
Hypothesis test
Hi all,
How do I conduct a hypothesis test as follow
$\displaystyle H_0: \mu_{trt,time(j+1)}\mu_{trt,time(j)} \ge \mu_{trt,time(j)}\mu_{trt,time(j1)}$
as opposed to
$\displaystyle H_0: \mu_{trt,time(j+1)}\mu_{trt,time(j)} < \mu_{trt,time(j)}\mu_{trt,time(j1)}$
To simplify this: Let look into 1 particular treatment with j=2:
$\displaystyle H_0: \mu_{time(3)}\mu_{time(2)} \ge \mu_{time(2)}\mu_{time(1)}$
Let say I have $\displaystyle \bar Y_1, \bar Y_2, \bar Y_3$ and $\displaystyle \sigma_{11},\sigma_{22}, \sigma_{33}, \sigma_{12}, \sigma_{23}, \sigma_{13}$
I was thinking maybe I can shift the equation into left hand side to form:
$\displaystyle \bar Y_32\bar Y_2+\bar Y_1$ as the estimate, but what about the S.E?
My guess is: $\displaystyle 1/n\sqrt{(\sigma_{11}+2*\sigma_{22}+\sigma_{33}+??)}$ These variables are correlated.