Introductory Biostatistics

z¼

ð 4  2 : 52 Þþð 25  18 : 70 Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 : 25 þ 14 : 00

p

¼ 1 : 93

which is significant at the 5% level (one-sided). The combined odds ratio esti-
mate is

ORMH¼

3 : 57 þ 18 : 84

2 : 09 þ 12 : 54

¼ 1 : 53

representing an approximate increase of 53% in myocardial infarction for oral
contraceptive users.

6.5 INFERENCES FOR GENERAL TWO-WAY TABLES

Data forming two-way contingency tables do not necessarily come from two
binomial samples. There may be more than two binomial samples to compare.
They may come from two independent samples, but the endpoint may have
more than two categories. They may come from a survey (i.e.,onesample), but
data are cross-tabulated based on two binary factors of interest (so we still have
a2
2 table as in the comparison of two proportions).
Consider the general case of anI
Jtable: say, resulting from a survey of
sizen. LetX 1 andX 2 denote two categorical variables,X 1 havingIlevels and
X 2 havingJlevels; there areIJcombinations of classifications. TheIJcells
represent theIJcombinations of classifications; their probabilities arefpijg,
wherepijdenotes the probability that the outcome (X 1 ;X 2 ) falls in the cell in
rowiand columnj. When two categorical variables forming the two-way table
are independent, allpij¼piþpþj. This is themultiplication rulefor probabili-
ties of independence events introduced in Chapter 3; herepiþandpþjare the
two marginal or univariate probabilities. The estimate ofpijunder this condi-
tion is

cppijij¼dppiiþþdppþþjj

¼piþpþj

¼

xiþ

n

xþj

n

¼

xiþxþj

n^2

where thex’s are the observed frequencies. Under the assumption of indepen-
dence, we would have in cellði;jÞ:

INFERENCES FOR GENERAL TWO-WAY TABLES 223