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

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, σ^{2}).

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

(b) Find **P _{X} , P_{1}, P_{A}P_{B}.**

(c) Use the decomposition

**P _{X} = P_{1} + 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}

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 Y_{ijk} = µ + β_{j} + γ_{ij} + ε_{ijk} as the reduced model and test H_{0}: α_{1} = α_{2}.

(c) Now consider Y_{ijk} = µ + β_{j} + ε_{ijk} as the reduced model and test H_{0}: α_{1} = α_{2}.

(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)σ^{2} + (N - (Σ_{i}n_{i}^{2}/N)) σ_{a}^{2}

E[SSW] = σ^{2}(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, σ^{2}) variables, a_{i} are iid N (0, σ_{a}^{2}) variables, b_{i} are iid N (0, σ_{b}^{2}) 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

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