.8%) was the big surprise in the group stage but otherwise almost all of the teams expected to proceed also did so. Afterwards, two of the other main favorites Brazil (16.6%) and Spain (12.5%) dropped out but eventually the fourth team with double-digit winning probability (France, 12.1%) prevailed.</p> <p><img src="https://eeecon.uibk.ac.at/~zeileis/assets/posts/2018-07-23-fifa2018eval/consensus.gif" alt="tournament animation" /></p> <h2 id="correlations">Correlations</h2> <p>Compared to other rankings of the teams in the tournament, the bookmaker consensus model did quite well. To illustrate this we compute the Spearman rank correlation of observed partial tournament ranking (1 FRA, 2 CRO, 3 BEL, 4 ENG, 6.5 URU, 6.5 BRA, …) with the bookmaker consensus model as well as Elo and FIFA rating.</p> <table> <thead> <tr> <th style="text-align: left">Method</th> <th style="text-align: right">Correlation</th> </tr> </thead> <tbody> <tr> <td style="text-align: left">Bookmaker consensus <br /> Elo rating <br /> FIFA rating</td> <td style="text-align: right">0.704 <br /> 0.592 <br /> 0.411</td> </tr> </tbody> </table> <h2 id="match-probabilities">Match probabilities</h2> <p>As there is no good way to assess the predicted winning probabilities for winning the title with only one realization of the tournament, we at least (roughly) assess the quality of the predicted probabilities for the individual matches. To do so, we split the 63 matches into three groups, depending on the winning probability of the stronger team.</p> <p><img src="https://eeecon.uibk.ac.at/~zeileis/assets/posts/2018-07-23-fifa2018eval/eval.png" alt="pairwise probability evaluation" /></p> <p>This gives us matches that were predicted to be almost even (50-58%), had moderate advantages for the stronger team (58-72%), or clear advantages for the stronger team (72-85%). It turns out that in the latter two groups the average predicted probabilities (dashed red line) match the actual observed proportions quite well. Only in the “almost even” group, the stronger teams won slightly more often than expected.</p> <h2 id="group-stage-probabilities">Group stage probabilities</h2> <p>As already mentioned above, there was only one big surprise in the group stage - with Germany being eliminated. As the tables below show, most other results from the group rankings conformed quite well with the predicted probabilities to “survive” the group stage.</p> <div class="row"> <div class="t20 small-6 large-3 columns"> <table> <thead> <tr> <th style="text-align: left">A <br /> Rank</th> <th style="text-align: left"> <br /> Team</th> <th style="text-align: right"> <br /> Prob. (in %)</th> </tr> </thead> <tbody> <tr> <td style="text-align: left"><strong>1</strong> <br /> <strong>2</strong> <br /> 3 <br /> 4</td> <td style="text-align: left"><strong>URU</strong> <br /> <strong>RUS</strong> <br /> KSA <br /> EGY</td> <td style="text-align: right"><strong>68.1</strong> <br /> <strong>64.2</strong> <br /> 19.2 <br /> 39.3</td> </tr> </tbody> </table> </div> <div class="t20 small-6 large-3 columns"> <table> <thead> <tr> <th style="text-align: left">B <br /> Rank</th> <th style="text-align: left"> <br /> Team</th> <th style="text-align: right"> <br /> Prob. (in %)</th> </tr> </thead> <tbody> <tr> <td style="text-align: left"><strong>1</strong> <br /> <strong>2</strong> <br /> 3 <br /> 4</td> <td style="text-align: left"><strong>ESP</strong> <br /> <strong>POR</strong> <br /> IRN <br /> MAR</td> <td style="text-align: right"><strong>85.9</strong> <br /> <strong>66.3</strong> <br /> 26.5 <br /> 27.3</td> </tr> </tbody> </table> </div> <div class="t20 small-6 large-3 columns"> <table> <thead> <tr> <th style="text-align: left">C <br /> Rank</th> <th style="text-align: left"> <br /> Team</th> <th style="text-align: right"> <br /> Prob. (in %)</th> </tr> </thead> <tbody> <tr> <td style="text-align: left"><strong>1</strong> <br /> <strong>2</strong> <br /> 3 <br /> 4</td> <td style="text-align: left"><strong>FRA</strong> <br /> <strong>DEN</strong> <br /> PER <br /> AUS</td> <td style="text-align: right"><strong>87.0</strong> <br /> <strong>46.7</strong> <br /> 31.7 <br /> 25.2</td> </tr> </tbody> </table> </div> <div class="t20 small-6 large-3 columns"> <table> <thead> <tr> <th style="text-align: left">D <br /> Rank</th> <th style="text-align: left"> <br /> Team</th> <th style="text-align: right"> <br /> Prob. (in %)</th> </tr> </thead> <tbody> <tr> <td style="text-align: left"><strong>1</strong> <br /> <strong>2</strong> <br /> 3 <br /> 4</td> <td style="text-align: left"><strong>CRO</strong> <br /> <strong>ARG</strong> <br /> NGA <br /> ISL</td> <td style="text-align: right"><strong>58.7</strong> <br /> <strong>78.7</strong> <br /> 41.2 <br /> 30.9</td> </tr> </tbody> </table> </div> </div> <div class="row"> <div class="t20 small-6 large-3 columns"> <table> <thead> <tr> <th style="text-align: left">E <br /> Rank</th> <th style="text-align: left"> <br /> Team</th> <th style="text-align: right"> <br /> Prob. (in %)</th> </tr> </thead> <tbody> <tr> <td style="text-align: left"><strong>1</strong> <br /> <strong>2</strong> <br /> 3 <br /> 4</td> <td style="text-align: left"><strong>BRA</strong> <br /> <strong>SUI</strong> <br /> SRB <br /> CRC</td> <td style="text-align: right"><strong>89.9</strong> <br /> <strong>45.4</strong> <br /> 39.0 <br /> 22.6</td> </tr> </tbody> </table> </div> <div class="t20 small-6 large-3 columns"> <table> <thead> <tr> <th style="text-align: left">F <br /> Rank</th> <th style="text-align: left"> <br /> Team</th> <th style="text-align: right"> <br /> Prob. (in %)</th> </tr> </thead> <tbody> <tr> <td style="text-align: left"><strong>1</strong> <br /> <strong>2</strong> <br /> 3 <br /> 4</td> <td style="text-align: left"><strong>SWE</strong> <br /> <strong>MEX</strong> <br /> KOR <br /> GER</td> <td style="text-align: right"><strong>44.5</strong> <br /> <strong>45.2</strong> <br /> 26.8 <br /> 89.1</td> </tr> </tbody> </table> </div> <div class="t20 small-6 large-3 columns"> <table> <thead> <tr> <th style="text-align: left">G <br /> Rank</th> <th style="text-align: left"> <br /> Team</th> <th style="text-align: right"> <br /> Prob. (in %)</th> </tr> </thead> <tbody> <tr> <td style="text-align: left"><strong>1</strong> <br /> <strong>2</strong> <br /> 3 <br /> 4</td> <td style="text-align: left"><strong>BEL</strong> <br /> <strong>ENG</strong> <br /> TUN <br /> PAN</td> <td style="text-align: right"><strong>81.7</strong> <br /> <strong>75.6</strong> <br /> 23.5 <br /> 23.2</td> </tr> </tbody> </table> </div> <div class="t20 small-6 large-3 columns"> <table> <thead> <tr> <th style="text-align: left">H <br /> Rank</th> <th style="text-align: left"> <br /> Team</th> <th style="text-align: right"> <br /> Prob. (in %)</th> </tr> </thead> <tbody> <tr> <td style="text-align: left"><strong>1</strong> <br /> <strong>2</strong> <br /> 3 <br /> 4</td> <td style="text-align: left"><strong>COL</strong> <br /> <strong>JPN</strong> <br /> SEN <br /> POL</td> <td style="text-align: right"><strong>64.6</strong> <br /> <strong>36.3</strong> <br /> 37.9 <br /> 57.9</td> </tr> </tbody> </table> </div> </div>
2018-07-23T00:00:00+02:00
https://eeecon.uibk.ac.at/~zeileis/news/fifa2018sankey/
Sankey Diagram for the 2018 FIFA World Cup Forecast
2018-06-11T00:00:00+02:00
Achim Zeileis
Achim.Zeileis@R-project.org
https://eeecon.uibk.ac.at/~zeileis/
The probabilistic forecast from the bookmaker consensus model for the 2018 FIFA World Cup is visualized in an interactive Sankey diagram, highlighting the teams' most likely progress through the tournament.
<p>The probabilistic forecast from the bookmaker consensus model for the 2018 FIFA World Cup is visualized in an interactive Sankey diagram, highlighting the teams' most likely progress through the tournament.</p> <h2 id="bookmaker-consensus-model">Bookmaker consensus model</h2> <p>Two weeks ago we published our <a href="https://eeecon.uibk.ac.at/~zeileis/news/fifa2018/">Probabilistic Forecast for the 2018 FIFA World Cup</a>: By adjusting quoted bookmakers’ odds for the profit margins of the bookmakers (also known as overrounds), transforming and averaging them, a <a href="https://eeecon.uibk.ac.at/~zeileis/assets/posts/2018-05-30-fifa2018/p_win.html">predicted winning probability</a> for each team was obtained. By employing millions of tournament simulations in combination with a model for <a href="https://eeecon.uibk.ac.at/~zeileis/assets/posts/2018-05-30-fifa2018/p_pair.html">pairwise comparisons</a> (or matches) we could also obtain forecasted probabilities for each team to progress through the tournament. In our original study, we visualized these by <a href="https://eeecon.uibk.ac.at/~zeileis/assets/posts/2018-05-30-fifa2018/p_surv.html">“survival” curves</a>. See the <a href="http://EconPapers.RePEc.org/RePEc:inn:wpaper:2018-09">working paper</a> for more details and references.</p> <h2 id="sankey-diagram">Sankey diagram</h2> <p>Here, we present another display that highlights the likely flow of all teams through the tournament simultaneously. Click on the image to obtain an <a href="https://eeecon.uibk.ac.at/~zeileis/assets/posts/2018-06-11-fifa2018sankey/p_sankey.html">interactive full-width version</a> of this Sankey diagram produced by <a href="https://plot.ly/r/">Plotly</a>.</p> <p><a href="https://eeecon.uibk.ac.at/~zeileis/assets/posts/2018-06-11-fifa2018sankey/p_sankey.html"><img src="https://eeecon.uibk.ac.at/~zeileis/assets/posts/2018-06-11-fifa2018sankey/p_sankey.png" alt="Sankey diagram" /></a></p> <p>Compared to the survival curves shown in our original study this visualization brings out more clearly at which stages of the tournament the strong teams are most likely to meet. Its usage was inspired by the nice working paper <a href="https://arxiv.org/abs/1806.01930">On Elo based prediction models for the FIFA Worldcup 2018</a> by <a href="https://www.researchgate.net/profile/Sebastian_Mueller10">Lorenz A. Gilch</a> and <a href="https://sites.google.com/site/drsebastianmueller/">Sebastian Müller</a>.</p> <p>In a few days we will start learning which of these paths will actually come true. Enjoy the 2018 FIFA World Cup!</p>
2018-06-11T00:00:00+02:00
https://eeecon.uibk.ac.at/~zeileis/news/fifa2018/
Probabilistic Forecasting for the 2018 FIFA World Cup
2018-05-30T00:00:00+02:00
Achim Zeileis
Achim.Zeileis@R-project.org
https://eeecon.uibk.ac.at/~zeileis/
Using a consensus model based on quoted bookmakers' odds winning probabilities for all competing teams in the FIFA World Cup are obtained: The favorite is Brazil, closely followed by the defending World Champion Germany.
<p>Using a consensus model based on quoted bookmakers' odds winning probabilities for all competing teams in the FIFA World Cup are obtained: The favorite is Brazil, closely followed by the defending World Champion Germany.</p> <div class="row t20 b20"> <div class="small-8 medium-9 large-10 columns"> Football fans worldwide anticipate the 2018 FIFA World Cup that will take place in Russia from 14 June to 15 July 2018. 32 of the best teams from 5 confederations compete to determine the new World Champion. Everybody is curious already now what the most likely outcome of the tournament will be. Hence, a predictive model is established by leveraging the expert knowledge of 26 bookmakers and betting exchanges using a model averaging approach. </div> <div class="small-4 medium-3 large-2 columns"> <a href="http://www.fifa.com/worldcup/" alt="2018 FIFA World Cup web page"><img src="https://upload.wikimedia.org/wikipedia/en/6/67/2018_FIFA_World_Cup.svg" alt="2018 FIFA World Cup logo" /></a> </div> </div> <h2 id="winning-probabilities">Winning probabilities</h2> <p>The model is the so-called bookmaker consensus model which has been proposed by Leitner, Hornik, and Zeileis (2010, <em>International Journal of Forecasting</em>, <a href="https://doi.org/10.1016/j.ijforecast.2009.10.001">https://doi.org/10.1016/j.ijforecast.2009.10.001</a>) and successfully applied in previous football tournaments, e.g., correctly predicting the winner of the <a href="http://epub.wu.ac.at/702/">2010 FIFA World Cup</a> and three out of four semifinalists at the <a href="http://EconPapers.RePEc.org/RePEc:inn:wpaper:2014-17">2014 FIFA World Cup</a>. This time the forecast shows that Brazil is the favorite with a forecasted winning probability of 16.6%, closely followed by the defending World Champion and 2017 FIFA Confederations Cup winner Germany with a winning probability of 15.8%. Two other teams also have double-digit winning probabilities: Spain and France with 12.5% and 12.1%, respectively. More details are displayed in the following barchart.</p> <p><a href="https://eeecon.uibk.ac.at/~zeileis/assets/posts/2018-05-30-fifa2018/p_win.html">Full-width graphic</a></p> <div id="htmlwidget_container"> <div id="48a031dd7474" style="width:100%;height:400px;" class="plotly html-widget"></div> </div> <script type="application/json" data-for="48a031dd7474">{"x":{"visdat":{"48a03c184145":["function () ","plotlyVisDat"]},"cur_data":"48a03c184145","attrs":{"48a03c184145":{"x":["Brazil","Germany","Spain","France","Argentina","Belgium","England","Portugal","Uruguay","Croatia","Colombia","Russia","Poland","Denmark","Mexico","Switzerland","Sweden","Egypt","Serbia","Senegal","Peru","Nigeria","Iceland","Japan","Australia","Morocco","Costa Rica","South Korea","Iran","Tunisia","Saudi 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/>Winning probability: 0.5%<br />Group: A<br />Ability: 0.049","Team: Serbia<br />Winning probability: 0.5%<br />Group: E<br />Ability: 0.048","Team: Senegal<br />Winning probability: 0.5%<br />Group: H<br />Ability: 0.047","Team: Peru<br />Winning probability: 0.4%<br />Group: C<br />Ability: 0.048","Team: Nigeria<br />Winning probability: 0.4%<br />Group: D<br />Ability: 0.047","Team: Iceland<br />Winning probability: 0.4%<br />Group: D<br />Ability: 0.047","Team: Japan<br />Winning probability: 0.3%<br />Group: H<br />Ability: 0.042","Team: Australia<br />Winning probability: 0.2%<br />Group: C<br />Ability: 0.039","Team: Morocco<br />Winning probability: 0.2%<br />Group: B<br />Ability: 0.038","Team: Costa Rica<br />Winning probability: 0.2%<br />Group: E<br />Ability: 0.036","Team: South Korea<br />Winning probability: 0.2%<br />Group: F<br />Ability: 0.037","Team: Iran<br />Winning probability: 0.2%<br />Group: B<br />Ability: 0.038","Team: Tunisia<br />Winning probability: 0.1%<br />Group: G<br />Ability: 0.034","Team: Saudi Arabia<br />Winning probability: 0.1%<br />Group: A<br />Ability: 0.030","Team: Panama<br />Winning probability: 0.1%<br />Group: G<br />Ability: 0.031"],"name":"FIFA 2018 Winner","hoverinfo":["text","text","text","text","text","text","text","text","text","text","text","text","text","text","text","text","text","text","text","text","text","text","text","text","text","text","text","text","text","text","text","text"],"legendgroup":["E","F","B","C","D","G","A","H"],"type":"bar","marker":{"fillcolor":"rgba(112,128,144,1)","color":"rgba(112,128,144,1)","line":{"color":"transparent"}},"xaxis":"x","yaxis":"y","frame":null}],"highlight":{"on":"plotly_click","persistent":false,"dynamic":false,"selectize":false,"opacityDim":0.2,"selected":{"opacity":1}},"base_url":"https://plot.ly"},"evals":["config.modeBarButtonsToAdd.0.click"],"jsHooks":{"render":[{"code":"function(el, x) { var ctConfig = 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More precisely, the odds are first adjusted for the bookmakers’ profit margins (“overrounds”, on average 15.2%), averaged on the log-odds scale to a consensus rating, and then transformed back to winning probabilities.</p> <p>A more detailed description of the model as well as its results for the 2018 FIFA World Cup are available in a new <a href="http://EconPapers.RePEc.org/RePEc:inn:wpaper:2018-09">working paper</a>. The raw bookmakers’ odds as well as the forecasts for all teams are also available in machine-readable form in <a href="https://eeecon.uibk.ac.at/~zeileis/assets/posts/2018-05-30-fifa2018/fifa2018.csv">fifa2018.csv</a>.</p> <p>Although forecasting the winning probabilities for the 2018 FIFA World Cup is probably of most interest, the bookmaker consensus forecasts can also be employed to infer team-specific abilities using an “inverse” tournament simulation:</p> <ol> <li>If team abilities are available, pairwise winning probabilities can be derived for each possible match (see below).</li> <li>Given pairwise winning probabilities, the whole tournament can be easily simulated to see which team proceeds to which stage in the tournament and which team finally wins.</li> <li>Such a tournament simulation can then be run sufficiently often (here 1,000,000 times) to obtain relative frequencies for each team winning the tournament.</li> </ol> <p>Using this idea, abilities in step 1 can be chosen such that the simulated winning probabilities in step 3 closely match those from the bookmaker consensus shown above.</p> <h2 id="pairwise-comparisons">Pairwise comparisons</h2> <p>A classical approach to obtain winning probabilities in pairwise comparisons (i.e., matches between teams/players) is the Bradley-Terry model, which is similar to the Elo rating, popular in sports. The Bradley-Terry approach models the probability that a Team A beats a Team B by their associated abilities (or strengths):</p> <math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true"><mrow><mi fontstyle="normal">Pr</mi><mo stretchy="false">(</mo><mi>A</mi><mtext> beats </mtext><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mrow><mi fontstyle="italic">ability</mi></mrow><mrow><mi>A</mi></mrow></msub></mrow><mrow><msub><mrow><mi fontstyle="italic">ability</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>+</mo><msub><mrow><mi fontstyle="italic">ability</mi></mrow><mrow><mi>B</mi></mrow></msub></mrow></mfrac><mo>.</mo></mrow></mstyle></math> <p>Coupled with the “inverse” simulation of the tournament, as described in step 1-3 above, this yields pairwise probabilities for each possible match. The following heatmap shows the probabilistic forecasts for each match with light gray signalling approximately equal chances and green vs. pink signalling advantages for Team A or B, respectively.