Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

9.4. Multiple Comparisons 529

> fit <- lm(speed~factor(car))

> anova(fit)

### F-Stat and p-value 25.145 1.903e-08

> aovfit <- aov(speed~factor(car))

> TukeyHSD(aovfit)

## Tukeys procedures of all pairwise comparisons are computed.

## Summary of a pertinent few

## Cars Mean-diff LB CI UB CI Sig??

## Porsche - Ferrari -2.6166667 -9.0690855 3.835752 NS

## Viper - Porsche -7.7333333 -14.1857522 -1.280914 Sig.

## Bonferroni

> mcpbon(speed,car)

## Porsche - Ferrari -2.6166667 -9.3795891 4.1462558 NS

## Viper - Porsche -7.7333333 -14.496255 -0.9704109 Sig.

2.197038 6.762922 0.9704109 14.49625578

Fisher

mcpfisher(speed,car)

ftest 2.514542e+01 1.903360e-08

Porsche - Ferrari -2.6166667 -7.141552 1.908219 NS

Viper - Porsche -7.7333333 -12.258219 -3.208448 Sig.

For discussion, we cite only two of Tukey’s confidence intervals. As the second in-
terval in the above printout shows, the mean speeds of both the Ferrari and Porsche
are significantly faster than the mean speeds of the other cars. The difference be-
tween the Ferrari’s and Porsche’s mean speeds, though, is insignificant. Below the
two Tukey confidence intervals, we display the results based on the Bonferroni and
Fisher procedures. Note that all three procedures result in the same conclusions for
these comparisons. The Bonferroni intervals are slightly larger than those of the
Tukey procedure. The Fisher procedure gives the shortest intervals as expected.

In practice, the Tukey-Kramer procedure is often used, but there are many other
multiple comparison procedures. A classical monograph on MCPs is Miller (1981)
while Hus (1996) offers a more recent discussion.

EXERCISES

9.4.1. For the study discussed in Exercise 9.2.8, obtain the results of Bonferroni
multiple comparison procedure usingα=0.10. Based on this procedure, which
brand of fuel if any is significantly best?

9.4.2.For the study discussed in Exercise 9.2.6, compute the Tukey-Kramer pro-
cedure. Are there any significant differences?

9.4.3. SupposeXandY are discrete random variables that have the common
range{ 1 , 2 ,...,k}.Letp 1 jandp 2 jbe the respective probabilitiesP(X=j)and