I'm currently trying to familiarize myself with linear mixed models (with normally distributed continuous outcome, no generalized versions for the time being). One particular question I'm currently struggling with is the following:
Given a dataset resulting from measuring both the continuous outcome and a continuous exposure at t1 and t2, a set of time-invariant covariates, and a set of time-varying covariates, the latter again measured at t1 and t2. Covariates might include both quantitative and categorical variables. Assume there are no missing values.
Now, I usually would use an ordinary baseline-adjusted linear regression of the outcome at t2 on all time-invariant covariates, the time-varying covariates at t1, the exposure at t1, and the outcome at t1.
If I am interested in estimating the causal effect of the exposure (and assuming my covariates include all relevant confounding factors), would it make sense to apply a mixed model instead? Is there a mixed model specification where the interpretation of the fixed effect regression coefficients would be similar to that of the ordinary baseline-adjusted model? Knowing this would help me get an understanding of the overlap / differences in what those models actually do. I calculated a random intercept model of the outcome on time and all covariates (assuming compound symmetry, i.e. no heterogeneous variances), however I suppose the coefficients do not represent the same comparison as in the baseline-adjusted model as the model as the mixed model includes additional information (time-varying covariates at t2)?
I'm familiar with the distinction between wide and long datasets and how to transform them so that is not an issue.
I would greatly appreciate any suggestions or hints about pertinent literature sources (specific to this question).
Thanks for your reply, Jochen. In terms of the simplified model you suggested, my baseline-adjusted model would be yFollowup = β0+β1*YBaseline+β2*XGroup (an ANCOVA in analysis of variance terms).
The last paragraph of your answer gets at the core of my problem: My exposure variable (what would be XGroup in the simplified model) is time-varying. Would it even make sense to include this exposure as a time-varying covariate for estimating its causal effect, or should I treat it as time-invariant by only using the baseline values in the mixed model?
Yes, my interest is in the case where the exposure of interest does change over time. For example it could be something like smoking status (effectively it doesn't matter whether it is an observed or manipulated exposure, the point is how to deal with these time-varying exposures in general if I am interested in estimating their causal effects).
I think these resources will be helpful with time varying and time invariant terms
http://eprints.lse.ac.uk/52199/1/Steele_Subject_specific_population_2013.pdf
Szmaragd, Camille, Clarke, Paul and Steele, Fiona (2013) Subject specific and population average models for binary longitudinal data: a tutorial. Longitudinal and life course studies, 42 (2), pp. 147-165.
and
https://www.researchgate.net/publication/233756428_Explaining_Fixed_Effects_Random_Effects_modelling_of_Time-Series_Cross-Sectional_and_Panel_Data
Explaining Fixed Effects: Random Effects modelling of Time-Series Cross-Sectional and Panel Data Andrew Bel and Kelvyn Jones
Fiona Steele has written online training material which includes a discussion of initial conditions
http://www.bristol.ac.uk/media-library/sites/cmm/migrated/documents/longitudinal.pdf
http://www.bristol.ac.uk/cmm/learning/online-course/course-topics.html#m15
Article Explaining Fixed Effects: Random Effects Modeling of Time-Se...
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