# Residual variance for glmer

I am running a glmer model and I want to determine the total variance. My data is for survival and it is coded as 0 and 1, where 1 represents that the individual survived and 0 represents that the individual died. My data represents offspring from a full factorial cross where some individuals are full sibs or half sibs.

When running a glmer model, and there is no residual variance in the summary output. I have read that the residual variance should be (π^2)/3 for generalized linear mixed models with binomial data and logit link function (Nakagawa, S., Schielzeth, H. 2010. Repeatability for Gaussian and non-Gaussian data: a practical guide for biologists. Biol. Rev. 85:935-956.).

Is this true? Or is there a different way to calculate the residual variance for glmer?

Here is my model and output:

model6 = glmer(X09.Nov~(1|Dam)+(1|Sire)+(1|Sire:Dam), family=binomial, data=data)
summary(model6)

Generalized linear mixed model fit by maximum likelihood (Laplace Approximation
[glmerMod]
Family: binomial  ( logit )
Formula: X09.Nov ~ (1 | Dam) + (1 | Sire) + (1 | Sire:Dam)
Data: data

AIC      BIC   logLik deviance df.resid
1274.4   1295.3   -633.2   1266.4     1375

Scaled residuals:
Min      1Q  Median      3Q     Max
-3.2747  0.3366  0.3931  0.4664  1.1090

Random effects:
Groups   Name        Variance  Std.Dev.
Sire:Dam (Intercept) 3.853e-01 6.207e-01
Sire     (Intercept) 4.181e-02 2.045e-01
Dam      (Intercept) 6.036e-09 7.769e-05
Number of obs: 1379, groups:  Sire:Dam, 49; Sire, 7; Dam, 7
Fixed effects:
Estimate Std. Error z value     Pr
(Intercept)   1.6456     0.1419    11.6 <2e-16 *


One possible interpretation of the logistic regression model is to state that there is an underlying score $$y^*_i = x_i'\beta + \epsilon_i,$$ with the observed variable being $$y_i = \left\{ \begin{array}{ll} 1, & y^*_i > 0 \\ 0, & y^*_i \le 0\end{array}\right.$$ This would be the way logistic regression would be introduced in social sciences, as opposed to biostatistics. In this formulation, $\epsilon$ follows a logistic distribution, which does have the variance of $\pi^2/3$. Mixed models stick an additional random effects term into the equation, and introduce the double subscripts, making it $$y^*_{ij} = x_{ij}'\beta + u_i + \epsilon_{ij},$$ where $u_i$ is assumed normal because the normal distribution is something that everybody understands. The variance of $u_i$ is estimated by your mixed model package (although without the standard errors; Douglas Bates has a pretty strong stand on it). So the total variance is then $\sigma^2_u + \mathbb{V}[\epsilon] = \sigma^2_u + \pi^2/3$.
• Hi @StasK, just a follow-up question, if I want to know how much variance (in percentage) that the random effect Sire account for. Is it calculated as 4.181e-02/(3.853e-01+4.181e-02+ 6.036e-09)? – Chloe 20 hours ago