[R-sig-ME] correspondence between intercept in a logit model and mean y response/probability

[Jarrod Hadfield]

Hi,

0.2173616 is the predicted mode. The inverse-logit transform is  
non-linear so f(E[x]) does not equal E[f(x)].

E[f(x)] can be approximated (well) as:

c2<-((16*sqrt(3))/(15*pi))^2
plogis(eta/sqrt(1+c2*v))

where eta is the linear predictor on the link scale (the intercept in  
your case), and v is the variation around the linear predictor on the  
link scale (probably the sum of your variance components).

Jarrod




Quoting Malcolm Fairbrother <m.fairbrother at bristol.ac.uk> on Wed, 29  
Jun 2011 10:31:25 +0100:

> Dear list,
>
> I'm fitting a mixed logit model with lme4, and finding something  
> that seems weird to me, but probably has a simple explanation. I  
> suspect someone on this list will be able to clarify what's going  
> on. In brief, the issue is the correspondence between the intercept  
> term in a mixed logit model and the mean response/probability of an  
> outcome across all units.
>
> The mean of my binary response variable is about 0.35:
>
>> mean(longdata$contact)
> [1] 0.3503684
>
> But when I fit mod1 below, the Intercept is estimated to be  
> -1.28111, which does NOT correspond to this mean response:
>
>> mod1 <- lmer(contact ~ 1 + (1 | group) + (1 | id), longdata,  
>> family=binomial)
>> plogis(fixef(mod1))
> (Intercept)
>  0.2173616
>
> Huh? Why is this happening? Is it something to do with the shrinkage  
> that occurs because of the clustering in higher-level units? I would  
> have expected an intercept term close to the log-odds equivalent of  
> a probability of 0.35. I presume the difference between empirical  
> and modelled mean probability isn't indicative of any big problems,  
> and indeed might be a useful result, but I'd like to know what I  
> should understand by it.
>
> Any help would be much appreciated (and apologies for posting a lot  
> to this list recently).
>
> - Malcolm
>
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>
>



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