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data was displayed in a rectangular table of cells. Each table consisted of one row
and a specified number of columns, where the columns corresponded to categories
into which the population had been divided. We now study problems in which the
data also consists of counts or frequencies, but the data table will now have / rows
(I > 2) and J columns, so /J cells. There are two commonly encountered situations
in which such data arises:
1. There are / populations of interest, each corresponding to a different row of the
table, and each population is divided into the same J categories. A sample is
taken from the ith population (i = 1, ..., J), and the counts are entered in the
cells in the ith row of the table. For example, customers of each of J = 3
department store chains might have available the same J =5 payment
categories: cash, check, store credit card, Visa, and MasterCard.
2. There is a single population of interest, with each individual in the population cate-
gorized with respect to two different factors. There are / categories associated
with the first factor, and J categories associated with the second factor. A single
sample is taken, and the number of individuals belonging in both category 7 of factor
1 and category j of factor 2 is entered in the cell in row i, column j (i = 1, ..., J;
j= 1,...,J). As an example, customers making a purchase might be classified
according to both department in which the purchase was made, with / = 6
departments, and according to method of payment, with J = 5 as in (1) above.
Let nj; denote the number of individuals in the sample(s) falling in the (i, j )th cell
(row i, column j) of the table—that is, the (i, )th cell count. The table displaying
the n,; s is called a two-way contingency table; a prototype is shown in Table 13.9.
Table 13.9 A two-way contingency table
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--- Trang 758 ---
13.3 Two-Way Contingency Tables 745
In situations of type 1, we want to investigate whether the proportions in the
different categories are the same for all populations. The null hypothesis states that
the populations are homogeneous with respect to these categories. In type 2 situa-
tions, we investigate whether the categories of the two factors occur independently
of each other in the population.
Testing for Homogeneity
We assume that each individual in every one of the / populations belongs in exactly
one of J categories. A sample of n; individuals is taken from the ith population; let
n= Zn,and
nj = the number of individuals in the ith sample who fall into category j
y the total number of individuals among
n= ny = A A
4 fal 1 the n sampled who fall into category j
The nj’s are recorded in a two-way contingency table with J rows and J columns.
The sum of the nj;’s in the ith row is n;, whereas the sum of entries in the jth column
is nj.
Let
__ the proportion of the individuals in
Py = population 7 who fall into category j
Thus, for population 1, the J proportions are pj, P12, ..., Piz (Which sum to 1) and
similarly for the other populations. The null hypothesis of homogeneity states that
the proportion of individuals in category j is the same for each population and
that this is true for every category; that is, for every j, pyj = Px = °** = Pyj-
When H) is true, we can use pj, >, ..., Py to denote the population propor-
tions in the J different categories; these proportions are common to all / popula-
tions. The expected number of individuals in the ith sample who fall in the jth
category when Hp is true is then E(N;j) = n;- p;. To estimate E(Nj;), we must first
estimate p;, the proportion in category j. Among the total sample of n individuals,
N, fall into category j, so we use pj = N.j/n as the estimator (this can be shown to
be the maximum likelihood estimator of p,). Substitution of the estimate p; for p; in
nip; yields a simple formula for estimated expected counts under Ho:
‘ ‘ Fi fn,
é;; = estimated expected count in cell (i,j) = nj:
n
__ (ith row total)(jth column total) (13.10)
7 n
The test statistic also has the same form as in previous problem situations. The
number of degrees of freedom comes from the general rule of thumb. In each row of
Table 13.9 there are J — 1 freely determined cell counts (each sample size n; is
fixed), so there are a total of /(J — 1) freely determined cells. Parameters p, .. ., py
are estimated, but because Xp; = 1, only J — | of these are independent. Thus
df = IJ -1)-V-1) = V-1)d-D.
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746 = carrer 13. Goodness-of-Fit Tests and Categorical Data Analysis
Null hypothesis: Ho : pry = px =+:- = py f= 1,2,-...5
Alternative hypothesis: H, : Ho is not true
Test statistic value:
2 y (observed — estimated expected)? _ y y (ny ei)
a ; estimated expected - ej
all cells i=l j=l
Rejection region: 77 > 73)1)—1
P-value information can be obtained as described in Section 13.1. The test