9.4. Multiple Comparisons 525There are R commands that compute the cdf of noncentralχ^2 andFrandom
variables. For example, suppose we want to compute P(Y ≤y), whereY has
aχ^2 -distribution withddegrees of freedom and noncentrality parameterb.This
probability is returned with the commandpchisq(y,d,b). The corresponding value
of the pdf atyis computed by the commanddchisq(y,d,b). As another exam-
ple, suppose we wantP(W≥w), whereWhas anF-distribution withn1andn2
degrees of freedom and noncentrality parametertheta. This is computed by the
command1-pf(w,n1,n2,theta), while the commanddf(w,n1,n2,theta)com-
putes the value of the density ofWatw. Tables of the noncentral chi-square and
noncentralF-distributions are available in the literature also.
EXERCISES
9.3.1. LetYi,i=1, 2 ,...,n, denote independent random variables that are, re-
spectively,χ^2 (ri,θi),i=1, 2 ,...,n.ProvethatZ=
∑n
1 Yiisχ(^2) (∑n
1 ri,
∑n
1 θi).
9.3.2.Compute the variance of a random variable that isχ^2 (r, θ).
9.3.3.Three different medical procedures (A, B, and C) for a certain disease are
under investigation. For the study, 3mpatients having this disease are to be selected
andmare to be assigned to each procedure. This common sample sizemmust be
determined. Letμ 1 ,μ 2 ,andμ 3 ,be the means of the response of interest under
treatments A, B, and C, respectively. The hypotheses are: H 0 :μ 1 =μ 2 =μ 3
versusH 1 :μj =μj′ for somej =j′. To determinem, from a pilot study the
experimenters use a guess of 30 ofσ^2 and they select the significance level of 0.05.
They are interested in detecting the pattern of means:μ 2 =μ 1 +5 andμ 3 =μ 1 +10.
(a)Determine the noncentrality parameter under the above pattern of means.
(b)Use the R functionpfto determine the powers of theF-test to detect the
above pattern of means form=5andm= 10.
(c)Determine the smallest value ofmso that the power of detection is at least
0.80.
(d)Answer (a)–(c)ifσ^2 = 40.
9.3.4.Show that the square of a noncentralTrandom variable is a noncentralF
random variable.
9.3.5. LetX 1 andX 2 be two independent random variables. LetX 1 andY =
X 1 +X 2 beχ^2 (r 1 ,θ 1 )andχ^2 (r, θ), respectively. Herer 1 <randθ 1 ≤θ. Show that
X 2 isχ^2 (r−r 1 ,θ−θ 1 ).
9.4 MultipleComparisons
For this section, consider the one-way ANONA model withbtreatments as de-
scribed in expression (9.2.1) of Section 9.2. In that section, we developed theF-test