Assessment of Regression Models Performance

Utilities for computing measures to assess model quality, which are not directly provided by R's 'base' or 'stats' packages. These include e.g. measures like r-squared, intraclass correlation coefficient (Nakagawa, Johnson & Schielzeth (2017) ), root mean squared error or functions to check models for overdispersion, singularity or zero-inflation and more. Functions apply to a large variety of regression models, including generalized linear models, mixed effects models and Bayesian models.

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Test if your model is a good model!

The primary goal of the performance package is to provide utilities for computing indices of model quality and goodness of fit. This includes measures like r-squared, root mean squared error or intraclass correlation coefficient (ICC) , but also functions to check (mixed) models for overdispersion, zero-inflation, convergence or singularity.


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Assessing model quality


performance has a generic r2() function, which computes the r-squared for many different models, including mixed effects and Bayesian regression models.

r2() returns a list containing values related to the “most appropriate” r-squared for the given model.

model <- lm(mpg ~ wt + cyl, data = mtcars)
#> # R2 for linear models
#>        R2: 0.830
#>   adj. R2: 0.819
model <- glm(am ~ wt + cyl, data = mtcars, family = binomial)
#> $R2_Tjur
#> Tjur's R2 
#>    0.7051
model <- polr(Sat ~ Infl + Type + Cont, weights = Freq, data = housing)
#> $R2_Nagelkerke
#> Nagelkerke's R2 
#>          0.1084

The different r-squared measures can also be accessed directly via functions like r2_bayes(), r2_coxsnell() or r2_nagelkerke() (see a full list of functions here).

For mixed models, the conditional and marginal r-squared are returned. The marginal r-squared considers only the variance of the fixed effects and indicates how much of the model’s variance is explained by the fixed effects part only. The conditional r-squared takes both the fixed and random effects into account and indicates how much of the model’s variance is explained by the “complete” model.

For frequentist mixed models, r2() (resp. r2_nakagawa()) computes the mean random effect variances, thus r2() is also appropriate for mixed models with more complex random effects structures, like random slopes or nested random effects (see Johnson 2014 and Nakagawa et al. 2017).

model <- stan_glmer(Petal.Length ~ Petal.Width + (1 | Species), 
    data = iris, cores = 4)
#> # Bayesian R2 with Standard Error
#>   Conditional R2: 0.954 [0.002]
#>      Marginal R2: 0.412 [0.119]
model <- lmer(Reaction ~ Days + (1 + Days | Subject), data = sleepstudy)
#> # R2 for mixed models
#>   Conditional R2: 0.799
#>      Marginal R2: 0.279

Intraclass Correlation Coefficient (ICC)

Similar to r-squared, the ICC provides information on the explained variance and can be interpreted as “the proportion of the variance explained by the grouping structure in the population” (Hox 2010: 15).

icc() calculates the ICC for various mixed model objects, including stanreg models.

model <- lmer(Reaction ~ Days + (1 + Days | Subject), data = sleepstudy)
#> # Intraclass Correlation Coefficient
#>      Adjusted ICC: 0.722
#>   Conditional ICC: 0.521

For models of class brmsfit, an ICC based on variance decomposition is returned (for details, see the documentation).

model <- brm(mpg ~ wt + (1 | cyl) + (1 + wt | gear), data = mtcars)
#> # Random Effect Variances and ICC
#> Conditioned on: all random effects
#> ## Variance Ratio (comparable to ICC)
#> Ratio: 0.39  CI 95%: [-0.57 0.78]
#> ## Variances of Posterior Predicted Distribution
#> Conditioned on fixed effects: 22.79  CI 95%: [ 8.48 57.73]
#> Conditioned on rand. effects: 37.70  CI 95%: [25.08 55.43]
#> ## Difference in Variances
#> Difference: 14.42  CI 95%: [-18.82 36.07]

Model diagnostics

Check for overdispersion

Overdispersion occurs when the observed variance in the data is higher than the expected variance from the model assumption (for Poisson, variance roughly equals the mean of an outcome). check_overdispersion() checks if a count model (including mixed models) is overdispersed or not.

model <- glm(count ~ spp + mined, family = poisson, data = Salamanders)
#> # Overdispersion test
#>        dispersion ratio =    2.946
#>   Pearson's Chi-Squared = 1873.710
#>                 p-value =  < 0.001
#> Overdispersion detected.

Overdispersion can be fixed by either modelling the dispersion parameter (not possible with all packages), or by choosing a different distributional family (like Quasi-Poisson, or negative binomial, see Gelman and Hill 2007).

Check for zero-inflation

Zero-inflation (in (Quasi-)Poisson models) is indicated when the amount of observed zeros is larger than the amount of predicted zeros, so the model is underfitting zeros. In such cases, it is recommended to use negative binomial or zero-inflated models.