</p> <p><a href="https://eeecon.uibk.ac.at/~zeileis/assets/posts/2018-05-30-fifa2018/p_pair.html">Full-width graphic</a></p> <div id="htmlwidget_container"> <div id="48a045a34037" style="width:80%;height:700px;" class="plotly html-widget"></div> </div> <script type="application/json" data-for="48a045a34037">{"x":{"visdat":{"48a029ac085":["function () ","plotlyVisDat"]},"cur_data":"48a029ac085","attrs":{"48a029ac085":{"x":["Panama","Saudi Arabia","Tunisia","Iran","South Korea","Costa Rica","Morocco","Australia","Japan","Iceland","Nigeria","Peru","Senegal","Serbia","Egypt","Sweden","Switzerland","Mexico","Denmark","Poland","Russia","Colombia","Croatia","Uruguay","Portugal","England","Belgium","Argentina","France","Spain","Germany","Brazil"],"y":["Panama","Saudi Arabia","Tunisia","Iran","South 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beats Saudi Arabia<br />Probability: 51.0%","Panama beats Tunisia<br />Probability: 47.9%","Panama beats Iran<br />Probability: 45.2%","Panama beats South Korea<br />Probability: 45.6%","Panama beats Costa Rica<br />Probability: 46.2%","Panama beats Morocco<br />Probability: 45.1%","Panama beats Australia<br />Probability: 44.4%","Panama beats Japan<br />Probability: 42.3%","Panama beats Iceland<br />Probability: 39.9%","Panama beats Nigeria<br />Probability: 40.0%","Panama beats Peru<br />Probability: 39.4%","Panama beats Senegal<br />Probability: 39.8%","Panama beats Serbia<br />Probability: 39.2%","Panama beats Egypt<br />Probability: 38.6%","Panama beats Sweden<br />Probability: 38.6%","Panama beats Switzerland<br />Probability: 36.7%","Panama beats Mexico<br />Probability: 36.2%","Panama beats Denmark<br />Probability: 36.0%","Panama beats Poland<br />Probability: 32.9%","Panama beats Russia<br />Probability: 30.5%","Panama beats Colombia<br />Probability: 30.0%","Panama beats Croatia<br />Probability: 28.3%","Panama beats Uruguay<br />Probability: 28.8%","Panama beats Portugal<br />Probability: 27.1%","Panama beats England<br />Probability: 25.1%","Panama beats Belgium<br />Probability: 21.9%","Panama beats Argentina<br />Probability: 20.0%","Panama beats France<br />Probability: 17.4%","Panama beats Spain<br />Probability: 17.5%","Panama beats Germany<br />Probability: 15.8%","Panama beats Brazil<br />Probability: 15.5%"],["Saudi Arabia beats Panama<br />Probability: 49.0%","","Saudi Arabia beats Tunisia<br />Probability: 46.9%","Saudi Arabia beats Iran<br />Probability: 44.2%","Saudi Arabia beats South Korea<br />Probability: 44.6%","Saudi Arabia beats Costa Rica<br />Probability: 45.2%","Saudi Arabia beats Morocco<br />Probability: 44.1%","Saudi Arabia beats Australia<br />Probability: 43.4%","Saudi Arabia beats Japan<br />Probability: 41.3%","Saudi Arabia beats Iceland<br />Probability: 38.9%","Saudi Arabia beats Nigeria<br />Probability: 39.0%","Saudi Arabia beats Peru<br />Probability: 38.4%","Saudi Arabia beats Senegal<br />Probability: 38.9%","Saudi Arabia beats Serbia<br />Probability: 38.2%","Saudi Arabia beats Egypt<br />Probability: 37.7%","Saudi Arabia beats Sweden<br />Probability: 37.6%","Saudi Arabia beats Switzerland<br />Probability: 35.8%","Saudi Arabia beats Mexico<br />Probability: 35.3%","Saudi Arabia beats Denmark<br />Probability: 35.0%","Saudi Arabia beats Poland<br />Probability: 32.0%","Saudi Arabia beats Russia<br />Probability: 29.7%","Saudi Arabia beats Colombia<br />Probability: 29.1%","Saudi Arabia beats Croatia<br />Probability: 27.5%","Saudi Arabia beats Uruguay<br />Probability: 27.9%","Saudi Arabia beats Portugal<br />Probability: 26.3%","Saudi Arabia beats England<br />Probability: 24.4%","Saudi Arabia beats Belgium<br />Probability: 21.2%","Saudi Arabia beats Argentina<br />Probability: 19.4%","Saudi Arabia beats France<br />Probability: 16.8%","Saudi Arabia beats Spain<br />Probability: 16.9%","Saudi Arabia beats Germany<br />Probability: 15.3%","Saudi Arabia beats Brazil<br />Probability: 15.0%"],["Tunisia beats Panama<br />Probability: 52.1%","Tunisia beats Saudi Arabia<br />Probability: 53.1%","","Tunisia beats Iran<br />Probability: 47.3%","Tunisia beats South Korea<br />Probability: 47.7%","Tunisia beats Costa Rica<br />Probability: 48.3%","Tunisia beats Morocco<br />Probability: 47.2%","Tunisia beats Australia<br />Probability: 46.5%","Tunisia beats Japan<br />Probability: 44.4%","Tunisia beats Iceland<br />Probability: 41.9%","Tunisia beats Nigeria<br />Probability: 42.0%","Tunisia beats Peru<br />Probability: 41.4%","Tunisia beats Senegal<br />Probability: 41.9%","Tunisia beats Serbia<br />Probability: 41.2%","Tunisia beats Egypt<br />Probability: 40.6%","Tunisia beats Sweden<br />Probability: 40.6%","Tunisia beats Switzerland<br />Probability: 38.7%","Tunisia beats Mexico<br />Probability: 38.2%","Tunisia beats Denmark<br />Probability: 37.9%","Tunisia beats Poland<br />Probability: 34.7%","Tunisia beats Russia<br />Probability: 32.3%","Tunisia beats Colombia<br />Probability: 31.8%","Tunisia beats Croatia<br />Probability: 30.1%","Tunisia beats Uruguay<br />Probability: 30.5%","Tunisia beats Portugal<br />Probability: 28.8%","Tunisia beats England<br />Probability: 26.7%","Tunisia beats Belgium<br />Probability: 23.4%","Tunisia beats Argentina<br />Probability: 21.4%","Tunisia beats France<br />Probability: 18.7%","Tunisia beats Spain<br />Probability: 18.8%","Tunisia beats Germany<br />Probability: 17.0%","Tunisia beats Brazil<br />Probability: 16.6%"],["Iran beats Panama<br />Probability: 54.8%","Iran beats Saudi Arabia<br />Probability: 55.8%","Iran beats Tunisia<br />Probability: 52.7%","","Iran beats South Korea<br />Probability: 50.4%","Iran beats Costa Rica<br />Probability: 51.0%","Iran beats Morocco<br />Probability: 49.9%","Iran beats Australia<br />Probability: 49.2%","Iran beats Japan<br />Probability: 47.1%","Iran beats Iceland<br />Probability: 44.6%","Iran beats Nigeria<br />Probability: 44.7%","Iran beats Peru<br />Probability: 44.1%","Iran beats Senegal<br />Probability: 44.5%","Iran beats Serbia<br />Probability: 43.8%","Iran beats Egypt<br />Probability: 43.3%","Iran beats Sweden<br />Probability: 43.2%","Iran beats Switzerland<br />Probability: 41.3%","Iran beats Mexico<br />Probability: 40.8%","Iran beats Denmark<br />Probability: 40.5%","Iran beats Poland<br />Probability: 37.2%","Iran beats Russia<br />Probability: 34.7%","Iran beats Colombia<br />Probability: 34.2%","Iran beats Croatia<br />Probability: 32.4%","Iran beats Uruguay<br />Probability: 32.9%","Iran beats Portugal<br />Probability: 31.1%","Iran beats England<br />Probability: 28.9%","Iran beats Belgium<br />Probability: 25.4%","Iran beats Argentina<br />Probability: 23.3%","Iran beats France<br />Probability: 20.4%","Iran beats Spain<br />Probability: 20.5%","Iran beats Germany<br />Probability: 18.5%","Iran beats Brazil<br />Probability: 18.2%"],["South Korea beats Panama<br />Probability: 54.4%","South Korea beats Saudi Arabia<br />Probability: 55.4%","South Korea beats Tunisia<br />Probability: 52.3%","South Korea beats Iran<br />Probability: 49.6%","","South Korea beats Costa Rica<br />Probability: 50.6%","South Korea beats Morocco<br />Probability: 49.5%","South Korea beats Australia<br />Probability: 48.8%","South Korea beats Japan<br />Probability: 46.6%","South Korea beats Iceland<br />Probability: 44.2%","South Korea beats Nigeria<br />Probability: 44.2%","South Korea beats Peru<br />Probability: 43.7%","South Korea beats Senegal<br />Probability: 44.1%","South Korea beats Serbia<br />Probability: 43.4%","South Korea beats Egypt<br />Probability: 42.8%","South Korea beats Sweden<br />Probability: 42.8%","South Korea beats Switzerland<br />Probability: 40.9%","South Korea beats Mexico<br />Probability: 40.4%","South Korea beats Denmark<br />Probability: 40.1%","South Korea beats Poland<br />Probability: 36.8%","South Korea beats Russia<br />Probability: 34.3%","South Korea beats Colombia<br />Probability: 33.8%","South Korea beats Croatia<br />Probability: 32.0%","South Korea beats Uruguay<br />Probability: 32.5%","South Korea beats Portugal<br />Probability: 30.7%","South Korea beats England<br />Probability: 28.6%","South Korea beats Belgium<br />Probability: 25.1%","South Korea beats Argentina<br />Probability: 23.0%","South Korea beats France<br />Probability: 20.1%","South Korea beats Spain<br />Probability: 20.2%","South Korea beats Germany<br />Probability: 18.3%","South Korea beats Brazil<br />Probability: 17.9%"],["Costa Rica beats Panama<br />Probability: 53.8%","Costa Rica beats Saudi Arabia<br />Probability: 54.8%","Costa Rica beats Tunisia<br />Probability: 51.7%","Costa Rica beats Iran<br />Probability: 49.0%","Costa Rica beats South Korea<br />Probability: 49.4%","","Costa Rica beats Morocco<br />Probability: 48.9%","Costa Rica beats Australia<br />Probability: 48.2%","Costa Rica beats Japan<br />Probability: 46.1%","Costa Rica beats Iceland<br />Probability: 43.6%","Costa Rica beats Nigeria<br />Probability: 43.7%","Costa Rica beats Peru<br />Probability: 43.1%","Costa Rica beats Senegal<br />Probability: 43.6%","Costa Rica beats Serbia<br />Probability: 42.9%","Costa Rica beats Egypt<br />Probability: 42.3%","Costa Rica beats Sweden<br />Probability: 42.3%","Costa Rica beats Switzerland<br />Probability: 40.3%","Costa Rica beats Mexico<br />Probability: 39.8%","Costa Rica beats Denmark<br />Probability: 39.6%","Costa Rica beats Poland<br />Probability: 36.3%","Costa Rica beats Russia<br />Probability: 33.8%","Costa Rica beats Colombia<br />Probability: 33.3%","Costa Rica beats Croatia<br />Probability: 31.5%","Costa Rica beats Uruguay<br />Probability: 32.0%","Costa Rica beats Portugal<br />Probability: 30.3%","Costa Rica beats England<br />Probability: 28.1%","Costa Rica beats Belgium<br />Probability: 24.6%","Costa Rica beats Argentina<br />Probability: 22.6%","Costa Rica beats France<br />Probability: 19.7%","Costa Rica beats Spain<br />Probability: 19.8%","Costa Rica beats Germany<br />Probability: 17.9%","Costa Rica beats Brazil<br />Probability: 17.6%"],["Morocco beats Panama<br />Probability: 54.9%","Morocco beats Saudi Arabia<br />Probability: 55.9%","Morocco beats Tunisia<br />Probability: 52.8%","Morocco beats Iran<br />Probability: 50.1%","Morocco beats South Korea<br />Probability: 50.5%","Morocco beats Costa Rica<br />Probability: 51.1%","","Morocco beats Australia<br />Probability: 49.3%","Morocco beats Japan<br />Probability: 47.1%","Morocco beats Iceland<br />Probability: 44.7%","Morocco beats Nigeria<br />Probability: 44.8%","Morocco beats Peru<br />Probability: 44.2%","Morocco beats Senegal<br />Probability: 44.6%","Morocco beats Serbia<br />Probability: 43.9%","Morocco beats Egypt<br />Probability: 43.4%","Morocco beats Sweden<br />Probability: 43.3%","Morocco beats Switzerland<br />Probability: 41.4%","Morocco beats Mexico<br />Probability: 40.9%","Morocco beats Denmark<br />Probability: 40.6%","Morocco beats Poland<br />Probability: 37.3%","Morocco beats Russia<br />Probability: 34.8%","Morocco beats Colombia<br />Probability: 34.3%","Morocco beats Croatia<br />Probability: 32.5%","Morocco beats Uruguay<br />Probability: 32.9%","Morocco beats Portugal<br />Probability: 31.2%","Morocco beats England<br />Probability: 29.0%","Morocco beats Belgium<br />Probability: 25.4%","Morocco beats Argentina<br />Probability: 23.3%","Morocco beats France<br />Probability: 20.4%","Morocco beats Spain<br />Probability: 20.5%","Morocco beats Germany<br />Probability: 18.6%","Morocco beats Brazil<br />Probability: 18.2%"],["Australia beats Panama<br />Probability: 55.6%","Australia beats Saudi Arabia<br />Probability: 56.6%","Australia beats Tunisia<br />Probability: 53.5%","Australia beats Iran<br />Probability: 50.8%","Australia beats South Korea<br />Probability: 51.2%","Australia beats Costa Rica<br />Probability: 51.8%","Australia beats Morocco<br />Probability: 50.7%","","Australia beats Japan<br />Probability: 47.8%","Australia beats Iceland<br />Probability: 45.4%","Australia beats Nigeria<br />Probability: 45.4%","Australia beats Peru<br />Probability: 44.9%","Australia beats Senegal<br />Probability: 45.3%","Australia beats Serbia<br />Probability: 44.6%","Australia beats Egypt<br />Probability: 44.0%","Australia beats Sweden<br />Probability: 44.0%","Australia beats Switzerland<br />Probability: 42.1%","Australia beats Mexico<br />Probability: 41.5%","Australia beats Denmark<br />Probability: 41.3%","Australia beats Poland<br />Probability: 38.0%","Australia beats Russia<br />Probability: 35.5%","Australia beats Colombia<br />Probability: 34.9%","Australia beats Croatia<br />Probability: 33.1%","Australia beats Uruguay<br />Probability: 33.6%","Australia beats Portugal<br />Probability: 31.8%","Australia beats England<br />Probability: 29.6%","Australia beats Belgium<br />Probability: 26.0%","Australia beats Argentina<br />Probability: 23.8%","Australia beats France<br />Probability: 20.9%","Australia beats Spain<br />Probability: 21.0%","Australia beats Germany<br />Probability: 19.0%","Australia beats Brazil<br />Probability: 18.7%"],["Japan beats Panama<br />Probability: 57.7%","Japan beats Saudi Arabia<br />Probability: 58.7%","Japan beats Tunisia<br />Probability: 55.6%","Japan beats Iran<br />Probability: 52.9%","Japan beats South Korea<br />Probability: 53.4%","Japan beats Costa Rica<br />Probability: 53.9%","Japan beats Morocco<br />Probability: 52.9%","Japan beats Australia<br />Probability: 52.2%","","Japan beats Iceland<br />Probability: 47.5%","Japan beats Nigeria<br />Probability: 47.6%","Japan beats Peru<br />Probability: 47.0%","Japan beats Senegal<br />Probability: 47.5%","Japan beats Serbia<br />Probability: 46.7%","Japan beats Egypt<br />Probability: 46.2%","Japan beats Sweden<br />Probability: 46.1%","Japan beats Switzerland<br />Probability: 44.2%","Japan beats Mexico<br />Probability: 43.6%","Japan beats Denmark<br />Probability: 43.4%","Japan beats Poland<br />Probability: 40.0%","Japan beats Russia<br />Probability: 37.5%","Japan beats Colombia<br />Probability: 36.9%","Japan beats Croatia<br />Probability: 35.0%","Japan beats Uruguay<br />Probability: 35.5%","Japan beats Portugal<br />Probability: 33.7%","Japan beats England<br />Probability: 31.4%","Japan beats Belgium<br />Probability: 27.7%","Japan beats Argentina<br />Probability: 25.4%","Japan beats France<br />Probability: 22.3%","Japan beats Spain<br />Probability: 22.5%","Japan beats Germany<br />Probability: 20.4%","Japan beats Brazil<br />Probability: 20.0%"],["Iceland beats Panama<br />Probability: 60.1%","Iceland beats Saudi Arabia<br />Probability: 61.1%","Iceland beats Tunisia<br />Probability: 58.1%","Iceland beats Iran<br />Probability: 55.4%","Iceland beats South Korea<br />Probability: 55.8%","Iceland beats Costa Rica<br />Probability: 56.4%","Iceland beats Morocco<br />Probability: 55.3%","Iceland beats Australia<br />Probability: 54.6%","Iceland beats Japan<br />Probability: 52.5%","","Iceland beats Nigeria<br />Probability: 50.1%","Iceland beats Peru<br />Probability: 49.5%","Iceland beats Senegal<br />Probability: 50.0%","Iceland beats Serbia<br />Probability: 49.2%","Iceland beats Egypt<br />Probability: 48.7%","Iceland beats Sweden<br />Probability: 48.6%","Iceland beats Switzerland<br />Probability: 46.7%","Iceland beats Mexico<br />Probability: 46.1%","Iceland beats Denmark<br />Probability: 45.8%","Iceland beats Poland<br />Probability: 42.4%","Iceland beats Russia<br />Probability: 39.8%","Iceland beats Colombia<br />Probability: 39.2%","Iceland beats Croatia<br />Probability: 37.3%","Iceland beats Uruguay<br />Probability: 37.8%","Iceland beats Portugal<br />Probability: 35.9%","Iceland beats England<br />Probability: 33.6%","Iceland beats Belgium<br />Probability: 29.7%","Iceland beats Argentina<br />Probability: 27.4%","Iceland beats France<br />Probability: 24.1%","Iceland beats Spain<br />Probability: 24.3%","Iceland beats Germany<br />Probability: 22.1%","Iceland beats Brazil<br />Probability: 21.7%"],["Nigeria beats Panama<br />Probability: 60.0%","Nigeria beats Saudi Arabia<br />Probability: 61.0%","Nigeria beats Tunisia<br />Probability: 58.0%","Nigeria beats Iran<br />Probability: 55.3%","Nigeria beats South Korea<br />Probability: 55.8%","Nigeria beats Costa Rica<br />Probability: 56.3%","Nigeria beats Morocco<br />Probability: 55.2%","Nigeria beats Australia<br />Probability: 54.6%","Nigeria beats Japan<br />Probability: 52.4%","Nigeria beats Iceland<br />Probability: 49.9%","","Nigeria beats Peru<br />Probability: 49.4%","Nigeria beats Senegal<br />Probability: 49.9%","Nigeria beats Serbia<br />Probability: 49.2%","Nigeria beats Egypt<br />Probability: 48.6%","Nigeria beats Sweden<br />Probability: 48.5%","Nigeria beats Switzerland<br />Probability: 46.6%","Nigeria beats Mexico<br />Probability: 46.0%","Nigeria beats Denmark<br />Probability: 45.8%","Nigeria beats Poland<br />Probability: 42.4%","Nigeria beats Russia<br />Probability: 39.7%","Nigeria beats Colombia<br />Probability: 39.2%","Nigeria beats Croatia<br />Probability: 37.3%","Nigeria beats Uruguay<br />Probability: 37.7%","Nigeria beats Portugal<br />Probability: 35.9%","Nigeria beats England<br />Probability: 33.5%","Nigeria beats Belgium<br />Probability: 29.6%","Nigeria beats Argentina<br />Probability: 27.3%","Nigeria beats France<br />Probability: 24.1%","Nigeria beats Spain<br />Probability: 24.2%","Nigeria beats Germany<br />Probability: 22.0%","Nigeria beats Brazil<br />Probability: 21.6%"],["Peru beats Panama<br />Probability: 60.6%","Peru beats Saudi Arabia<br />Probability: 61.6%","Peru beats Tunisia<br />Probability: 58.6%","Peru beats Iran<br />Probability: 55.9%","Peru beats South Korea<br />Probability: 56.3%","Peru beats Costa Rica<br />Probability: 56.9%","Peru beats Morocco<br />Probability: 55.8%","Peru beats Australia<br />Probability: 55.1%","Peru beats Japan<br />Probability: 53.0%","Peru beats Iceland<br />Probability: 50.5%","Peru beats Nigeria<br />Probability: 50.6%","","Peru beats Senegal<br />Probability: 50.5%","Peru beats Serbia<br />Probability: 49.8%","Peru beats Egypt<br />Probability: 49.2%","Peru beats Sweden<br />Probability: 49.1%","Peru beats Switzerland<br />Probability: 47.2%","Peru beats Mexico<br />Probability: 46.6%","Peru beats Denmark<br />Probability: 46.3%","Peru beats Poland<br />Probability: 42.9%","Peru beats Russia<br />Probability: 40.3%","Peru beats Colombia<br />Probability: 39.7%","Peru beats Croatia<br />Probability: 37.8%","Peru beats Uruguay<br />Probability: 38.3%","Peru beats Portugal<br />Probability: 36.4%","Peru beats England<br />Probability: 34.0%","Peru beats Belgium<br />Probability: 30.1%","Peru beats Argentina<br />Probability: 27.8%","Peru beats France<br />Probability: 24.5%","Peru beats Spain<br />Probability: 24.6%","Peru beats Germany<br />Probability: 22.4%","Peru beats Brazil<br />Probability: 22.0%"],["Senegal beats Panama<br />Probability: 60.2%","Senegal beats Saudi Arabia<br />Probability: 61.1%","Senegal beats Tunisia<br />Probability: 58.1%","Senegal beats Iran<br />Probability: 