Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

526 Inferences About Normal Linear Models

of the hypotheses of equal means, (9.2.2). In practice, besides this test, statisticians
usually want to make pairwise comparisons of the formμj−μj′. Thisisoftencalled
theSecond Stage Analysis, while theF-test is consider theFirst Stage Anal-
ysis. The analysis for such comparisons usually consists of confidence intervals for
the differencesμj−μj′ andμjisdeclared differentfromμj′if 0 is not in the
confidence interval. The random samples for treatmentsjandj′are:X 1 j,...,Xnjj
from theN(μj,σ^2 ) distribution andX 1 j′,...,Xnj′j′ from theN(μj′,σ^2 ) distribu-
tion, which are independent random samples. Based on these samples the estimator
ofμj−μj′isX·j−X·j′. Further in the one-way analysis, an estimator ofσ^2 is the
full model estimatorσˆ^2 Ωdefined in expression (9.2.7). As discussed in Section 9.2,

(n−b)σˆ^2 Ω/σ^2 has aχ^2 (n−b) distribution which is independent of all the sample
meansX·j. Hence, for a specifiedαit follows as in (4.2.13) of Chapter 4 that

X·j−X·j′±tα/ 2 ,n−bˆσΩ

√

1

nj

+

1

nj′

(9.4.1)

is a (1−α)100% confidence interval forμj−μj′.

We often want to make many pairwise comparisons, though. For example, the

first treatment might be a placebo or represent the standard treatment. In this case,

there areb−1 pairwise comparisons of interest. On the other hand, we may want

to make all

(b

2

)
pairwise comparisons. In making so many comparisons, while each
confidence interval, (9.4.1), has confidence (1−α), it would seem that the overall
confidence diminishes. As we next show, thisslippageof overall confidence is true.
These problems are often calledMultiple Comparison Problems(MCP). In
this section, we present several MCP procedures.

Bonferroni Multiple Comparison Procedure

It is easy to motivate theBonferroni Procedurewhile, at the same time, showing
the slippage of confidence. This procedure is quite general and can be used in many
settings not just the one-way design. So suppose we havekparametersθiwith
(1−α)100% confidence intervalsIi,i=1,...,k,where0<α<1isgiven. Then
the overall confidence isP(θ 1 ∈I 1 ,...,θk∈Ik). Using the method of complements,
DeMorgan’s Laws, and Boole’s inequality, expression (1.3.7) of Chapter 1, we have

P(θ 1 ∈I 1 ,...,θk∈Ik)=1−P

(

∪ki=1θi ∈Ii

)

≥ 1 −

∑k

i=1

P(θi ∈Ii)=1−kα. (9.4.2)

The quantity 1−kαis the lower bound on the slippage of confidence. For example,

ifk=20andα=0.05 then the overall confidence may be 0. The Bonferroni

procedure follows from expression (9.4.2). Simply change the confidence level of

each confidence interval to [1−(α/k)]. Then the overall confidence is at least 1−α.

For our one-way analysis, suppose we havekdifferences of interest. Then the