You have a completely random experimental design with t treatments and r experimental units per treatment. The response of each experimental unit was determined by measuring the response variable on...

You have a completely random experimental design with t treatments and r experimental units per treatment. The response of each experimental unit was determined by measuring the response variable on each of s random samples. This gives the model

where τ_i
are fixed treatment effects and γ_ij
and ϵ_ijk are random experimental unit and sampling unit effects with zero means and variances σ²
_γ
and σ², respectively.

(a) What is σ²(Y_ijk)? What is Cov(Y_ijk, Y_ijk)? Show the form of the variance–covariance matrix Var(Y).

(b) What is the form of Var(Y) if the mean of all samples within each experimental unit is used as the response variable?

(c) If the

_ijk are used in the analysis using PROC GLM in SAS, how are the standard errors of the treatment means given by GLM computed? Are they correct? If not, how can they be corrected?

(d) If the experimental unit means are used in the analysis, how are the standard errors of the treatment means computed in PROC GLM? Are they correct? What if the numbers of samples per experimental unit are not constant?

(e) Explain the differences in assumptions between doing the analysis with a general linear models program such as PROC GLM and with a program such as PROC MIXED.