55.5%","Senegal beats South Korea<br />Probability: 55.9%","Senegal beats Costa Rica<br />Probability: 56.4%","Senegal beats Morocco<br />Probability: 55.4%","Senegal beats Australia<br />Probability: 54.7%","Senegal beats Japan<br />Probability: 52.5%","Senegal beats Iceland<br />Probability: 50.0%","Senegal beats Nigeria<br />Probability: 50.1%","Senegal beats Peru<br />Probability: 49.5%","","Senegal beats Serbia<br />Probability: 49.3%","Senegal beats Egypt<br />Probability: 48.7%","Senegal beats Sweden<br />Probability: 48.7%","Senegal beats Switzerland<br />Probability: 46.7%","Senegal beats Mexico<br />Probability: 46.2%","Senegal beats Denmark<br />Probability: 45.9%","Senegal beats Poland<br />Probability: 42.5%","Senegal beats Russia<br />Probability: 39.9%","Senegal beats Colombia<br />Probability: 39.3%","Senegal beats Croatia<br />Probability: 37.4%","Senegal beats Uruguay<br />Probability: 37.9%","Senegal beats Portugal<br />Probability: 36.0%","Senegal beats England<br />Probability: 33.6%","Senegal beats Belgium<br />Probability: 29.8%","Senegal beats Argentina<br />Probability: 27.4%","Senegal beats France<br />Probability: 24.2%","Senegal beats Spain<br />Probability: 24.3%","Senegal beats Germany<br />Probability: 22.1%","Senegal beats Brazil<br />Probability: 21.7%"],["Serbia beats Panama<br />Probability: 60.8%","Serbia beats Saudi Arabia<br />Probability: 61.8%","Serbia beats Tunisia<br />Probability: 58.8%","Serbia beats Iran<br />Probability: 56.2%","Serbia beats South Korea<br />Probability: 56.6%","Serbia beats Costa Rica<br />Probability: 57.1%","Serbia beats Morocco<br />Probability: 56.1%","Serbia beats Australia<br />Probability: 55.4%","Serbia beats Japan<br />Probability: 53.3%","Serbia beats Iceland<br />Probability: 50.8%","Serbia beats Nigeria<br />Probability: 50.8%","Serbia beats Peru<br />Probability: 50.2%","Serbia beats Senegal<br />Probability: 50.7%","","Serbia beats Egypt<br />Probability: 49.4%","Serbia beats Sweden<br />Probability: 49.4%","Serbia beats Switzerland<br />Probability: 47.4%","Serbia beats Mexico<br />Probability: 46.9%","Serbia beats Denmark<br />Probability: 46.6%","Serbia beats Poland<br />Probability: 43.2%","Serbia beats Russia<br />Probability: 40.5%","Serbia beats Colombia<br />Probability: 40.0%","Serbia beats Croatia<br />Probability: 38.1%","Serbia beats Uruguay<br />Probability: 38.5%","Serbia beats Portugal<br />Probability: 36.6%","Serbia beats England<br />Probability: 34.3%","Serbia beats Belgium<br />Probability: 30.4%","Serbia beats Argentina<br />Probability: 28.0%","Serbia beats France<br />Probability: 24.7%","Serbia beats Spain<br />Probability: 24.8%","Serbia beats Germany<br />Probability: 22.6%","Serbia beats Brazil<br />Probability: 22.2%"],["Egypt beats Panama<br />Probability: 61.4%","Egypt beats Saudi Arabia<br />Probability: 62.3%","Egypt beats Tunisia<br />Probability: 59.4%","Egypt beats Iran<br />Probability: 56.7%","Egypt beats South Korea<br />Probability: 57.2%","Egypt beats Costa Rica<br />Probability: 57.7%","Egypt beats Morocco<br />Probability: 56.6%","Egypt beats Australia<br />Probability: 56.0%","Egypt beats Japan<br />Probability: 53.8%","Egypt beats Iceland<br />Probability: 51.3%","Egypt beats Nigeria<br />Probability: 51.4%","Egypt beats Peru<br />Probability: 50.8%","Egypt beats Senegal<br />Probability: 51.3%","Egypt beats Serbia<br />Probability: 50.6%","","Egypt beats Sweden<br />Probability: 50.0%","Egypt beats Switzerland<br />Probability: 48.0%","Egypt beats Mexico<br />Probability: 47.4%","Egypt beats Denmark<br />Probability: 47.2%","Egypt beats Poland<br />Probability: 43.8%","Egypt beats Russia<br />Probability: 41.1%","Egypt beats Colombia<br />Probability: 40.5%","Egypt beats Croatia<br />Probability: 38.6%","Egypt beats Uruguay<br />Probability: 39.1%","Egypt beats Portugal<br />Probability: 37.2%","Egypt beats England<br />Probability: 34.8%","Egypt beats Belgium<br />Probability: 30.8%","Egypt beats Argentina<br />Probability: 28.4%","Egypt beats France<br />Probability: 25.1%","Egypt beats Spain<br />Probability: 25.2%","Egypt beats Germany<br />Probability: 23.0%","Egypt beats Brazil<br />Probability: 22.6%"],["Sweden beats Panama<br />Probability: 61.4%","Sweden beats Saudi Arabia<br />Probability: 62.4%","Sweden beats Tunisia<br />Probability: 59.4%","Sweden beats Iran<br />Probability: 56.8%","Sweden beats South Korea<br />Probability: 57.2%","Sweden beats Costa Rica<br />Probability: 57.7%","Sweden beats Morocco<br />Probability: 56.7%","Sweden beats Australia<br />Probability: 56.0%","Sweden beats Japan<br />Probability: 53.9%","Sweden beats Iceland<br />Probability: 51.4%","Sweden beats Nigeria<br />Probability: 51.5%","Sweden beats Peru<br />Probability: 50.9%","Sweden beats Senegal<br />Probability: 51.3%","Sweden beats Serbia<br />Probability: 50.6%","Sweden beats Egypt<br />Probability: 50.0%","","Sweden beats Switzerland<br />Probability: 48.0%","Sweden beats Mexico<br />Probability: 47.5%","Sweden beats Denmark<br />Probability: 47.2%","Sweden beats Poland<br />Probability: 43.8%","Sweden beats Russia<br />Probability: 41.1%","Sweden beats Colombia<br />Probability: 40.5%","Sweden beats Croatia<br />Probability: 38.6%","Sweden beats Uruguay<br />Probability: 39.1%","Sweden beats Portugal<br />Probability: 37.2%","Sweden beats England<br />Probability: 34.8%","Sweden beats Belgium<br />Probability: 30.9%","Sweden beats Argentina<br />Probability: 28.5%","Sweden beats France<br />Probability: 25.1%","Sweden beats Spain<br />Probability: 25.3%","Sweden beats Germany<br />Probability: 23.0%","Sweden beats Brazil<br />Probability: 22.6%"],["Switzerland beats Panama<br />Probability: 63.3%","Switzerland beats Saudi Arabia<br />Probability: 64.2%","Switzerland beats Tunisia<br />Probability: 61.3%","Switzerland beats Iran<br />Probability: 58.7%","Switzerland beats South Korea<br />Probability: 59.1%","Switzerland beats Costa Rica<br />Probability: 59.7%","Switzerland beats Morocco<br />Probability: 58.6%","Switzerland beats Australia<br />Probability: 57.9%","Switzerland beats Japan<br />Probability: 55.8%","Switzerland beats Iceland<br />Probability: 53.3%","Switzerland beats Nigeria<br />Probability: 53.4%","Switzerland beats Peru<br />Probability: 52.8%","Switzerland beats Senegal<br />Probability: 53.3%","Switzerland beats Serbia<br />Probability: 52.6%","Switzerland beats Egypt<br />Probability: 52.0%","Switzerland beats Sweden<br />Probability: 52.0%","","Switzerland beats Mexico<br />Probability: 49.5%","Switzerland beats Denmark<br />Probability: 49.2%","Switzerland beats Poland<br />Probability: 45.7%","Switzerland beats Russia<br />Probability: 43.1%","Switzerland beats Colombia<br />Probability: 42.5%","Switzerland beats Croatia<br />Probability: 40.5%","Switzerland beats Uruguay<br />Probability: 41.0%","Switzerland beats Portugal<br />Probability: 39.1%","Switzerland beats England<br />Probability: 36.6%","Switzerland beats Belgium<br />Probability: 32.6%","Switzerland beats Argentina<br />Probability: 30.1%","Switzerland beats France<br />Probability: 26.7%","Switzerland beats Spain<br />Probability: 26.8%","Switzerland beats Germany<br />Probability: 24.4%","Switzerland beats Brazil<br />Probability: 24.0%"],["Mexico beats Panama<br />Probability: 63.8%","Mexico beats Saudi Arabia<br />Probability: 64.7%","Mexico beats Tunisia<br />Probability: 61.8%","Mexico beats Iran<br />Probability: 59.2%","Mexico beats South Korea<br />Probability: 59.6%","Mexico beats Costa Rica<br />Probability: 60.2%","Mexico beats Morocco<br />Probability: 59.1%","Mexico beats Australia<br />Probability: 58.5%","Mexico beats Japan<br />Probability: 56.4%","Mexico beats Iceland<br />Probability: 53.9%","Mexico beats Nigeria<br />Probability: 54.0%","Mexico beats Peru<br />Probability: 53.4%","Mexico beats Senegal<br />Probability: 53.8%","Mexico beats Serbia<br />Probability: 53.1%","Mexico beats Egypt<br />Probability: 52.6%","Mexico beats Sweden<br />Probability: 52.5%","Mexico beats Switzerland<br />Probability: 50.5%","","Mexico beats Denmark<br />Probability: 49.7%","Mexico beats Poland<br />Probability: 46.3%","Mexico beats Russia<br />Probability: 43.6%","Mexico beats Colombia<br />Probability: 43.0%","Mexico beats Croatia<br />Probability: 41.1%","Mexico beats Uruguay<br />Probability: 41.5%","Mexico beats Portugal<br />Probability: 39.6%","Mexico beats England<br />Probability: 37.1%","Mexico beats Belgium<br />Probability: 33.1%","Mexico beats Argentina<br />Probability: 30.6%","Mexico beats France<br />Probability: 27.1%","Mexico beats Spain<br />Probability: 27.2%","Mexico beats Germany<br />Probability: 24.8%","Mexico beats Brazil<br />Probability: 24.4%"],["Denmark beats Panama<br />Probability: 64.0%","Denmark beats Saudi Arabia<br />Probability: 65.0%","Denmark beats Tunisia<br />Probability: 62.1%","Denmark beats Iran<br />Probability: 59.5%","Denmark beats South Korea<br />Probability: 59.9%","Denmark beats Costa Rica<br />Probability: 60.4%","Denmark beats Morocco<br />Probability: 59.4%","Denmark beats Australia<br />Probability: 58.7%","Denmark beats Japan<br />Probability: 56.6%","Denmark beats Iceland<br />Probability: 54.2%","Denmark beats Nigeria<br />Probability: 54.2%","Denmark beats Peru<br />Probability: 53.7%","Denmark beats Senegal<br />Probability: 54.1%","Denmark beats Serbia<br />Probability: 53.4%","Denmark beats Egypt<br />Probability: 52.8%","Denmark beats Sweden<br />Probability: 52.8%","Denmark beats Switzerland<br />Probability: 50.8%","Denmark beats Mexico<br />Probability: 50.3%","","Denmark beats Poland<br />Probability: 46.6%","Denmark beats Russia<br />Probability: 43.9%","Denmark beats Colombia<br />Probability: 43.3%","Denmark beats Croatia<br />Probability: 41.3%","Denmark beats Uruguay<br />Probability: 41.8%","Denmark beats Portugal<br />Probability: 39.9%","Denmark beats England<br />Probability: 37.4%","Denmark beats Belgium<br />Probability: 33.3%","Denmark beats Argentina<br />Probability: 30.8%","Denmark beats France<br />Probability: 27.3%","Denmark beats Spain<br />Probability: 27.5%","Denmark beats Germany<br />Probability: 25.0%","Denmark beats Brazil<br />Probability: 24.6%"],["Poland beats Panama<br />Probability: 67.1%","Poland beats Saudi Arabia<br />Probability: 68.0%","Poland beats Tunisia<br />Probability: 65.3%","Poland beats Iran<br />Probability: 62.8%","Poland beats South Korea<br />Probability: 63.2%","Poland beats Costa Rica<br />Probability: 63.7%","Poland beats Morocco<br />Probability: 62.7%","Poland beats Australia<br />Probability: 62.0%","Poland beats Japan<br />Probability: 60.0%","Poland beats Iceland<br />Probability: 57.6%","Poland beats Nigeria<br />Probability: 57.6%","Poland beats Peru<br />Probability: 57.1%","Poland beats Senegal<br />Probability: 57.5%","Poland beats Serbia<br />Probability: 56.8%","Poland beats Egypt<br />Probability: 56.2%","Poland beats Sweden<br />Probability: 56.2%","Poland beats Switzerland<br />Probability: 54.3%","Poland beats Mexico<br />Probability: 53.7%","Poland beats Denmark<br />Probability: 53.4%","","Poland beats Russia<br />Probability: 47.3%","Poland beats Colombia<br />Probability: 46.7%","Poland beats Croatia<br />Probability: 44.7%","Poland beats Uruguay<br />Probability: 45.2%","Poland beats Portugal<br />Probability: 43.2%","Poland beats England<br />Probability: 40.7%","Poland beats Belgium<br />Probability: 36.4%","Poland beats Argentina<br />Probability: 33.8%","Poland beats France<br />Probability: 30.1%","Poland beats Spain<br />Probability: 30.3%","Poland beats Germany<br />Probability: 27.7%","Poland beats Brazil<br />Probability: 27.3%"],["Russia beats Panama<br />Probability: 69.5%","Russia beats Saudi Arabia<br />Probability: 70.3%","Russia beats Tunisia<br />Probability: 67.7%","Russia beats Iran<br />Probability: 65.3%","Russia beats South Korea<br />Probability: 65.7%","Russia beats Costa Rica<br />Probability: 66.2%","Russia beats Morocco<br />Probability: 65.2%","Russia beats Australia<br />Probability: 64.5%","Russia beats Japan<br />Probability: 62.5%","Russia beats Iceland<br />Probability: 60.2%","Russia beats Nigeria<br />Probability: 60.3%","Russia beats Peru<br />Probability: 59.7%","Russia beats Senegal<br />Probability: 60.1%","Russia beats Serbia<br />Probability: 59.5%","Russia beats Egypt<br />Probability: 58.9%","Russia beats Sweden<br />Probability: 58.9%","Russia beats Switzerland<br />Probability: 56.9%","Russia beats Mexico<br />Probability: 56.4%","Russia beats Denmark<br />Probability: 56.1%","Russia beats Poland<br />Probability: 52.7%","","Russia beats Colombia<br />Probability: 49.4%","Russia beats Croatia<br />Probability: 47.4%","Russia beats Uruguay<br />Probability: 47.9%","Russia beats Portugal<br />Probability: 45.9%","Russia beats England<br />Probability: 43.3%","Russia beats Belgium<br />Probability: 39.0%","Russia beats Argentina<br />Probability: 36.3%","Russia beats France<br />Probability: 32.4%","Russia beats Spain<br />Probability: 32.6%","Russia beats Germany<br />Probability: 29.9%","Russia beats Brazil<br />Probability: 29.5%"],["Colombia beats Panama<br />Probability: 70.0%","Colombia beats Saudi Arabia<br />Probability: 70.9%","Colombia beats Tunisia<br />Probability: 68.2%","Colombia beats Iran<br />Probability: 65.8%","Colombia beats South Korea<br />Probability: 66.2%","Colombia beats Costa Rica<br />Probability: 66.7%","Colombia beats Morocco<br />Probability: 65.7%","Colombia beats Australia<br />Probability: 65.1%","Colombia beats Japan<br />Probability: 63.1%","Colombia beats Iceland<br />Probability: 60.8%","Colombia beats Nigeria<br />Probability: 60.8%","Colombia beats Peru<br />Probability: 60.3%","Colombia beats Senegal<br />Probability: 60.7%","Colombia beats Serbia<br />Probability: 60.0%","Colombia beats Egypt<br />Probability: 59.5%","Colombia beats Sweden<br />Probability: 59.5%","Colombia beats Switzerland<br />Probability: 57.5%","Colombia beats Mexico<br />Probability: 57.0%","Colombia beats Denmark<br />Probability: 56.7%","Colombia beats Poland<br />Probability: 53.3%","Colombia beats Russia<br />Probability: 50.6%","","Colombia beats Croatia<br />Probability: 48.0%","Colombia beats Uruguay<br />Probability: 48.5%","Colombia beats Portugal<br />Probability: 46.5%","Colombia beats England<br />Probability: 43.9%","Colombia beats Belgium<br />Probability: 39.6%","Colombia beats Argentina<br />Probability: 36.9%","Colombia beats France<br />Probability: 33.0%","Colombia beats Spain<br />Probability: 33.2%","Colombia beats Germany<br />Probability: 30.5%","Colombia beats Brazil<br />Probability: 30.0%"],["Croatia beats Panama<br />Probability: 71.7%","Croatia beats Saudi Arabia<br />Probability: 72.5%","Croatia beats Tunisia<br />Probability: 69.9%","Croatia beats Iran<br />Probability: 67.6%","Croatia beats South Korea<br />Probability: 68.0%","Croatia beats Costa Rica<br />Probability: 68.5%","Croatia beats Morocco<br />Probability: 67.5%","Croatia beats Australia<br />Probability: 66.9%","Croatia beats Japan<br />Probability: 65.0%","Croatia beats Iceland<br />Probability: 62.7%","Croatia beats Nigeria<br />Probability: 62.7%","Croatia beats Peru<br />Probability: 62.2%","Croatia beats Senegal<br />Probability: 62.6%","Croatia beats Serbia<br />Probability: 61.9%","Croatia beats Egypt<br />Probability: 61.4%","Croatia beats Sweden<br />Probability: 61.4%","Croatia beats Switzerland<br />Probability: 59.5%","Croatia beats Mexico<br />Probability: 58.9%","Croatia beats Denmark<br />Probability: 58.7%","Croatia beats Poland<br />Probability: 55.3%","Croatia beats Russia<br />Probability: 52.6%","Croatia beats Colombia<br />Probability: 52.0%","","Croatia beats Uruguay<br />Probability: 50.5%","Croatia beats Portugal<br />Probability: 48.5%","Croatia beats England<br />Probability: 45.9%","Croatia beats Belgium<br />Probability: 41.5%","Croatia beats Argentina<br />Probability: 38.7%","Croatia beats France<br />Probability: 34.8%","Croatia beats Spain<br />Probability: 35.0%","Croatia beats Germany<br />Probability: 32.2%","Croatia beats Brazil<br />Probability: 31.7%"],["Uruguay beats Panama<br />Probability: 71.2%","Uruguay beats Saudi Arabia<br />Probability: 72.1%","Uruguay beats Tunisia<br />Probability: 69.5%","Uruguay beats Iran<br />Probability: 67.1%","Uruguay beats South Korea<br />Probability: 67.5%","Uruguay beats Costa Rica<br />Probability: 68.0%","Uruguay beats Morocco<br />Probability: 67.1%","Uruguay beats Australia<br />Probability: 66.4%","Uruguay beats Japan<br />Probability: 64.5%","Uruguay beats Iceland<br />Probability: 62.2%","Uruguay beats Nigeria<br />Probability: 62.3%","Uruguay beats Peru<br />Probability: 61.7%","Uruguay beats Senegal<br />Probability: 62.1%","Uruguay beats Serbia<br />Probability: 61.5%","Uruguay beats Egypt<br />Probability: 60.9%","Uruguay beats Sweden<br />Probability: 60.9%","Uruguay beats Switzerland<br />Probability: 59.0%","Uruguay beats Mexico<br />Probability: 58.5%","Uruguay beats Denmark<br />Probability: 58.2%","Uruguay beats Poland<br />Probability: 54.8%","Uruguay beats Russia<br />Probability: 52.1%","Uruguay beats Colombia<br />Probability: 51.5%","Uruguay beats Croatia<br />Probability: 49.5%","","Uruguay beats Portugal<br />Probability: 48.0%","Uruguay beats England<br />Probability: 45.4%","Uruguay beats Belgium<br />Probability: 41.0%","Uruguay beats Argentina<br />Probability: 38.3%","Uruguay beats France<br />Probability: 34.3%","Uruguay beats Spain<br />Probability: 34.5%","Uruguay beats Germany<br />Probability: 31.7%","Uruguay beats Brazil<br />Probability: 31.2%"],["Portugal beats Panama<br />Probability: 72.9%","Portugal beats Saudi Arabia<br />Probability: 73.7%","Portugal beats Tunisia<br />Probability: 71.2%","Portugal beats Iran<br />Probability: 68.9%","Portugal beats South Korea<br />Probability: 69.3%","Portugal beats Costa Rica<br />Probability: 69.7%","Portugal beats Morocco<br />Probability: 68.8%","Portugal beats Australia<br />Probability: 68.2%","Portugal beats Japan<br />Probability: 66.3%","Portugal beats Iceland<br />Probability: 64.1%","Portugal beats Nigeria<br />Probability: 64.1%","Portugal beats Peru<br />Probability: 63.6%","Portugal beats Senegal<br />Probability: 64.0%","Portugal beats Serbia<br />Probability: 63.4%","Portugal beats Egypt<br />Probability: 62.8%","Portugal beats Sweden<br />Probability: 62.8%","Portugal beats Switzerland<br />Probability: 60.9%","Portugal beats Mexico<br />Probability: 60.4%","Portugal beats Denmark<br />Probability: 60.1%","Portugal beats Poland<br />Probability: 56.8%","Portugal beats Russia<br />Probability: 