In terms of the model, the difference between the sample mean and the sample mean ith kth is . Clearly the common random term has disappeared from this difference and you would not expect that would feature in the sed value. The mathematics is as follows. Define a contrast between the sample mean and the sample mean as the vector ith kth having a of +1 alongside the position of the mean and a -1 alongside the position of ith the mean. Note that and . kth Then which is identical to the sed under a fixed block assumption. Approach 2 Random Model is simply with the var(Error) an uniform Error correlation matrix. The assumptions for a random effect are that for each , and is independent of the error j, variates which are all independent, . Hence for each observation in every block, . If we now take two observations in the same block (say block we have the random effect j) in common. Hence for the and observations in block we have ith kth j all terms being uncorrelated.
In terms of the model, the difference between the ith sample mean and the kth sample mean is . Clearly the common random term has disappeared from this difference and you would not expect that would feature in the sed value. The mathematics is as follows. Define a contrast between the ith sample mean and the kth sample mean as the vector having a of +1 alongside the position of the ith mean and a -1 alongside the position of the kth mean. Note that and . Then which is identical to the sed under a fixed block assumption. Approach 2 Random Model is simply Error with the var(Error) an uniform correlation matrix. The assumptions for a random effect are that for each j, , and is independent of the error variates which are all independent, . Hence for each observation in every block, . If we now take two observations in the same block (say block j) we have the random effect in common. Hence for the ith and kth observations in block j we have all terms being uncorrelated.