I'm a beginner in statistics and I have to run multilevel logistic regressions. I am confused with the results as they differ from logistic regression with just one level.

I don't know how to interpret the variance and correlation of the random variables. And I wonder how to compute the ICC.

For example : I have a dependent variable about the protection friendship ties give to individuals (1 is for individuals who can rely a lot on their friends, 0 is for the others). There are 50 geographic clusters of respondant and one random variable which is a factor about the social situation of the neighborhood. Upper/middle class is the reference, the other modalities are working class and underprivileged neighborhoods.

I get these results :

`> summary(RLM3) Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod'] Family: binomial ( logit ) Formula: Arp ~ Densite2 + Sexe + Age + Etudes + pcs1 + Enfants + Origine3 + Sante + Religion + LPO + Sexe * Enfants + Rev + (1 + Strate | Quartier) Data: LPE Weights: PONDERATION Control: glmerControl(optimizer = "bobyqa") AIC BIC logLik deviance df.resid 3389.9 3538.3 -1669.9 3339.9 2778 Scaled residuals: Min 1Q Median 3Q Max -3.2216 -0.7573 -0.3601 0.8794 2.7833 Random effects: Groups Name Variance Std.Dev. Corr Neighb. (Intercept) 0.2021 0.4495 Working Cl. 0.2021 0.4495 -1.00 Underpriv. 0.2021 0.4495 -1.00 1.00 Number of obs: 2803, groups: Neigh., 50 Fixed effects: `

The differences with the "call" part is due to the fact I translated some words.

I think I understand the relation between the random intercept and the random slope for linear regressions but it is more difficult for logistics ones. I guess that when the correlation is positive, I can conclude that the type of neighborhood (social context) has a positive impact on the protectiveness of friendship ties, and conversely. But how do I quantify that ?

Moreover, I find it odd to get correlation of 1 or -1 and nothing more intermediate.

As for the ICC I am puzzled because I have seen a post about lmer regression that indicates that intraclass correlation can be computed by dividing the variance of the random intercept by the variance of the random intercept, plus the variance the random variables, plus the residuals.

But there are no residuals in the results of a glmer. I have read in a book that ICC must be computed by dividing the random intercept variance by the random intercept variance plus 2.36 (pi²/3). But in another book, 2.36 was replaced by the inter-group variance (the first level variance I guess).

What is the good solution ?

I hope these questions are not too confused.

Thank you for your attention !

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#### Best Answer

The ICC in a generalised linear model (binomial here) varies according to the values of the covariates. Therefore this is no single ICC, and you only have a conditional ICC.

An alternative to ICC that you might want to consider is the Median Odds Ratio which gives you an idea how much two identical individuals (with the same covariates) will vary when they belong to different clusters.

See pp. 256-257 in Rabe-Hesketh & Skrondal (2nd ed) "Multilevel and Longitudinal Modelling using Stata"

Also see this … http://www.bristol.ac.uk/media-library/sites/cmm/migrated/documents/pvmm.pdf which suggests some strategies for coming up with measures to partition variance in multilevel models

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