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If we live with a deep sense of gratitude, our life will be greatly embellished.

If we live with a deep sense of gratitude, our life will be greatly embellished. Categorical Data Analysis. Chapter 10: Tests for Matched Pairs. Meta Analysis. Also known as stratified analysis Section 6.3.2: Cochran-Mantel-Haenszel test; test for conditional independence

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If we live with a deep sense of gratitude, our life will be greatly embellished.

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  1. If we live with a deep sense of gratitude, our life will be greatly embellished.

  2. Categorical Data Analysis Chapter 10: Tests for Matched Pairs

  3. Meta Analysis • Also known as stratified analysis • Section 6.3.2: Cochran-Mantel-Haenszel test; test for conditional independence Situation: When another variable (strata Z) may “pollute” the effect of a categorical explanatory variable X on a categorical response Y Goal: Study the effect of X on Y while controlling the stratification variable Z without assuming a model

  4. Example: Respiratory Improvement (SAS textbook, P. 46)

  5. SAS Output Summary Statistics for trtmnt by response Controlling for center Cochran-Mantel-Haenszel Statistics (Based on Table Scores) Statistic Alternative Hypothesis DF Value Prob ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ 1 Nonzero Correlation 1 18.4106 <.0001 2 Row Mean Scores Differ 1 18.4106 <.0001 3 General Association 1 18.4106 <.0001

  6. What to Do if Dependent • (Section 6.3.5) When X and Y are NOT conditionally independent given Z, we would like to test for homogeneous association • (Section 6.3.6) If X, Y, Z have homogeneous association, we would like to estimate the common conditional odds ratio for X, Y given Z

  7. SAS Output Estimates of the Common Relative Risk (Row1/Row2) Type of Study Method Value 95% Confidence Limits ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Case-Control Mantel-Haenszel 4.0288 2.1057 7.7084 (Odds Ratio) Logit 4.0286 2.1057 7.7072 Cohort Mantel-Haenszel 1.7368 1.3301 2.2680 (Col1 Risk) Logit 1.6760 1.2943 2.1703 Cohort Mantel-Haenszel 0.4615 0.3162 0.6737 (Col2 Risk) Logit 0.4738 0.3264 0.6877 Breslow-Day Test for Homogeneity of the Odds Ratios ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ Chi-Square 0.0002 DF 1 Pr > ChiSq 0.9900 Total Sample Size = 180

  8. Matched-pair Data • Comparing categorical responses for two “paired” samples When either • Each sample has the same subjects (or say subjects are measured twice) Or • A natural pairing exists between each subject in one sample and a subject from the other sample (eg. Twins)

  9. Example: Rating for Prime Minister

  10. Marginal Homogeneity • The probabilities of “success” for both samples are identical (The data table shows “symmetry” across the main diagonal) • Eg. The probability of approve at the first and 2nd surveys are identical

  11. Estimating Differences of Proportions • Sample estimate: P+1-P1+ • Standard error of P+1-P1+ (based on the multinomial distribution of data): • Asymptotical (1-a) confidence interval:

  12. McNemar Test (for 2x2 Tables only) • See SAS textbook Sec 3.7 (p. 40) • Ho: marginal homogeneity Ha: no marginal homogeneity • A special case of C-M-H test; an approximate test (when n*=n12+n21>10) • Exact test (when n*=n12+n21<10)

  13. Level of Agreement: Kappa Coefficient • The larger the Kappa coefficient is; the stronger the agreement is • The difference between observed agreement and that expected under independence compared to the maximum possible difference is called Kappa coefficient

  14. SAS Output McNemar's Test Statistic (S) 17.3559 DF 1 Asymptotic Pr > S <.0001 Exact Pr >= S 3.716E-05 Simple Kappa Coefficient Kappa 0.6996 ASE 0.0180 95% Lower Conf Limit 0.6644 95% Upper Conf Limit 0.7348 Sample Size = 1600 Level of agreement

  15. Chi-square Test for Square Tables Consider a IxI table • Marginal homogeneity: • Symmetry: for all pairs of cells, Symmetry => marginal homogeneity <=

  16. Chi-square Test for Square Tables Ho: symmetry vs. Ha: not symmetry • Fitted values: • Standardized Pearson residuals: • Pearson Chi-square Test statistic: X^2 follows approximately Chi-square with df = I(I-1)/2

  17. Example: Coffee Purchase

  18. Example: Coffee Purchase • X^2 = 20.4 and df is 5(5-1)/2=10  lack of fit (reject Ho: symmetry)  which pairs of cells cause the lack of fit? Examine their standardized Pearson residuals  The pair (1,3) and (3,1) contribute the most; other pairs are fine (rij^2 is around 1 or less)

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