# variance & covariance of orthogonal contrasts

• Jun 12th 2011, 09:16 AM
Sambit
variance & covariance of orthogonal contrasts
Consider an RBD with 5 replications and 4 treatments $t_1,t_2,t_3,t_4$. We are given the following three treatment-contrasts:
$\frac{t_1-t_2}{\sqrt{2}}$
$\frac{t_1+t_2-2t_3}{\sqrt{6}}$
$\frac{t_1+t_2+t_3-t_4}{\sqrt{12}}$
Find the covariances of all possible pairs of BLUEs and variances of the BLUEs of the three contrasts.

What I can see is they are mutually orthogonal. Then?
• Jun 12th 2011, 12:08 PM
theodds
Where exactly are you having problems? A precise statement of what the model is would be useful - it isn't clear to me where the blocks are showing up, given the way the question is phrased.

The anticipation is that the orthogonality is going to cause the contrasts to be uncorrelated. If $\hat{\bm t}$ is the BLUE of $t = (t_1, t_2, t_3, t_4)^T$ and $\bm w_1$ and $\bm w_2$ are orthogonal contrasts, you are going to want to see what happens when you calculate

$\mbox{Cov}[\bm w_1 ^T \hat{\bm t}, \bm w_2 ^T \hat{\bm t}] = \bm w_1 ^T \mbox{Var}[\hat {\bm t}] \bm w_2$.

If $\mbox{Var}[\hat t]$ is proportional to the identity matrix, this is going to give you everything immediately.
• Jun 12th 2011, 11:17 PM
Sambit
Suppose the BLUE of $c_it_i$ is $c_i(\bar{y_{io}}-\bar{y_{oo}})$. Then variance of 1st contrast becomes $\frac{1}{2}Var(\bar{y_{1o}}-\bar{y_{2o}})$ = $\frac{1}{2}[Var(\bar{y_{1o}})+Var(\bar{y_{2o}})]$ = $\frac{1}{2}[\frac{\sigma^2}{5}+\frac{\sigma^2}{5}] = \frac{\sigma^2}{5}$. But ideally it should be 1...
• Jun 13th 2011, 09:13 AM
theodds
Quote:

Originally Posted by Sambit
Suppose the BLUE of $c_it_i$ is $c_i(\bar{y_{io}}-\bar{y_{oo}})$. Then variance of 1st contrast becomes $\frac{1}{2}Var(\bar{y_{1o}}-\bar{y_{2o}})$ = $\frac{1}{2}[Var(\bar{y_{1o}})+Var(\bar{y_{2o}})]$ = $\frac{1}{2}[\frac{\sigma^2}{5}+\frac{\sigma^2}{5}] = \frac{\sigma^2}{5}$. But ideally it should be 1...

No it shouldn't. The variance of the contrast (1) needs to depend on $\sigma^2$ for obvious reasons, and (2) needs to go to 0 as we increase the amount of information we have. The number 1 does neither of these things.