Pseudo-R-squared for Generalized Mixed-Effect models

Calculate conditional and marginal coefficient of determination for Generalized mixed-effect models (GLMM).

r.squaredGLMM(object, null, ...)
# S3 method for class 'merMod'
r.squaredGLMM(object, null, envir = parent.frame(), pj2014 = FALSE, ...)

Arguments

object: a fitted linear model object.
null: optionally, a null model, including only random effects. See ‘Details’.
envir: optionally, the environment in which the null model is to be evaluated. Defaults to the current frame. See eval.
pj2014: logical, if TRUE and object is of poisson family, the result will include GLMM using original formulation of Johnson (2014). This requires fitting object with an observation-level random effect term added.
...: additional arguments, ignored.

Value

r.squaredGLMM returns a two-column numeric matrix, each (possibly named) row holding values for marginal and conditional GLMM calculated with different methods, such as “delta”, “log-normal”, “trigamma”, or “theoretical” for models of binomial family.

Details

There are two types of GLMM: marginal and conditional.

Marginal GLMM represents the variance explained by the fixed effects, and is defined as:

R_GLMM(m)^2= _f^2_f^2 + _^2 + _ ^2 R_GLMM(m)² = (_f²) / (_f² + _² + _²) R_GLMM(m)^2 = (sigma_f^2) / (sigma_f^2 + sigma_alpha^2 + sigma_epsilon^2)

Conditional GLMM represents the variance explained by the entire model, including both fixed and random effects. It is calculated by the equation:

R_GLMM(c)^2= _f^2 + _^2_f^2 + _^2 + _ ^2 R_GLMM(c)² = (_f² + _²) / (_f² + _² + _²) R_GLMM(c)^2 = (sigma_f^2 + sigma_alpha^2) / (sigma_f^2 + sigma_alpha^2 + sigma_epsilon^2)

where _f^2_f²sigma_f^2 is the variance of the fixed effect components, __²sigma_alpha^2 is the variance of the random effects, and _^2_²sigma_epsilon^2 is the “observation-level” variance.

Three methods are available for deriving the observation-level variance \(\sigma_\varepsilon\): the delta method, lognormal approximation and using the trigamma function.

The delta method can be used with for all distributions and link functions, while lognormal approximation and trigamma function are limited to distributions with logarithmic link. Trigamma-estimate is recommended whenever available. Additionally, for binomial distributions, theoretical variances exist specific for each link function distribution.

Null model. Calculation of the observation-level variance involves in some cases fitting a null model containing no fixed effects other than intercept, otherwise identical to the original model (including all the random effects). When using r.squaredGLMM for several models differing only in their fixed effects, in order to avoid redundant calculations, the null model object can be passed as the argument null. Otherwise, a null model will be fitted via updating the original model. This assumes that all the variables used in the original model call have the same values as when the model was fitted. The function warns about this when fitting the null model is required. This warnings can be disabled by setting options(MuMIn.noUpdateWarning = TRUE).

Note

Important: as of MuMIn version 1.41.0, r.squaredGLMM returns a revised statistics based on Nakagawa et al. (2017) paper. The returned value's format also has changed (it is a matrix rather than a numeric vector as before). Pre-1.41.0 version of the function calculated the “theoretical” GLMM for binomial models.

GLMM can be calculated also for fixed-effect only models. In the simpliest case of OLS it reduces to Var()Var() + D/2Var(mu) / (Var(mu) + D(y, mu) / 2), where Var()Var(mu) is the variance of fitted values, and D is the model deviance. Unlike likelihood-ratio based for OLS, value of this statistic differs from that of the classical .

Currently methods exist for classes: merMod, lme, glmmTMB, glmmADMB, glmmPQL, cpglm(m) and (g)lm.

For families other than gaussian, Gamma, Poisson, binomial and negative binomial, the residual variance is obtained using get_varianceinsight from package insight.

See note in r.squaredLR help page for comment on using in model selection.

Author

Kamil Bartoń. This implementation is based on the ‘Supporting Information’ for Nakagawa et al. (2014), (the extension for random-slopes) Johnson (2014), and includes developments from Nakagawa et al. (2017).

References

Nakagawa, S., Schielzeth, H. 2013 A general and simple method for obtaining from Generalized Linear Mixed-effects Models. Methods in Ecology and Evolution 4, 133–142.

Johnson, P. C. D. 2014 Extension of Nakagawa & Schielzeth’s GLMM to random slopes models. Methods in Ecology and Evolution 5, 44–946.

Nakagawa, S., Johnson, P. C. D., Schielzeth, H. 2017 The coefficient of determination and intra-class correlation coefficient from generalized linear mixed-effects models revisited and expanded. J. R. Soc. Interface 14, 20170213.