# Confusing generalized linear mixed effects model

I am working on a clinical trial where the key question is “does treatment 1 work better than treatment 2 on Eyes”. In this study 90% of patients received treatment on both eyes and 10% only on one eye (planned missing values). For whom received treatment on both eyes, there is a correlation between first and second eyes meaning that the treatment of the first eye affects the second eye.

I stored the result of the experiment under the binary variable Response (1 is success and 0 is failure). I can simply do a chi-square test to see whether there is a meaningful difference between the two treatments but, I need to consider the effect of the first eye on the treatment of the second eye as well. Therefore, a simple chi-square does not work. I read a couple of papers and finally, I found that the best solution is generalized linear mixed effects model. Here is one paper: link.

Now the problem is that I am not sure how to formalize the model. I considered “subject ID” and “eye” as random effects, “treatment” as fixed effect and “response” as the dependent variable. But I think the result does not reflect real difference between the two treatments because the success rate for the first and second treatments are 94.6% and 85.3% respectively. As the distribution is binary, I expect that the estimate for the fixed effect treatment equals 9.3 (94.6-85.3). But I could never get this number even in absence of the random effect “eye”. Here is a link to my dataset (simulated data).

I really appreciate any help (with R or SAS would be perfect). Regards Sina

• Your model sounds reasonable, in R it would be something like this: glmer(y~treatment + (1|ID/eye), family = "binomial", data = mydata). This fits so called nested random effects. Please note that the coefficient for treatment will not be a percentage difference but a difference on the log-odds scale. Exponentiated coefficients are odds ratios. – COOLSerdash Apr 26 at 7:05
• Thanks for your comment, but there are some concerns about our study that I just understood, 1- Eye is not nested in ID (subject) because then we should not have the same subject having both levels of binary variable eye, or simply speaking none of the subjects should have submitted both eyes. However, in our data 90% of subjects submitted both eyes, therefore, the variable “Eye” might be a cross effect. – Sina ALizadeh Apr 26 at 13:08
• 2- The mixed effects model consider interdependency between variables. For example, how do the categories “male” and “female” in the variable gender affects the treatment. Here there is not any correlation between male and female. In our case, however, there is intra-dependency in variable eye. Eye 1 is correlated with eye 2. The treatment does not depend on the left or right eye but on the correlation or interaction between them. So, I think they must be crossed. But still I am not sure if we will include the correlation between two eyes if we cross them. What do you think? – Sina ALizadeh Apr 26 at 13:08