1. Section 4.4:Contingency Tables and Association Contingency table – What and why a contingency table – Marginal distribution – Conditional distribution Simpson’s Paradox – What is it? – What causes it?

2. Contingency tables are for summarizing bivariate (or multivariate) qualitative data. sexheightshoeeyeshairhand male709brownbrownright male7111blueblondleft male7311.5blueblondright female647brownblackright male667.5brownlightbrownright female636.5brownblackright female646.5blueredright male7210brownblondleft male668.5greenlightbrownright female678brownlightbrownright male7411.5brownbrownleft male7212bluebrownright female688.5bluelightbrownright male7812blueblondright male7012greenblondright female688blueredboth female689.5greenbrownleft female667blueblondright male6610brownbrownright :::: :: :: ::::: ::::: :::::

3. blackblondbrownlightbrownredTotal blue01932327 brown52154127 green11022015 Total631208469 Contingency table results: Rows: eyes Columns: hair

4. bluebrowngreenTotal black0516 blond1921031 brown315220 lightbrown2428 red3104 Total27 1569 Contingency table results: Rows: hair Columns: eyes Often it is arbitrary which variable gets to be the row variable.

5. blackblondbrownlightbrownredTotal blue01001213 brown3182115 green041106 Total31594334 blackblondbrownlightbrownredTotal blue0931114 brown2172012 green161109 Total316114135 Contingency table results for sex=female: Rows: eyes Columns: hair Contingency table results for sex=male Displaying three variables (sex, eye color, hair color). We will focus on two variables.

6. The 793 adult male passenger survival, by 1st class, 2nd class, and 3rd class fares: http://www.encyclopedia-titanica.org/titanic-statistics.html Status \ Class1st Class2nd Class3rd ClassTotal Saved581360131 Lost118154390662 Total176167450793

7. Status \ Class1st Class2nd Class3rd ClassTotal Saved58 (7.31%) 13 (1.64%) 60 (7.57%) 131 (16.52%) Lost118 (14.88%) 154 (19.42%) 390 (49.18%) 662 (83.48%) Total176 (22.19%) 167 (21.06%) 450 (56.75%) 793 (100%) Relative Frequency marginal distribution: (in parentheses) Margins show relative amount in each row or column Add to one.

8. Status \ Class1st Class2nd Class3rd ClassTotal Saved58 (44.27%) 13 (9.92%) 60 (45.8%) 131 (100%) Lost118 (17.82%) 154 (23.26%) 390 (58.91%) 662 (100%) Total176 (22.19%) 167 (21.06%) 450 (56.75%) 793 (100%) Conditional Distribution Either rows or columns add to one (100%). Percentages conditioned on survival status

9. Status \ Class1st Class2nd Class3rd ClassTotal Saved58 (32.95%) 13 (7.78%) 60 (13.33%) 131 (16.52%) Lost118 (67.05%) 154 (92.22%) 390 (86.67%) 662 (83.48%) Total176 (100%) 167 (100%) 450 (100%) 793 (100%) Percentages conditioned on passenger class

10. Women & children Mentotal saved369131500 lost155662817 total5247931317 1.What proportion of passengers were women & children? 2.What proportion of the passengers were lost? 3.What proportion of the women & children were lost? 4.Of the passengers who were lost, what proportion of the passengers were women and children?

11. AcceptedRejectedtotal% Accepted Male620380100062 Female480520100048 Simpson’s Paradox: Example Hypothetical graduate school acceptance data: Men do better

12. But if a third variable is accounted for the story changes… MajorAMajorB Accept Rejecte dtotal%AcceptedAccept Rejecte dtotal% Accepted Male60030090066.6667208010020 Female160402008032048080040 Women actually do better

13. Both majorsMajorAMajorB AcceptedRejectedtotal%AcceptedAcceptRejectedtotal%AcceptedAcceptRejectedtotal% Accepted Male62038010006260030090066.66667208010020 Female480520100048160402008032048080040 Why the change?

14. Simpson’s Paradox represents a situation in which an association between two variables inverts or goes away when a third variable is introduced to the analysis. See: http://users.humboldt.edu/rizzardi/Handouts.dir/SimpsonParadoxExample.xlsx