# How to control for age in analyzing longitudinal data using mixed effects regression

I am analyzing a longitudinal dataset. Elderly subjects perform a cognitive test once a year, for five consecutive years. I want to know if there is a decline in their performance through the study period (due to aging, no treatment) while controlling for differences in the ages at the first year of study. I thought of three options:

A. Testing the effect of Time and adding a covariate of Age.initial (age of the subjects at the beginning of the study):

lmer(Score ~ Time + Age.initial + (Time | Subject), data = data, REML = FALSE)


B. It then occurred to me that model A has a random intercept for Time, which corresponds to the effect of Age.initial. Does this make sense? Maybe a model without a random intercept is better:

lmer(Score ~ Time + Age.initial + (- 1 + Time | Subject), data = data, REML = FALSE)


I get that Age.initial is non-significant in A but significant in B, and vice-versa for Time.

C. A third option is to unify Time and Age.initial to one predictor, the age at the time of measurement (there is no treatment other than measuring Score in different Times):

lmer(Score ~ Age.at.Time + (Age.at.Time | Subject), data = data, REML = FALSE)


What is the difference between them? Which one should I choose?

P.S. Here is a related question in which the author chose option A.

• What is the purpose of your analysis: to say something about the individuals in your analysis or to say something about the population as a whole? Jan 9, 2019 at 21:05
• The difference between the models A/B and C is in the time scale. In particular, you first model is $$\texttt{Score}_{ij} = \beta_0 + \beta_1 \texttt{Time}_{ij} + \beta_2 \texttt{Age0}_i + b_{i0} + b_{i1} \texttt{Time}_{ij} + \varepsilon_{ij},$$ where $$\texttt{Time}_{ij}$$ denotes the follow-up time for subject $$i$$ and visit $$j$$, $$\texttt{Age0}_i$$ denotes the baseline age for the subjects, and $$b_i = (b_{i0}, b_{i1})$$ denote the random effects. Likewise, your second model is $$\texttt{Score}_{ij} = \gamma_0 + \gamma_1 \texttt{Age}_{ij} + u_{i0} + u_{i1} \texttt{Age}_{ij} + \epsilon_{ij},$$ where $$\texttt{Age}_{ij}$$ denotes the age of the $$i$$-th subject at the $$j$$-th visit, and the random effects are now $$u_i = (u_{i0}, u_{i1})$$.
• The coefficients $$\beta_1$$ and $$\gamma_1$$ in the two models have a different interpretation. Namely, coefficients $$\beta_1$$ denotes the average change in the score for every year of follow-up for subjects with the same baseline age. The interpretation of $$\gamma_1$$ is the average change in the score for every year increase of their age. Hence, it is not that one formulation is wrong and the other correct, but rather you need to select which of the two answers your research question (i.e., perhaps forcing you to rethink it and formulate it more accurately).
• In model A**/**B the intercept $\beta_0$ denotes the average score at the beginning of the follow-up, i.e., for $\texttt{Time}_{ij} = 0$ and for baseline age equal to zero, i.e., $\texttt{Age0}_i = 0$. In model C the intercept $\gamma_0$ denotes the average score at follow-up age equal to zero, i.e., $\texttt{Age}_{ij} = 0$. Jan 13, 2019 at 18:32