My question is regarding the minimum sample size needed in a mixed effects approach for analyzing a main effect.

Lets say that a sample is taken of n=8 subjects with k=10 measurements of a continuous variable on each subject. Additionally, each subject is measured at a different point in time (but only one point in time for each subject). So the data structure could be written:

$Y_{1,1}$, $X_{1,1}$, $T_1$

$Y_{1,2}$, $X_{1,2}$, $T_1$

... ...

$Y_{1,10}$, $X_{1,10}$, $T_1$

$Y_{2,1}$, $X_{2,1}$, $T_2$

$Y_{2,2}$, $X_{2,2}$, $T_2$

... ...

$Y_{8,10}$, $X_{8,10}$, $T_8$

where $Y_{n,k}$ is the dependent variable for the $n$th subject and the $k$th measurement, $X_{n,k}$ is the independent variable, and $T_n$ is the time at wich each subject was measured.

My question is: Can this data be used to say anything about time? The concern is that there is only one observation at level 1 for each subject.

So more specifically, could I use the mixed model (in R format):

$lmer(Y_{n,k} \sim X_{n,k} + T_n + (1 | Subject_k)) $

and reasonably say there is an effect of time?


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