# 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.

2. This is an approach one might take. Knowing more about the design can help others answer this more intelligently. It depends on power, design, and model accuracy.

3. If you fit a fixed or random effect for ligand, you account for their mean differences. Whether there are interactions between ligand and other factors can be tested. Otherwise, I don't know what effect you mean.

• Thank you Adam. That was very helpful. I fit a GLMM using glmer function in R with binomial family. But the response variable has many zeroes. Should I use a zero inflated model in this case? – AMC Apr 12 '18 at 18:09
• @AMC There is no zero-inflated model for binomial outcomes of which I'm aware. I don't think the zeros are a cause for concern either. If your goal is modeling the S-shaped curve that relates exposure to an outcome, then a zero is a real observation that need not be explained away by a probabilistic mixture. I would still inspect the deviance residuals to be sure there are no crazy violations of model assumptions. – AdamO Apr 12 '18 at 18:19
• Thanks @AdamO. The GLMM model with binomial family is overdispersed. So I included an observational random effect to deal with overdispersion. Is this a right approach? – AMC Apr 12 '18 at 18:28