I am a bit puzzled by a result I got in linear mixed effect regression. Let the systematic part of the linear mixed effect model be: $$ Y = b_0 + b_1X_1 + b_2X_2 + b_3D_2 + b_4D_3 + b_{12}X_1X_2 + b_{13} X_1 D_2 + b_{14} X_1 D_3 + b_{23} X_2 D_2 + b_{24} X_2 D_3 + b_{123} X_1 X_2 D_2 + b_{124} X_1 X_2 D_3 $$ Where $D_2$ and $D_3$ are two binary variables representing the second and third tertile of a given variable. $X_1$ is time and $X_2$ is a binary variable representing treatment.
I am interested in testing the null hypothesis $H_0: b_{123} = b_{124} = 0$. This null hypothesis is stipulating that the treatment effect is the same according to tertiles $D_2, D_3$ and $D_1$ (the reference).
When I test $H_0$ using LRT or the F test for linear combination I cannot reject $H_0$. However when I use the union intersection test I reject the null hypothesis. The union intersection test can be done by noticing that $H_0$ is the intersection of $H_{01}$ and $H_{02}$ where $H_{01}: b_{123} =0$ and $H_{02}: b_{124} =0$.
Is there anything wrong in the way I am dealing with it?
