Introduction to Categorical Data Analysis

38 CONTINGENCY TABLES
Table 2.5. Cross Classification of Party Identification by Gender
Party Identification
Gender Democrat Independent Republican Total
Females 762 327 468 1557
(703.7) (319.6) (533.7)
Males 484 239 477 1200
(542.3) (246.4) (411.3)
Total 1246 566 945 2757
Note: Estimated expected frequencies for hypothesis of independence in parentheses. Data
from 2000 General Social Survey.
would be rather unusual if the variables were truly independent. Both test statistics
suggest that political party identification and gender are associated.
2.4.5 Residuals for Cells in a Contingency Table
A test statistic and its P -value describe the evidence against the null hypothesis.
A cell-by-cell comparison of observed and estimated expected frequencies helps us
better understand the nature of the evidence. Larger differences between nij and ˆμij
tend to occur for cells that have larger expected frequencies, so the raw difference
nij −ˆμij is insufficient. For the test of independence, a useful cell residual is
nij −ˆμij
� ˆμij (1 − pi+)(1 − p+j )
(2.9)
The denominator is the estimated standard error of nij −ˆμij , under H0. The ratio
(2.9) is called a standardized residual, because it divides nij −ˆμij by its SE.
When H0 is true, each standardized residual has a large-sample standard normal
distribution. A standardized residual having absolute value that exceeds about 2 when
there are few cells or about 3 when there are many cells indicates lack of fit of H0 in
that cell. (Under H0, we expect about 5% of the standardized residuals to be farther
from 0 than ±2 by chance alone.)
Table 2.6 shows the standardized residuals for testing independence in Table 2.5.
For the first cell, for instance, n11 = 762 and ˆμ11 = 703.7. The first row and
first column marginal proportions equal p1+ = 1557/2757 = 0.565 and p+1 =
1246/2757 = 0.452. Substituting into (2.9), the standardized residual for this cell
equals
(762 − 703.7)/ � 703.7(1 − 0.565)(1 − 0.452) = 4.50
This cell shows a greater discrepancy between n11 and ˆμ11 than we would expect if
the variables were truly independent.