54.1%","Portugal beats Colombia<br />Probability: 53.5%","Portugal beats Croatia<br />Probability: 51.5%","Portugal beats Uruguay<br />Probability: 52.0%","","Portugal beats England<br />Probability: 47.4%","Portugal beats Belgium<br />Probability: 43.0%","Portugal beats Argentina<br />Probability: 40.2%","Portugal beats France<br />Probability: 36.2%","Portugal beats Spain<br />Probability: 36.3%","Portugal beats Germany<br />Probability: 33.5%","Portugal beats Brazil<br />Probability: 33.0%"],["England beats Panama<br />Probability: 74.9%","England beats Saudi Arabia<br />Probability: 75.6%","England beats Tunisia<br />Probability: 73.3%","England beats Iran<br />Probability: 71.1%","England beats South Korea<br />Probability: 71.4%","England beats Costa Rica<br />Probability: 71.9%","England beats Morocco<br />Probability: 71.0%","England beats Australia<br />Probability: 70.4%","England beats Japan<br />Probability: 68.6%","England beats Iceland<br />Probability: 66.4%","England beats Nigeria<br />Probability: 66.5%","England beats Peru<br />Probability: 66.0%","England beats Senegal<br />Probability: 66.4%","England beats Serbia<br />Probability: 65.7%","England beats Egypt<br />Probability: 65.2%","England beats Sweden<br />Probability: 65.2%","England beats Switzerland<br />Probability: 63.4%","England beats Mexico<br />Probability: 62.9%","England beats Denmark<br />Probability: 62.6%","England beats Poland<br />Probability: 59.3%","England beats Russia<br />Probability: 56.7%","England beats Colombia<br />Probability: 56.1%","England beats Croatia<br />Probability: 54.1%","England beats Uruguay<br />Probability: 54.6%","England beats Portugal<br />Probability: 52.6%","","England beats Belgium<br />Probability: 45.5%","England beats Argentina<br />Probability: 42.7%","England beats France<br />Probability: 38.6%","England beats Spain<br />Probability: 38.8%","England beats Germany<br />Probability: 35.9%","England beats Brazil<br />Probability: 35.4%"],["Belgium beats Panama<br />Probability: 78.1%","Belgium beats Saudi Arabia<br />Probability: 78.8%","Belgium beats Tunisia<br />Probability: 76.6%","Belgium beats Iran<br />Probability: 74.6%","Belgium beats South Korea<br />Probability: 74.9%","Belgium beats Costa Rica<br />Probability: 75.4%","Belgium beats Morocco<br />Probability: 74.6%","Belgium beats Australia<br />Probability: 74.0%","Belgium beats Japan<br />Probability: 72.3%","Belgium beats Iceland<br />Probability: 70.3%","Belgium beats Nigeria<br />Probability: 70.4%","Belgium beats Peru<br />Probability: 69.9%","Belgium beats Senegal<br />Probability: 70.2%","Belgium beats Serbia<br />Probability: 69.6%","Belgium beats Egypt<br />Probability: 69.2%","Belgium beats Sweden<br />Probability: 69.1%","Belgium beats Switzerland<br />Probability: 67.4%","Belgium beats Mexico<br />Probability: 66.9%","Belgium beats Denmark<br />Probability: 66.7%","Belgium beats Poland<br />Probability: 63.6%","Belgium beats Russia<br />Probability: 61.0%","Belgium beats Colombia<br />Probability: 60.4%","Belgium beats Croatia<br />Probability: 58.5%","Belgium beats Uruguay<br />Probability: 59.0%","Belgium beats Portugal<br />Probability: 57.0%","Belgium beats England<br />Probability: 54.5%","","Belgium beats Argentina<br />Probability: 47.1%","Belgium beats France<br />Probability: 42.9%","Belgium beats Spain<br />Probability: 43.1%","Belgium beats Germany<br />Probability: 40.1%","Belgium beats Brazil<br />Probability: 39.5%"],["Argentina beats Panama<br />Probability: 80.0%","Argentina beats Saudi Arabia<br />Probability: 80.6%","Argentina beats Tunisia<br />Probability: 78.6%","Argentina beats Iran<br />Probability: 76.7%","Argentina beats South Korea<br />Probability: 77.0%","Argentina beats Costa Rica<br />Probability: 77.4%","Argentina beats Morocco<br />Probability: 76.7%","Argentina beats Australia<br />Probability: 76.2%","Argentina beats Japan<br />Probability: 74.6%","Argentina beats Iceland<br />Probability: 72.6%","Argentina beats Nigeria<br />Probability: 72.7%","Argentina beats Peru<br />Probability: 72.2%","Argentina beats Senegal<br />Probability: 72.6%","Argentina beats Serbia<br />Probability: 72.0%","Argentina beats Egypt<br />Probability: 71.6%","Argentina beats Sweden<br />Probability: 71.5%","Argentina beats Switzerland<br />Probability: 69.9%","Argentina beats Mexico<br />Probability: 69.4%","Argentina beats Denmark<br />Probability: 69.2%","Argentina beats Poland<br />Probability: 66.2%","Argentina beats Russia<br />Probability: 63.7%","Argentina beats Colombia<br />Probability: 63.1%","Argentina beats Croatia<br />Probability: 61.3%","Argentina beats Uruguay<br />Probability: 61.7%","Argentina beats Portugal<br />Probability: 59.8%","Argentina beats England<br />Probability: 57.3%","Argentina beats Belgium<br />Probability: 52.9%","","Argentina beats France<br />Probability: 45.7%","Argentina beats Spain<br />Probability: 45.9%","Argentina beats Germany<br />Probability: 42.9%","Argentina beats Brazil<br />Probability: 42.3%"],["France beats Panama<br />Probability: 82.6%","France beats Saudi Arabia<br />Probability: 83.2%","France beats Tunisia<br />Probability: 81.3%","France beats Iran<br />Probability: 79.6%","France beats South Korea<br />Probability: 79.9%","France beats Costa Rica<br />Probability: 80.3%","France beats Morocco<br />Probability: 79.6%","France beats Australia<br />Probability: 79.1%","France beats Japan<br />Probability: 77.7%","France beats Iceland<br />Probability: 75.9%","France beats Nigeria<br />Probability: 75.9%","France beats Peru<br />Probability: 75.5%","France beats Senegal<br />Probability: 75.8%","France beats Serbia<br />Probability: 75.3%","France beats Egypt<br />Probability: 74.9%","France beats Sweden<br />Probability: 74.9%","France beats Switzerland<br />Probability: 73.3%","France beats Mexico<br />Probability: 72.9%","France beats Denmark<br />Probability: 72.7%","France beats Poland<br />Probability: 69.9%","France beats Russia<br />Probability: 67.6%","France beats Colombia<br />Probability: 67.0%","France beats Croatia<br />Probability: 65.2%","France beats Uruguay<br />Probability: 65.7%","France beats Portugal<br />Probability: 63.8%","France beats England<br />Probability: 61.4%","France beats Belgium<br />Probability: 57.1%","France beats Argentina<br />Probability: 54.3%","","France beats Spain<br />Probability: 50.2%","France beats Germany<br />Probability: 47.1%","France beats Brazil<br />Probability: 46.5%"],["Spain beats Panama<br />Probability: 82.5%","Spain beats Saudi Arabia<br />Probability: 83.1%","Spain beats Tunisia<br />Probability: 81.2%","Spain beats Iran<br />Probability: 79.5%","Spain beats South Korea<br />Probability: 79.8%","Spain beats Costa Rica<br />Probability: 80.2%","Spain beats Morocco<br />Probability: 79.5%","Spain beats Australia<br />Probability: 79.0%","Spain beats Japan<br />Probability: 77.5%","Spain beats Iceland<br />Probability: 75.7%","Spain beats Nigeria<br />Probability: 75.8%","Spain beats Peru<br />Probability: 75.4%","Spain beats Senegal<br />Probability: 75.7%","Spain beats Serbia<br />Probability: 75.2%","Spain beats Egypt<br />Probability: 74.8%","Spain beats Sweden<br />Probability: 74.7%","Spain beats Switzerland<br />Probability: 73.2%","Spain beats Mexico<br />Probability: 72.8%","Spain beats Denmark<br />Probability: 72.5%","Spain beats Poland<br />Probability: 69.7%","Spain beats Russia<br />Probability: 67.4%","Spain beats Colombia<br />Probability: 66.8%","Spain beats Croatia<br />Probability: 65.0%","Spain beats Uruguay<br />Probability: 65.5%","Spain beats Portugal<br />Probability: 63.7%","Spain beats England<br />Probability: 61.2%","Spain beats Belgium<br />Probability: 56.9%","Spain beats Argentina<br />Probability: 54.1%","Spain beats France<br />Probability: 49.8%","","Spain beats Germany<br />Probability: 46.9%","Spain beats Brazil<br />Probability: 46.3%"],["Germany beats Panama<br />Probability: 84.2%","Germany beats Saudi Arabia<br />Probability: 84.7%","Germany beats Tunisia<br />Probability: 83.0%","Germany beats Iran<br />Probability: 81.5%","Germany beats South Korea<br />Probability: 81.7%","Germany beats Costa Rica<br />Probability: 82.1%","Germany beats Morocco<br />Probability: 81.4%","Germany beats Australia<br />Probability: 81.0%","Germany beats Japan<br />Probability: 79.6%","Germany beats Iceland<br />Probability: 77.9%","Germany beats Nigeria<br />Probability: 78.0%","Germany beats Peru<br />Probability: 77.6%","Germany beats Senegal<br />Probability: 77.9%","Germany beats Serbia<br />Probability: 77.4%","Germany beats Egypt<br />Probability: 77.0%","Germany beats Sweden<br />Probability: 77.0%","Germany beats Switzerland<br />Probability: 75.6%","Germany beats Mexico<br />Probability: 75.2%","Germany beats Denmark<br />Probability: 75.0%","Germany beats Poland<br />Probability: 72.3%","Germany beats Russia<br />Probability: 70.1%","Germany beats Colombia<br />Probability: 69.5%","Germany beats Croatia<br />Probability: 67.8%","Germany beats Uruguay<br />Probability: 68.3%","Germany beats Portugal<br />Probability: 66.5%","Germany beats England<br />Probability: 64.1%","Germany beats Belgium<br />Probability: 59.9%","Germany beats Argentina<br />Probability: 57.1%","Germany beats France<br />Probability: 52.9%","Germany beats Spain<br />Probability: 53.1%","","Germany beats Brazil<br />Probability: 49.4%"],["Brazil beats Panama<br />Probability: 84.5%","Brazil beats Saudi Arabia<br />Probability: 85.0%","Brazil beats Tunisia<br />Probability: 83.4%","Brazil beats Iran<br />Probability: 81.8%","Brazil beats South Korea<br />Probability: 82.1%","Brazil beats Costa Rica<br />Probability: 82.4%","Brazil beats Morocco<br />Probability: 81.8%","Brazil beats Australia<br />Probability: 81.3%","Brazil beats Japan<br />Probability: 80.0%","Brazil beats Iceland<br />Probability: 78.3%","Brazil beats Nigeria<br />Probability: 78.4%","Brazil beats Peru<br />Probability: 78.0%","Brazil beats Senegal<br />Probability: 78.3%","Brazil beats Serbia<br />Probability: 77.8%","Brazil beats Egypt<br />Probability: 77.4%","Brazil beats Sweden<br />Probability: 77.4%","Brazil beats Switzerland<br />Probability: 76.0%","Brazil beats Mexico<br />Probability: 75.6%","Brazil beats Denmark<br />Probability: 75.4%","Brazil beats Poland<br />Probability: 72.7%","Brazil beats Russia<br />Probability: 70.5%","Brazil beats Colombia<br />Probability: 70.0%","Brazil beats Croatia<br />Probability: 68.3%","Brazil beats Uruguay<br />Probability: 68.8%","Brazil beats Portugal<br />Probability: 67.0%","Brazil beats England<br />Probability: 64.6%","Brazil beats Belgium<br />Probability: 60.5%","Brazil beats Argentina<br />Probability: 57.7%","Brazil beats France<br />Probability: 53.5%","Brazil beats Spain<br />Probability: 53.7%","Brazil beats Germany<br />Probability: 50.6%",""]],"hoverinfo":"text","colors":["function (x) ","roundcolor(cbind(palette[[1L]](x), palette[[2L]](x), palette[[3L]](x), "," if (alpha) palette[[4L]](x))) * 255"],"alpha":1,"sizes":[10,100],"type":"heatmap"}},"layout":{"margin":{"b":40,"l":60,"t":25,"r":10},"xaxis":{"domain":[0,1],"scaleanchor":"y","scaleratio":1,"constrain":"domain","automargin":true,"mirror":true},"yaxis":{"domain":[0,1],"constrain":"domain","automargin":true},"hovermode":"closest","showlegend":false,"legend":{"y":0.5,"yanchor":"top"}},"source":"A","config":{"modeBarButtonsToAdd":[{"name":"Collaborate","icon":{"width":1000,"ascent":500,"descent":-50,"path":"M487 375c7-10 9-23 5-36l-79-259c-3-12-11-23-22-31-11-8-22-12-35-12l-263 0c-15 0-29 5-43 15-13 10-23 23-28 37-5 13-5 25-1 37 0 0 0 3 1 7 1 5 1 8 1 11 0 2 0 4-1 6 0 3-1 5-1 6 1 2 2 4 3 6 1 2 2 4 4 6 2 3 4 5 5 7 5 7 9 16 13 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beats Saudi Arabia<br />Probability: 51.0%","Panama beats Tunisia<br />Probability: 47.9%","Panama beats Iran<br />Probability: 45.2%","Panama beats South Korea<br />Probability: 45.6%","Panama beats Costa Rica<br />Probability: 46.2%","Panama beats Morocco<br />Probability: 45.1%","Panama beats Australia<br />Probability: 44.4%","Panama beats Japan<br />Probability: 42.3%","Panama beats Iceland<br />Probability: 39.9%","Panama beats Nigeria<br />Probability: 40.0%","Panama beats Peru<br />Probability: 39.4%","Panama beats Senegal<br />Probability: 39.8%","Panama beats Serbia<br />Probability: 39.2%","Panama beats Egypt<br />Probability: 38.6%","Panama beats Sweden<br />Probability: 38.6%","Panama beats Switzerland<br />Probability: 36.7%","Panama beats Mexico<br />Probability: 36.2%","Panama beats Denmark<br />Probability: 36.0%","Panama beats Poland<br />Probability: 32.9%","Panama beats Russia<br />Probability: 30.5%","Panama beats Colombia<br />Probability: 30.0%","Panama beats Croatia<br />Probability: 28.3%","Panama beats Uruguay<br />Probability: 28.8%","Panama beats Portugal<br />Probability: 27.1%","Panama beats England<br />Probability: 25.1%","Panama beats Belgium<br />Probability: 21.9%","Panama beats Argentina<br />Probability: 20.0%","Panama beats France<br />Probability: 17.4%","Panama beats Spain<br />Probability: 17.5%","Panama beats Germany<br />Probability: 15.8%","Panama beats Brazil<br />Probability: 15.5%"],["Saudi Arabia beats Panama<br />Probability: 49.0%","","Saudi Arabia beats Tunisia<br />Probability: 46.9%","Saudi Arabia beats Iran<br />Probability: 44.2%","Saudi Arabia beats South Korea<br />Probability: 44.6%","Saudi Arabia beats Costa Rica<br />Probability: 45.2%","Saudi Arabia beats Morocco<br />Probability: 44.1%","Saudi Arabia beats Australia<br />Probability: 43.4%","Saudi Arabia beats Japan<br />Probability: 41.3%","Saudi Arabia beats Iceland<br />Probability: 38.9%","Saudi Arabia beats Nigeria<br />Probability: 39.0%","Saudi Arabia beats Peru<br />Probability: 38.4%","Saudi Arabia beats Senegal<br />Probability: 38.9%","Saudi Arabia beats Serbia<br />Probability: 38.2%","Saudi Arabia beats Egypt<br />Probability: 37.7%","Saudi Arabia beats Sweden<br />Probability: 37.6%","Saudi Arabia beats Switzerland<br />Probability: 35.8%","Saudi Arabia beats Mexico<br />Probability: 35.3%","Saudi Arabia beats Denmark<br />Probability: 35.0%","Saudi Arabia beats Poland<br />Probability: 32.0%","Saudi Arabia beats Russia<br />Probability: 29.7%","Saudi Arabia beats Colombia<br />Probability: 29.1%","Saudi Arabia beats Croatia<br />Probability: 27.5%","Saudi Arabia beats Uruguay<br />Probability: 27.9%","Saudi Arabia beats Portugal<br />Probability: 26.3%","Saudi Arabia beats England<br />Probability: 24.4%","Saudi Arabia beats Belgium<br />Probability: 21.2%","Saudi Arabia beats Argentina<br />Probability: 19.4%","Saudi Arabia beats France<br />Probability: 16.8%","Saudi Arabia beats Spain<br />Probability: 16.9%","Saudi Arabia beats Germany<br />Probability: 15.3%","Saudi Arabia beats Brazil<br />Probability: 15.0%"],["Tunisia beats Panama<br />Probability: 52.1%","Tunisia beats Saudi Arabia<br />Probability: 53.1%","","Tunisia beats Iran<br />Probability: 47.3%","Tunisia beats South Korea<br />Probability: 47.7%","Tunisia beats Costa Rica<br />Probability: 48.3%","Tunisia beats Morocco<br />Probability: 47.2%","Tunisia beats Australia<br />Probability: 46.5%","Tunisia beats Japan<br />Probability: 44.4%","Tunisia beats Iceland<br />Probability: 41.9%","Tunisia beats Nigeria<br />Probability: 42.0%","Tunisia beats Peru<br />Probability: 41.4%","Tunisia beats Senegal<br />Probability: 41.9%","Tunisia beats Serbia<br />Probability: 41.2%","Tunisia beats Egypt<br />Probability: 40.6%","Tunisia beats Sweden<br />Probability: 40.6%","Tunisia beats Switzerland<br />Probability: 38.7%","Tunisia beats Mexico<br />Probability: 38.2%","Tunisia beats Denmark<br />Probability: 37.9%","Tunisia beats Poland<br />Probability: 34.7%","Tunisia beats Russia<br />Probability: 32.3%","Tunisia beats Colombia<br />Probability: 31.8%","Tunisia beats Croatia<br />Probability: 30.1%","Tunisia beats Uruguay<br />Probability: 30.5%","Tunisia beats Portugal<br />Probability: 28.8%","Tunisia beats England<br />Probability: 26.7%","Tunisia beats Belgium<br />Probability: 23.4%","Tunisia beats Argentina<br />Probability: 21.4%","Tunisia beats France<br />Probability: 18.7%","Tunisia beats Spain<br />Probability: 18.8%","Tunisia beats Germany<br />Probability: 17.0%","Tunisia beats Brazil<br />Probability: 16.6%"],["Iran beats Panama<br />Probability: 54.8%","Iran beats Saudi Arabia<br />Probability: 55.8%","Iran beats Tunisia<br />Probability: 52.7%","","Iran beats South Korea<br />Probability: 50.4%","Iran beats Costa Rica<br />Probability: 51.0%","Iran beats Morocco<br />Probability: 49.9%","Iran beats Australia<br />Probability: 49.2%","Iran beats Japan<br />Probability: 47.1%","Iran beats Iceland<br />Probability: 44.6%","Iran beats Nigeria<br />Probability: 44.7%","Iran beats Peru<br />Probability: 44.1%","Iran beats Senegal<br />Probability: 44.5%","Iran beats Serbia<br />Probability: 43.8%","Iran beats Egypt<br />Probability: 43.3%","Iran beats Sweden<br />Probability: 43.2%","Iran beats Switzerland<br />Probability: 41.3%","Iran beats Mexico<br />Probability: 40.8%","Iran beats Denmark<br />Probability: 40.5%","Iran beats Poland<br />Probability: 37.2%","Iran beats Russia<br />Probability: 34.7%","Iran beats Colombia<br />Probability: 34.2%","Iran beats Croatia<br />Probability: 32.4%","Iran beats Uruguay<br />Probability: 32.9%","Iran beats Portugal<br />Probability: 31.1%","Iran beats England<br />Probability: 28.9%","Iran beats Belgium<br />Probability: 25.4%","Iran beats Argentina<br />Probability: 23.3%","Iran beats France<br />Probability: 20.4%","Iran beats Spain<br />Probability: 20.5%","Iran beats Germany<br />Probability: 18.5%","Iran beats Brazil<br />Probability: 18.2%"],["South Korea beats Panama<br />Probability: 54.4%","South Korea beats Saudi Arabia<br />Probability: 55.4%","South Korea beats Tunisia<br />Probability: 52.3%","South Korea beats Iran<br />Probability: 49.6%","","South Korea beats Costa Rica<br />Probability: 50.6%","South Korea beats Morocco<br />Probability: 49.5%","South Korea beats Australia<br />Probability: 48.8%","South Korea beats Japan<br />Probability: 46.6%","South Korea beats Iceland<br />Probability: 44.2%","South Korea beats Nigeria<br />Probability: 44.2%","South Korea beats Peru<br />Probability: 43.7%","South Korea beats Senegal<br />Probability: 44.1%","South Korea beats Serbia<br />Probability: 43.4%","South Korea beats Egypt<br />Probability: 42.8%","South Korea beats Sweden<br />Probability: 42.8%","South Korea beats Switzerland<br />Probability: 40.9%","South Korea beats Mexico<br />Probability: 40.4%","South Korea beats