Singer (1998, p. 327, Eq. 4) gave a big covariance matrix as the following --

...if we combine the variance components for the two random effects together into a single matrix, we would find a highly structured block diagonal matrix. For example, if there were three students in each class, we would have:

If the number of students per class varied, the size of each of these submatrices would also vary, although they would still have this common structure. The variance in MATHACH for any given student is assumed to be \tau_{00}+\sigma^{2} ...

Quiz:

1. Sampling *K* repetitions with replacement, Y_{k} denotes MATHACH of the *k*-th random student within a given class, say, the first class. What is the population variance of the Y series? A: \sigma^{2}, B: \tau_{00}+\sigma^{2}.

2. Sampling 2\times K repetitions with replacement, \left(Y_{1st,k},Y_{2nd,k}\right) denotes MATHACH(s) of the *k*-th pair of random students within a given class, say, the first class. Sometime the 2nd sampled student is just the 1st one by chance. What is the population covariance of the Y_{1st} and Y_{2nd} series? A: 0, B: \tau_{00}.

3. Sampling *K* repetitions with replacement, Y_{k} denotes MATHACH of the *k*-th random student within one randomly sampled class for each repetition. What is the population variance of the *Y* series? A: \sigma^{2}, B: \tau_{00}+\sigma^{2}.

4. Sampling 2\times K repetitions with replacement, \left(Y_{1st,k},Y_{2nd,k}\right) denotes MATHACH(s) of the *k*-th pair of random students within one random class sampled for each pair of students. Sometime the 2nd sampled student is just the 1st one by chance. What is the population covariance of the Y_{1st} and Y_{2nd} series? A: 0, B: \tau_{00}.

Correct answers: A A B B.

Your plausible incorrect answers for Quiz 1 and 2 just tell how the matrix caused my misunderstanding when I read the paper the first time. It is difficult to give names for the randomly sampled classes, and the matrix columns and rows.

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Singer, J. D. (1998). Using SAS PROC MIXED to fit multilevel models, hierarchical models, and individual growth models. *Journal of Educational and Behavioral Statistics. 24.* 323-355.