Modeling repeated measures data in R - Interpretation and Validation

Question

I am currently working on a control-case study where patients and controls are assessed at five different time intervals. The aim of the study is to assess possible differences in the response variable between Healthy and Patients at each phase of the experiment. I have implemented a linear mixed model using the lme4 package in R to analyze the data. I have a few questions regarding the interpretation of the results.

Find below the code to reproduce the results

library(lme4)
library(emmeans)

set.seed(534)

n_subjects <- 50  # number of subjects
n_timepoints <- 5  # number of repeated measurements per subject
subject_ids <- rep(1:n_subjects, each = n_timepoints)
timepoints <- rep(1:n_timepoints, times = n_subjects)

group <- character(length = length(subject_ids))

group <- rep(sample(c("Healthy", "Patient"), size = n_subjects, replace = TRUE), each = n_timepoints)

random_intercepts <- rnorm(n_subjects, mean = 0, sd = 2)
response <- rnorm(n_subjects * n_timepoints) + random_intercepts

# Create a data frame
simulated_data <- data.frame(
  SubjectID = factor(subject_ids),
  Timepoint = factor(timepoints),
  Group = factor(group),
  Response = response
)

# Fit a linear mixed model
lmm_model <- lmer(Response ~ Group*Timepoint + (1|SubjectID), data = simulated_data)
summary(lmm_model)
emmip(lmm_model,  Group ~ Timepoint , data = simulated_data, CIs = TRUE, xlab="PHASE")

# Display contrasts for PHASE within each level of Sailors
emm <- emmeans(lmm_model, ~ Group * Timepoint)
contrasts_phases <- pairs(emm, simple="each", adjust="Bonferroni")
print(contrasts_phases)

I'm struggling interpreting the results as I've not found similar examples online.

Looking at the results I only have a significant interaction, namely GroupPatient:Timepoint4, which is telling me how much greater is the difference between Healthy and Patients in the phase 4 compared to the phase 1 (interpretation of interaction-term in linear regression, with and without main-effect).

> summary(lmm_model)
...
Fixed effects:
                        Estimate Std. Error       df t value Pr(>|t|)  
(Intercept)               0.1503     0.4807 240.0000   0.313   0.7548  
GroupPatient              0.7639     0.6798 240.0000   1.124   0.2622  
Timepoint2               -0.9397     0.6798 240.0000  -1.382   0.1681  
Timepoint3               -0.7947     0.6613 240.0000  -1.202   0.2307  
Timepoint4                0.6256     0.6732 240.0000   0.929   0.3537  
Timepoint5                0.7200     0.6868 240.0000   1.048   0.2955  
GroupPatient:Timepoint2  -0.8411     0.9613 240.0000  -0.875   0.3825  
GroupPatient:Timepoint3  -0.2299     0.9648 240.0000  -0.238   0.8119  
GroupPatient:Timepoint4  -2.3312     0.9617 240.0000  -2.424   0.0161 *
GroupPatient:Timepoint5  -0.9183     0.9617 240.0000  -0.955   0.3406 

Examining the marginal means reveals a significant stacked difference between the Healthy and Patient groups specifically in phase 4 of the experiment. Can I confidently infer that the data demonstrate a distinction between the two groups solely in this phase, or should additional checks be conducted to validate this conclusion? Also, I am a bit lost concerning if the contrast method is correct, as many times I see this in pairwise comparison, which are displaying all possible contrasts.

> contrasts_phases <- pairs(emm, simple="each", adjust="Bonferroni")
> print(contrasts_phases)
$`simple contrasts for Group`
Timepoint = 1:
 contrast          estimate    SE  df t.ratio p.value
 Healthy - Patient  -0.7639 0.685 240  -1.114  0.2662

Timepoint = 2:
 contrast          estimate    SE  df t.ratio p.value
 Healthy - Patient   0.0771 0.685 240   0.113  0.9105

Timepoint = 3:
 contrast          estimate    SE  df t.ratio p.value
 Healthy - Patient  -0.5340 0.690 240  -0.774  0.4400

Timepoint = 4:
 contrast          estimate    SE  df t.ratio p.value
 Healthy - Patient   1.5673 0.686 240   2.285  0.0232

Timepoint = 5:
 contrast          estimate    SE  df t.ratio p.value
 Healthy - Patient   0.1543 0.686 240   0.225  0.8222

Answer 1

To help make your original regression results more interpretable, I suggest that you code timepoint such that the first occasion is given a value of 0. This is because in regression models, the intercept is the mean outcome value when all predictors are at 0. In the simulated data, timepoint==0 is outside your data, so it is an extrapolation. To change this in your simulation setup, I just changed line 9:

timepoints <- rep(0:4, times = n_subjects)

This change makes it such that the interpretation you put forward in your original question is nearly consistent with model results. Specifically, the interaction between GroupPatient and Timepoint3 gives you a parameter estimate and standard error for the test of the null hypothesis that there is no difference between healthy people (the reference group) and patients at timepoint==3 (the fourth measurement occasion).

It is also worth testing whether the coefficients for all interactions are simultaneously different from 0. We can do that lots of ways. With mixed models, it is common to use likelihood ratio tests. In this case, we compare the model with all the interactions to a model with no interactions, just the main effects of GroupPatient and Timepoint.

lmm_model_1 <- lmer(Response ~ Group + Timepoint + (1|SubjectID), data = simulated_data)
summary(lmm_model_1)
anova(lmm_model_1, lmm_model)

The lrtest is significant, suggesting that there is evidence for an interaction between GroupPatient and Timepoint:

refitting model(s) with ML (instead of REML)
Data: simulated_data
Models:
lmm_model_1: Response ~ Group + Timepoint + (1 | SubjectID)
lmm_model: Response ~ Group * Timepoint + (1 | SubjectID)
            npar    AIC    BIC  logLik deviance  Chisq Df Pr(>Chisq)  
lmm_model_1    8 1102.7 1130.9 -543.36   1086.7                       
lmm_model     12 1100.0 1142.3 -538.03   1076.0 10.671  4    0.03053 *

From here, you can use emmeans to do the contrasts and pairwise comparisons. Personally, I would show the same graph you did and report the simple contrasts.