Denmark<br />Probability: 40.1%","South Korea beats Poland<br />Probability: 36.8%","South Korea beats Russia<br />Probability: 34.3%","South Korea beats Colombia<br />Probability: 33.8%","South Korea beats Croatia<br />Probability: 32.0%","South Korea beats Uruguay<br />Probability: 32.5%","South Korea beats Portugal<br />Probability: 30.7%","South Korea beats England<br />Probability: 28.6%","South Korea beats Belgium<br />Probability: 25.1%","South Korea beats Argentina<br />Probability: 23.0%","South Korea beats France<br />Probability: 20.1%","South Korea beats Spain<br />Probability: 20.2%","South Korea beats Germany<br />Probability: 18.3%","South Korea beats Brazil<br />Probability: 17.9%"],["Costa Rica beats Panama<br />Probability: 53.8%","Costa Rica beats Saudi Arabia<br />Probability: 54.8%","Costa Rica beats Tunisia<br />Probability: 51.7%","Costa Rica beats Iran<br />Probability: 49.0%","Costa Rica beats South Korea<br />Probability: 49.4%","","Costa Rica beats Morocco<br />Probability: 48.9%","Costa Rica beats Australia<br />Probability: 48.2%","Costa Rica beats Japan<br />Probability: 46.1%","Costa Rica beats Iceland<br />Probability: 43.6%","Costa Rica beats Nigeria<br />Probability: 43.7%","Costa Rica beats Peru<br />Probability: 43.1%","Costa Rica beats Senegal<br />Probability: 43.6%","Costa Rica beats Serbia<br />Probability: 42.9%","Costa Rica beats Egypt<br />Probability: 42.3%","Costa Rica beats Sweden<br />Probability: 42.3%","Costa Rica beats Switzerland<br />Probability: 40.3%","Costa Rica beats Mexico<br />Probability: 39.8%","Costa Rica beats Denmark<br />Probability: 39.6%","Costa Rica beats Poland<br />Probability: 36.3%","Costa Rica beats Russia<br />Probability: 33.8%","Costa Rica beats Colombia<br />Probability: 33.3%","Costa Rica beats Croatia<br />Probability: 31.5%","Costa Rica beats Uruguay<br />Probability: 32.0%","Costa Rica beats Portugal<br />Probability: 30.3%","Costa Rica beats England<br />Probability: 28.1%","Costa Rica beats Belgium<br />Probability: 24.6%","Costa Rica beats Argentina<br />Probability: 22.6%","Costa Rica beats France<br />Probability: 19.7%","Costa Rica beats Spain<br />Probability: 19.8%","Costa Rica beats Germany<br />Probability: 17.9%","Costa Rica beats Brazil<br />Probability: 17.6%"],["Morocco beats Panama<br />Probability: 54.9%","Morocco beats Saudi Arabia<br />Probability: 55.9%","Morocco beats Tunisia<br />Probability: 52.8%","Morocco beats Iran<br />Probability: 50.1%","Morocco beats South Korea<br />Probability: 50.5%","Morocco beats Costa Rica<br />Probability: 51.1%","","Morocco beats Australia<br />Probability: 49.3%","Morocco beats Japan<br />Probability: 47.1%","Morocco beats Iceland<br />Probability: 44.7%","Morocco beats Nigeria<br />Probability: 44.8%","Morocco beats Peru<br />Probability: 44.2%","Morocco beats Senegal<br />Probability: 44.6%","Morocco beats Serbia<br />Probability: 43.9%","Morocco beats Egypt<br />Probability: 43.4%","Morocco beats Sweden<br />Probability: 43.3%","Morocco beats Switzerland<br />Probability: 41.4%","Morocco beats Mexico<br />Probability: 40.9%","Morocco beats Denmark<br />Probability: 40.6%","Morocco beats Poland<br />Probability: 37.3%","Morocco beats Russia<br />Probability: 34.8%","Morocco beats Colombia<br />Probability: 34.3%","Morocco beats Croatia<br />Probability: 32.5%","Morocco beats Uruguay<br />Probability: 32.9%","Morocco beats Portugal<br />Probability: 31.2%","Morocco beats England<br />Probability: 29.0%","Morocco beats Belgium<br />Probability: 25.4%","Morocco beats Argentina<br />Probability: 23.3%","Morocco beats France<br />Probability: 20.4%","Morocco beats Spain<br />Probability: 20.5%","Morocco beats Germany<br />Probability: 18.6%","Morocco beats Brazil<br />Probability: 18.2%"],["Australia beats Panama<br />Probability: 55.6%","Australia beats Saudi Arabia<br />Probability: 56.6%","Australia beats Tunisia<br />Probability: 53.5%","Australia beats Iran<br />Probability: 50.8%","Australia beats South Korea<br />Probability: 51.2%","Australia beats Costa Rica<br />Probability: 51.8%","Australia beats Morocco<br />Probability: 50.7%","","Australia beats Japan<br />Probability: 47.8%","Australia beats Iceland<br />Probability: 45.4%","Australia beats Nigeria<br />Probability: 45.4%","Australia beats Peru<br />Probability: 44.9%","Australia beats Senegal<br />Probability: 45.3%","Australia beats Serbia<br />Probability: 44.6%","Australia beats Egypt<br />Probability: 44.0%","Australia beats Sweden<br />Probability: 44.0%","Australia beats Switzerland<br />Probability: 42.1%","Australia beats Mexico<br />Probability: 41.5%","Australia beats Denmark<br />Probability: 41.3%","Australia beats Poland<br />Probability: 38.0%","Australia beats Russia<br />Probability: 35.5%","Australia beats Colombia<br />Probability: 34.9%","Australia beats Croatia<br />Probability: 33.1%","Australia beats Uruguay<br />Probability: 33.6%","Australia beats Portugal<br />Probability: 31.8%","Australia beats England<br />Probability: 29.6%","Australia beats Belgium<br />Probability: 26.0%","Australia beats Argentina<br />Probability: 23.8%","Australia beats France<br />Probability: 20.9%","Australia beats Spain<br />Probability: 21.0%","Australia beats Germany<br />Probability: 19.0%","Australia beats Brazil<br />Probability: 18.7%"],["Japan beats Panama<br />Probability: 57.7%","Japan beats Saudi Arabia<br />Probability: 58.7%","Japan beats Tunisia<br />Probability: 55.6%","Japan beats Iran<br />Probability: 52.9%","Japan beats South Korea<br />Probability: 53.4%","Japan beats Costa Rica<br />Probability: 53.9%","Japan beats Morocco<br />Probability: 52.9%","Japan beats Australia<br />Probability: 52.2%","","Japan beats Iceland<br />Probability: 47.5%","Japan beats Nigeria<br />Probability: 47.6%","Japan beats Peru<br />Probability: 47.0%","Japan beats Senegal<br />Probability: 47.5%","Japan beats Serbia<br />Probability: 46.7%","Japan beats Egypt<br />Probability: 46.2%","Japan beats Sweden<br />Probability: 46.1%","Japan beats Switzerland<br />Probability: 44.2%","Japan beats Mexico<br />Probability: 43.6%","Japan beats Denmark<br />Probability: 43.4%","Japan beats Poland<br />Probability: 40.0%","Japan beats Russia<br />Probability: 37.5%","Japan beats Colombia<br />Probability: 36.9%","Japan beats Croatia<br />Probability: 35.0%","Japan beats Uruguay<br />Probability: 35.5%","Japan beats Portugal<br />Probability: 33.7%","Japan beats England<br />Probability: 31.4%","Japan beats Belgium<br />Probability: 27.7%","Japan beats Argentina<br />Probability: 25.4%","Japan beats France<br />Probability: 22.3%","Japan beats Spain<br />Probability: 22.5%","Japan beats Germany<br />Probability: 20.4%","Japan beats Brazil<br />Probability: 20.0%"],["Iceland beats Panama<br />Probability: 60.1%","Iceland beats Saudi Arabia<br />Probability: 61.1%","Iceland beats Tunisia<br />Probability: 58.1%","Iceland beats Iran<br />Probability: 55.4%","Iceland beats South Korea<br />Probability: 55.8%","Iceland beats Costa Rica<br />Probability: 56.4%","Iceland beats Morocco<br />Probability: 55.3%","Iceland beats Australia<br />Probability: 54.6%","Iceland beats Japan<br />Probability: 52.5%","","Iceland beats Nigeria<br />Probability: 50.1%","Iceland beats Peru<br />Probability: 49.5%","Iceland beats Senegal<br />Probability: 50.0%","Iceland beats Serbia<br />Probability: 49.2%","Iceland beats Egypt<br />Probability: 48.7%","Iceland beats Sweden<br />Probability: 48.6%","Iceland beats Switzerland<br />Probability: 46.7%","Iceland beats Mexico<br />Probability: 46.1%","Iceland beats Denmark<br />Probability: 45.8%","Iceland beats Poland<br />Probability: 42.4%","Iceland beats Russia<br />Probability: 39.8%","Iceland beats Colombia<br />Probability: 39.2%","Iceland beats Croatia<br />Probability: 37.3%","Iceland beats Uruguay<br />Probability: 37.8%","Iceland beats Portugal<br />Probability: 35.9%","Iceland beats England<br />Probability: 33.6%","Iceland beats Belgium<br />Probability: 29.7%","Iceland beats Argentina<br />Probability: 27.4%","Iceland beats France<br />Probability: 24.1%","Iceland beats Spain<br />Probability: 24.3%","Iceland beats Germany<br />Probability: 22.1%","Iceland beats Brazil<br />Probability: 21.7%"],["Nigeria beats Panama<br />Probability: 60.0%","Nigeria beats Saudi Arabia<br />Probability: 61.0%","Nigeria beats Tunisia<br />Probability: 58.0%","Nigeria beats Iran<br />Probability: 55.3%","Nigeria beats South Korea<br />Probability: 55.8%","Nigeria beats Costa Rica<br />Probability: 56.3%","Nigeria beats Morocco<br />Probability: 55.2%","Nigeria beats Australia<br />Probability: 54.6%","Nigeria beats Japan<br />Probability: 52.4%","Nigeria beats Iceland<br />Probability: 49.9%","","Nigeria beats Peru<br />Probability: 49.4%","Nigeria beats Senegal<br />Probability: 49.9%","Nigeria beats Serbia<br />Probability: 49.2%","Nigeria beats Egypt<br />Probability: 48.6%","Nigeria beats Sweden<br />Probability: 48.5%","Nigeria beats Switzerland<br />Probability: 46.6%","Nigeria beats Mexico<br />Probability: 46.0%","Nigeria beats Denmark<br />Probability: 45.8%","Nigeria beats Poland<br />Probability: 42.4%","Nigeria beats Russia<br />Probability: 39.7%","Nigeria beats Colombia<br />Probability: 39.2%","Nigeria beats Croatia<br />Probability: 37.3%","Nigeria beats Uruguay<br />Probability: 37.7%","Nigeria beats Portugal<br />Probability: 35.9%","Nigeria beats England<br />Probability: 33.5%","Nigeria beats Belgium<br />Probability: 29.6%","Nigeria beats Argentina<br />Probability: 27.3%","Nigeria beats France<br />Probability: 24.1%","Nigeria beats Spain<br />Probability: 24.2%","Nigeria beats Germany<br />Probability: 22.0%","Nigeria beats Brazil<br />Probability: 21.6%"],["Peru beats Panama<br />Probability: 60.6%","Peru beats Saudi Arabia<br />Probability: 61.6%","Peru beats Tunisia<br />Probability: 58.6%","Peru beats Iran<br />Probability: 55.9%","Peru beats South Korea<br />Probability: 56.3%","Peru beats Costa Rica<br />Probability: 56.9%","Peru beats Morocco<br />Probability: 55.8%","Peru beats Australia<br />Probability: 55.1%","Peru beats Japan<br />Probability: 53.0%","Peru beats Iceland<br />Probability: 50.5%","Peru beats Nigeria<br />Probability: 50.6%","","Peru beats Senegal<br />Probability: 50.5%","Peru beats Serbia<br />Probability: 49.8%","Peru beats Egypt<br />Probability: 49.2%","Peru beats Sweden<br />Probability: 49.1%","Peru beats Switzerland<br />Probability: 47.2%","Peru beats Mexico<br />Probability: 46.6%","Peru beats Denmark<br />Probability: 46.3%","Peru beats Poland<br />Probability: 42.9%","Peru beats Russia<br />Probability: 40.3%","Peru beats Colombia<br />Probability: 39.7%","Peru beats Croatia<br />Probability: 37.8%","Peru beats Uruguay<br />Probability: 38.3%","Peru beats Portugal<br />Probability: 36.4%","Peru beats England<br />Probability: 34.0%","Peru beats Belgium<br />Probability: 30.1%","Peru beats Argentina<br />Probability: 27.8%","Peru beats France<br />Probability: 24.5%","Peru beats Spain<br />Probability: 24.6%","Peru beats Germany<br />Probability: 22.4%","Peru beats Brazil<br />Probability: 22.0%"],["Senegal beats Panama<br />Probability: 60.2%","Senegal beats Saudi Arabia<br />Probability: 61.1%","Senegal beats Tunisia<br />Probability: 58.1%","Senegal beats Iran<br />Probability: 55.5%","Senegal beats South Korea<br />Probability: 55.9%","Senegal beats Costa Rica<br />Probability: 56.4%","Senegal beats Morocco<br />Probability: 55.4%","Senegal beats Australia<br />Probability: 54.7%","Senegal beats Japan<br />Probability: 52.5%","Senegal beats Iceland<br />Probability: 50.0%","Senegal beats Nigeria<br />Probability: 50.1%","Senegal beats Peru<br />Probability: 49.5%","","Senegal beats Serbia<br />Probability: 49.3%","Senegal beats Egypt<br />Probability: 48.7%","Senegal beats Sweden<br />Probability: 48.7%","Senegal beats Switzerland<br />Probability: 46.7%","Senegal beats Mexico<br />Probability: 46.2%","Senegal beats Denmark<br />Probability: 45.9%","Senegal beats Poland<br />Probability: 42.5%","Senegal beats Russia<br />Probability: 39.9%","Senegal beats Colombia<br />Probability: 39.3%","Senegal beats Croatia<br />Probability: 37.4%","Senegal beats Uruguay<br />Probability: 37.9%","Senegal beats Portugal<br />Probability: 36.0%","Senegal beats England<br />Probability: 33.6%","Senegal beats Belgium<br />Probability: 29.8%","Senegal beats Argentina<br />Probability: 27.4%","Senegal beats France<br />Probability: 24.2%","Senegal beats Spain<br />Probability: 24.3%","Senegal beats Germany<br />Probability: 22.1%","Senegal beats Brazil<br />Probability: 21.7%"],["Serbia beats Panama<br />Probability: 60.8%","Serbia beats Saudi Arabia<br />Probability: 61.8%","Serbia beats Tunisia<br />Probability: 58.8%","Serbia beats Iran<br />Probability: 56.2%","Serbia beats South Korea<br />Probability: 56.6%","Serbia beats Costa Rica<br />Probability: 57.1%","Serbia beats Morocco<br />Probability: 56.1%","Serbia beats Australia<br />Probability: 55.4%","Serbia beats Japan<br />Probability: 53.3%","Serbia beats Iceland<br />Probability: 50.8%","Serbia beats Nigeria<br />Probability: 50.8%","Serbia beats Peru<br />Probability: 50.2%","Serbia beats Senegal<br />Probability: 50.7%","","Serbia beats Egypt<br />Probability: 49.4%","Serbia beats Sweden<br />Probability: 49.4%","Serbia beats Switzerland<br />Probability: 47.4%","Serbia beats Mexico<br />Probability: 46.9%","Serbia beats Denmark<br />Probability: 46.6%","Serbia beats Poland<br />Probability: 43.2%","Serbia beats Russia<br />Probability: 40.5%","Serbia beats Colombia<br />Probability: 40.0%","Serbia beats Croatia<br />Probability: 38.1%","Serbia beats Uruguay<br />Probability: 38.5%","Serbia beats Portugal<br />Probability: 36.6%","Serbia beats England<br />Probability: 34.3%","Serbia beats Belgium<br />Probability: 30.4%","Serbia beats Argentina<br />Probability: 28.0%","Serbia beats France<br />Probability: 24.7%","Serbia beats Spain<br />Probability: 24.8%","Serbia beats Germany<br />Probability: 22.6%","Serbia beats Brazil<br />Probability: 22.2%"],["Egypt beats Panama<br />Probability: 61.4%","Egypt beats Saudi Arabia<br />Probability: 62.3%","Egypt beats Tunisia<br />Probability: 59.4%","Egypt beats Iran<br />Probability: 56.7%","Egypt beats South Korea<br />Probability: 57.2%","Egypt beats Costa Rica<br />Probability: 57.7%","Egypt beats Morocco<br />Probability: 56.6%","Egypt beats Australia<br />Probability: 56.0%","Egypt beats Japan<br />Probability: 53.8%","Egypt beats Iceland<br />Probability: 51.3%","Egypt beats Nigeria<br />Probability: 51.4%","Egypt beats Peru<br />Probability: 50.8%","Egypt beats Senegal<br />Probability: 51.3%","Egypt beats Serbia<br />Probability: 50.6%","","Egypt beats Sweden<br />Probability: 50.0%","Egypt beats Switzerland<br />Probability: 48.0%","Egypt beats Mexico<br />Probability: 47.4%","Egypt beats Denmark<br />Probability: 47.2%","Egypt beats Poland<br />Probability: 43.8%","Egypt beats Russia<br />Probability: 41.1%","Egypt beats Colombia<br />Probability: 40.5%","Egypt beats Croatia<br />Probability: 38.6%","Egypt beats Uruguay<br />Probability: 39.1%","Egypt beats Portugal<br />Probability: 37.2%","Egypt beats England<br />Probability: 34.8%","Egypt beats Belgium<br />Probability: 30.8%","Egypt beats Argentina<br />Probability: 28.4%","Egypt beats France<br />Probability: 25.1%","Egypt beats Spain<br />Probability: 25.2%","Egypt beats Germany<br />Probability: 23.0%","Egypt beats Brazil<br />Probability: 22.6%"],["Sweden beats Panama<br />Probability: 61.4%","Sweden beats Saudi Arabia<br />Probability: 62.4%","Sweden beats Tunisia<br />Probability: 59.4%","Sweden beats Iran<br />Probability: 56.8%","Sweden beats South Korea<br />Probability: 57.2%","Sweden beats Costa Rica<br />Probability: 57.7%","Sweden beats Morocco<br />Probability: 56.7%","Sweden beats Australia<br />Probability: 56.0%","Sweden beats Japan<br />Probability: 53.9%","Sweden beats Iceland<br />Probability: 51.4%","Sweden beats Nigeria<br />Probability: 51.5%","Sweden beats Peru<br />Probability: 50.9%","Sweden beats Senegal<br />Probability: 51.3%","Sweden beats Serbia<br />Probability: 50.6%","Sweden beats Egypt<br />Probability: 50.0%","","Sweden beats Switzerland<br />Probability: 48.0%","Sweden beats Mexico<br />Probability: 47.5%","Sweden beats Denmark<br />Probability: 47.2%","Sweden beats Poland<br />Probability: 43.8%","Sweden beats Russia<br />Probability: 41.1%","Sweden beats Colombia<br />Probability: 40.5%","Sweden beats Croatia<br />Probability: 38.6%","Sweden beats Uruguay<br />Probability: 39.1%","Sweden beats Portugal<br />Probability: 37.2%","Sweden beats England<br />Probability: 34.8%","Sweden beats Belgium<br />Probability: 30.9%","Sweden beats Argentina<br />Probability: 28.5%","Sweden beats France<br />Probability: 25.1%","Sweden beats Spain<br />Probability: 25.3%","Sweden beats Germany<br />Probability: 23.0%","Sweden beats Brazil<br />Probability: 22.6%"],["Switzerland beats Panama<br />Probability: 63.3%","Switzerland beats Saudi Arabia<br />Probability: 64.2%","Switzerland beats Tunisia<br />Probability: 61.3%","Switzerland beats Iran<br />Probability: 58.7%","Switzerland beats South Korea<br />Probability: 59.1%","Switzerland beats Costa Rica<br />Probability: 59.7%","Switzerland beats Morocco<br />Probability: 58.6%","Switzerland beats Australia<br />Probability: 57.9%","Switzerland beats Japan<br />Probability: 55.8%","Switzerland beats Iceland<br />Probability: 53.3%","Switzerland beats Nigeria<br />Probability: 53.4%","Switzerland beats Peru<br />Probability: 52.8%","Switzerland beats Senegal<br />Probability: 53.3%","Switzerland beats Serbia<br />Probability: 52.6%","Switzerland beats Egypt<br />Probability: 52.0%","Switzerland beats Sweden<br />Probability: 52.0%","","Switzerland beats Mexico<br />Probability: 49.5%","Switzerland beats Denmark<br />Probability: 49.2%","Switzerland beats Poland<br />Probability: 45.7%","Switzerland beats Russia<br />Probability: 43.1%","Switzerland beats Colombia<br />Probability: 42.5%","Switzerland beats Croatia<br />Probability: 40.5%","Switzerland beats Uruguay<br />Probability: 41.0%","Switzerland beats Portugal<br />Probability: 39.1%","Switzerland beats England<br />Probability: 36.6%","Switzerland beats Belgium<br />Probability: 32.6%","Switzerland beats Argentina<br />Probability: 30.1%","Switzerland beats France<br />Probability: 26.7%","Switzerland beats Spain<br />Probability: 26.8%","Switzerland beats Germany<br />Probability: 24.4%","Switzerland beats Brazil<br />Probability: 24.0%"],["Mexico beats Panama<br />Probability: 63.8%","Mexico beats Saudi