Question

2. (a) Let us consider a full model of a balanced (all t treatments have equal number of observations r) CRD design with t treatments and r replications of each treatment, hence having n-rt observations i. Minimizing sum of square error Δfull(μ, Tỉ)-Σι-12jai (Vij-l-ri)2 with respect to μ and Ti find the least square estimators of μ and Te as μ and Ti Hint: Take derivative of the objective function with respect to u and Ti and equate then to zero to get two equations. These are called Normal equations. Solve the normal equations to find the least square estimators while utilizing the fact that Σ.1 Ti-0.1 ii. Show that the minimized value of the sum of squared errors is full (, Ti) - yij - yi (b) Now let us reduce this model with the constraint Ti = τ, for all i and consider the reduced model i. Minimizing the sum of squared error of the reduced model Δreduced(μ, r)-Σι-Tr-1(Vij ,1-r)2 show that it is not possible to find least square estimators of μ and τ separately. Find the least square estimator of μ + T. 11. Show that the minimized value of the error sum of squares is ΔΤ'educed(AT)-ΣΖΊ Σ]-1 (3/ij (c) Which one among the two error sum of squares Δ full (μ. Τ.) and ΔΤ educed(μ. τ) is larger? Find their difference. Can you recognize the difference as a known sum of squares from ANOVA sum of squares decomposition?

Answer 1