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