Hi, I have recently analyzed an unbalanced panel data, but since it is my first time to work with such data, I have some questions and I hope you can help me.

First of all, my data is very unbalanced panel data. Subjects were defined as those who participated in any of the 2nd, 3rd, 5th, and 7th waves of the survey (please see below). Each subject was measured once for the exposure variable (DHEA-S; continuous) and the outcome variable (fr_0_12; 0 or 1, dichotomous) each time they took the survey. I have pooled all the data together and tertile the exposure variable (dhea_s_3cn; 1, 2, or 3, categorical). I would like to ask if the random effects model analysis I performed with this data can be called a longitudinal analysis?
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Secondly, before proceeding to the formal analysis, what would you recommend in terms of data processing?

Third, I have analyzed the data using the following model, and what do you see as the problem with regard to that result?
I am interested in the association between dhea_s_3cn and fr_0_12.
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Finally, what would be a good way to use to characterize subjects based on the exposure variables (dhea_s_3cn; 1, 2, or 3, categorical)? Because my data are unbalanced panel data, each subject has a different number of repeated measures of characteristics. Using a general linear model or χ2 test is obviously not a good way to measure between-group differences in characteristics of exposure variables.
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Thank you all for your help.