[R-sig-ME] correspondence between intercept in a logit model and mean y response/probability
j.hadfield at ed.ac.uk
Wed Jun 29 16:52:40 CEST 2011
Its perhaps easier to understand in terms of the odds ratios.
The expected odds ratio for the observations might be:
E[exp(Xb+Zu)] = exp(XB)E[exp(Zu)]
since exp(XB) is a constant and so E[exp(XB)] = exp(XB) and
In your case you also treated exp(Zu) as fixed (u=0, and so
E[exp(Xb+Zu)] = exp(XB)
conditional on u=0.
However, averaging over the population - treating Zu as a random
variable - is different. Imagine Z=I, then
E[exp(Zu)] = E[exp(u)] = exp(VAR(u)/2)
since exp(u) is log normal if u is normal on the link scale (with mean
zero and variance var(u)).
In this case E[exp(Xb+Zu)] depends on the variance of the random effects:
E[exp(Xb+Zu)] = exp(Xb)*exp(VAR(u)/2) = exp(Xb+VAR(u)/2)
Quoting Malcolm Fairbrother <m.fairbrother at bristol.ac.uk> on Wed, 29
Jun 2011 15:23:09 +0100:
> Thanks Jarrod, I figured it would be something elementary like that.
> Also, I see that:
> mean(plogis(mod1 at eta))
> mean(mod1 at mu)
> both yield 0.3473491--very close to the mean of longdata$contact (0.3503684).
> However, I don't really know what to make of the "predicted mode".
> The usual explanation of logit models says something like: (a) we're
> interested in probabilities; (b) we model the log-odds by necessity;
> and (c) having fitted a logit model it's useful to reconvert the
> expected values for different combinations of covariates back to
> probabilities, using prob = exp(XB)/(1+exp(XB). If "prob" in this
> case is estimated to be 0.2173616, whereas we know that the overall
> average probability in the dataset is 0.3503684... what gives?
> What's the relationship between the "predicted mode" and the
> "expected probability"?
> Much appreciated,
> On 29 Jun 2011, at 13:48, Jarrod Hadfield wrote:
>> 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:
>> 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).
>> 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:
>>>  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,
>>> 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
>>> R-sig-mixed-models at r-project.org mailing list
>> The University of Edinburgh is a charitable body, registered in
>> Scotland, with registration number SC005336.
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