# Opposite sign for fixed effect using population averaged model and mixed model

I'm looking at the longitudinal outcome for two treatment groups using both population averaged model and mixed effect (random intercept and slope) model. I've got opposite results on the fixed effect of treatment. I thought the covariance structure could bias the fixed effect estimates as indicated in Gruka et al. (2011). But I've tried unstructured and many other structures, still the results are opposite. Do any of you know the reason?

Below are brief SAS code and results:

* population averaged model:
proc mixed data=test;
class id interval;
model y=treatment|time/solution;
repeated interval/subject=id type=ar(1);
run;

Effect            Estimate       Error   Pr > |t|
Intercept           0.53         0.02      <.0001
trt                 -0.10        0.03      0.0004
time               -0.005        0.001     0.0017
trt*time           -0.003        0.002     0.185

* mixed model;
proc mixed data=test;
class id;
model y=treatment|time/solution;
random intercept time/subject=id type=un;
run;

Effect            Estimate       Error       Pr > |t|
Intercept           0.1602         3.91      <.0001
trt                 0.4869          8.58     <.0001
time                0.003110        0.95     0.3443
trt*time           -0.01894         -4.17      <.0001


## 1 Answer

Two thoughts and one question:

1) I presume the interval variable equivalent to time, but incorporated in the model as a categorical variable whereas time is continuous?

2) The treatment variable is not in the CLASS statement in either model, so each model is regressing y on treatment. If treatment is a categorical variable, add it to the CLASS statement and see what happens. There could be some sort order issue.

3) The standard errors in the second model are quite large relative to the parameter estimates, which suggests that there are serious estimation issues for this model. Could be that the data structure is not compatible with the model. Could be too few observations to support the unstructured covariance matrix. Could be that id is not unique (e.g., id=1 for multiple levels of treatment). Could be something else. Regardless, the results from this model are probably wrong, so the comparison between the two models is moot.