Three Flavors of Chi-Square: Pearson, Likelihood Ratio, and Wald
Here is a short SAS program and annotated output.
options pageno=min nodate formdlim='-';
proc format; value yn 1='Yes' 2='No'; value ww 1='Alone' 2='Partner';
data duh; input Interest WithWhom count;
weight count; cards;
1 1 51
1 2 16
2 1 21
2 2 1
proc freq; format Interest yn. WithWhom ww.;
table Interest*WithWhom / chisq nopercent nocol relrisk; run;
proc logistic; model WithWhom = Interest; run;
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The SAS System 1
The FREQ Procedure
Table of Interest by WithWhom
Interest WithWhom
Frequency‚
Row Pct ‚Alone ‚Partner ‚ Total
ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ
Yes ‚ 51 ‚ 16 ‚ 67
‚ 76.12 ‚ 23.88 ‚
ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ
No ‚ 21 ‚ 1 ‚ 22
‚ 95.45 ‚ 4.55 ‚
ƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒˆ
Total 72 17 89
Statistics for Table of Interest by WithWhom
Statistic DF Value Prob
ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ
Chi-Square (Pearson) 1 4.0068 0.0453
Likelihood Ratio Chi-Square 1 5.0124 0.0252
Notice that the relationship is significant with both the Pearson and LR Chi-Square.
WARNING: 25% of the cells have expected counts less
than 5. Chi-Square may not be a valid test.
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The SAS System 2
The FREQ Procedure
Statistics for Table of Interest by WithWhom
Estimates of the Relative Risk (Row1/Row2)
Type of Study Value 95% Confidence Limits
ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ
Case-Control (Odds Ratio) 0.1518 0.0189 1.2189
Cohort (Col1 Risk) 0.7974 0.6781 0.9378
Cohort (Col2 Risk) 5.2537 0.7385 37.3741
Sample Size = 89
Notice that although the Pearson and LR Chi-Square statistics were significant beyond .05, the 95% confidence interval for the odds ratio includes the value one. As you will soon see, this is because a more conservative Chi-Square, the Wald Chi-Square, is used in constructing that confidence interval.
Since most people are uncomfortable with odds ratios between 0 and 1, I shall invert the odds ratio, to 6.588, with a confidence interval extending from 0.820 to 52.910.
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The SAS System 3
The LOGISTIC Procedure
Testing Global Null Hypothesis: BETA=0
Test Chi-Square DF Pr > ChiSq
Likelihood Ratio 5.0124 1 0.0252
Score (Pearson) 4.0068 1 0.0453
Wald 3.1461 1 0.0761
The values of the Pearson and the LR Chi-Square statistics are the same as reported with Proc Freq. Notice that here we also get the conservative Wald Chi-Square, and it falls short of significance. The Wald Chi-square is essentially a squared t, where t = the value of the slope in the logistic regression divided by its standard error.
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Odds Ratio Estimates
Point 95% Wald
Effect Estimate Confidence Limits
Interest 6.588 0.820 52.900
So we should not be surprised that the confidence interval, based upon the Wald Chi-Square statistic, does include one.
What should one do when the results are significant via the Pearson or Likelihood Chi-Square test, but the confidence interval includes the value 1? Reporting such a confidence interval while claiming the test to be significant would be confusing. In such a case I have recommended reporting the value of the Chi-Square statistics, exact p value, and odds ratio, but not the confidence interval.
Karl L. Wuensch, January, 2017