10: Crosstabs & Independent Proportions

Transcription

1 10: Crosstabs & Independent Proportions p P Background < Two independent groups < Binary outcome < Compare binomial proportions P Illustrative example ( oswege.sav ) < Food poisoning following church picnic in Oswego, NY < Two groups based on exposure to vanilla ice cream Variable vanilla: 1 yes; 2 no < Outcome is sickness Variable ill: 1 yes; 2 no < Quantify association between vanilla and ill 10: Crosstabs & Independent Proportions 1 Notation p P Count number of cases and non-cases in each group P a / number w/ postive outcome ( cases ) < a 1 / number of exposed cases < a 0 / number of non-exposed cases P b / number w/ negative outcome ( non-cases ) < b 1 / number of exposed non-cases < b 0 / number of non-exposed non-cases Notation 2

2 2-by-2 Cross-Tabulation of Counts pp Outcome + Outcome - Group 1 a 1 b 1 Group 0 a 0 b 0 P For uniformity < Groups status along rows < Outcome status down columns P Tabulation options < By hand (rare) < By computer (e.g., SPSS > Descriptive statistics > Crosstabs) 2-by-2 Cross-Tabulation of Counts 3 Table Margins ( Totals ) p P Tables often show totals along margins P Row margins < n 1 / number in exposed group a 1 + b 1 < n 0 / number in non-exposed group a 0 + b 0 P Column margins < m 1 / number of cases a 1 + a 0 < m 0 / number of non-cases b 1 + b 0 Outcome + Outcome - Total Group 1 a 1 b 1 n 1 Group 0 a 0 b 0 n 0 Total m 1 m 0 N Table Margins ( Totals ) 4

3 Illustrative Data (oswego.sav, vanilla * ill ) p ILL+ ILL- Total VANILLA VANILLA Total Illustrative Data (oswego.sav, vanilla * ill) 5 Calculate Binomial Proportions in Groups p P Recall: binomial proportions / incidences or prevalences (discussed last week) +! Total Group Group Total a1 43 a p p. n n Calculate Binomial Proportions in Groups 6

4 Risk Difference (DO NOT COVER) p P Risk difference a single number that quantifies average increase in risk on absolute scale < Use with care P Risk difference parameter (p 1!p 0 ) is unknown P Risk difference statistic ( p hat 1! p hat 0 ) is calculated in sample P Illustrative example < Risk difference.763! Ñ 65% < ˆ Eating the vanilla ice cream increased risk by 65% Risk Difference (DO NOT COVER) 7 Point estimate: p 1 p se p p q p q (. 796)( ) (. 143)( ) p + + n n ( p 1 p 0 ) ± ( 196. )( se p p ) ( ) ± ( 196. )(. 094). 653±. 184 (. 469,. 837) 95% CI for Risk Difference Parameter (DO NOT COVER) pp Standard error: 95% CI: % CI for Risk Difference Parameter 8

5 P Well-suited for count data Chi-Square (P²) Test p P P² distributions < Family of distributions < Have degrees of freedom (like t distributions) < Asymmetrical w/ long right tails (not like t distributions) df 1 df 2 df Chi-square Chi-Square (P²) Test 9 Chi-Square Probabilities p P Use P² table to determine P² probabilities P Area under curve to left / cumulative probability P Area under curve to right / tail probability P Notation: P² df, cum. probability P e.g., P² 1, is: P² Chi-Square Probabilities 10

6 Another example not in Reader P Use P² table to determine that P² 3, P Visually:.90 P² Another example 11 P² Test for Two Groups pp H 0 : p 1 p 0 vs. H 1 : p 1 p 0 " Calculate P² stat Conclusion Convert P² stat to p value Reject H 0 when p # " Otherwise retain H 0 Retention implies there is not enough evidence to reject the H 0. This is not the same this as accepting the null hypothesis, because retention does not imply H 0 is true. P² Test for Two Groups 12

7 P² stat Calculation Formulas 10.6 and 10.7 (p. 10.6) 2 χ stat ( O E ) i i E i 2 where: O i / observed (from crosstab data) E i / expected if H 0 true and E i row total column total total sample size P²stat Calculation 13 Calculation of Expected Values OBSERVED +! Total Group Group Total EXPECTED +! Total Group a Group b 21 Total a Example of a calculation: b Example of a calculation: E E Calculation of Expected Values 14

8 P² stat Calculation 2 χ stat 2 χ stat ( O E ) i i E ( ) i ( ) ( ) ( ) by-2 crosstabs always have 1 df P²stat Calculation 15 Convesion of P² stat to p value p P Reasoning < If H 0 were true, P² stat would have a (random) P² distribution < Probability of observing P² stat or greater area to right of P² stat p < Use P² table and wedgie technique to determine approx. tail area /p value P Illustrative example: P² stat with 1 df 6 p < Test statistic way out in tail (landmark from P² table) Convesion of P²stat to p value 16

9 R-by-C Crosstab p P Chi-square test applied to nominal data with 2 or more categories P Illustrative example (sessmoke.sav ) < Row variable socio-economic status (SES) 1 very high SES 2 high SES 3 intermediate SES 4 low SES 5 very low SES < Column variable current smoker 1 yes 2 no R-by-C Crosstab 17 R-by-C Crosstab (cont.) p P Cross-tabulating reveals R rows and C columns P Illustrative example: 5-by-2 table SMOKE + SMOKE! SES SES SES SES SES R-by-C Crosstab (cont.) 18

10 Describe Proportions Within Groups p p i no. positive_ for_ attribute row_ total SMOKE + SMOKE! Total SES SES SES SES SES p % p % 50 Describe Proportions Within Groups 19 Illustrative Example: Prevalence by SES p p p p p p In the population, are proportions homogenous? Illustrative Example: Prevalence by SES 20

11 Testing R-by-C Data p P Hypotheses < H 0 : no association in population (proportions homogenous) < H 1 : H 0 is false P " P Test statistics < Same chi-square formula < df (R!1)(C!1) where R number of row categories and C number of column categories P p value and conclusion Testing R-by-C Data 21 R-by-C Observed and Expected pp OBSERVED SMOKE + SMOKE! SES SES SES SES SES EXPECTED SMOKE + SMOKE! SES a SES SES SES SES a Example of a calculation: E R-by-C Observed and Expected 22

12 Calculation of P² stat p P Application of 3 (O!E)²/E formula shown on p < P² stat 5.32 for illustrative example < df (5-1)(2-1) (4)(1) 4 P Convert P² stat to p value using P² table and landmark(s) < : p >.20 [see figure] < ˆ retain H 0 ; association is not significant P² 4 p (landmark from chi-square table) Calculation of P²stat 23 Assumptions Required for Valid P² Test p P Expected values $ 5 P Samples are simple random samples from populations P Data are not misclassified Assumptions Required for Valid P² Test 24