I have fitted a general linear model $$y=\beta_0+\beta_1x_1+\beta_2x_2+\beta_3x_3,$$ whose log likelihood is $L_u$.
Now I wish to test if the coefficients are the same.
- First, overall test: the log likelihood of the reduced model $y=\beta_0+\beta_1\cdot(x_1+x_2+x_3)$ is $L_r$. By likelihood ratio test, the full model is significantly better than the reduced one with $p=0.02$.
- Next, $\beta_1=\beta_2$? The reduced model is $y=\beta_0+\beta_1\cdot(x_1+x_2)+\beta_2x_3$. The result is, $\beta_1$ is NOT different from $\beta_2$ with $p=0.15$.
- Similarly, $\beta_1=\beta_3$? They are different with $p=0.007$.
- Finally, $\beta_2=\beta_3$? They are NOT different with $p=0.12$.
This is quite confusing to me, because I expect the overall $p$ to be smaller than $0.007$, since obviously $\beta_1=\beta_2=\beta_3$ is a much stricter criterion than $\beta_1=\beta_3$ (who generates $p=0.007$).
That is, since I am already "$0.007$ confident" that $\beta_1=\beta_3$ does not hold, I should be "more confident" that $\beta_1=\beta_2=\beta_3$ does not hold. So my $p$ should go down.
Am I testing them wrongly? Otherwise, where am I wrong in the reasoning above?