# Solved – Logistic linear mixed effects model with categorical predictors

I've read a lot of threads on stack exchange but haven't exactly found what I'm looking for. Everyone seems to have a slightly different problem/issue.

First, lets have a look at my data:

• 120 Users
• 80 Items per User
• Between factor with 4 levels
• Within factor with 4 levels
• Binary response variable

Now, usually one would perform a logistic regression. However, there are several issues with that:

• Categorical predictors have to be dummy coded?
• How do deal with the within-subject fator? -> lmm?
• How would you plot this data?

As I think (but I'm not sure? correct me if I'm wrong) that one assumption of binary logistic regression is the independence of errors which is – again not sure – violated with within subject factor I'm trying to perform a linear-mixed effects model for my data. Now here begins the actual problem:

First: I know, that I want to model the between-subjects and within-subject factors as fixed effects.
However: which random effects should I implement?

• Random intercept of each subject as it can be assumed that they differ in they're apriori knowledge of the items (it's a performance test).
• Random intercept of the within-subject factor – as this is the reason for performing lmm in the first place?
• My data is actually not nested, so there is no sense in creating a random effect like all the examples "school", "county" and so on…
• other suggestions?

Okay my suggestion is to assume random intercepts of subjects (the first one):

``lmm3 <- glmer(y ~ between * within + (1|user), data, family = binomial(link = "logit")) ``

But, first I would have to calculate the ICC 1 and ICC2 to support the use uf lmm.

for the ICC 1 I use the nullmodel:

``lmm0 <- glmer(y ~ (1|user), data, family = binomial(link = "logit")) tau2<-lme4::VarCorr(lmm0)[] icc1 <- tau2/(tau2+pi^2/3) ``

No, again two questions arise:

• How do I calculate ICC2 for the logistic model? I know that there is a function for linear mixed models, but this is not the case here.
• However, my ICC1 is only 0.03760069 so it seems that this above model doesn't make a lot of sense. What kind of model should I try then?

I thank you a lot for your inputs. You need more specific information I would be willing to prepare some data for you. I know that this is a rather theoretical issue so I'm looking forward to a discussion.

Kind regards,
David

Contents

``(1|user) + (1|item) ``