1. Consider the following data set,

and the Balanced Factor- effects model with interactions

Y_ijk = μ + α_i + β_j + γ_ij + ε_ijk

i = 1,2 ,j= 1,2,3, k = 1,2

where  ε_ijk ∼ N (0, σ²).

(a) Write this model in the General Linear Model form Y = Xβ + ε.

(b) Find  P_X , P₁, P_AP_B.

(c) Use the decomposition

P_X = P₁ + P_A + P_B + P_AB,

to construct the ANOVA table for this model.

(d)Verify your results in (c) using R.

2. Consider the following data set, and Factor- effects model with interactions

Y_ijk = µ + α_i + β_j + γ_ij + ε_ijk

i = 1, 2,  j = 1, 3   k = 1, . . . , n_ij

(a) Write this model in the General Linear Model form Y = Xβ + ε.

(b) Consider Y_ijk = µ + β_j + γ_ij + ε_ijk as the reduced model and test H₀: α₁ = α₂.

(c) Now consider Y_ijk = µ + β_j + ε_ijk as the reduced model and test H₀: α₁ = α₂.

(d) Is there any discrepancy between your conclusions in (b) and (c)? Explain why.

3. For the balanced one-Factor random effects model discussed in class show

E[SSB] =   (A - 1)σ² + (N - (Σ_in_i²/N)) σ_a²

E[SSW] =  σ²(N - A)

4. Consider the two-factor balanced additive random-effects model without interaction

Y_ijk  = µ + a_i + b_j  + ε_ijk

i = 1, 2,  j = 1, 2,  k = 1, 2.

Suppose ε_ijk are iid N (0, σ²) variables, a_i are iid N (0, σ_a²) variables, b_i are iid N (0, σ_b²) variables, Cov(ε_ijk, a_i) = 0, Cov(ε_ijk, b_j ) = 0, and Cov(a_i, b_j) = 0.

i. Write this model in the General Linear Mixed Model form Y = Xβ + Zu + ε.

ii. Find an expression for V = Cov(Y).

Attachment:- Notes.rar