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I am currently using a mixed binomial model with the following specification in a paper I recently submitted (using lme4):

m1<-glmer(y~X1*X2*X3+(1|Subject.ID),data=data,family="binomial")

X1 and X2 are binary predictors

However, a reviewer has asked me to also include a random-effects probit model for the same analysis. What is the difference between a mixed effects model and a random effects model? Am I correct in thinking that a random-effects model would have no fixed effects and would thus be specified by:

m1<-glmer(y~1+(1|Subject.ID)+(1|X1)+(1|X2)+(1|X1:X2),data=data,family=binomial(link=probit))

If so how do I interpret the output. All I get is the Std.Dev. for each random effect.

Alternatively does it refer to a model where I would have a random slope for each of my predictors like:

m1<-glmer(y~1+(X1|Subject.ID)+(X2|Subject.ID)+(X1:X2|Subject.ID),data=data,family=binomial(link=probit))
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Random effects probit model might refer to econometric jargon. In econometrics fixed-effect and random effect models have different meaning. If we use mixed effect model terminology the random effect model in econometrics means that the intercept is random, i.e. it varies between groups but it is constant within the group. It is then called unobserved individual effect. So in your case (if I'm correct in interpreting the request of reviewer) you need to fit the model:

m1<-glmer(y~X1*X2*X3+(1|Subject.ID),data=data,family=binomial(link="probit"))

as in your case only intercept is allowed to be random. I would make sure though, by searching whether random effects probit model can really be estimated with glmer, there might be some caveats related to exogeneity of regressors, which are ignored in glmer case.

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    $\begingroup$ Thanks, this is what I suspected after reading a few papers which use 'random-effects probit models' $\endgroup$ – Jonathan Bone Dec 9 '14 at 14:20

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