Arabia<br />Probability: 64.7%","Mexico beats Tunisia<br />Probability: 61.8%","Mexico beats Iran<br />Probability: 59.2%","Mexico beats South Korea<br />Probability: 59.6%","Mexico beats Costa Rica<br />Probability: 60.2%","Mexico beats Morocco<br />Probability: 59.1%","Mexico beats Australia<br />Probability: 58.5%","Mexico beats Japan<br />Probability: 56.4%","Mexico beats Iceland<br />Probability: 53.9%","Mexico beats Nigeria<br />Probability: 54.0%","Mexico beats Peru<br />Probability: 53.4%","Mexico beats Senegal<br />Probability: 53.8%","Mexico beats Serbia<br />Probability: 53.1%","Mexico beats Egypt<br />Probability: 52.6%","Mexico beats Sweden<br />Probability: 52.5%","Mexico beats Switzerland<br />Probability: 50.5%","","Mexico beats Denmark<br />Probability: 49.7%","Mexico beats Poland<br />Probability: 46.3%","Mexico beats Russia<br />Probability: 43.6%","Mexico beats Colombia<br />Probability: 43.0%","Mexico beats Croatia<br />Probability: 41.1%","Mexico beats Uruguay<br />Probability: 41.5%","Mexico beats Portugal<br />Probability: 39.6%","Mexico beats England<br />Probability: 37.1%","Mexico beats Belgium<br />Probability: 33.1%","Mexico beats Argentina<br />Probability: 30.6%","Mexico beats France<br />Probability: 27.1%","Mexico beats Spain<br />Probability: 27.2%","Mexico beats Germany<br />Probability: 24.8%","Mexico beats Brazil<br />Probability: 24.4%"],["Denmark beats Panama<br />Probability: 64.0%","Denmark beats Saudi Arabia<br />Probability: 65.0%","Denmark beats Tunisia<br />Probability: 62.1%","Denmark beats Iran<br />Probability: 59.5%","Denmark beats South Korea<br />Probability: 59.9%","Denmark beats Costa Rica<br />Probability: 60.4%","Denmark beats Morocco<br />Probability: 59.4%","Denmark beats Australia<br />Probability: 58.7%","Denmark beats Japan<br />Probability: 56.6%","Denmark beats Iceland<br />Probability: 54.2%","Denmark beats Nigeria<br />Probability: 54.2%","Denmark beats Peru<br />Probability: 53.7%","Denmark beats Senegal<br />Probability: 54.1%","Denmark beats Serbia<br />Probability: 53.4%","Denmark beats Egypt<br />Probability: 52.8%","Denmark beats Sweden<br />Probability: 52.8%","Denmark beats Switzerland<br />Probability: 50.8%","Denmark beats Mexico<br />Probability: 50.3%","","Denmark beats Poland<br />Probability: 46.6%","Denmark beats Russia<br />Probability: 43.9%","Denmark beats Colombia<br />Probability: 43.3%","Denmark beats Croatia<br />Probability: 41.3%","Denmark beats Uruguay<br />Probability: 41.8%","Denmark beats Portugal<br />Probability: 39.9%","Denmark beats England<br />Probability: 37.4%","Denmark beats Belgium<br />Probability: 33.3%","Denmark beats Argentina<br />Probability: 30.8%","Denmark beats France<br />Probability: 27.3%","Denmark beats Spain<br />Probability: 27.5%","Denmark beats Germany<br />Probability: 25.0%","Denmark beats Brazil<br />Probability: 24.6%"],["Poland beats Panama<br />Probability: 67.1%","Poland beats Saudi Arabia<br />Probability: 68.0%","Poland beats Tunisia<br />Probability: 65.3%","Poland beats Iran<br />Probability: 62.8%","Poland beats South Korea<br />Probability: 63.2%","Poland beats Costa Rica<br />Probability: 63.7%","Poland beats Morocco<br />Probability: 62.7%","Poland beats Australia<br />Probability: 62.0%","Poland beats Japan<br />Probability: 60.0%","Poland beats Iceland<br />Probability: 57.6%","Poland beats Nigeria<br />Probability: 57.6%","Poland beats Peru<br />Probability: 57.1%","Poland beats Senegal<br />Probability: 57.5%","Poland beats Serbia<br />Probability: 56.8%","Poland beats Egypt<br />Probability: 56.2%","Poland beats Sweden<br />Probability: 56.2%","Poland beats Switzerland<br />Probability: 54.3%","Poland beats Mexico<br />Probability: 53.7%","Poland beats Denmark<br />Probability: 53.4%","","Poland beats Russia<br />Probability: 47.3%","Poland beats Colombia<br />Probability: 46.7%","Poland beats Croatia<br />Probability: 44.7%","Poland beats Uruguay<br />Probability: 45.2%","Poland beats Portugal<br />Probability: 43.2%","Poland beats England<br />Probability: 40.7%","Poland beats Belgium<br />Probability: 36.4%","Poland beats Argentina<br />Probability: 33.8%","Poland beats France<br />Probability: 30.1%","Poland beats Spain<br />Probability: 30.3%","Poland beats Germany<br />Probability: 27.7%","Poland beats Brazil<br />Probability: 27.3%"],["Russia beats Panama<br />Probability: 69.5%","Russia beats Saudi Arabia<br />Probability: 70.3%","Russia beats Tunisia<br />Probability: 67.7%","Russia beats Iran<br />Probability: 65.3%","Russia beats South Korea<br />Probability: 65.7%","Russia beats Costa Rica<br />Probability: 66.2%","Russia beats Morocco<br />Probability: 65.2%","Russia beats Australia<br />Probability: 64.5%","Russia beats Japan<br />Probability: 62.5%","Russia beats Iceland<br />Probability: 60.2%","Russia beats Nigeria<br />Probability: 60.3%","Russia beats Peru<br />Probability: 59.7%","Russia beats Senegal<br />Probability: 60.1%","Russia beats Serbia<br />Probability: 59.5%","Russia beats Egypt<br />Probability: 58.9%","Russia beats Sweden<br />Probability: 58.9%","Russia beats Switzerland<br />Probability: 56.9%","Russia beats Mexico<br />Probability: 56.4%","Russia beats Denmark<br />Probability: 56.1%","Russia beats Poland<br />Probability: 52.7%","","Russia beats Colombia<br />Probability: 49.4%","Russia beats Croatia<br />Probability: 47.4%","Russia beats Uruguay<br />Probability: 47.9%","Russia beats Portugal<br />Probability: 45.9%","Russia beats England<br />Probability: 43.3%","Russia beats Belgium<br />Probability: 39.0%","Russia beats Argentina<br />Probability: 36.3%","Russia beats France<br />Probability: 32.4%","Russia beats Spain<br />Probability: 32.6%","Russia beats Germany<br />Probability: 29.9%","Russia beats Brazil<br />Probability: 29.5%"],["Colombia beats Panama<br />Probability: 70.0%","Colombia beats Saudi Arabia<br />Probability: 70.9%","Colombia beats Tunisia<br />Probability: 68.2%","Colombia beats Iran<br />Probability: 65.8%","Colombia beats South Korea<br />Probability: 66.2%","Colombia beats Costa Rica<br />Probability: 66.7%","Colombia beats Morocco<br />Probability: 65.7%","Colombia beats Australia<br />Probability: 65.1%","Colombia beats Japan<br />Probability: 63.1%","Colombia beats Iceland<br />Probability: 60.8%","Colombia beats Nigeria<br />Probability: 60.8%","Colombia beats Peru<br />Probability: 60.3%","Colombia beats Senegal<br />Probability: 60.7%","Colombia beats Serbia<br />Probability: 60.0%","Colombia beats Egypt<br />Probability: 59.5%","Colombia beats Sweden<br />Probability: 59.5%","Colombia beats Switzerland<br />Probability: 57.5%","Colombia beats Mexico<br />Probability: 57.0%","Colombia beats Denmark<br />Probability: 56.7%","Colombia beats Poland<br />Probability: 53.3%","Colombia beats Russia<br />Probability: 50.6%","","Colombia beats Croatia<br />Probability: 48.0%","Colombia beats Uruguay<br />Probability: 48.5%","Colombia beats Portugal<br />Probability: 46.5%","Colombia beats England<br />Probability: 43.9%","Colombia beats Belgium<br />Probability: 39.6%","Colombia beats Argentina<br />Probability: 36.9%","Colombia beats France<br />Probability: 33.0%","Colombia beats Spain<br />Probability: 33.2%","Colombia beats Germany<br />Probability: 30.5%","Colombia beats Brazil<br />Probability: 30.0%"],["Croatia beats Panama<br />Probability: 71.7%","Croatia beats Saudi Arabia<br />Probability: 72.5%","Croatia beats Tunisia<br />Probability: 69.9%","Croatia beats Iran<br />Probability: 67.6%","Croatia beats South Korea<br />Probability: 68.0%","Croatia beats Costa Rica<br />Probability: 68.5%","Croatia beats Morocco<br />Probability: 67.5%","Croatia beats Australia<br />Probability: 66.9%","Croatia beats Japan<br />Probability: 65.0%","Croatia beats Iceland<br />Probability: 62.7%","Croatia beats Nigeria<br />Probability: 62.7%","Croatia beats Peru<br />Probability: 62.2%","Croatia beats Senegal<br />Probability: 62.6%","Croatia beats Serbia<br />Probability: 61.9%","Croatia beats Egypt<br />Probability: 61.4%","Croatia beats Sweden<br />Probability: 61.4%","Croatia beats Switzerland<br />Probability: 59.5%","Croatia beats Mexico<br />Probability: 58.9%","Croatia beats Denmark<br />Probability: 58.7%","Croatia beats Poland<br />Probability: 55.3%","Croatia beats Russia<br />Probability: 52.6%","Croatia beats Colombia<br />Probability: 52.0%","","Croatia beats Uruguay<br />Probability: 50.5%","Croatia beats Portugal<br />Probability: 48.5%","Croatia beats England<br />Probability: 45.9%","Croatia beats Belgium<br />Probability: 41.5%","Croatia beats Argentina<br />Probability: 38.7%","Croatia beats France<br />Probability: 34.8%","Croatia beats Spain<br />Probability: 35.0%","Croatia beats Germany<br />Probability: 32.2%","Croatia beats Brazil<br />Probability: 31.7%"],["Uruguay beats Panama<br />Probability: 71.2%","Uruguay beats Saudi Arabia<br />Probability: 72.1%","Uruguay beats Tunisia<br />Probability: 69.5%","Uruguay beats Iran<br />Probability: 67.1%","Uruguay beats South Korea<br />Probability: 67.5%","Uruguay beats Costa Rica<br />Probability: 68.0%","Uruguay beats Morocco<br />Probability: 67.1%","Uruguay beats Australia<br />Probability: 66.4%","Uruguay beats Japan<br />Probability: 64.5%","Uruguay beats Iceland<br />Probability: 62.2%","Uruguay beats Nigeria<br />Probability: 62.3%","Uruguay beats Peru<br />Probability: 61.7%","Uruguay beats Senegal<br />Probability: 62.1%","Uruguay beats Serbia<br />Probability: 61.5%","Uruguay beats Egypt<br />Probability: 60.9%","Uruguay beats Sweden<br />Probability: 60.9%","Uruguay beats Switzerland<br />Probability: 59.0%","Uruguay beats Mexico<br />Probability: 58.5%","Uruguay beats Denmark<br />Probability: 58.2%","Uruguay beats Poland<br />Probability: 54.8%","Uruguay beats Russia<br />Probability: 52.1%","Uruguay beats Colombia<br />Probability: 51.5%","Uruguay beats Croatia<br />Probability: 49.5%","","Uruguay beats Portugal<br />Probability: 48.0%","Uruguay beats England<br />Probability: 45.4%","Uruguay beats Belgium<br />Probability: 41.0%","Uruguay beats Argentina<br />Probability: 38.3%","Uruguay beats France<br />Probability: 34.3%","Uruguay beats Spain<br />Probability: 34.5%","Uruguay beats Germany<br />Probability: 31.7%","Uruguay beats Brazil<br />Probability: 31.2%"],["Portugal beats Panama<br />Probability: 72.9%","Portugal beats Saudi Arabia<br />Probability: 73.7%","Portugal beats Tunisia<br />Probability: 71.2%","Portugal beats Iran<br />Probability: 68.9%","Portugal beats South Korea<br />Probability: 69.3%","Portugal beats Costa Rica<br />Probability: 69.7%","Portugal beats Morocco<br />Probability: 68.8%","Portugal beats Australia<br />Probability: 68.2%","Portugal beats Japan<br />Probability: 66.3%","Portugal beats Iceland<br />Probability: 64.1%","Portugal beats Nigeria<br />Probability: 64.1%","Portugal beats Peru<br />Probability: 63.6%","Portugal beats Senegal<br />Probability: 64.0%","Portugal beats Serbia<br />Probability: 63.4%","Portugal beats Egypt<br />Probability: 62.8%","Portugal beats Sweden<br />Probability: 62.8%","Portugal beats Switzerland<br />Probability: 60.9%","Portugal beats Mexico<br />Probability: 60.4%","Portugal beats Denmark<br />Probability: 60.1%","Portugal beats Poland<br />Probability: 56.8%","Portugal beats Russia<br />Probability: 54.1%","Portugal beats Colombia<br />Probability: 53.5%","Portugal beats Croatia<br />Probability: 51.5%","Portugal beats Uruguay<br />Probability: 52.0%","","Portugal beats England<br />Probability: 47.4%","Portugal beats Belgium<br />Probability: 43.0%","Portugal beats Argentina<br />Probability: 40.2%","Portugal beats France<br />Probability: 36.2%","Portugal beats Spain<br />Probability: 36.3%","Portugal beats Germany<br />Probability: 33.5%","Portugal beats Brazil<br />Probability: 33.0%"],["England beats Panama<br />Probability: 74.9%","England beats Saudi Arabia<br />Probability: 75.6%","England beats Tunisia<br />Probability: 73.3%","England beats Iran<br />Probability: 71.1%","England beats South Korea<br />Probability: 71.4%","England beats Costa Rica<br />Probability: 71.9%","England beats Morocco<br />Probability: 71.0%","England beats Australia<br />Probability: 70.4%","England beats Japan<br />Probability: 68.6%","England beats Iceland<br />Probability: 66.4%","England beats Nigeria<br />Probability: 66.5%","England beats Peru<br />Probability: 66.0%","England beats Senegal<br />Probability: 66.4%","England beats Serbia<br />Probability: 65.7%","England beats Egypt<br />Probability: 65.2%","England beats Sweden<br />Probability: 65.2%","England beats Switzerland<br />Probability: 63.4%","England beats Mexico<br />Probability: 62.9%","England beats Denmark<br />Probability: 62.6%","England beats Poland<br />Probability: 59.3%","England beats Russia<br />Probability: 56.7%","England beats Colombia<br />Probability: 56.1%","England beats Croatia<br />Probability: 54.1%","England beats Uruguay<br />Probability: 54.6%","England beats Portugal<br />Probability: 52.6%","","England beats Belgium<br />Probability: 45.5%","England beats Argentina<br />Probability: 42.7%","England beats France<br />Probability: 38.6%","England beats Spain<br />Probability: 38.8%","England beats Germany<br />Probability: 35.9%","England beats Brazil<br />Probability: 35.4%"],["Belgium beats Panama<br />Probability: 78.1%","Belgium beats Saudi Arabia<br />Probability: 78.8%","Belgium beats Tunisia<br />Probability: 76.6%","Belgium beats Iran<br />Probability: 74.6%","Belgium beats South Korea<br />Probability: 74.9%","Belgium beats Costa Rica<br />Probability: 75.4%","Belgium beats Morocco<br />Probability: 74.6%","Belgium beats Australia<br />Probability: 74.0%","Belgium beats Japan<br />Probability: 72.3%","Belgium beats Iceland<br />Probability: 70.3%","Belgium beats Nigeria<br />Probability: 70.4%","Belgium beats Peru<br />Probability: 69.9%","Belgium beats Senegal<br />Probability: 70.2%","Belgium beats Serbia<br />Probability: 69.6%","Belgium beats Egypt<br />Probability: 69.2%","Belgium beats Sweden<br />Probability: 69.1%","Belgium beats Switzerland<br />Probability: 67.4%","Belgium beats Mexico<br />Probability: 66.9%","Belgium beats Denmark<br />Probability: 66.7%","Belgium beats Poland<br />Probability: 63.6%","Belgium beats Russia<br />Probability: 61.0%","Belgium beats Colombia<br />Probability: 60.4%","Belgium beats Croatia<br />Probability: 58.5%","Belgium beats Uruguay<br />Probability: 59.0%","Belgium beats Portugal<br />Probability: 57.0%","Belgium beats England<br />Probability: 54.5%","","Belgium beats Argentina<br />Probability: 47.1%","Belgium beats France<br />Probability: 42.9%","Belgium beats Spain<br />Probability: 43.1%","Belgium beats Germany<br />Probability: 40.1%","Belgium beats Brazil<br />Probability: 39.5%"],["Argentina beats Panama<br />Probability: 80.0%","Argentina beats Saudi Arabia<br />Probability: 80.6%","Argentina beats Tunisia<br />Probability: 78.6%","Argentina beats Iran<br />Probability: 76.7%","Argentina beats South Korea<br />Probability: 77.0%","Argentina beats Costa Rica<br />Probability: 77.4%","Argentina beats Morocco<br />Probability: 76.7%","Argentina beats Australia<br />Probability: 76.2%","Argentina beats Japan<br />Probability: 74.6%","Argentina beats Iceland<br />Probability: 72.6%","Argentina beats Nigeria<br />Probability: 72.7%","Argentina beats Peru<br />Probability: 72.2%","Argentina beats Senegal<br />Probability: 72.6%","Argentina beats Serbia<br />Probability: 72.0%","Argentina beats Egypt<br />Probability: 71.6%","Argentina beats Sweden<br />Probability: 71.5%","Argentina beats Switzerland<br />Probability: 69.9%","Argentina beats Mexico<br />Probability: 69.4%","Argentina beats Denmark<br />Probability: 69.2%","Argentina beats Poland<br />Probability: 66.2%","Argentina beats Russia<br />Probability: 63.7%","Argentina beats Colombia<br />Probability: 63.1%","Argentina beats Croatia<br />Probability: 61.3%","Argentina beats Uruguay<br />Probability: 61.7%","Argentina beats Portugal<br />Probability: 59.8%","Argentina beats England<br />Probability: 57.3%","Argentina beats Belgium<br />Probability: 52.9%","","Argentina beats France<br />Probability: 45.7%","Argentina beats Spain<br />Probability: 45.9%","Argentina beats Germany<br />Probability: 42.9%","Argentina beats Brazil<br />Probability: 42.3%"],["France beats Panama<br />Probability: 82.6%","France beats Saudi Arabia<br />Probability: 83.2%","France beats Tunisia<br />Probability: 81.3%","France beats Iran<br />Probability: 79.6%","France beats South Korea<br />Probability: 79.9%","France beats Costa Rica<br />Probability: 80.3%","France beats Morocco<br />Probability: 79.6%","France beats Australia<br />Probability: 79.1%","France beats Japan<br />Probability: 77.7%","France beats Iceland<br />Probability: 75.9%","France beats Nigeria<br />Probability: 75.9%","France beats Peru<br />Probability: 75.5%","France beats Senegal<br />Probability: 75.8%","France beats Serbia<br />Probability: 75.3%","France beats Egypt<br />Probability: 74.9%","France beats Sweden<br />Probability: 74.9%","France beats Switzerland<br />Probability: 73.3%","France beats Mexico<br />Probability: 72.9%","France beats Denmark<br />Probability: 72.7%","France beats Poland<br />Probability: 69.9%","France beats Russia<br />Probability: 67.6%","France beats Colombia<br />Probability: 67.0%","France beats Croatia<br />Probability: 65.2%","France beats Uruguay<br />Probability: 65.7%","France beats Portugal<br />Probability: 63.8%","France beats England<br />Probability: 61.4%","France beats Belgium<br />Probability: 57.1%","France beats Argentina<br />Probability: 54.3%","","France beats Spain<br />Probability: 50.2%","France beats Germany<br />Probability: 47.1%","France beats Brazil<br />Probability: 46.5%"],["Spain beats Panama<br />Probability: 82.5%","Spain beats Saudi Arabia<br />Probability: 83.1%","Spain beats Tunisia<br />Probability: 81.2%","Spain beats Iran<br />Probability: 79.5%","Spain beats South Korea<br />Probability: 79.8%","Spain beats Costa Rica<br />Probability: 80.2%","Spain beats Morocco<br />Probability: 79.5%","Spain beats Australia<br />Probability: 79.0%","Spain beats Japan<br />Probability: 77.5%","Spain beats Iceland<br />Probability: 75.7%","Spain beats Nigeria<br />Probability: 75.8%","Spain beats Peru<br />Probability: 75.4%","Spain beats Senegal<br />Probability: 75.7%","Spain beats Serbia<br />Probability: 75.2%","Spain beats Egypt<br />Probability: 74.8%","Spain beats Sweden<br />Probability: 74.7%","Spain beats Switzerland<br />Probability: 73.2%","Spain beats Mexico<br />Probability: 72.8%","Spain beats