r.squaredGLMM: Pseudo-R-squared for Generalized Mixed-Effect models

Description

Calculate conditional and marginal coefficient of determination for Generalized mixed-effect models ($R_{GLMM}^{2}$).

Usage

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

Value

r.squaredGLMM returns a two-column numeric matrix, each (possibly named) row holding values for marginal and conditional $R_{GLMM}^{2}$

calculated with different methods, such as “delta”,

“log-normal”, “trigamma”, or “theoretical” for models of binomial family.

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 $R_{GLMM}^{2}$ using original formulation of Johnson (2014). This requires fitting object with an observation-level random effect term added.
...: additional arguments, ignored.

Author

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

Details

For mixed-effects models, $R_{GLMM}^{2}$ comes in two types: marginal and conditional.

Marginal $R_{GLMM}^{2}$ represents the variance explained by the fixed effects, and is defined as:

$$R_{GLMM(m)}^{2}= \frac{\sigma_f^2}{\sigma_f^2 + \sigma_{\alpha}^2 + \sigma_{\varepsilon }^2} $$

Conditional $R_{GLMM}^{2}$ is interpreted as a variance explained by the entire model, including both fixed and random effects, and is calculated according to the equation:

$$R_{GLMM(c)}^{2}= \frac{\sigma_f^2 + \sigma_{\alpha}^2}{\sigma_f^2 + \sigma_{\alpha}^2 + \sigma_{\varepsilon }^2} $$

where $\sigma_f^2$ is the variance of the fixed effect components, $\sigma_{\alpha}$ is the variance of the random effects, and $\sigma_\epsilon^2$ is the “observation-level” variance.

Three different 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).

References

Nakagawa, S., Schielzeth, H. 2013 A general and simple method for obtaining $R^{2}$ from Generalized Linear Mixed-effects Models. Methods in Ecology and Evolution 4, 133--142.

Johnson, P. C. D. 2014 Extension of Nakagawa & Schielzeth’s $R_{GLMM}^{2}$ 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 $R^{2}$ and intra-class correlation coefficient from generalized linear mixed-effects models revisited and expanded. J. R. Soc. Interface 14, 20170213.

Examples

Run this code


 if(require(nlme)) {   
data(Orthodont, package = "nlme")

fm1 <- lme(distance ~ Sex * age, ~ 1 | Subject, data = Orthodont)

fmnull <- lme(distance ~ 1, ~ 1 | Subject, data = Orthodont)

r.squaredGLMM(fm1)
r.squaredGLMM(fm1, fmnull)
r.squaredGLMM(update(fm1, . ~ Sex), fmnull)

r.squaredLR(fm1)
r.squaredLR(fm1, null.RE = TRUE)
r.squaredLR(fm1, fmnull) # same result

if (FALSE) {
if(require(MASS)) {
    fm <- glmmPQL(y ~ trt + I(week > 2), random = ~ 1 | ID, 
        family = binomial, data = bacteria, verbose = FALSE)
    fmnull <- update(fm, . ~ 1)
    r.squaredGLMM(fm)

    # Include R2GLMM (delta method estimates) in a model selection table:
    # Note the use of a common null model
    dredge(fm, extra = list(R2 = function(x) r.squaredGLMM(x, fmnull)["delta", ]))
    
}
}
  }

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