[SOLVED] MAST30025: Linear Statistical Models: Lab 9

Description

1. Recall Question 5 from the Week 8 lab. In a manufacturing plant, filters are used to remove pollutants. We are interested in comparing the lifespan of 5 different types of filters. Six filters of each type are tested, and the time to failure in hours is given in the dataset filters (on the website, in csv format).
   • Is µ − τ₁ + τ₅ estimable?
   • Is estimable?
   • In the week 8 lab you were asked to find two solutions to the normal equations. Verify thatthey produce the same estimate of τ₄− τ₅.
   • Do your two solutions produce the same estimate of 2µ + τ₁?
   • Write down the quantities corresponding to: (i) the lifespan of type 1 filters; (ii) the differencebetween the lifespans of type 2 and type 3 filters; (iii) the amount by which type 4 filters outlive the average filter; (iv) the expected total time to failure of a set of filters containing one of each type.

Verify directly that all of these quantities are estimable, and estimate them.

• Fit a lm model using treatment contrasts (the default). This gives estimates of µ₁,µ₂− µ₁,…,µ₅− µ₁. Use these to estimate ¯µ,µ₁− µ,…,µ¯ ₅− µ¯. Check your answers by fitting a contr.sum model.

2. According to the Gauss-Markov theorem, the estimator for t^Tβ with the lowest variance is t^Tb. Assuming that t^Tβ is estimable, show that this variance is σ²t^T(X^TX)^ct. 3. For the one-way classification model, with n_i observations in group i, show that

k

SS_Reg := yˆ^Tyˆ = y^TX(X^TX)^cX^Ty = ^X(¯y_i)²n_i.

i=1

4. Consider the one-way classification model with 3 levels (k = 3). Find all estimable quantities of the form.
5. Consider the two-way classification model

yij = µ + τi + βj + εij.

Suppose that you have at least one sample from each combination of factor levels.

Treatment contrasts for the first factor are defined here as ^P_i a_iτ_i, where ^P_i a_i = 0. Show that these treatment contrasts are estimable.

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