Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

9.4. Multiple Comparisons 527

Bonferroni confidence interval forμj−μj′is

X·j−X·j′±tα/(2k),n−bσˆΩ

√

1

nj

+

1

nj′

(9.4.3)

While the overall confidence of the Bonferroni procedure is at least (1−α), for a
large number of comparisons, the lengths of its intervals are wide; i.e., a loss in
precision. We offer two other procedures that, generally, lessen this effect.
The R functionmcpbon.R^2 computes the Bonferroni procedure for all pairwise
comparisons for a one-way design. The call ismcpbon(y,ind,alpha=0.05)where
yis the vector of the combined samples andindis the corresponding treatment
vector. See Example 9.4.1 below.

Tukey’s Multiple Comparison Procedure

To stateTukey’s procedure, we first need to define the Studentized range distri-
bution.
Definition 9.4.1.LetY 1 ,...,Ykbe iidN(μ, σ^2 ). Denote the range of these vari-
ables byR=max{Yi}−min{Yi}.SupposemS^2 /σ^2 has aχ^2 (m)distribution which
is independent ofY 1 ,...,Yk. Then we say thatQ=R/Shas aStudentized range
distribution with parameterskandm.

The distribution ofQcannot be obtained in close form but packages such as R

have functions that compute the cdf and quantiles. In R, the callptukey(x,k,m)

computes the cdf ofQatx, while the callqtukey(p,k,m)returns thepth quantile.

Consider the one-way design. First, assume that all the sample sizes are the

same; i.e., for some positive integera,nj =a, for allj =1,...,b.LetR=

Range{X· 1 −μ 1 ,...,X·b−μb}.ThensinceX· 1 −μ 1 ,...,X·b−μbare iidN(0,σ^2 /a),

the random variableQ=R/(ˆσΩ/

√

a) has a Studentized range distribution with

parametersbandn−b.Letqc=q 1 −α,b,n−b.

1 −α = P(Q≤qc)=P

(

max{X·j−μj}−min{X·j−μj}≤qcˆσΩ/

√

a

)

= P

(

|(μj−μj′)−(X·j−X·j′)|≤qcσˆΩ/

√

a,for allj, j′

)

If we expand the inequality in the last statement, we obtain the (1−α)100% simul-
taneous confidence intervals for all pairwise differences given by

X·j−X·j′±q 1 −α,b,n−b

ˆσΩ

√

a

, for allj, j′in 1,...b. (9.4.4)

The statistician John Tukey developed these simultaneous confidence intervals for

the balanced case. For the unbalanced case, first write the error term in (9.4.4) as

q 1 −α,b,n−b

√

2

σˆΩ

√

1

a

+

1

a

.

(^2) Downloadable at the site listed in the Preface.