In a logistic Generalized Linear Mixed Model (family = binomial), I don't know how to interpret the random effects variance:
Random effects: Groups Name Variance Std.Dev. HOSPITAL (Intercept) 0.4295 0.6554 Number of obs: 2275, groups: HOSPITAL, 14
How do I interpret this numerical result?
I have a sample of renal trasplanted patients in a multicenter study. I was testing if the probability of a patient being treated with a specific antihypertensive treatment is the same among centers. The proportion of patients treated varies greatly between centers, but may be due to differences in basal characteristics of the patients. So I estimated a generalized linear mixed model (logistic), adjusting for the principal features of the patiens. This are the results:
Generalized linear mixed model fit by maximum likelihood ['glmerMod'] Family: binomial ( logit ) Formula: HTATTO ~ AGE + SEX + BMI + INMUNOTTO + log(SCR) + log(PROTEINUR) + (1 | CENTER) Data: DATOS AIC BIC logLik deviance 1815.888 1867.456 -898.944 1797.888 Random effects: Groups Name Variance Std.Dev. CENTER (Intercept) 0.4295 0.6554 Number of obs: 2275, groups: HOSPITAL, 14 Fixed effects: Estimate Std. Error z value Pr(>|z|) (Intercept) -1.804469 0.216661 -8.329 < 2e-16 *** AGE -0.007282 0.004773 -1.526 0.12712 SEXFemale -0.127849 0.134732 -0.949 0.34267 BMI 0.015358 0.014521 1.058 0.29021 INMUNOTTOB 0.031134 0.142988 0.218 0.82763 INMUNOTTOC -0.152468 0.317454 -0.480 0.63102 log(SCR) 0.001744 0.195482 0.009 0.99288 log(PROTEINUR) 0.253084 0.088111 2.872 0.00407 **
The quantitative variables are centered. I know that the among-hospital standard deviation of the intercept is 0.6554, in log-odds scale. Because the intercept is -1.804469, in log-odds scale, then probability of being treated with the antihypertensive of a man, of average age, with average value in all variables and inmuno treatment A, for an "average" center, is 14.1 %. And now begins the interpretation: under the assumption that the random effects follow a normal distribution, we would expect approximately 95% of centers to have a value within 2 standard deviations of the mean of zero, so the probability of being treated for the average man will vary between centers with coverage interval of:
Is this correct?
Also, how can I test in glmer if the variability between centers is statistically significant? I used to work with MIXNO, an excellent software of Donald Hedeker, and there I have an standard error of the estimate variance, that I don't have in glmer. How can I have the probability of being treated for the "average" man in each center, with a confidene interval?