Introduction to Categorical Data Analysis

8.3 COMPARING MARGINS OF SQUARE CONTINGENCY TABLES 255
is a generalization of binary model (8.4) that expresses each cumulative logit in terms
of subject effects and a margin effect. Like the cumulative logit models of Section 6.2,
it makes the proportional odds assumption by which the effect β is assumed to be the
same for each cumulative probability. The model states that, for each matched pair,
the odds that observation 1 falls in category j or below (instead of above category j)
are exp(β) times the odds for observation 2.
An estimate of β in this model is
���
ˆβ = log
ij
�
(j − i)nij
(i − j)nij
(8.7)
The numerator sum weights each cell count above the main diagonal by its distance
(j − i) from that diagonal. The denominator sum refers to cells below the main
diagonal. An ordinal test of marginal homogeneity (β = 0) uses this effect. Estimator
(8.7) of β has
�
� ��
�
SE = � ij
��� � +
2
ij (i − j)nij
The ratio ˆβ/SE is an approximate standard normal test statistic.
A simple alternative test compares the sample means for the two margins, for
ordered category scores {ui}. Denote the sample means for the rows (X) and columns
(Y )by¯x = �
i uipi+ and ¯y = �
i uip+i. The difference ( ¯x −¯y) divided by its
estimated standard error under marginal homogeneity, which is the square root of
�
� �
(1/n) (ui − uj ) 2 �
pij
i
j
has an approximate null standard normal distribution. This test is designed to detect
differences between true marginal means.
8.3.4 Example: Recycle or Drive Less to Help Environment?
Table 8.6 is from a General Social Survey. Subjects were asked “How often do you
cut back on driving a car for environmental reasons?” and “How often do you make
a special effort to sort glass or cans or plastic or papers and so on for recycling?”
For Table 8.6, the numerator of (8.7) equals
1(43 + 99 + 230) + 2(163 + 185) + 3(233) = 1767
� 2