Chapter 10 Hypothesis testing Categorical Data Analysis EPI

Chapter 10 Hypothesis testing: Categorical Data Analysis EPI 809/Spring 2008 1

Learning Objectives 1. Comparison of binomial proportion using Z and 2 Test. 2. Explain 2 Test for Independence of 2 variables 3. Explain The Fisher’s test for independence 4. Mc. Nemar’s tests for correlated data 5. Kappa Statistic 6. Use of SAS Proc FREQ EPI 809/Spring 2008 2

Data Types EPI 809/Spring 2008 3

Qualitative Data 1. 2. 3. 4. Qualitative Random Variables Yield Responses That Can Be Put In Categories. Example: Gender (Male, Female) Measurement or Count Reflect # in Category Nominal (no order) or Ordinal Scale (order) Data can be collected as continuous but recoded to categorical data. Example (Systolic Blood Pressure - Hypotension, Normal tension, hypertension ) EPI 809/Spring 2008 4

Hypothesis Tests Qualitative Data EPI 809/Spring 2008 5

Z Test for Differences in Two Proportions EPI 809/Spring 2008 6

Hypotheses for Two Proportions EPI 809/Spring 2008 7

Hypotheses for Two Proportions EPI 809/Spring 2008 8

Hypotheses for Two Proportions EPI 809/Spring 2008 9

Hypotheses for Two Proportions EPI 809/Spring 2008 10

Hypotheses for Two Proportions EPI 809/Spring 2008 11

Hypotheses for Two Proportions EPI 809/Spring 2008 12

Z Test for Difference in Two Proportions 1. Assumptions l l l 2. Populations Are Independent Populations Follow Binomial Distribution Normal Approximation Can Be Used for large samples (All Expected Counts 5) Z-Test Statistic for Two Proportions EPI 809/Spring 2008 13

Sample Distribution for Difference Between Proportions EPI 809/Spring 2008 14

Z Test for Two Proportions Thinking Challenge MA Ø You’re an epidemiologist for the US Department of Health and Human Services. You’re studying the prevalence of disease X in two states (MA and CA). In MA, 74 of 1500 people surveyed were diseased and in CA, 129 of 1500 were diseased. At. 05 level, does MA have a lower prevalence rate? EPI 809/Spring 2008 CA 15

Z Test for Two Proportions Solution* EPI 809/Spring 2008 16

Z Test for Two Proportions Solution* H 0: H a: = n. MA = n. CA = Critical Value(s): Test Statistic: Decision: Conclusion: EPI 809/Spring 2008 17

Z Test for Two Proportions Solution* H 0: p. MA - p. CA = 0 Ha: p. MA - p. CA < 0 = n. MA = n. CA = Critical Value(s): Test Statistic: Decision: Conclusion: EPI 809/Spring 2008 18

Z Test for Two Proportions Solution* H 0: p. MA - p. CA = 0 Ha: p. MA - p. CA < 0 =. 05 n. MA = 1500 n. CA = 1500 Critical Value(s): Test Statistic: Decision: Conclusion: EPI 809/Spring 2008 19

Z Test for Two Proportions Solution* H 0: p. MA - p. CA = 0 Ha: p. MA - p. CA < 0 =. 05 n. MA = 1500 n. CA = 1500 Critical Value(s): Test Statistic: Decision: Conclusion: EPI 809/Spring 2008 20

Z Test for Two Proportions Solution* EPI 809/Spring 2008 21

Z Test for Two Proportions Solution* H 0: p. MA - p. CA = 0 Ha: p. MA - p. CA < 0 =. 05 n. MA = 1500 n. CA = 1500 Critical Value(s): Test Statistic: Z = -4. 00 Decision: Conclusion: EPI 809/Spring 2008 22

Z Test for Two Proportions Solution* H 0: p. MA - p. CA = 0 Ha: p. MA - p. CA < 0 =. 05 n. MA = 1500 n. CA = 1500 Critical Value(s): Test Statistic: Z = -4. 00 Decision: Reject at =. 05 Conclusion: EPI 809/Spring 2008 23

Z Test for Two Proportions Solution* H 0: p. MA - p. CA = 0 Ha: p. MA - p. CA < 0 =. 05 n. MA = 1500 n. CA = 1500 Critical Value(s): Test Statistic: Z = -4. 00 Decision: Reject at =. 05 Conclusion: There is evidence MA is less than CA EPI 809/Spring 2008 24

2 Test of Independence Between 2 Categorical Variables EPI 809/Spring 2008 25

Hypothesis Tests Qualitative Data EPI 809/Spring 2008 26

2 Test of Independence 1. Shows If a Relationship Exists Between 2 Qualitative Variables, but does Not Show Causality 2. Assumptions Multinomial Experiment All Expected Counts 5 3. Uses Two-Way Contingency Table EPI 809/Spring 2008 27

2 Test of Independence Contingency Table Ø 1. Shows # Observations From 1 Sample Jointly in 2 Qualitative Variables EPI 809/Spring 2008 28

2 Test of Independence Contingency Table 1. Shows # Observations From 1 Sample Jointly in 2 Qualitative Variables Levels of variable 2 Levels of variable 1 EPI 809/Spring 2008 29

2 Test of Independence Hypotheses & Statistic 1. Hypotheses l l H 0: Variables Are Independent Ha: Variables Are Related (Dependent) EPI 809/Spring 2008 30

2 Test of Independence Hypotheses & Statistic 1. Hypotheses H 0: Variables Are Independent Ha: Variables Are Related (Dependent) 2. Test Statistic Observed count Expected count EPI 809/Spring 2008 31

2 Test of Independence Hypotheses & Statistic 1. Hypotheses H 0: Variables Are Independent Ha: Variables Are Related (Dependent) 2. Test Statistic Observed count Expected count Rows Columns Degrees of Freedom: (r - 1)(c - 1) EPI 809/Spring 2008 32

2 Test of Independence Expected Counts 1. Statistical Independence Means Joint Probability Equals Product of Marginal Probabilities 2. Compute Marginal Probabilities & Multiply for Joint Probability 3. Expected Count Is Sample Size Times Joint Probability EPI 809/Spring 2008 33

Expected Count Example EPI 809/Spring 2008 34

Expected Count Example Marginal probability = 112 160 EPI 809/Spring 2008 35

Expected Count Example Marginal probability = 112 160 Marginal probability = 78 160 EPI 809/Spring 2008 36

Expected Count Example 112 78 Joint probability = 160 Marginal probability = 112 160 78 160 EPI 809/Spring 2008 37

Expected Count Example 112 78 Joint probability = 160 Marginal probability = 112 160 78 160 112 78 Expected count = 160· 160 = 54. 6 Marginal probability = EPI 809/Spring 2008 38

Expected Count Calculation EPI 809/Spring 2008 39

Expected Count Calculation EPI 809/Spring 2008 40

Expected Count Calculation 112 x 78 160 112 x 82 160 48 x 78 160 EPI 809/Spring 2008 48 x 82 160 41

2 Test of Independence Example on HIV Ø You randomly sample 286 sexually active individuals and collect information on their HIV status and History of STDs. At the. 05 level, is there evidence of a relationship? EPI 809/Spring 2008 42

2 Test of Independence Solution EPI 809/Spring 2008 43

2 Test of Independence Solution H 0: H a: = df = Critical Value(s): Test Statistic: Decision: Conclusion: EPI 809/Spring 2008 44

2 Test of Independence Solution H 0: No Relationship Ha: Relationship = df = Critical Value(s): Test Statistic: Decision: Conclusion: EPI 809/Spring 2008 45

2 Test of Independence Solution H 0: No Relationship Ha: Relationship =. 05 df = (2 - 1) = 1 Critical Value(s): Test Statistic: Decision: Conclusion: EPI 809/Spring 2008 46

2 Test of Independence Solution H 0: No Relationship Ha: Relationship =. 05 df = (2 - 1) = 1 Critical Value(s): =. 05 Test Statistic: Decision: Conclusion: EPI 809/Spring 2008 47

2 Test of Independence Solution E(nij) 5 in all cells 116 x 132 286 154 x 116 286 170 x 132 286 EPI 809/Spring 2008 170 x 154 286 48

2 Test of Independence Solution EPI 809/Spring 2008 49

2 Test of Independence Solution H 0: No Relationship Ha: Relationship =. 05 df = (2 - 1) = 1 Critical Value(s): =. 05 Test Statistic: 2 = 54. 29 Decision: Conclusion: EPI 809/Spring 2008 50

2 Test of Independence Solution H 0: No Relationship Ha: Relationship =. 05 df = (2 - 1) = 1 Critical Value(s): =. 05 Test Statistic: 2 = 54. 29 Decision: Reject at =. 05 Conclusion: EPI 809/Spring 2008 51

2 Test of Independence Solution H 0: No Relationship Ha: Relationship =. 05 df = (2 - 1) = 1 Critical Value(s): =. 05 Test Statistic: 2 = 54. 29 Decision: Reject at =. 05 Conclusion: There is evidence of a relationship EPI 809/Spring 2008 52

2 Test of Independence SAS CODES Data dis; input STDs HIV count; cards; 1 1 84 1 2 32 2 1 48 2 2 122 ; run; Proc freq data=dis order=data; weight Count; tables STDs*HIV/chisq; run; EPI 809/Spring 2008 53

2 Test of Independence SAS OUTPUT Statistics for Table of STDs by HIV Statistic DF Value Prob ---------------------------Chi-Square 1 54. 1502 <. 0001 Likelihood Ratio Chi-Square 1 55. 7826 <. 0001 Continuity Adj. Chi-Square 1 52. 3871 <. 0001 Mantel-Haenszel Chi-Square 1 53. 9609 <. 0001 Phi Coefficient 0. 4351 Contingency Coefficient 0. 3990 Cramer's V 0. 4351 EPI 809/Spring 2008 54