Multivariate Analysis ch5

?—Reach valid conclusions concerning a population on the basis of information from a sample. We shall concentrate on inferences about a mean vector and its componentfor a normal population meandenote a random sample from a normal population, the appropriate test statistic is1μ−2 1() 1njjX X=−∑1 at significance level ,2), where (2) denotesth percentile of th n t αα−10e -distribution|| is large is equivalent () (5-1) in (5-1) is the square X t t μ−−0 of the to the test value .μConsider the problem of determining whether a givenis a plausible value for the mean of a multivariate normal distribution.A natural generalization of the squared0distance in)) (5-2)μand ,)')(−j X X X Hotelling’s T2inhonor of Harold Hotelling. Here (1/n )S is the estimated covariance matrix of .X1, be a random sample from an1) population. Then with and ) (5-3)e nj j p n n F =−⎞⎟=⎠∑X X () is the upper distribution.p α001,Statement (5-3) leads immediately to a test of the : . At the in favor of if the 1))p n p F n H H p α−≠−μμ() (5-4)n p α−Perspiration from 20 healthy females was analyzed. Three components, X 1=sweat rate, =potassium content, were measured, and the results,sweat data , are presented in '=[4, 50, 10]against [4, 50, 10] at significance α=.10.We assume the sweat data are multivariate normal. This can be tested by the methods ofof level 10% at the see we ,18.8)10(.74.9 observed the 3.628 5.6402T p ==⎥⎥⎦will be rejected if one or more of the component means, or some combination of means, differs too much from the hypothesized values [4, 50, 10]. At this point, we have no idea which of these hypothesized values may not be supported byratio testsThere is a general principle for constructing test likelihood ratio method , -statistics can be derived as the μ= μ0. Likelihood ratio tests have several optimal properties for reasonably large samples, and they are particularly convenient for hypotheses formulated in terms of multivariate normalover their varied is the of maximum the ,estimates.1ˆ }2/exp{bygiven is values possible 1Σx x μ==−∑=n np n j jis a plausible value of , the ) is compared with the unrestricted μ/200). The resulting ratio is called the ,)ˆ||ˆ(,)||n ⎛⎞=⎜⎟⎜⎟⎝⎠μΣΣμΣΣ/20)th perce )'|nti le of th )'|e n j j c α⎞−⎟⎟<⎟−⎟⎠x x x μ. Because of the following relation , we need not the distribution of the00 be a random sample from an ) population. Then the test in (5-4) based on is equivalent to the likehood ratio test of : H =μμbe a vector of unknown populationbe the set of all possibleconfidence region is a regionvalues. It is a generalization of theconfidence interval. This region is determined by the data, we shall=[X1, X2,…, Xn]'is) is said to be a 100(1-α)%if, before the sample is(X) will cover the true θ]= 1-αThis probability is calculated under the true,12 sample observations from an ) population. Then a 100(1)% 2))(/j confidence x αα−−⎤⎥⎦1, and (/2)2)th percentile of distribution with n t t α−, sample observations from an ) population. Then a 100(1)% is the ellipsoid determined by all p n confidence F α−2,,), and ())th percentile of distributio () (5-n.5)j p n p n p p p F F αα−−−−xat of favor in reject for vectors all of consists that the see we ,analogous is this Since region.,)(])/()1 the If ).(]))()'(1000,,010ααH H F p n n F p p n p p n p μμμμx S μx ≠−−−−−−−62.6)05(.402(41)603.564..0146 .0117 values all of consists .population ),(.0146 .0117 40,2211=⎥⎦⎤⎢⎣⎡−−⎥⎦⎤⎥⎦⎤−F μμμΣμ5.1. Figure in plotted .05 at the .589]' be not would .589]' ,562 region. in the is 62.61.30 589.603.562.564..0146.01171=≤=⎥⎦⎤⎢⎣⎡−−⎥⎦⎤−αabout a population mean vector When the sample size is large, tests of hypotheses and confidence regions for μcan be constructed without the assumption of a normal population. In fact,serious departures from a normal population can be overcome by large sample sizes. Both tests of hypotheses and confidence regions will then possess (approximately) their nominal levels.The advantages associated with large samples may be partially offset by a loss in sample information caused byx Susing only the summary statistics and .On the othersummary fornormal populations, the closer the underlying population is to multivariate normal, the more efficiently the sample information will be utilized in making inferences.2All large-sample inferences about are based on - be a random sample from and positive definite covariance is large,by the r χμ)1 (5-6esu )lts in Chapter 4,)th percentile of the αα⎤≈−⎦μis the ellipsoid determined by all such thate the sample observations.is rejected in favor of, at a level of significance approximately ,αsample observations.Chap. 5 Exercises 5.1, 5.3。