Denmark<br />Probability: 72.5%","Spain beats Poland<br />Probability: 69.7%","Spain beats Russia<br />Probability: 67.4%","Spain beats Colombia<br />Probability: 66.8%","Spain beats Croatia<br />Probability: 65.0%","Spain beats Uruguay<br />Probability: 65.5%","Spain beats Portugal<br />Probability: 63.7%","Spain beats England<br />Probability: 61.2%","Spain beats Belgium<br />Probability: 56.9%","Spain beats Argentina<br />Probability: 54.1%","Spain beats France<br />Probability: 49.8%","","Spain beats Germany<br />Probability: 46.9%","Spain beats Brazil<br />Probability: 46.3%"],["Germany beats Panama<br />Probability: 84.2%","Germany beats Saudi Arabia<br />Probability: 84.7%","Germany beats Tunisia<br />Probability: 83.0%","Germany beats Iran<br />Probability: 81.5%","Germany beats South Korea<br />Probability: 81.7%","Germany beats Costa Rica<br />Probability: 82.1%","Germany beats Morocco<br />Probability: 81.4%","Germany beats Australia<br />Probability: 81.0%","Germany beats Japan<br />Probability: 79.6%","Germany beats Iceland<br />Probability: 77.9%","Germany beats Nigeria<br />Probability: 78.0%","Germany beats Peru<br />Probability: 77.6%","Germany beats Senegal<br />Probability: 77.9%","Germany beats Serbia<br />Probability: 77.4%","Germany beats Egypt<br />Probability: 77.0%","Germany beats Sweden<br />Probability: 77.0%","Germany beats Switzerland<br />Probability: 75.6%","Germany beats Mexico<br />Probability: 75.2%","Germany beats Denmark<br />Probability: 75.0%","Germany beats Poland<br />Probability: 72.3%","Germany beats Russia<br />Probability: 70.1%","Germany beats Colombia<br />Probability: 69.5%","Germany beats Croatia<br />Probability: 67.8%","Germany beats Uruguay<br />Probability: 68.3%","Germany beats Portugal<br />Probability: 66.5%","Germany beats England<br />Probability: 64.1%","Germany beats Belgium<br />Probability: 59.9%","Germany beats Argentina<br />Probability: 57.1%","Germany beats France<br />Probability: 52.9%","Germany beats Spain<br />Probability: 53.1%","","Germany beats Brazil<br />Probability: 49.4%"],["Brazil beats Panama<br />Probability: 84.5%","Brazil beats Saudi Arabia<br />Probability: 85.0%","Brazil beats Tunisia<br />Probability: 83.4%","Brazil beats Iran<br />Probability: 81.8%","Brazil beats South Korea<br />Probability: 82.1%","Brazil beats Costa Rica<br />Probability: 82.4%","Brazil beats Morocco<br />Probability: 81.8%","Brazil beats Australia<br />Probability: 81.3%","Brazil beats Japan<br />Probability: 80.0%","Brazil beats Iceland<br />Probability: 78.3%","Brazil beats Nigeria<br />Probability: 78.4%","Brazil beats Peru<br />Probability: 78.0%","Brazil beats Senegal<br />Probability: 78.3%","Brazil beats Serbia<br />Probability: 77.8%","Brazil beats Egypt<br />Probability: 77.4%","Brazil beats Sweden<br />Probability: 77.4%","Brazil beats Switzerland<br />Probability: 76.0%","Brazil beats Mexico<br />Probability: 75.6%","Brazil beats Denmark<br />Probability: 75.4%","Brazil beats Poland<br />Probability: 72.7%","Brazil beats Russia<br />Probability: 70.5%","Brazil beats Colombia<br />Probability: 70.0%","Brazil beats Croatia<br />Probability: 68.3%","Brazil beats Uruguay<br />Probability: 68.8%","Brazil beats Portugal<br />Probability: 67.0%","Brazil beats England<br />Probability: 64.6%","Brazil beats Belgium<br />Probability: 60.5%","Brazil beats Argentina<br />Probability: 57.7%","Brazil beats France<br />Probability: 53.5%","Brazil beats Spain<br />Probability: 53.7%","Brazil beats Germany<br />Probability: 50.6%",""]],"hoverinfo":"text","type":"heatmap","xaxis":"x","yaxis":"y","frame":null}],"highlight":{"on":"plotly_click","persistent":false,"dynamic":false,"selectize":false,"opacityDim":0.2,"selected":{"opacity":1}},"base_url":"https://plot.ly"},"evals":["config.modeBarButtonsToAdd.0.click"],"jsHooks":{"render":[{"code":"function(el, x) { var ctConfig = crosstalk.var('plotlyCrosstalkOpts').set({\"on\":\"plotly_click\",\"persistent\":false,\"dynamic\":false,\"selectize\":false,\"opacityDim\":0.2,\"selected\":{\"opacity\":1}}); }","data":null}]}}</script> <h2 id="performance-throughout-the-tournament">Performance throughout the tournament</h2> <p>As every single match can be simulated with the pairwise probabilities above, it is also straightfoward to simulate the entire tournament (here: 1,000,000 times) providing “survival” probabilities for each team across the different stages.</p> <p><a href="https://eeecon.uibk.ac.at/~zeileis/assets/posts/2018-05-30-fifa2018/p_surv.html">Full-width graphic</a></p> <div id="htmlwidget_container"> <div id="48a035e8cf47" style="width:120%;height:700px;" class="plotly html-widget"></div> </div> <script type="application/json" data-for="48a035e8cf47">{"x":{"visdat":{"48a04f2973da":["function () ","plotlyVisDat"],"48a06bf74709":["function () ","data"],"48a072b727f9":["function () ","data"],"48a07e33283":["function () ","data"],"48a0397e6d8c":["function () ","data"],"48a0299459e6":["function () ","data"],"48a0797426e0":["function () ","data"],"48a02e1cd387":["function () ","data"],"48a04b83c6b2":["function () 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/>Probability: 0.2%<br />Group: B"],"type":"scatter","mode":"lines+markers","marker":{"size":10},"legendgroup":"B","hoverinfo":["text","text","text","text","text"],"name":"Iran","line":{"fillcolor":"rgba(230,171,2,0.5)","color":"rgba(230,171,2,1)"},"xaxis":"x","yaxis":"y","frame":null},{"x":["Round of 16","Quarterfinal","Semifinal","Final","Win"],"y":[0.870423,0.564965,0.363181,0.212606,0.123041],"connectgaps":false,"text":["Team: France<br />Stage: Round of 16<br />Probability: 87.0%<br />Group: C","Team: France<br />Stage: Quarterfinal<br />Probability: 56.5%<br />Group: C","Team: France<br />Stage: Semifinal<br />Probability: 36.3%<br />Group: C","Team: France<br />Stage: Final<br />Probability: 21.3%<br />Group: C","Team: France<br />Stage: Win<br />Probability: 12.3%<br />Group: 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C"],"type":"scatter","mode":"lines+markers","marker":{"size":10},"legendgroup":"C","hoverinfo":["text","text","text","text","text"],"name":"Denmark","line":{"fillcolor":"rgba(117,112,179,0.5)","color":"rgba(117,112,179,1)"},"xaxis":"x","yaxis":"y","frame":null},{"x":["Round of 16","Quarterfinal","Semifinal","Final","Win"],"y":[0.317116,0.119602,0.044694,0.014793,0.004488],"connectgaps":false,"text":["Team: Peru<br />Stage: Round of 16<br />Probability: 31.7%<br />Group: C","Team: Peru<br />Stage: Quarterfinal<br />Probability: 12.0%<br />Group: C","Team: Peru<br />Stage: Semifinal<br />Probability: 4.5%<br />Group: C","Team: Peru<br />Stage: Final<br />Probability: 1.5%<br />Group: C","Team: Peru<br />Stage: Win<br />Probability: 0.4%<br />Group: C"],"type":"scatter","mode":"lines+markers","marker":{"size":10},"legendgroup":"C","hoverinfo":["text","text","text","text","text"],"name":"Peru","line":{"fillcolor":"rgba(117,112,179,0.5)","color":"rgba(117,112,179,1)"},"xaxis":"x","yaxis":"y","frame":null},{"x":["Round of 16","Quarterfinal","Semifinal","Final","Win"],"y":[0.25237,0.097993,0.03009,0.008655,0.002204],"connectgaps":false,"text":["Team: Australia<br />Stage: Round of 16<br />Probability: 25.2%<br />Group: C","Team: Australia<br />Stage: Quarterfinal<br />Probability: 9.8%<br />Group: C","Team: Australia<br />Stage: Semifinal<br />Probability: 3.0%<br />Group: C","Team: Australia<br />Stage: Final<br />Probability: 0.9%<br />Group: C","Team: Australia<br />Stage: Win<br />Probability: 0.2%<br />Group: C"],"type":"scatter","mode":"lines+markers","marker":{"size":10},"legendgroup":"C","hoverinfo":["text","text","text","text","text"],"name":"Australia","line":{"fillcolor":"rgba(117,112,179,0.5)","color":"rgba(117,112,179,1)"},"xaxis":"x","yaxis":"y","frame":null},{"x":["Round of 16","Quarterfinal","Semifinal","Final","Win"],"y":[0.786987,0.48559,0.286893,0.157035,0.08334],"connectgaps":false,"text":["Team: Argentina<br />Stage: Round of 16<br />Probability: 78.7%<br />Group: D","Team: Argentina<br />Stage: Quarterfinal<br />Probability: 48.6%<br />Group: D","Team: Argentina<br />Stage: Semifinal<br />Probability: 28.7%<br />Group: D","Team: Argentina<br />Stage: Final<br />Probability: 15.7%<br />Group: D","Team: Argentina<br />Stage: Win<br />Probability: 8.3%<br />Group: D"],"type":"scatter","mode":"lines+markers","marker":{"size":10},"legendgroup":"D","hoverinfo":["text","text","text","text","text"],"name":"Argentina","line":{"fillcolor":"rgba(102,166,30,0.5)","color":"rgba(102,166,30,1)"},"xaxis":"x","yaxis":"y","frame":null},{"x":["Round of 16","Quarterfinal","Semifinal","Final","Win"],"y":[0.587267,0.291807,0.142287,0.062552,0.025634],"connectgaps":false,"text":["Team: Croatia<br />Stage: Round of 16<br />Probability: 58.7%<br />Group: D","Team: Croatia<br />Stage: Quarterfinal<br />Probability: 29.2%<br />Group: D","Team: Croatia<br />Stage: Semifinal<br />Probability: 14.2%<br />Group: D","Team: Croatia<br />Stage: Final<br />Probability: 6.3%<br />Group: D","Team: Croatia<br />Stage: Win<br />Probability: 2.6%<br />Group: D"],"type":"scatter","mode":"lines+markers","marker":{"size":10},"legendgroup":"D","hoverinfo":["text","text","text","text","text"],"name":"Croatia","line":{"fillcolor":"rgba(102,166,30,0.5)","color":"rgba(102,166,30,1)"},"xaxis":"x","yaxis":"y","frame":null},{"x":["Round of 16","Quarterfinal","Semifinal","Final","Win"],"y":[0.411639,0.153422,0.049749,0.01656,0.004982],"connectgaps":false,"text":["Team: Nigeria<br />Stage: Round of 16<br />Probability: 41.2%<br />Group: D","Team: Nigeria<br />Stage: Quarterfinal<br />Probability: 15.3%<br />Group: D","Team: Nigeria<br />Stage: Semifinal<br />Probability: 5.0%<br />Group: D","Team: Nigeria<br />Stage: Final<br />Probability: 1.7%<br />Group: D","Team: Nigeria<br />Stage: Win<br />Probability: 0.5%<br />Group: D"],"type":"scatter","mode":"lines+markers","marker":{"size":10},"legendgroup":"D","hoverinfo":["text","text","text","text","text"],"name":"Nigeria","line":{"fillcolor":"rgba(102,166,30,0.5)","color":"rgba(102,166,30,1)"},"xaxis":"x","yaxis":"y","frame":null},{"x":["Round of 16","Quarterfinal","Semifinal","Final","Win"],"y":[0.30863,0.115495,0.042904,0.013961,0.004164],"connectgaps":false,"text":["Team: Iceland<br />Stage: Round of 16<br />Probability: 30.9%<br />Group: D","Team: Iceland<br />Stage: Quarterfinal<br />Probability: 11.5%<br />Group: D","Team: Iceland<br />Stage: Semifinal<br />Probability: 4.3%<br />Group: D","Team: Iceland<br />Stage: Final<br />Probability: 1.4%<br />Group: D","Team: Iceland<br />Stage: Win<br />Probability: 0.4%<br />Group: D"],"type":"scatter","mode":"lines+markers","marker":{"size":10},"legendgroup":"D","hoverinfo":["text","text","text","text","text"],"name":"Iceland","line":{"fillcolor":"rgba(102,166,30,0.5)","color":"rgba(102,166,30,1)"},"xaxis":"x","yaxis":"y","frame":null},{"x":["Round of 16","Quarterfinal","Semifinal","Final","Win"],"y":[0.898605,0.61244,0.419597,0.265781,0.162689],"connectgaps":false,"text":["Team: Brazil<br />Stage: Round of 16<br />Probability: 89.9%<br />Group: E","Team: Brazil<br />Stage: Quarterfinal<br />Probability: 61.2%<br />Group: E","Team: Brazil<br />Stage: Semifinal<br />Probability: 42.0%<br />Group: E","Team: Brazil<br />Stage: Final<br />Probability: 26.6%<br />Group: E","Team: Brazil<br />Stage: Win<br />Probability: 16.3%<br />Group: E"],"type":"scatter","mode":"lines+markers","marker":{"size":10},"legendgroup":"E","hoverinfo":["text","text","text","text","text"],"name":"Brazil","line":{"fillcolor":"rgba(231,41,138,0.5)","color":"rgba(231,41,138,1)"},"xaxis":"x","yaxis":"y","frame":null},{"x":["Round of 16","Quarterfinal","Semifinal","Final","Win"],"y":[0.4541,0.172601,0.073381,0.026675,0.008529],"connectgaps":false,"text":["Team: Switzerland<br />Stage: Round of 16<br />Probability: 45.4%<br />Group: E","Team: Switzerland<br />Stage: Quarterfinal<br />Probability: 17.3%<br />Group: E","Team: Switzerland<br />Stage: Semifinal<br />Probability: 7.3%<br />Group: E","Team: Switzerland<br />Stage: Final<br />Probability: 2.7%<br />Group: E","Team: Switzerland<br />Stage: Win<br />Probability: 0.9%<br />Group: E"],"type":"scatter","mode":"lines+markers","marker":{"size":10},"legendgroup":"E","hoverinfo":["text","text","text","text","text"],"name":"Switzerland","line":{"fillcolor":"rgba(231,41,138,0.5)","color":"rgba(231,41,138,1)"},"xaxis":"x","yaxis":"y","frame":null},{"x":["Round of 16","Quarterfinal","Semifinal","Final","Win"],"y":[0.389605,0.145828,0.054123,0.018234,0.005503],"connectgaps":false,"text":["Team: Serbia<br />Stage: Round of 16<br />Probability: 39.0%<br />Group: E","Team: Serbia<br />Stage: Quarterfinal<br />Probability: 14.6%<br />Group: E","Team: Serbia<br />Stage: Semifinal<br />Probability: 5.4%<br />Group: E","Team: Serbia<br />Stage: Final<br />Probability: 1.8%<br />Group: E","Team: Serbia<br />Stage: Win<br />Probability: 0.6%<br />Group: E"],"type":"scatter","mode":"lines+markers","marker":{"size":10},"legendgroup":"E","hoverinfo":["text","text","text","text","text"],"name":"Serbia","line":{"fillcolor":"rgba(231,41,138,0.5)","color":"rgba(231,41,138,1)"},"xaxis":"x","yaxis":"y","frame":null},{"x":["Round of 16","Quarterfinal","Semifinal","Final","Win"],"y":[0.225854,0.083826,0.024619,0.006662,0.001643],"connectgaps":false,"text":["Team: Costa Rica<br />Stage: Round of 16<br />Probability: 22.6%<br />Group: E","Team: Costa Rica<br />Stage: Quarterfinal<br />Probability: 8.4%<br />Group: E","Team: Costa Rica<br />Stage: Semifinal<br />Probability: 2.5%<br />Group: E","Team: Costa Rica<br />Stage: Final<br />Probability: 0.7%<br />Group: E","Team: Costa Rica<br />Stage: Win<br />Probability: 0.2%<br />Group: E"],"type":"scatter","mode":"lines+markers","marker":{"size":10},"legendgroup":"E","hoverinfo":["text","text","text","text","text"],"name":"Costa Rica","line":{"fillcolor":"rgba(231,41,138,0.5)","color":"rgba(231,41,138,1)"},"xaxis":"x","yaxis":"y","frame":null},{"x":["Round of 16","Quarterfinal","Semifinal","Final","Win"],"y":[0.890941,0.604435,0.416032,0.260752,0.157647],"connectgaps":false,"text":["Team: Germany<br />Stage: Round of 16<br />Probability: 89.1%<br />Group: F","Team: Germany<br />Stage: Quarterfinal<br />Probability: 60.4%<br />Group: F","Team: Germany<br />Stage: Semifinal<br />Probability: 41.6%<br />Group: F","Team: Germany<br />Stage: Final<br />Probability: 26.1%<br />Group: F","Team: Germany<br />Stage: Win<br />Probability: 15.8%<br />Group: F"],"type":"scatter","mode":"lines+markers","marker":{"size":10},"legendgroup":"F","hoverinfo":["text","text","text","text","text"],"name":"Germany","line":{"fillcolor":"rgba(27,158,119,0.5)","color":"rgba(27,158,119,1)"},"xaxis":"x","yaxis":"y","frame":null},{"x":["Round of 16","Quarterfinal","Semifinal","Final","Win"],"y":[0.451636,0.173899,0.074006,0.027401,0.008806],"connectgaps":false,"text":["Team: Mexico<br />Stage: Round of 16<br />Probability: 45.2%<br />Group: F","Team: Mexico<br />Stage: Quarterfinal<br />Probability: 17.4%<br />Group: F","Team: Mexico<br />Stage: Semifinal<br />Probability: 7.4%<br />Group: F","Team: Mexico<br />Stage: Final<br />Probability: 2.7%<br />Group: F","Team: Mexico<br />Stage: Win<br />Probability: 0.9%<br />Group: F"],"type":"scatter","mode":"lines+markers","marker":{"size":10},"legendgroup":"F","hoverinfo":["text","text","text","text","text"],"name":"Mexico","line":{"fillcolor":"rgba(27,158,119,0.5)","color":"rgba(27,158,119,1)"},"xaxis":"x","yaxis":"y","frame":null},{"x":["Round of 16","Quarterfinal","Semifinal","Final","Win"],"y":[0.445216,0.16131,0.058714,0.019912,0.006128],"connectgaps":false,"text":["Team: Sweden<br />Stage: Round of 16<br />Probability: 44.5%<br />Group: F","Team: Sweden<br />Stage: Quarterfinal<br />Probability: 16.1%<br />Group: F","Team: Sweden<br />Stage: Semifinal<br />Probability: 5.9%<br />Group: F","Team: Sweden<br />Stage: Final<br />Probability: 2.0%<br />Group: F","Team: Sweden<br />Stage: Win<br />Probability: 0.6%<br />Group: F"],"type":"scatter","mode":"lines+markers","marker":{"size":10},"legendgroup":"F","hoverinfo":["text","text","text","text","text"],"name":"Sweden","line":{"fillcolor":"rgba(27,158,119,0.5)","color":"rgba(27,158,119,1)"},"xaxis":"x","yaxis":"y","frame":null},{"x":["Round of 16","Quarterfinal","Semifinal","Final","Win"],"y":[0.267926,0.080664,0.0275,0.007874,0.001963],"connectgaps":false,"text":["Team: South Korea<br />Stage: Round of 16<br />Probability: 26.8%<br />Group: F","Team: South Korea<br />Stage: Quarterfinal<br />Probability: 8.1%<br />Group: F","Team: South Korea<br />Stage: Semifinal<br />Probability: 2.8%<br />Group: F","Team: South Korea<br />Stage: Final<br />Probability: 0.8%<br />Group: F","Team: South Korea<br />Stage: Win<br />Probability: 0.2%<br />Group: F"],"type":"scatter","mode":"lines+markers","marker":{"size":10},"legendgroup":"F","hoverinfo":["text","text","text","text","text"],"name":"South Korea","line":{"fillcolor":"rgba(27,158,119,0.5)","color":"rgba(27,158,119,1)"},"xaxis":"x","yaxis":"y","frame":null},{"x":["Round of 16","Quarterfinal","Semifinal","Final","Win"],"y":[0.816643,0.535588,0.275355,0.14769,0.073953],"connectgaps":false,"text":["Team: Belgium<br />Stage: Round of 16<br />Probability: 81.7%<br />Group: G","Team: Belgium<br />Stage: Quarterfinal<br />Probability: 53.6%<br />Group: G","Team: Belgium<br />Stage: Semifinal<br />Probability: 27.5%<br />Group: G","Team: Belgium<br />Stage: Final<br />Probability: 14.8%<br />Group: G","Team: Belgium<br />Stage: Win<br />Probability: 7.4%<br />Group: G"],"type":"scatter","mode":"lines+markers","marker":{"size":10},"legendgroup":"G","hoverinfo":["text","text","text","text","text"],"name":"Belgium","line":{"fillcolor":"rgba(217,95,2,0.5)","color":"rgba(217,95,2,1)"},"xaxis":"x","yaxis":"y","frame":null},{"x":["Round of 16","Quarterfinal","Semifinal","Final","Win"],"y":[0.756058,0.464149,0.219589,0.108332,0.048997],"connectgaps":false,"text":["Team: England<br />Stage: Round of 16<br />Probability: 75.6%<br />Group: G","Team: England<br />Stage: Quarterfinal<br />Probability: 46.4%<br />Group: G","Team: England<br />Stage: Semifinal<br />Probability: 22.0%<br />Group: G","Team: England<br />Stage: Final<br />Probability: 10.8%<br />Group: G","Team: England<br />Stage: Win<br />Probability: 4.9%<br />Group: G"],"type":"scatter","mode":"lines+markers","marker":{"size":10},"legendgroup":"G","hoverinfo":["text","text","text","text","text"],"name":"England","line":{"fillcolor":"rgba(217,95,2,0.5)","color":"rgba(217,95,2,1)"},"xaxis":"x","yaxis":"y","frame":null},{"x":["Round of 16","Quarterfinal","Semifinal","Final","Win"],"y":[0.234823,0.086461,0.022832,0.006228,0.001494],"connectgaps":false,"text":["Team: