# Decipher driver of overtime changes of response variable in linear mixed model

I am analysing patients cohort, all of them were treated with the same medication and 2 parameters were measured 3 times: before treatment and twice after treatment. Both parameters improve over time and associated with each other. I have a hypothesis that the treatment improves one parameter and this parameter improvement (predictor) explains the improvement of another parameter (response). I tried to use a linear mixed model to prove it:

model <- lmer(response parameter ~ predictor parameter*predictor state before treatment + time point + desease severity + age + sex +(1+time point|subject), data = data, REML = FALSE)

both response and predictor parameters are continuous, predictor parameter has an interaction term with dummy variable, which shows if predictor was at normal range or not before treatment to account for different relationship between the predictor and response variable in these groups. Random effect is to account for repeated measurements of each subject and by time point random slope is to allow indivdual overtime changes for each subject. Other fixed effects are just for controlling of their confounding.

I have three questions:

1. If the model is valid for this set up and research question? If it is right to include time point as a fixed effect and a random slope?

2. When I compare this model to the model without predictor parameter, it is significant, so predictor parameter significantly explains the changes in response variable. Can I say that the predictor parameter overtime changes explain the response variable overtime changes based on this model or it is rather the association between predictor and response variables independent of their over time changes? How should I build the model if I want to answer the questions if the predictor variable overtime changes explains the response variable overtime changes? Should I add time varying variables?

3. How can I prove that these are the overtime changes in predictor variable which explain the overtime changes in response variable and not that the treatment with the medication explains overtime changes of both predictor and response variable independently? Is inclusion of time point as fixed effect represent the treatment effect in the model?

I would be grateful for any thoughts and suggestions.

Thank you.

• I don't have answers for all your questions, but regarding the first one - maybe add why you picked the model? Since you ask if it is suitable, you either have doubts about some of the assumptions or you don't really know why you picked it in the first place. (this possibly sounds harsher than I mean it, I just want to understand your reasoning why you picked the model in the first place.) Oct 28, 2020 at 14:21

If the model is valid for this set up and research question? If it is right to include time point as a fixed effect and a random slope?

Rememeber that all models are wrong, but some are useful. This seems like a useful approach to the analysis of your study. Random slopes for time are often a good idea in longitudinal studies, provided that such a model is supported by the data.

When I compare this model to the model without predictor parameter, it is significant, so predictor parameter significantly explains the changes in response variable. Can I say that the predictor parameter overtime changes explain the response variable overtime changes based on this model or it is rather the association between predictor and response variables independent of their over time changes? How should I build the model if I want to answer the questions if the predictor variable overtime changes explains the response variable overtime changes? Should I add time varying variables?

There are a lot of questions here. Each independent variable is interpreted in terms of it's association with the outcome - that is a 1 unit change in the preditor is associated with a change in the outcome, leaving other variables unchanged, except for the variables involved in an interaction, where they are interpreted conditional on the other variable being zero. The question about changes over time may be answered by including an interaction with time for that variable.

How can I prove that these are the overtime changes in predictor variable which explain the overtime changes in response variable and not that the treatment with the medication explains overtime changes of both predictor and response variable independently? Is inclusion of time point as fixed effect represent the treatment effect in the model?

First, you can't prove anything with statistics. You may find some evidence that supports a particular theory but you can't prove the theory is correct. Interacting a variable with time will tell you if the effect of time is different among groups, or equivalently whether the groups have a different response over time. However, you seem to be interested in whether there is mediation of the treatment effect by the predictor, which you can read up about in many posts on this site and others

• Thank you very much for your answer, it helped a lot! I did not have a chance to further work on this problem so far and to read on the "mediation", therefore, it is possible that I might come back with further questions about this model. Nov 12, 2020 at 12:04
• You're welcome. Feel free to ask a new question, and if you want me to take a look, you can reply to this answer and I will get a notification. Nov 12, 2020 at 12:06