538 Inferences About Normal Linear Models0.780.800.820.840.86Bindermean of ConductivityPG58 PG64 PG70Aggregrate(^3841)
44
Figure 9.5.1: Mean profile plot for the study discussed in Example 9.5.1. The
profiles are nearly parallel, indicating little interaction between the factors.
9.5.2.Consider the discussion above expression (9.5.14). Show that the random
variables
√
aX· 1 ,...,
√
aX·bare independent with the commonN(
√
aμ, σ^2 ) distri-
bution.
9.5.3.For the two-way interaction model, (9.5.15), show that the noncentrality
parameter of the test statisticFABis equal toc
∑b
j=1
∑a
i=1γ
2
ij/σ
(^2).
9.5.4.Using the background of the two-way classification with one observation per
cell, determine the distribution of the maximum likelihood estimators ofαi,βj,and
μ.
9.5.5. Prove that the linear functionsXij−Xi.−X.j+X..andX.j−X..are
uncorrelated, under the assumptions of this section.
9.5.6. Given the following observations associated with a two-way classification
witha=3andb= 4, use R or another statistical package to compute theF-
statistic used to test the equality of the column means (β 1 =β 2 =β 3 =β 4 =0)
and the equality of the row means (α 1 =α 2 =α 3 = 0), respectively.
Row/Column 1 2 3 4
1 3.1 4.2 2.7 4.9
2 2.7 2.9 1.8 3.0
3 4.0 4.6 3.0 3.9
9.5.7.With the background of the two-way classification withc>1observations
per cell, determine the distribution of the mles ofαi,βj,andγij.