Non-parametric test (distribution-free) Nominal level dependent measure

1. Chi-Square Non-parametric test (distribution-free) Nominal level dependent measure

2. Categorical Variables • Generally the count of objects falling in each of several categories. • Examples: • number of fraternity, sorority, and nonaffiliated members of a class • number of students choosing answers: 1, 2, 3, 4, or 5 • Emphasis on frequency in each category

3. One-way Classification • Observations sorted on only one dimension • Example: • Observe children and count red, green, yellow, or orange Jello choices • Are these colors chosen equally often, or is there a preference for one over the other? Cont.

4. One-way--cont. • Want to compare observed frequencies with frequencies predicted by null hypothesis • Chi-square test used to compare expected and observed • Called goodness-of-fit chi-square (c2)

5. Goodness-of-Fit Chi-square • Fombonne (1989) Season of birth and childhood psychosis • Are children born at particular times of year more likely to be diagnosed with childhood psychosis • He knew the % normal children born in each month • e.g. .8.4% normal children born in January

6. Fombonne’s Data

7. Chi-Square (c2) • Compare Observed (O) with Expected (E) • Take size of E into account • With large E, a large (O-E) is not unusual. • With small E, a large (O-E) is unusual.

8. Calculation of c2 2.05(11) = 19.68

10. Conclusions • Obtained 2= 14.58 • df = c - 1, where c = # categories • Critical value of 2 on 11 df = 19.68 • Since 19.68 > 14.58, do not reject H0 • Conclude that birth month distribution of children with psychoses doesn’t differ from normal.

11. Jello Choices • Red Green Yellow Orange • 35 20 25 20 • Is there a significant preference for one color of jello over other colors?

12. Red Green Yellow Orange O: 35 20 25 20 E: 25 25 25 25 X2 = (35-25)2/25 + (20-25)2/25 + (25-25)2/25 + (20-25)2/25= 6 There was not one jello color chosen significantly more often than any other jello color, X2 (3, N= 100) = 6, p > .05

13. Contingency Tables • Two independent variables • Are men happier than women? • Male vs. Female X Happy vs Not Happy • Intimacy (Yes/No) X Depression/Nondepression

14. Intimacy and Depression • Everitt & Smith (1979) • Asked depressed and non-depressed women about intimacy with boyfriend/husband • Data on next slide

15. Data

16. Chi-Square on Contingency Table • Same formula • Expected frequencies • E = RT X CT GT • RT = Row total, CT = Column total, GT = Grand total

17. Expected Frequencies • E11 = 37*138/419 = 12.19 • E12 = 37*281/419 = 24.81 • E21 = 382*138/419 = 125.81 • E22 = 382*281/419 = 256.19 • Enter on following table

18. Observed and Expected Freq.

19. Chi-Square Calculation

20. Degrees of Freedom • For contingency table, df = (R - 1)(C - 1) • For our example this is (2 - 1)(2 - 1) = 1 • Note that knowing any one cell and the marginal totals, you could reconstruct all other cells.

21. Conclusions • Since 25.61 > 3.84, reject H0 • Conclude that depression and intimacy are not independent. • How one responds to “satisfaction with intimacy” depends on whether they are depressed. • Could be depression-->dissatisfaction, lack of intimacy --> depression, depressed people see world as not meeting needs, etc.

22. Larger Contingency Tables • Jankowski & Leitenberg (pers. comm.) • Does abuse continue? • Do adults who are, and are not, being abused differ in childhood history of abuse? • One variable = adult abuse (yes or no) • Other variable = number of abuse categories (out of 4) suffered as children • Sexual, Physical, Alcohol, or Personal violence

24. Chi-Square Calculation

25. Conclusions • 29.62 > 7.82 • Reject H0 • Conclude that adult abuse is related to childhood abuse • Increasing levels of childhood abuse are associated with greater levels of adult abuse. • e.g. Approximately 10% of nonabused children are later abused as adults. Cont.

26. Nonindependent Observations • We require that observations be independent. • Only one score from each respondent • Sum of frequencies must equal number of respondents • If we don’t have independence of observations, test is not valid.

27. Small Expected Frequencies • Rule of thumb: E> 5 in each cell • Not firm rule • Violated in earlier example, but probably not a problem • More of a problem in tables with few cells. • Never have expected frequency of 0. • Collapse adjacent cells if necessary. Cont.

28. Expected Frequencies--cont. • More of a problem in tables with few cells. • Never have expected frequency of 0. • Collapse adjacent cells if necessary.

29. Effect Size • Phi and Cramer’s Phi • Define N and k • Not limited to 2X2 tables Cont.

30. Effect Size—cont. • Everitt & cc data Cont.

31. Effect Size—Odss Ratio. • Odds Dep|Lack Intimacy • 26/112 = .232 • Odds Dep | No Lack • 11/270 = .041 • Odds Ratio = .232/.041 = 5.69 • Odds Depressed = 5.69 times great if experiencing lack of intimacy.

32. Effect Size—Risk Ratio. • Risk Depression/Lack Intimacy • 26/138 = .188 • Risk Depression | No Lack • 11/281 = .039 • Odds Ratio = .188/.039 = 4.83 • Risk of Depressed = 4.83 times greater if experiencing lack of intimacy.