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Contingency analysis

Contingency analysis. Sample. Null hypothesis. Test statistic. Null distribution. compare. How unusual is this test statistic?. P > 0.05. P < 0.05. Reject H o. Fail to reject H o. Using one tail in the c 2. We always use only one tail for a c 2 test Why?.

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Contingency analysis

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  1. Contingency analysis

  2. Sample Null hypothesis Test statistic Null distribution compare How unusual is this test statistic? P > 0.05 P < 0.05 Reject Ho Fail to reject Ho

  3. Using one tail in the c2 • We always use only one tail for a c2 test • Why?

  4. Data match null expectation exactly 0 Data deviate from null expectation in some way

  5. Reality Ho true Ho false Result correct Reject Ho Type I error Do not reject Ho correct Type II error

  6. If null hypothesis is really true… Do not reject Ho Correct answer Reject Ho Type I error Test statistic

  7. If null hypothesis is really false… Do not reject Ho Type II error Reject Ho correct Test statistic

  8. Errors and statistics • These are theoretical - you usually don’t know for sure if you’ve made an error • Pr[Type I error] =  • Pr[Type II error] = … • Requires power analysis • Depends on sample size

  9. Contingency analysis • Estimates and tests for an association between two or more categorical variables

  10. Music and wine buying

  11. Mosaic plot

  12. Odds ratio • Odds of success = probability of success divided by the probability of failure

  13. Estimating the Odds ratio • Odds of success = probability of success divided by the probability of failure

  14. Music and wine buying

  15. Example • Out of 48 bottles of wine, 40 were French

  16. Example • Out of 48 bottles of wine, 40 were French Interpretation: people are about 5 times more likely to buy a French wine

  17. Failure more likely Success and failure equally likely Success more likely O=1

  18. Odds ratio • The odds of success in one group divided by the odds of success in a second group

  19. Estimating the Odds ratio • The odds of success in one group divided by the odds of success in a second group

  20. Music and wine buying • Group 1 = French music, Group 2 = German music • Success = French wine

  21. Group 2 • Out of 34 bottles of wine, 12 were French

  22. Music and wine buying • Group 1 = French music, Group 2 = German music • Success = French wine

  23. Music and wine buying • Group 1 = French music, Group 2 = German music • Success = French wine Interpretation: people are about 9 times more likely to buy French wine in Group 1 compared to Group 2

  24. Success more likely in Group 2 Success equally likely in both groups Success more likely in Group 1 OR=1

  25. Hypothesis testing • Contingency analysis • Is there a difference in odds between two groups?

  26. Hypothesis testing • Contingency analysis • Is there an association between two categorical variables?

  27. Music and wine buying

  28. Contingency analysis • Is there a difference in the odds of buying French wine depending on the music that is playing? • Is there an association between wine bought and music playing? • Is the nationality of the wine independent of the music playing when it is sold?

  29. Hypotheses • H0: The nationality of the bottle of wine is independent of the nationality of the music played when it is sold. • HA: The nationality of the bottle of wine sold depends on the nationality of the music being played when it is sold.

  30. Calculating the expectations With independence, Pr[ French wine AND French music] = Pr[French wine]  Pr[French music]

  31. Calculating the expectations Pr[French wine] = 52/82=0.634 Pr[French music] = 48/82= 0.585 By H0, Pr[French wine AND French music] = (0.634)(0.585)=0.37112

  32. Calculating the expectations By H0, Pr[French wine AND French music] = (0.634)(0.585)=0.37112

  33. Calculating the expectations

  34. c2

  35. Degrees of freedom For a 2 Contingency test, df = # categories -1- # parameters df= (# columns -1)(# rows -1) For music/wine example, df = (2-1)(2-1) = 1

  36. Conclusion c2 = 20.0 >> c21,a=0.05 = 3.84, So we can reject the null hypothesis of independence, and say that the nationality of the wine sold did depend on what music was played.

  37. Assumptions • This c2 test is just a special case of the c2 goodness-of-fit test, so the same rules apply. • You can’t have any expectation less than 1, and no more than 20% < 5

  38. Fisher’s exact test • For 2 x 2 contingency analysis • Does not make assumptions about the size of expectations • JMP will do it, but cumbersome to do by hand

  39. Other extensions you might see • Yates correction for continuity • G-test • Read about these in your book

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