Use check_zeroinflation() to check if zero-inflation is present in the fitted model.

model <- glm(count ~ spp + mined, family = poisson, data = Salamanders)
#> # Check for zero-inflation
#>    Observed zeros: 387
#>   Predicted zeros: 298
#>             Ratio: 0.77
#> Model is underfitting zeros (probable zero-inflation).

Check for singular model fits

A “singular” model fit means that some dimensions of the variance-covariance matrix have been estimated as exactly zero. This often occurs for mixed models with overly complex random effects structures.

check_singularity() checks mixed models (of class lme, merMod, glmmTMB or MixMod) for singularity, and returns TRUE if the model fit is singular.

# prepare data
sleepstudy$mygrp <- sample(1:5, size = 180, replace = TRUE)
sleepstudy$mysubgrp <- NA
for (i in 1:5) {
    filter_group <- sleepstudy$mygrp == i
    sleepstudy$mysubgrp[filter_group] <- sample(1:30, size = sum(filter_group), 
        replace = TRUE)
# fit strange model
model <- lmer(Reaction ~ Days + (1 | mygrp/mysubgrp) + (1 | Subject), 
    data = sleepstudy)
#> [1] TRUE

Remedies to cure issues with singular fits can be found here.

Model performance summaries

model_performance() computes indices of model performance for regression models. Depending on the model object, typical indices might be r-squared, AIC, BIC, RMSE, ICC or LOOIC.

Linear model

m1 <- lm(mpg ~ wt + cyl, data = mtcars)
AIC BIC R2 R2_adjusted RMSE
156 161.9 0.83 0.82 2.44

Logistic regression

m2 <- glm(vs ~ wt + mpg, data = mtcars, family = "binomial")
31.3 35.7 0.48

Linear mixed model

m3 <- lmer(Reaction ~ Days + (1 + Days | Subject), data = sleepstudy)
AIC BIC R2_conditional R2_marginal ICC_adjusted ICC_conditional RMSE
1756 1775 0.8 0.28 0.72 0.52 23.44

Comparing different models

compare_performance(m1, m2, m3)
name class AIC BIC RMSE R2 R2_adjusted R2_Tjur R2_conditional R2_marginal ICC_adjusted ICC_conditional
m1 glm 31.3 35.7 0.48
m2 lm 156.0 161.9 2.44 0.83 0.82
m3 lmerMod 1755.6 1774.8 23.44 0.8 0.28 0.72 0.52


Gelman, A., & Hill, J. (2007). Data analysis using regression and multilevel/hierarchical models. Cambridge; New York: Cambridge University Press.

Hox, J. J. (2010). Multilevel analysis: techniques and applications (2nd ed). New York: Routledge.

Johnson, P. C. D. (2014). Extension of Nakagawa & Schielzeth’s R2 GLMM to random slopes models. Methods in Ecology and Evolution, 5(9), 944–946.

Nakagawa, S., Johnson, P. C. D., & Schielzeth, H. (2017). The coefficient of determination R2 and intra-class correlation coefficient from generalized linear mixed-effects models revisited and expanded. Journal of The Royal Society Interface, 14(134), 20170213.


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0.8.0 by Daniel Lüdecke, a month ago

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Authors: Daniel Lüdecke [aut, cre] , @strengejacke) , Dominique Makowski [aut, ctb] , @Dom_Makowski) , Mattan S. Ben-Shachar [aut, ctb] , @mattansb) , Indrajeet Patil [aut, ctb] , @patilindrajeets) , Philip Waggoner [aut, ctb] , Brenton M. Wiernik [aut, ctb] , @bmwiernik) , Vincent Arel-Bundock [ctb] , Martin Jullum [rev] , gjo11 [rev]

Documentation:   PDF Manual  

GPL-3 license

Imports bayestestR, insight, datawizard, methods, stats, utils

Suggests AER, afex, BayesFactor, betareg, bigutilsr, brms, car, CompQuadForm, correlation, cplm, dbscan, dplyr, fixest, forecast, gamm4, ggplot2, glmmTMB, ICS, ICSOutlier, ISLR, lavaan, lme4, lmtest, loo, magrittr, MASS, Matrix, mclust, metafor, mgcv, mlogit, multimode, nlme, nonnest2, ordinal, parallel, parameters, patchwork, pscl, psych, randomForest, rmarkdown, rstanarm, rstantools, sandwich, see, spelling, survey, survival, testthat, tweedie, VGAM

Imported by broomExtra, bruceR, drhur, effectsize, ggstatsplot, modelbased, modelsummary, psycModel, report, sjPlot, sjstats, statsExpressions.

Suggested by JSmediation, afex, archetyper, bayestestR, datawizard, dominanceanalysis, insight, parameters, pubh, see, specr.

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