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I have a multinomial logistic GLMM with one random intercept. The number of response categories $C = 4$. Since a multinomial logit model consists of $C-1$ binomial logit models -- each pairing one non-baseline response category with an arbitrarily chosen baseline category -- there are three random-effect parameters.

Hosmer et al. (2013: 327) instruct that the proportion of the overall variation that is accounted for by the random component in a binary logistic GLMM can be estimated by $$\frac{\theta^2}{\frac{\pi^2}{3}+\theta^2}$$ where $\theta$ is the estimated standard deviation of the random effect, and $\pi$ (unless I'm gravely mistaken) is simply pi.

How do we generalize this formula to a multinomial setting, where there are $C-1$ logits instead of one, and the random effect of each logit has a different $\theta$ and a different $n$?

Note that I'm not asking how to test the significance of the random effect -- I already know roughly how that is done. I'm specifically interested in quantifying the proportion of the overall variability explained by the random effect.

I imagine the answer might be some kind of mean of the result of the above formula over the $C-1$ logits. But if so, how should that mean be weighted? The response categories $c$ are not evenly distributed.

FYI:

$c1$ (the baseline category) has $n = 688$.

$c2$ has $n = 747$.

$c3$ has $n = 667$.

$c4$ has $n = 437$.


References:

Hosmer, D. W., Lemeshow, S. & Sturdivant, R. X. (2013). Applied logistic regression (3rd ed.). Hoboken, N.J.: Wiley.

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