Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

528 Inferences About Normal Linear Models

For the unbalanced case, this suggests the following intervals

X·j−X·j′±

q 1 −α,b,n−b

√

2

ˆσΩ

√

1

nj

+

1

nj′

, for allj, j′in 1,...b. (9.4.5)

This correction is due to Kramer and these intervals are often referred to as the
Tukey-Kramer multiple comparison procedure; see Miller (1981) for discussion.
These intervals do not have exact confidence (1−α) but studies have indicated
that if the unbalance is not severe the confidence is close to (1−α); see Dunnett
(1980). Corresponding R code is shown in Example 9.4.1.

Fisher’s PLSD Multiple Comparison Procedure

The final procedure we discuss isFisher’s Protected Least Significance Dif-
ference (PLSD). The setting is the general (unbalanced) one-way design (9.2.1).
This procedure is a two-stage procedure. It can be used for an arbitrary umber of
comparisons but we state it for all comparisons. For a specified level of significance
α, Stage 1 consists of theF-test of the hypotheses of equal means, (9.2.2). If the test
rejects at levelαthen Stage 2 consists of the usual pairwise (1−α)100% confidence
intervals, i.e.,

X·j−X·j′±tα/ 2 ,n−bˆσΩ

√

1

nj

+

1

nj′

, for allj, j′in 1,...,b. (9.4.6)

If the test in Stage 1 fails to reject, users sometimes perform Stage 2 using the

Bonferroni procedure. Fisher’s procedure does not have overall coverage 1−α, but

the initialF-test offers protection. Simulation studies have shown that Fisher’s

procedure performs well in terms of power and level; see, for instance, Carmer and

Swanson (1973) and McKean et al. (1989). The R function^3 mcpfisher.Rcomputes

this procedure as discussed in the next example.

Example 9.4.1(Fast Cars). Kitchens (1997) discusses an experiment concern-

ing the speed of cars. Five cars are considered: Acura (1), Ferrari (2), Lotus (3),

Porsche (4), and Viper (5). For each car, 6 runs were made, 3 in each direction. For

each run, the speed recorded is the maximum speed on the run achieved without

exceeding the engine’s redline. The data are in the filefastcars.rda. Figure 9.4.1

displays the comparison boxplots of the speeds versus the cars, which shows clearly

that there are differences in speed due to the car. Ferrari and Porsche seem to be

the fastest but are the differences significant? We assume the one-way design (9.2.1)

and use R to do the computations. Key commands and corresponding results are

given next. The overallF-test of the hypotheses of equal means, (9.2.2), is quite

significant:F=25.15 with thep-value 0.0000. We selected the Tukey MCP at level

0.05. The command below returns all

( 5

2

)
= 10 pairwise comparisons, but in our
summary we only list two.

Code assumes that fastcars.rda has been loaded in R

(^3) Down loadable at the site listed in the Preface.