In Exercise 17.2 fixed block effects were assumed. Show how the expectations change if block effects are assumed to be random variables with zero mean and variance σ 2 b . Show how this changes your...

In Exercise 17.2 fixed block effects were assumed. Show how the expectations change if block effects are assumed to be random variables with zero mean and variance σ²
_b. Show how this changes your conclusions when the numbers are unequal as in Exercise 17.3.

Exercise 17.2

The partial (Type III) sums of squares from the analysis of variance, Table 19.2, and the mean square expectations, Table 19.10, have been used to estimate the two components of variance σ²
and σ²
_δ. Compute standard errors for each. (Assume each mean square is distributed as a chi-squared random variable scaled by E(MS)/d.f. so that its variance is 2[E(MS)]2/d.f and that the two mean squares are independent. The estimate of the variance of a chi-squared random variable is obtained by substituting the observed mean square for its expectation.) Compare these estimated standard errors with those given for the PROC MIXED solution, Section 19.4.

Exercise 17.3

Verify that the constructed error mean square for testing HERB effects in the analysis of variance approach is E’ = 166.31 and that its approximate degrees of freedom are f = 14.