Tunisia<br />Stage: Round of 16<br />Probability: 23.5%<br />Group: G","Team: Tunisia<br />Stage: Quarterfinal<br />Probability: 8.6%<br />Group: G","Team: Tunisia<br />Stage: Semifinal<br />Probability: 2.3%<br />Group: G","Team: Tunisia<br />Stage: Final<br />Probability: 0.6%<br />Group: G","Team: Tunisia<br />Stage: Win<br />Probability: 0.1%<br />Group: G"],"type":"scatter","mode":"lines+markers","marker":{"size":10},"legendgroup":"G","hoverinfo":["text","text","text","text","text"],"name":"Tunisia","line":{"fillcolor":"rgba(217,95,2,0.5)","color":"rgba(217,95,2,1)"},"xaxis":"x","yaxis":"y","frame":null},{"x":["Round of 16","Quarterfinal","Semifinal","Final","Win"],"y":[0.232191,0.06224,0.016963,0.004221,0.000956],"connectgaps":false,"text":["Team: Panama<br />Stage: Round of 16<br />Probability: 23.2%<br />Group: G","Team: Panama<br />Stage: Quarterfinal<br />Probability: 6.2%<br />Group: G","Team: Panama<br />Stage: Semifinal<br />Probability: 1.7%<br />Group: G","Team: Panama<br />Stage: Final<br />Probability: 0.4%<br />Group: G","Team: Panama<br />Stage: Win<br />Probability: 0.1%<br />Group: G"],"type":"scatter","mode":"lines+markers","marker":{"size":10},"legendgroup":"G","hoverinfo":["text","text","text","text","text"],"name":"Panama","line":{"fillcolor":"rgba(217,95,2,0.5)","color":"rgba(217,95,2,1)"},"xaxis":"x","yaxis":"y","frame":null},{"x":["Round of 16","Quarterfinal","Semifinal","Final","Win"],"y":[0.646099,0.309478,0.129922,0.056719,0.022363],"connectgaps":false,"text":["Team: Colombia<br />Stage: Round of 16<br />Probability: 64.6%<br />Group: H","Team: Colombia<br />Stage: Quarterfinal<br />Probability: 30.9%<br />Group: H","Team: Colombia<br />Stage: Semifinal<br />Probability: 13.0%<br />Group: H","Team: Colombia<br />Stage: Final<br />Probability: 5.7%<br />Group: H","Team: Colombia<br />Stage: Win<br />Probability: 2.2%<br />Group: H"],"type":"scatter","mode":"lines+markers","marker":{"size":10},"legendgroup":"H","hoverinfo":["text","text","text","text","text"],"name":"Colombia","line":{"fillcolor":"rgba(102,102,102,0.5)","color":"rgba(102,102,102,1)"},"xaxis":"x","yaxis":"y","frame":null},{"x":["Round of 16","Quarterfinal","Semifinal","Final","Win"],"y":[0.579227,0.257849,0.100903,0.040849,0.014835],"connectgaps":false,"text":["Team: Poland<br />Stage: Round of 16<br />Probability: 57.9%<br />Group: H","Team: Poland<br />Stage: Quarterfinal<br />Probability: 25.8%<br />Group: H","Team: Poland<br />Stage: Semifinal<br />Probability: 10.1%<br />Group: H","Team: Poland<br />Stage: Final<br />Probability: 4.1%<br />Group: H","Team: Poland<br />Stage: Win<br />Probability: 1.5%<br />Group: H"],"type":"scatter","mode":"lines+markers","marker":{"size":10},"legendgroup":"H","hoverinfo":["text","text","text","text","text"],"name":"Poland","line":{"fillcolor":"rgba(102,102,102,0.5)","color":"rgba(102,102,102,1)"},"xaxis":"x","yaxis":"y","frame":null},{"x":["Round of 16","Quarterfinal","Semifinal","Final","Win"],"y":[0.379369,0.132769,0.052009,0.017469,0.005068],"connectgaps":false,"text":["Team: Senegal<br />Stage: Round of 16<br />Probability: 37.9%<br />Group: H","Team: Senegal<br />Stage: Quarterfinal<br />Probability: 13.3%<br />Group: H","Team: Senegal<br />Stage: Semifinal<br />Probability: 5.2%<br />Group: H","Team: Senegal<br />Stage: Final<br />Probability: 1.7%<br />Group: H","Team: Senegal<br />Stage: Win<br />Probability: 0.5%<br />Group: H"],"type":"scatter","mode":"lines+markers","marker":{"size":10},"legendgroup":"H","hoverinfo":["text","text","text","text","text"],"name":"Senegal","line":{"fillcolor":"rgba(102,102,102,0.5)","color":"rgba(102,102,102,1)"},"xaxis":"x","yaxis":"y","frame":null},{"x":["Round of 16","Quarterfinal","Semifinal","Final","Win"],"y":[0.363035,0.127391,0.039065,0.012353,0.003368],"connectgaps":false,"text":["Team: Japan<br />Stage: Round of 16<br />Probability: 36.3%<br />Group: H","Team: Japan<br />Stage: Quarterfinal<br />Probability: 12.7%<br />Group: H","Team: Japan<br />Stage: Semifinal<br />Probability: 3.9%<br />Group: H","Team: Japan<br />Stage: Final<br />Probability: 1.2%<br />Group: H","Team: Japan<br />Stage: Win<br />Probability: 0.3%<br />Group: H"],"type":"scatter","mode":"lines+markers","marker":{"size":10},"legendgroup":"H","hoverinfo":["text","text","text","text","text"],"name":"Japan","line":{"fillcolor":"rgba(102,102,102,0.5)","color":"rgba(102,102,102,1)"},"xaxis":"x","yaxis":"y","frame":null}],"highlight":{"on":"plotly_click","persistent":false,"dynamic":false,"selectize":false,"opacityDim":0.2,"selected":{"opacity":1}},"base_url":"https://plot.ly"},"evals":["config.modeBarButtonsToAdd.0.click"],"jsHooks":{"render":[{"code":"function(el, x) { var ctConfig = crosstalk.var('plotlyCrosstalkOpts').set({\"on\":\"plotly_click\",\"persistent\":false,\"dynamic\":false,\"selectize\":false,\"opacityDim\":0.2,\"selected\":{\"opacity\":1}}); }","data":null}]}}</script> <p>This also shows that indeed the most likely final is a match of the top favorites Brazil and Germany (with a probability of 5.5%) where Brazil has the chance to compensate the dramatic semifinal in Belo Horizonte, four years ago. However, given that it comes to this final, the chances are almost even (50.6% for Brazil vs. 49.4% for Germany). For the semifinals it is most likely (with a probability of 9.4%) that Brazil and France meet in the first semifinal (with chances slightly in favor of Brazil in such a match, 53.5%) while Germany and Spain most likely (with 9.2%) play the second semifinal (with chances slightly in favor of Germany with 53.1%).</p> <h2 id="odds-and-ends">Odds and ends</h2> <p>The bookmaker consensus model has performed well in previous tournaments, often predicting winners or finalists correctly. However, all forecasts are probabilistic, clearly below 100%, and thus by no means certain. This showed prominently at the <a href="http://EconPapers.RePEc.org/RePEc:inn:wpaper:2016-15">UEFA Euro 2016</a>:</p> <ul> <li>The model correctly predicted that France would beat Germany in the semifinal.</li> <li>For the final, France had a predicted 68.8% probability to beat Portugal, i.e., being expected to win about 2 out of every 3 matches between these two teams.</li> <li>But in the actual final Gignac failed to seal the deal in added time and Portugal was able to take the victory in overtime.</li> </ul> <p>This illustrates that small things can often make the decisive difference in football, which is why predictions with high probabilities cannot be made. Moreover, it is in the very nature of predictions that they can be wrong, otherwise football tournaments would be very boring. The only forecast that can be made with certainty is that the World Cup will be an exciting tournament that football fans worldwide look forward to.</p> <p>In addition to this forecast, other interesting approaches will surely also be published in the next days, e.g., using the ideas of <a href="https://doi.org/10.1515/jqas-2014-0051">Groll, Schauberger, Tutz (2016)</a>. Also, Claus Ekstrøm will evaluate and compare predictions for the 2018 FIFA World Cup, see his <a href="http://biostatistics.dk/talks/eRum2018">slides</a>, <a href="https://www.youtube.com/watch?v=urJ1obHPsV8">video</a>, <a href="https://github.com/ekstroem/socceR2018">code</a>.</p> <p>As a final remark: Betting on the outcome based on the results presented here is not recommended. Not only because the winning probabilities are clearly far below 100% but, more importantly, because the bookmakers have a sizeable profit margin of about 15.2% which assures that the best chances of making money based on sports betting lie with them!</p> <h2 id="working-paper">Working paper</h2> <p>Zeileis A, Leitner C, Hornik K (2018). <em>“Probabilistic Forecasts for the 2018 FIFA World Cup Based on the Bookmaker Consensus Model”</em>, Working Paper 2018-09, Working Papers in Economics and Statistics, Research Platform Empirical and Experimental Economics, Universität Innsbruck. <a href="http://EconPapers.RePEc.org/RePEc:inn:wpaper:2018-09">http://EconPapers.RePEc.org/RePEc:inn:wpaper:2018-09</a></p>
2018-05-30T00:00:00+02:00
https://eeecon.uibk.ac.at/~zeileis/news/distforest/
Distributional regression forests on arXiv
2018-04-10T00:00:00+02:00
Achim Zeileis
Achim.Zeileis@R-project.org
https://eeecon.uibk.ac.at/~zeileis/
Distributional regression trees and forests provide flexible data-driven probabilistic forecasts by blending distributional models (for location, scale, shape, and beyond) with regression trees and random forests. Accompanied by the R package disttree.
<p>Distributional regression trees and forests provide flexible data-driven probabilistic forecasts by blending distributional models (for location, scale, shape, and beyond) with regression trees and random forests. Accompanied by the R package disttree.</p> <h3 id="citation">Citation</h3> <p>Lisa Schlosser, Torsten Hothorn, Reto Stauffer, Achim Zeileis (2018). “Distributional Regression Forests for Probabilistic Precipitation Forecasting in Complex Terrain.” <em>arXiv.org E-Print Archive</em> arXiv:1804.02921 [stat.ME]. <a href="https://arxiv.org/abs/1804.02921">https://arxiv.org/abs/1804.02921</a></p> <h3 id="abstract">Abstract</h3> <p>To obtain a probabilistic model for a dependent variable based on some set of explanatory variables, a distributional approach is often adopted where the parameters of the distribution are linked to regressors. In many classical models this only captures the location of the distribution but over the last decade there has been increasing interest in distributional regression approaches modeling all parameters including location, scale, and shape. Notably, so-called non-homogenous Gaussian regression (NGR) models both mean and variance of a Gaussian response and is particularly popular in weather forecasting. More generally, the GAMLSS framework allows to establish generalized additive models for location, scale, and shape with smooth linear or nonlinear effects. However, when variable selection is required and/or there are non-smooth dependencies or interactions (especially unknown or of high-order), it is challenging to establish a good GAMLSS. A natural alternative in these situations would be the application of regression trees or random forests but, so far, no general distributional framework is available for these. Therefore, a framework for distributional regression trees and forests is proposed that blends regression trees and random forests with classical distributions from the GAMLSS framework as well as their censored or truncated counterparts. To illustrate these novel approaches in practice, they are employed to obtain probabilistic precipitation forecasts at numerous sites in a mountainous region (Tyrol, Austria) based on a large number of numerical weather prediction quantities. It is shown that the novel distributional regression forests automatically select variables and interactions, performing on par or often even better than GAMLSS specified either through prior meteorological knowledge or a computationally more demanding boosting approach.</p> <h3 id="software">Software</h3> <p>R package <code class="highlighter-rouge">disttree</code> at <a href="https://R-Forge.R-project.org/R/?group_id=261">https://R-Forge.R-project.org/R/?group_id=261</a></p> <h3 id="illustration">Illustration</h3> <p>Distributional trees as part of the parametric and recursive partitioning modeling toolbox.</p> <p><a href="https://eeecon.uibk.ac.at/~zeileis/assets/posts/2018-04-10-distforest/distforest-motivation.png"><img src="https://eeecon.uibk.ac.at/~zeileis/assets/posts/2018-04-10-distforest/distforest-motivation.png" alt="Modeling toolbox" /></a></p> <p>Total precipitation predictions by a distributional forest at station Axams for July 24 in 2009, 2010, 2011 and 2012 learned on data from 1985-2008. Observations are left-censored at 0.</p> <p><a href="https://eeecon.uibk.ac.at/~zeileis/assets/posts/2018-04-10-distforest/distforest-axams.png"><img src="https://eeecon.uibk.ac.at/~zeileis/assets/posts/2018-04-10-distforest/distforest-axams.png" alt="Precipitation forecast Axams" /></a></p> <p>Map of Tyrol coding the best-performing model for each station (type of symbol). The color codes whether the distributional forest had higher (green) or lower (red) CRPS compared to the best of the other three models. Station Axams is highlighted in bold.</p> <p><a href="https://eeecon.uibk.ac.at/~zeileis/assets/posts/2018-04-10-distforest/distforest-tyrol.png"><img src="https://eeecon.uibk.ac.at/~zeileis/assets/posts/2018-04-10-distforest/distforest-tyrol.png" alt="Precipitation forecast Tyrol" /></a></p>
2018-04-10T00:00:00+02:00
https://eeecon.uibk.ac.at/~zeileis/news/bamlss/
BAMLSS paper published in JCGS
2018-01-30T00:00:00+01:00
Achim Zeileis
Achim.Zeileis@R-project.org
https://eeecon.uibk.ac.at/~zeileis/
Bayesian additive models for location, scale, and shape (and beyond) provide a general framework for distributional regression. Accompanied by the R package bamlss.
<p>Bayesian additive models for location, scale, and shape (and beyond) provide a general framework for distributional regression. Accompanied by the R package bamlss.</p> <h3 id="citation">Citation</h3> <p>Nikolaus Umlauf, Nadja Klein, Achim Zeileis (2018). “BAMLSS: Bayesian Additive Models for Location, Scale and Shape (and Beyond).” <em>Journal of Computational and Graphical Statistics</em>. Forthcoming. <a href="https://dx.doi.org/10.1080/10618600.2017.1407325">doi:10.1080/10618600.2017.1407325</a> [ <a href="https://eeecon.uibk.ac.at/~zeileis/papers/Umlauf+Klein+Zeileis-2017.pdf">pdf</a> ]</p> <h3 id="abstract">Abstract</h3> <p>Bayesian analysis provides a convenient setting for the estimation of complex generalized additive regression models (GAMs). Since computational power has tremendously increased in the past decade it is now possible to tackle complicated inferential problems, e.g., with Markov chain Monte Carlo simulation, on virtually any modern computer. This is one of the reasons why Bayesian methods have become increasingly popular, leading to a number of highly specialized and optimized estimation engines and with attention shifting from conditional mean models to probabilistic distributional models capturing location, scale, shape (and other aspects) of the response distribution. In order to embed many different approaches suggested in literature and software, a unified modeling architecture for distributional GAMs is established that exploits distributions, estimation techniques (posterior mode or posterior mean), and model terms (fixed, random, smooth, spatial, …). It is shown that within this framework implementing algorithms for complex regression problems, as well as the integration of already existing software, is relatively straightforward. The usefulness is emphasized with two complex and computationally demanding application case studies: a large daily precipitation climatology, as well as a Cox model for continuous time with space-time interactions.</p> <h3 id="software">Software</h3> <p><a href="https://CRAN.R-project.org/package=bamlss">https://CRAN.R-project.org/package=bamlss</a></p> <h3 id="illustration">Illustration</h3> <p>Censored heteroscedastic precepitation climatology, with spatially-varying seasonal effects, spatial main effects, and predicted average precipitation for target date.</p> <p><a href="https://eeecon.uibk.ac.at/~zeileis/assets/posts/2018-01-30-bamlss/bamlss.png"><img src="https://eeecon.uibk.ac.at/~zeileis/assets/posts/2018-01-30-bamlss/bamlss.png" alt="BAMLSS precipitation" /></a></p>
2018-01-30T00:00:00+01:00
https://eeecon.uibk.ac.at/~zeileis/news/glmertree/
GLMM trees published in BRM
2018-01-29T00:00:00+01:00
Achim Zeileis
Achim.Zeileis@R-project.org
https://eeecon.uibk.ac.at/~zeileis/
Generalized linear mixed-effects model trees, especially for detecting treatment-subgroup interactions in clustered data. Accompanied by the R package glmertree, combining partykit::glmtree and lme4::glmer.
<p>Generalized linear mixed-effects model trees, especially for detecting treatment-subgroup interactions in clustered data. Accompanied by the R package glmertree, combining partykit::glmtree and lme4::glmer.</p> <h3 id="citation">Citation</h3> <p>Marjolein Fokkema, Niels Smits, Achim Zeileis, Torsten Hothorn, Henk Kelderman (2018). “Detecting Treatment-Subgroup Interactions in Clustered Data with Generalized Linear Mixed-Effects Model Trees.” <em>Behavior Research Methods</em>. Forthcoming. <a href="https://dx.doi.org/10.3758/s13428-017-0971-x">doi:10.3758/s13428-017-0971-x</a></p> <h3 id="abstract">Abstract</h3> <p>Identification of subgroups of patients for whom treatment A is more effective than treatment B, and vice versa, is of key importance to the development of personalized medicine. Tree-based algorithms are helpful tools for the detection of such interactions, but none of the available algorithms allow for taking into account clustered or nested dataset structures, which are particularly common in psychological research. Therefore, we propose the generalized linear mixed-effects model tree (GLMM tree) algorithm, which allows for the detection of treatment-subgroup interactions, while accounting for the clustered structure of a dataset. The algorithm uses model-based recursive partitioning to detect treatment-subgroup interactions, and a GLMM to estimate the random-effects parameters. In a simulation study, GLMM trees show higher accuracy in recovering treatment-subgroup interactions, higher predictive accuracy, and lower type II error rates than linear-model-based recursive partitioning and mixed-effects regression trees. Also, GLMM trees show somewhat higher predictive accuracy than linear mixed-effects models with pre-specified interaction effects, on average. We illustrate the application of GLMM trees on an individual patient-level data meta-analysis on treatments for depression. We conclude that GLMM trees are a promising exploratory tool for the detection of treatment-subgroup interactions in clustered datasets.</p> <h3 id="software">Software</h3> <p><a href="https://CRAN.R-project.org/package=glmertree">https://CRAN.R-project.org/package=glmertree</a></p> <h3 id="illustration">Illustration</h3> <p>GLMM tree for treatment-subgroup interaction in a motivating artificial dataset.</p> <p><a href="https://eeecon.uibk.ac.at/~zeileis/assets/posts/2018-01-29-glmertree/glmertree.png"><img src="https://eeecon.uibk.ac.at/~zeileis/assets/posts/2018-01-29-glmertree/glmertree.png" alt="GLMM tree" /></a></p>
2018-01-29T00:00:00+01:00