Basic Analysis and Graphing - SAS

Chapter 6 Performing Contingency Analysis 213
Statistical Details for the Contingency Platform
C
=
κˆ
– P e
( 1 – κˆ
)
2
See Fleiss, Cohen, and Everitt (1969).
For Bowker’s test of symmetry, the null hypothesis is that the probabilities in the two-by-two table satisfy
symmetry (p ij =p ji ).
Statistical Details for the Odds Ratio Option
The Odds Ratio is calculated as follows:
where p ij is the count in the i th row and j th column of the 2x2 table.
Statistical Details for the Tests Report
Rsquare (U)
Rsquare (U) is computed as follows:
The total negative log-likelihood is found by fitting fixed response rates across the total sample.
Test
p 11
× p
---------------------- 22
p 12
× p 21
---------------------------------------------------------------------------------
–log likelihood for Model
–log likelihood for Corrected Total
The two Chi-square tests are as follows:
The Likelihood Ratio Chi-square test is computed as twice the negative log-likelihood for Model in the
Tests table. Some books use the notation G 2 for this statistic. The difference of two negative log-likelihoods,
one with whole-population response probabilities and one with each-population response rates, is written as
follows:
G 2 = 2 ( – n ) ln( p ) ij
– j –
n ij
ln( p ij
) where p n N
----- ij and j
= p ij ij
ij
N j
= -----
N
This formula can be more compactly written as follows:
G 2 = 2
i
n ij
n ij
ln-----
e j ij
The Pearson Chi-square is calculated by summing the squares of the differences between the observed and
expected cell counts. The Pearson Chi-square exploits the property that frequency counts tend to a normal
distribution in very large samples. The familiar form of this Chi-square statistic is as follows: