Write this model in the general linear model form

Assignment Help Mathematics
Reference no: EM131034896

1. Consider the following data set,


B
1 2 3
A 1 74 71 99
64 68 104
2 99 108 114
98 110 111

and the Balanced Factor- effects model with interactions

Yijk = μ + αi + βj + γij + εijk

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

where  εijk ∼ N (0, σ2).

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

(b) Find  PX , P1, PAPB.

(c) Use the decomposition

PX = P1 + PA + PB + PAB,

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

Yijk = µ + αi + βj + γij + εijk

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


B
1 2 3
A 1 74 71 99
64 68 104
60 75 93
2 99 108 114
98 110 111

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

(b) Consider Yijk = µ + βj + γij + εijk as the reduced model and test H0: α1 = α2.

(c) Now consider Yijk = µ + βj + εijk 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 - (Σini2/N)) σa2

E[SSW] =  σ2(N - A)

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

Yijk  = µ + ai + bj  + εijk

i = 1, 2,  j = 1, 2,  k = 1, 2.

Suppose εijk are iid N (0, σ2) variables, ai are iid N (0, σa2) variables, bi are iid N (0, σb2) variables, Cov(εijk, ai) = 0, Cov(εijk, bj ) = 0, and Cov(ai, bj) = 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

Reference no: EM131034896

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