I have one independent variable (X) and three dependent variables (W, Y and Z). I am fitting a generalized linear model to each:
$Y = g(\alpha X + \epsilon)$
$Z = g(\beta X + \epsilon)$
$W = g(\gamma X + \epsilon)$
I solve the above in R and get estimates for $\alpha, \beta\, \gamma$ along with p-values that they are not zero.
However, what I am really interested in is a complex hypothesis such as the following: $ H_0 : |\alpha - \gamma| < |\beta - \gamma|$ and $(\alpha - \gamma)(\beta - \gamma) > 0$
$H_1 : |\alpha - \gamma| > |\beta - \gamma|$ if $(\alpha - \gamma)(\beta - \gamma) > 0 $ or $(\alpha - \gamma)(\beta - \gamma) < 0 $
- I'd like to get a p-value for the above hypothesis, and if possible
an "effect size" for something like the estimate of
$s = (\alpha - \beta)$ if $H_1$, 0 otherwise
How would I go about doing this? If this is too complicated to explain, how would I test any non-trivial hypothesis involving glm coefficients? Could we use the estimates and their distributions? Or could we transform it into an equivalent model whose coefficient has the same p-value?
ps. I am actually solving the glms in LIMMA.