Why is x significant in linear regression but not in mixed model analysis? I have a dataset ($n=700$) where I measured $x$ and $y$ (continuous) at three points in time ($T_0$, $T_1$, $T_2$). I am interested in running a mixed model analysis in SPSS.
When I run normal linear regression analysis on my three separate datasets ($T_0$, $T_1$ and $T_2$) with $x_1$ and $y_1$, $x_2$ and $y_2$ and $x_3$ and $y_3$ separately, I get three highly significant results. However, when I run the mixed model analysis in my long (combined) data structure, $x$ is no longer significant.
Why does this happen?
 A: Assuming all things are setup correctly, there are 3 avenues of investigation.
Case 1: the trend of X and Y is non-homogeneous.
Explanation: the mixed model pools the separate cross-sectional analyses of time point 1, 2, and 3. However, if the trend is non-homogeneous, as evidenced by highly inconsistent cross-sectional slopes and estimates, the power of the mixed analysis is diminished.
Solution: provide forest plot of estimates and 95% CIS from cross sectional models. Consider using a GEE, or adjust for time and it's interaction with "X" in the mixed model. Note: the correlation structure of the mixed model should be exchangeable to prevent singularity.
Case 2: the correlation structure is misspecified
Explanation: In a panel design, there is correlation within participants and within similar time frames within a participant. Adjusting for "X" or time or other blocking factors may reduce the residual correlations so that a weaker correlation structure (even independence) is justified.
Solution: obtain estimates of intraclass correlation and plot variograms
Case 3: the intraclass correlation is too high
Explanation: it takes a little extra power to estimate the correlation structure. Consider that if three replications of a design were performed, and the repeated measures were perfectly correlated, the conservative assumption would be that the correlation was too high to obtain any additional precision from the repeated experiments, so they are effectively thrown away. Of course, the regression for time point 1 of X on Y was significant, so we have to consider that the "price" of fitting a more complex model has reduced the overall precision.
I may update the answer if more detail is provided.
