covariance between two observations belonging to the same block

#1
Hi, does anyone know how to find the
expression for the covariance between two observations belonging to the same
block for a balanced complete block planning?
(block has random effect and treatment has fixed effect).
 

Jake

Cookie Scientist
#2
Our model for an observation with the ith treatment in the jth block is
\(y_{ij}=\mu+\theta_i+u_j+e_{ij}\)
where the \(\theta_i\) are the treatment effects and the \(u_j\) are the block effects.

So let's set up the expression for the covariance between two observations from the same block, say block j, but different treatments, say 1 and 2, as:
\(\text{cov}(y_{1j}, y_{2j})=\text{cov}(\mu+\theta_1+u_j+e_{1j}, \mu+\theta_2+u_j+e_{2j})\)
And now you can use the rules about covariances of linear combinations to simplify this expression.