Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

4.7. Chi-Square Tests 287

is somewhat greater than it would be if minimum chi-square estimates are used.
Hence, when comparing it to a critical value listed in the chi-square table withk− 3
degrees of freedom, there is a greater chance of rejection than there would be if the
actual minimum ofQk− 1 is used. Accordingly, the approximate significance level of
such a test may be higher than thep-value as calculated in theχ^2 -analysis. This
modification should be kept in mind and, if at all possible, eachpishould be esti-
mated using the frequenciesX 1 ,...,Xkrather than directly using the observations
Y 1 ,Y 2 ,...,Ynof the random sample.

Example 4.7.3.In this example, we consider two multinomial distributions with

parametersnj,p 1 j,p 2 j,...,pkjandj=1,2, respectively. LetXij,i=1, 2 ,...,k,

j=1,2, represent the corresponding frequencies. Ifn 1 andn 2 are large and the

observations from one distribution are independent of those from the other, the

random variable

∑^2

j=1

∑k

i=1

(Xij−njpij)^2

njpij

is the sum of two independent random variables each of which we treat as though it
wereχ^2 (k−1); that is, the random variable is approximatelyχ^2 (2k−2). Consider
the hypothesis
H 0 :p 11 =p 12 ,p 21 =p 22 ,...,pk 1 =pk 2 ,

where eachpi 1 =pi 2 ,i=1, 2 ,...,k, is unspecified. Thus we need point estimates
of these parameters. The maximum likelihood estimator ofpi 1 =pi 2 , based upon
the frequenciesXij,is(Xi 1 +Xi 2 )/(n 1 +n 2 ),i=1, 2 ,...,k. Note that we need
onlyk−1 point estimates, because we have a point estimate ofpk 1 =pk 2 once we
have point estimates of the firstk−1 probabilities. In accordance with the fact
that has been stated, the random variable

Qk− 1 =

∑^2

j=1

∑k

i=1

{Xij−nj[(Xi 1 +Xi 2 )/(n 1 +n 2 )]}^2

nj[(Xi 1 +Xi 2 )/(n 1 +n 2 )]

has an approximateχ^2 distribution with 2k− 2 −(k−1) =k−1 degrees of freedom.
Thus we are able to test the hypothesis that two multinomial distributions are the
same. For a specified levelα,thehypothesisH 0 is rejected when the computed
value ofQk− 1 exceeds the 1−αquantile of aχ^2 -distribution withk−1 degrees of
freedom. This test is often called the chi-square test forhomogeneity(the null is
equivalent tohomogeneous distributions).

The second example deals with the subject ofcontingency tables.

Example 4.7.4.Let the result of a random experiment be classified by two at-
tributes (such as the color of the hair and the color of the eyes). That is, one
attribute of the outcome is one and only one of certain mutually exclusive and
exhaustive events, sayA 1 ,A 2 ,...,Aa; and the other attribute of the outcome is
also one and only one of certain mutually exclusive and exhaustive events, say
B 1 ,B 2 ,...,Bb.Letpij=P(Ai∩Bj),i=1, 2 ,...,a;j=1, 2 ,...,b. The random