# 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? – StatsStudent Jan 9 at 21:05
• Without any data to work with, it is difficult to advise on your query. I would suggest adding more information. – Michael Grogan Jan 9 at 21:11
• I want to know whether there is a decline in Score over time in general population, and also to assess individual effects (pressumably, I would take the random slopes). – Galit Jan 9 at 21:12
• @MichaelGrogan, I added some information which hopefully will clarify. Let me know if you feel anything else is missing. – Galit Jan 9 at 21:24
• @Galit: Well, I meant a dataset of some sort to work with, otherwise it's hard to judge which regression would be the most suitable in this situation. – Michael Grogan Jan 9 at 21:52

A couple of points:

• In the case of linear mixed models inclusion or exclusion of random-effects terms does not change the interpretation of the fixed effects coefficients. That is, the interpretation of the fixed effects in models A and B is the same. However, as you observed, the inclusion or exclusion of random effects can affect the estimated value of the coefficients, their standard errors, and p-values. In general, the random effects are used to model correlations you have in the repeated measurements of each subject. If these correlations are not modeled appropriately, then this may affect the estimation and inference for the fixed effects. This is even more so if you have missing data / dropout in your longitudinal outcome, which is of the missing at random type.
• 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).
• Thank you, Dimitris! I am still puzzled though - in models A/B, the intercept isn't exactly like the predictor Age0? That's what made me think of model C. (I see this question now is unrelated to random effects, but to the interpretation of the predictors in general). – Galit Jan 13 at 13:22
• 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., $\texttt{Age}_{ij} = 0$. – Dimitris Rizopoulos Jan 13 at 18:32