Multiple regression or linear mixed effect model?

Question

(UPDATED) I wonder if someone could advise on the correct model to be used ot analyse the longitudinal data defined as datafrane, DF, as follows:

• ID: subject ID
• SEX: categorical covariate, M/F
• TRT: categorical covariate representing drug treatment, A, B, C or D
• TIME: time of the measurement of each covariate and ENDPOINT
• ENDPOINT: response variable
• AGE: age of subjects, constant covariate
• BW: body weight, constant covariate
• HT: body height, constant covariate
• BIOMERKER1: time varying biomarker 1
• BIOMERKER2: time varying biomarker 2

BIOMARKERS1/2 are measured along with ENDPOINT and recorded at TIME points.

The model should answer the question whether any and/or which of the constant covariates (SEX, TRT, AGE, BW, HT) and time varying covariates BIOMARKER1, BIOMARKER2 have impact on the ENDPOINT. I see drug treatment effect to some extend but would like to know if it significant.

See also screenshot of the dataset structure. The real data have a trend which admittedly is not visible here, sorry for that.

Multiple regression reads

model <- lm(ENDPOINT ~., data = DF)

Linear mixed effect model, using TRT as grouping variable, would read

model <- lmer(ENDPOINT ~ TIME + WT + HT + BIOMARKER1 + BIOMARKER2 + (TIME | ID) + (1 | TRT), data = DF)

So the question is, which model is appropriate? Multiple regression or linear mixed effect model? Multiple regresssion assumes constant predictors if I understand correctly so that means it is not suitable in this case, am I right?

Any comments would be very appreciated.

UPDATE2: In the meantime I ran the linear mixed effect model on real data, lmer, and get this output:

UPDATE3: I have changed the model formulation as suggested by @Sointu and it runs even with additional variables, now using 'lmer' from lmerTest package.

model <- lmer(ENDPOINT ~ TIME + TRT + SEX + AGE + WT + BIOMARKER1 + BIOMARKER2 + (1 | ID), data = DF)

However, it looks like I have nothing significant if I interpert it correctly when looking at the p-values.

UPDATE4: now with additional interaction term 'TRT*TIME'

Answer 1

Single-level multiple regression is not suitable because your outcome variable observations are not independent (they are clustered within participants). lmer is appropriate as such but:

1. Typically, you should cluster-mean center your time-varying predictor variables prior to running the model to get interpretable estimates (regression coefficients). See, for instance, this article:

Enders, C. K., & Tofighi, D. (2007). Centering predictor variables in cross-sectional multilevel models: A new look at an old issue. Psychological Methods, 12(2), 121–138. https://doi.org/10.1037/1082-989X.12.2.121

Or this one: http://www.statmodel.com/download/HamakerMuthenFixedVSRandom.pdf

Now, your estimates are a mix of within- and between-participant effects.

I don't work with bioscience data so you may have some own statistical conventions but generally, centering is recommended. You'd do this by computing the mean of each biomarker for each participant and then would subtract this mean from each individual biomarker value and then use these participant-mean-centered values as predictors. Their coefficients tell you how much the biomarker's deviations from each person's own mean are related to the outcome.

If you want, you can also include the participant-specific mean as a separate predictor. This way you also get the average between-participant biomarker effect. But you already have a quite large model relative to the amount of data so that might lead to overfitting.

2. You probably have too little data to estimate the random slope of time. I think the non-convergence warning in the end relates to this (not necessarily, but that would be my guess). You may have to remove the random slope and go just with (1|ID).

3. An alternative for you might be a regular regression with participant-clustered standard errors, or a Generalized estimating equations (GEE) model, which also handles the non-independence of observations without using a multilevel framework. This doesn't allow for random slopes though.

EDITED TO ADD: Is it really so that you don't want the actual treatment fixed effect? Now, you have TRT modeled as random effect, which only gives you the estimate of the amount of variance in ENDPOINT attributable to the TRT levels, not the differences between different treatments? I understand with TIME fixed effect you get the change over time but it seems like TIME*TRT interaction effect would be relevant here - it would tell you whether change over time is different in different TRT groups.

(An additional concern may be that your residual covariance structure is not adequately captured by the lmer default, compound symmetry, given that the ENDPOINT observations measured close in time are likely to be more similar, within participant, than observations measured with longer time intervals. For instance an autoregressive residual structure might fit better. However, this is a problem in R (I believe) because lme4 does not allow fitting other residual structures, and the alternative, nlme package's lme function, does not seem to allow crossed random effects. SPSS can do both.)