Question

(A) Consider the analysis of covariance (ANCOVA) model: \[ y_{i j}=\mu+\alpha_{i}+x_{i j} \beta+e_{i j}, \] for \( i=1,2,3 \) and \( j=1, \ldots, n \). Write the model in matrix form, clearly specifying all model components. (B) Is the model matrix \( X \) full column rank? Explain.

          (A) Consider the analysis of covariance (ANCOVA) model:
\[
y_{i j}=\mu+\alpha_{i}+x_{i j} \beta+e_{i j},
\]
for \( i=1,2,3 \) and \( j=1, \ldots, n \). Write the model in matrix form, clearly specifying all model components.
(B) Is the model matrix \( X \) full column rank? Explain.

(A) Consider the analysis of covariance (ANCOVA) model:

yi j=μ+αi+xi jβ+ei j,

for i=1,2,3 and j=1, …, n. Write the model in matrix form, clearly specifying all model components.
(B) Is the model matrix X full column rank? Explain.

Added by Kristina G.

Applied Statistics and Probability for Engineers

Douglas C. Montgomery 6th Edition

Chapter 11

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Solved by Expert Keerti J

Step 1

### Part (a): Mean and Variance of an Observation, Yij ** Show more…

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Consider a random effects model to analyze data from a completely randomized experimental design, Yij = µ + Ti + Eij , with i = 1, . . . , t and j = 1, . . . , n where Ti iid∼ N(0, σ2 T ) and Eij iid∼ N(0, σ2 ) with Ti ⊥⊥ Eij . (a) Give the mean and variance of an observation, Yij . (b) Write a treatment mean, Y i· as a function of µ and (averages of) random effects. (c) Give the mean and variance of a sample treatment mean, Y i· .