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.