Three Flavors of Chi-Square: Pearson, Likelihood Ratio ...



Three Flavors of Chi-Square: Pearson, Likelihood Ratio, and WaldHere 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 1 511 2 162 1 212 2 1proc freq; format Interest yn. WithWhom ww.;table Interest*WithWhom / chisq nopercent nocol relrisk; run;proc logistic; model WithWhom = Interest; run;-------------------------------------------------------------------------------------------------- The SAS System 1 The FREQ Procedure Table of Interest by WithWhom Interest WithWhom Frequency? Row Pct ?Alone ?Partner ? Total ???????????????????????????? Yes ? 51 ? 16 ? 67 ? 76.12 ? 23.88 ? ???????????????????????????? No ? 21 ? 1 ? 22 ? 95.45 ? 4.55 ? ???????????????????????????? Total 72 17 89 Statistics for Table of Interest by WithWhom Statistic DF Value Prob ?????????????????????????????????????????????????????? Chi-Square (Pearson) 1 4.0068 0.0453 Likelihood Ratio Chi-Square 1 5.0124 0.0252Notice 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.-------------------------------------------------------------------------------------------------- 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 ????????????????????????????????????????????????????????????????? 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 = 89Notice 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.-------------------------------------------------------------------------------------------------- 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 0.0761The 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.-------------------------------------------------------------------------------------------------- Odds Ratio Estimates Point 95% Wald Effect Estimate Confidence Limits Interest 6.588 0.820 52.900So 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 ................
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