# Mixed effects model for multiple time points and drugs

The data I am working on has multiple time points and multiple ligand effects. The data looks like this with percentage values in concentration being measured at each time point and ligand:

Sample Day ligand Condition Conc1 Conc2 .... Conc10

1    1    A     Mild      99    86.6 ....  0.58

1    1    B     Mild      96    85.4 ....  0.24

1    1    C     Mild      92.56 88.23....  0.22


There are 100 samples 1 through 100, three time points: day 1, 15 and 30; five ligands: A,B,C,D and E; two conditions: Mild and Severe.

I am trying to check for each conc, if there is a significant difference between mild group and severe group. In addition to this, I also need to check for a significant difference in samples with respect to time points and lignads. I have several questions regarding the approach to follow:

1. Can I use a linear mixed model or a generalized linear mixed model or any other method since the response variable is in percentages?

2. If I use a linear mixed model, can I suppose ligands, day, and condition to be fixed effects and the sample to be a random effect?

3. Would there be any effect or variation between ligands?

1. Yes, although there are some possible issues of heteroscedasticity. You might consider some suitable transformation of the outcome, like a log which is useful for modeling concentrations. You can also use a generalize linear mixed model. Probit and logit models with a binomial link can be used to model "S"-shaped curves that relate exposure to your outcomes (the %s) and take the variance to be the mean * (1-mean). Non-linear least squares can also be used, a common thing in pharmacokinetic studies.