1. header= Verdana 28 pt., Red 1 STA 517 – Chapter 3: Inference for Contingency Tables 3. Inference for Contingency Tables 3.1 Confidence Intervals for Association Parameters, 70  confidence intervals for measures of association 3.2 Testing Independence in Two-Way Contingency Tables, 78  chi-squared tests of the hypothesis of independence 3.3 Following-Up Chi-Squared Tests, 80  using residuals or the partitioning property of chi-squared to extract components 3.4 Two-Way Tables with Ordered Classifications, 86  inference applicable with ordered categories. 3.5 Small-Sample Tests of Independence, 91  Fisher’s Exact Test

2. header= Verdana 28 pt., Red 2 STA 517 – Chapter 3: Inference for Contingency Tables 3.1 CONFIDENCE INTERVALS FOR ASSOCIATION PARAMETERS  The accuracy of estimators of association parameters is characterized by standard errors of their sampling distributions.  Odds Ratios  Interval Estimation of Difference of Proportions  Interval Estimation of Relative Risk  Deriving Standard Errors with the Delta Method*

3. header= Verdana 28 pt., Red 3 STA 517 – Chapter 3: Inference for Contingency Tables 3.1.1 Interval Estimation of Odds Ratios  The sample odds ratio  equals 0 or  if any  the amended estimators

4. header= Verdana 28 pt., Red 4 STA 517 – Chapter 3: Inference for Contingency Tables standard error  The estimators and have the same asymptotic normal distribution around . Unless n is quite large, however, their distributions are highly skewed.  The log transform, having an additive rather than multiplicative structure, converges more rapidly to normality.  Wald CI:

5. header= Verdana 28 pt., Red 5 STA 517 – Chapter 3: Inference for Contingency Tables Why distributions are highly skewed? (1)Theoretically derive the skewness of (2)Simulation study: Let’s simulate one case where true =1, n=100, and 11=0.3, 12=0.2, 21=0.3, 22=0.2 Repeat 5,000 simulations to find out the theta hat distribution

6. header= Verdana 28 pt., Red 6 STA 517 – Chapter 3: Inference for Contingency Tables Data simulation; n=100; p11=0.3; p12=0.2; p21=0.3; p22=0.2; do i=1 to 5000; n11=0; n12=0;n21=0;n22=0; do j=1 to n; x=UNIFORM(-1); if x<p11 then n11=n11+1; else if x<(p11+p12) then n12=n12+1; else if x<(p11+p12+p21) then n21=n21+1; else n22=n22+1; end; OR=n11*n22/(n12*n21); r1=n11/(n11+n12); r2=n21/(n21+n22); RR=r1/r2; logOR=log(OR); logRR=log(RR); output; end; keep n11 n12 n21 n22 OR logOR RR logRR; run; /*histogram*/ proc univariate; var OR logOR RR logRR; histogram OR logOR RR logRR; run;

7. header= Verdana 28 pt., Red 7 STA 517 – Chapter 3: Inference for Contingency Tables Simulated data

8. header= Verdana 28 pt., Red 8 STA 517 – Chapter 3: Inference for Contingency Tables OR and log(OR)

9. header= Verdana 28 pt., Red 9 STA 517 – Chapter 3: Inference for Contingency Tables

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11. header= Verdana 28 pt., Red 11 STA 517 – Chapter 3: Inference for Contingency Tables RR and log(RR)

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14. header= Verdana 28 pt., Red 14 STA 517 – Chapter 3: Inference for Contingency Tables 3.1.2 Aspirin and Myocardial Infarction Example  The study randomly assigned 1360 patients who had already suffered a stroke to an aspirin treatment (one low-dose tablet a day) or to a placebo treatment.  follow-up 3 years

15. header= Verdana 28 pt., Red 15 STA 517 – Chapter 3: Inference for Contingency Tables OR  =1.56  =1.55   95% confidence interval  The corresponding interval for  is

16. header= Verdana 28 pt., Red 16 STA 517 – Chapter 3: Inference for Contingency Tables  Since the confidence interval for contains 1.0, it is plausible that the true odds of death due to myocardial infarction are equal for aspirin and placebo.  If there truly is a beneficial effect of aspirin but the odds ratio is not large, it may require a large sample size to show that benefit because of the relatively small number of myocardial infarction cases

17. header= Verdana 28 pt., Red 17 STA 517 – Chapter 3: Inference for Contingency Tables 3.1.3 Interval Estimation of Difference of Proportions  The difference of proportions and the relative risk compare conditional distributions of a response variable for two groups.  For these measures, we treat the samples as independent binomials.  For group i, y i has a binomial distribution with sample size n i and a probability  i of a ‘‘success’’ response.  and variance  Since independence between two groups

18. header= Verdana 28 pt., Red 18 STA 517 – Chapter 3: Inference for Contingency Tables  =0.0409 =0.0266  =0.0143 =0.00978  95% CI =(-0.00487, 0.0335)

19. header= Verdana 28 pt., Red 19 STA 517 – Chapter 3: Inference for Contingency Tables 3.1.4 Interval Estimation of Relative Risk  The sample relative risk is  Like the odds ratio, it converges to normality faster on the log scale.  The asymptotic standard error of log r is  The Wald interval  =0.0409/0.0266=1.54, =0.297  CI or (0.86, 2.75)  We infer that the death rate for those taking placebo was between 0.86 and 2.75 times that for those taking aspirin.

20. header= Verdana 28 pt., Red 20 STA 517 – Chapter 3: Inference for Contingency Tables SAS code data table3_1; input treatment $ y n; MI='yes'; count=y; output; /*death due to myocardial */ MI='no '; count=n; output; cards; Placebo 28 656 Aspirin 18 658 ; proc freq data=table3_1; WHERE treatment ='Placebo' OR treatment='Aspirin'; weight Count; tables treatment*MI / chisq relrisk RISKDIFF MEASURES ; title 'Swedish Study on Aspirin Use and Myocardial Infarction'; run;

21. header= Verdana 28 pt., Red 21 STA 517 – Chapter 3: Inference for Contingency Tables

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24. header= Verdana 28 pt., Red 24 STA 517 – Chapter 3: Inference for Contingency Tables OR