Multilevel model for repeated measures data using lme4 in R - r

Question

I have a data frame with post and follow-up measurements for approximately 200 people. In the study, we try to find out if there is a correlation between sports participation and distress symptoms. We have two measurement periods (post and follow-up) that are conducted after a workshop about health and sports. Post was conducted 6 months after the Workshop and followup one year after the workshop. We formed the following hypothesis: „Participation in sport for obese people within one year after a workshop correlates significantly positively with psychological distress symptoms at follow up.“ I assume, the dependent variable is psychological distress and the independent is the participation in sports activities. The data structure looks like:
Df
$ measurement_period : Factor w/ 2 levels "0","1": 1 1 1 1
$ psychological_distress ; int 12 45 32 85
$ participation : Factor w/ 2 levels "0","1": 1 1 1 1
$ id : num 1 2 3 4
After reading some posts here, we believe that there are 2 levels in the model: 1 ) measurement period (post and follow up) 2) id
At first we conductet a unconditiional Model (intercept only Model for confirming if a multilevel Model fits, hope that this is right) with following code:
test <-lmer(psychological_distress ~1+(1|id),data=Df
But we are not sure if the model is appropriate given the data structure and, whether the level 1 and level 2 classification is correct.
Thank you very much in advance!

Answer 1

Your model:
lmer(psychological_distress ~ 1 + (1|id) , data = Df)
is a variance components model. It will tell you how much of the variation in psychological_distress is attributable to the id level, and how much is attributable to the unit/residual level. That isn't going to answer your research question:
we try to find out if there is a correlation between sports participation and distress symptoms
To look into this, you need to include the participation variable as a fixed effect, and also the time variable, and their interaction. So in the first instance I would consider this:
lmer(psychological_distress ~ measurement_period*participation + (1|id) , data = Df)

Answer 2

A good website on how to fit longitudinal and growth models using lme4 is https://rpsychologist.com/r-guide-longitudinal-lme-lmer
As Robert pointed out, and as demonstrated on the website, it is often useful to fit an interaction between "time" and "group" (e.g., treatment vs. control), to see how the outcome changes for each group over time. You can see this change by looking at the coefficients, but it's usually easier to plot (adjusted) predictions.
Here's a toy example:
library(parameters)
library(datawizard)
library(lme4)
library(ggeffects)
data("qol_cancer")
# filter two time points
qol_cancer <- data_filter(qol_cancer, time %in% c(1, 2))
# create fake treatment/control variable
set.seed(123)
treatment <- sample(unique(qol_cancer$ID), size = length(unique(qol_cancer$ID)) / 2, replace = FALSE)
qol_cancer$treatment <- 0
qol_cancer$treatment[qol_cancer$ID %in% treatment] <- 1
qol_cancer$time <- as.factor(qol_cancer$time)
qol_cancer$treatment <- factor(qol_cancer$treatment, labels = c("control", "treatment"))
m <- lmer(QoL ~ time * treatment + (1 + time | ID),
data = qol_cancer,
control = lmerControl(check.nobs.vs.nRE = "ignore"))
model_parameters(m)
#> # Fixed Effects
#>
#> Parameter | Coefficient | SE | 95% CI | t(368) | p
#> ----------------------------------------------------------------------------------------
#> (Intercept) | 70.74 | 2.15 | [66.52, 74.97] | 32.90 | < .001
#> time [2] | 0.27 | 2.22 | [-4.10, 4.64] | 0.12 | 0.905
#> treatment [treatment] | 4.88 | 3.04 | [-1.10, 10.86] | 1.60 | 0.110
#> time [2] * treatment [treatment] | 1.95 | 3.14 | [-4.23, 8.13] | 0.62 | 0.535
#>
#> # Random Effects
#>
#> Parameter | Coefficient
#> ---------------------------------------
#> SD (Intercept: ID) | 15.14
#> SD (time2: ID) | 7.33
#> Cor (Intercept~time2: ID) | -0.62
#> SD (Residual) | 14.33
#>
#> Uncertainty intervals (equal-tailed) and p-values (two-tailed) computed
#> using a Wald t-distribution approximation.
ggpredict(m, c("time", "treatment")) |> plot()
Regarding the statistical significance of the interaction term: the p-values from the summary might be misleading. If you're really interested in statistically significant differences either between time points, or between groups (treatment vs. control), it is recommended to calculate pairwise contrasts including p-values. You can do this, e.g., with the emmeans-package.
library(emmeans)
emmeans(m, c("time", "treatment")) |> contrast(method = "pairwise", adjust = "none")
#> contrast estimate SE df t.ratio p.value
#> time1 control - time2 control -0.266 2.22 186 -0.120 0.9049
#> time1 control - time1 treatment -4.876 3.04 186 -1.604 0.1105
#> time1 control - time2 treatment -7.092 2.89 316 -2.453 0.0147
#> time2 control - time1 treatment -4.610 2.89 316 -1.594 0.1118
#> time2 control - time2 treatment -6.826 2.73 186 -2.497 0.0134
#> time1 treatment - time2 treatment -2.216 2.22 186 -0.997 0.3199
#>
#> Degrees-of-freedom method: kenward-roger
Created on 2022-05-22 by the reprex package (v2.0.1)
Here you can see, e.g., that treatment and control do not differ regarding their QoL at time point 1, but they do at time point 2.