Introduction to Categorical Data Analysis

6.4 TESTS OF CONDITIONAL INDEPENDENCE 195
model-based statistics without interaction terms, these statistics perform well when
the conditional association is similar in each partial table. There are three versions,
according to whether both, one, or neither of Y and the predictor are treated as
ordinal.
When both variables are ordinal, the test statistic generalizes the correlation statistic
(2.10) for two-way tables. It is designed to detect a linear trend in the association that
has the same direction in each partial table. The generalized correlation statistic
has approximately a chi-squared distribution with df = 1. Its formula is complex
and we omit computational details. It is available in standard software (e.g., PROC
FREQ in SAS).
For Table 6.12 with the scores {3, 10, 20, 35} for income and {1, 3, 4, 5} for satisfaction,
the sample correlation between income and job satisfaction equals 0.16 for
females and 0.37 for males. The generalized correlation statistic equals 6.2 with
df = 1(P = 0.01). This gives the same conclusion as the ordinal-model-based
likelihood-ratio test of the previous subsection.
When in doubt about scoring, perform a sensitivity analysis by using a few different
choices that seem sensible. Unless the categories exhibit severe imbalance in their
totals, the choice of scores usually has little impact on the results. With the row
and column numbers as the scores, the sample correlation equals 0.17 for females
and 0.38 for males, and the generalized correlation statistic equals 6.6 with df = 1
(P = 0.01), giving the same conclusion.
6.4.3 Detecting Nominal–Ordinal Conditional Association ∗
When the predictor is nominal and Y is ordinal, scores are relevant only for levels of
Y . We summarize the responses of subjects within a given row by the mean of their
scores on Y , and then average this row-wise mean information across the K strata.
The test of conditional independence compares the I rows using a statistic based on
the variation among the I averaged row mean responses. This statistic is designed to
detect differences among their true mean values. It has a large-sample chi-squared
distribution with df = (I − 1).
For Table 6.12, this test treats job satisfaction as ordinal and income as nominal.
The test searches for differences among the four income levels in their mean job
satisfaction. Using scores {1, 2, 3, 4}, the mean job satisfaction at the four levels
of income equal (2.82, 2.84, 3.29, 3.00) for females and (2.60, 2.78, 3.30, 3.31) for
males. For instance, the mean for the 17 females with income