Chapter 19 Chi-Square Fundamental Statistics for the Behavioral Sciences, 5th edition David C. Howell © 2003 Brooks/Cole Publishing Company/ITP

2Chapter 19 Chi-Square Major Points Categorical variablesCategorical variables One-way classificationOne-way classification XAn example Contingency tablesContingency tables X2X2 tables XLarger tables Cont.

3Chapter 19 Chi-Square Major Points--cont. Tests on proportionsTests on proportions Non-independent observationsNon-independent observations Small expected frequenciesSmall expected frequencies Effect sizesEffect sizes XOdds ratios and Phi Review questionsReview questions

4Chapter 19 Chi-Square Categorical Variables Generally the count of objects falling in each of several categories.Generally the count of objects falling in each of several categories. Examples:Examples: Xnumber of fraternity, sorority, and nonaffiliated members of a class Xnumber of students choosing answers: 1, 2, 3, 4, or 5 Emphasis on frequency in each categoryEmphasis on frequency in each category

5Chapter 19 Chi-Square One-way Classification Observations sorted on only one dimensionObservations sorted on only one dimension Example:Example: XObserve children and count red, green, yellow, or orange Jello choices XAre these colors chosen equally often, or is there a preference for one over the other? Cont.

6Chapter 19 Chi-SquareOne-way--cont. Want to compare observed frequencies with frequencies predicted by null hypothesisWant to compare observed frequencies with frequencies predicted by null hypothesis Chi-square test used to compare expected and observedChi-square test used to compare expected and observed  Called goodness-of-fit chi-square (  2 )

7Chapter 19 Chi-Square Goodness-of-Fit Chi-square Fombonne (1989) Season of birth and childhood psychosisFombonne (1989) Season of birth and childhood psychosis Are children born at particular times of year more likely to be diagnosed with childhood psychosisAre children born at particular times of year more likely to be diagnosed with childhood psychosis He knew the % normal children born in each monthHe knew the % normal children born in each month Xe.g..8.4% normal children born in January

8Chapter 19 Chi-Square Fombonne’s Data

9Chapter 19 Chi-Square Chi-Square (  2 ) Compare Observed (O) with Expected (E)Compare Observed (O) with Expected (E) Take size of E into accountTake size of E into account XWith large E, a large (O-E) is not unusual. XWith small E, a large (O-E) is unusual.

10Chapter 19 Chi-Square Calculation of  2  2.05 (11) = 19.68

11Chapter 19 Chi-Square

12Chapter 19 Chi-SquareConclusions Obtained  2 = 14.58Obtained  2 = df = c - 1, where c = # categoriesdf = c - 1, where c = # categories Critical value of  2 on 11 df = 19.68Critical value of  2 on 11 df = Since > 14.58, do not reject H 0Since > 14.58, do not reject H 0 Conclude that birth month distribution of children with psychoses doesn’t differ from normal.Conclude that birth month distribution of children with psychoses doesn’t differ from normal.

13Chapter 19 Chi-SquareElaboration Degrees of freedomDegrees of freedom Why formula makes logical senseWhy formula makes logical sense How to read critical values from tableHow to read critical values from table Why reject for only large positive  2Why reject for only large positive  2 XWhat would “significantly small” mean?

14Chapter 19 Chi-Square Contingency Tables Two independent variablesTwo independent variables XAre men happier than women? Male vs. Female X Happy vs Not HappyMale vs. Female X Happy vs Not Happy XIntimacy (Yes/No) X Depression/Nondepression

15Chapter 19 Chi-Square Intimacy and Depression Everitt & Smith (1979)Everitt & Smith (1979) Asked depressed and non-depressed women about intimacy with boyfriend/husbandAsked depressed and non-depressed women about intimacy with boyfriend/husband Data on next slideData on next slide

16Chapter 19 Chi-SquareData

17Chapter 19 Chi-Square What Do Data Say? It looks as if depressed women are more likely to report lack of intimacy.It looks as if depressed women are more likely to report lack of intimacy. XWhat direction does the causation run? XDo we know it is causal? What alternative explanations?What alternative explanations? XIs the relationship reliably different from chance? Chi-square testChi-square test

18Chapter 19 Chi-Square Chi-Square on Contingency Table Same formulaSame formula Expected frequenciesExpected frequencies XE = RT X CT GT RT = Row total, CT = Column total, GT = Grand totalRT = Row total, CT = Column total, GT = Grand total

19Chapter 19 Chi-Square Expected Frequencies E 11 = 37*138/419 = 12.19E 11 = 37*138/419 = E 12 = 37*281/419 = 24.81E 12 = 37*281/419 = E 21 = 382*138/419 = E 21 = 382*138/419 = E 22 = 382*281/419 = E 22 = 382*281/419 = Enter on following tableEnter on following table

20Chapter 19 Chi-Square Observed and Expected Freq.

21Chapter 19 Chi-Square Chi-Square Calculation

22Chapter 19 Chi-Square Degrees of Freedom For contingency table, df = (R - 1)(C - 1)For contingency table, df = (R - 1)(C - 1) For our example this is (2 - 1)(2 - 1) = 1For our example this is (2 - 1)(2 - 1) = 1 XNote that knowing any one cell and the marginal totals, you could reconstruct all other cells.

23Chapter 19 Chi-SquareConclusions Since > 3.84, reject H 0Since > 3.84, reject H 0 Conclude that depression and intimacy are not independent.Conclude that depression and intimacy are not independent. XHow one responds to “satisfaction with intimacy” depends on whether they are depressed. XCould be depression-->dissatisfaction, lack of intimacy --> depression, depressed people see world as not meeting needs, etc.

24Chapter 19 Chi-Square Larger Contingency Tables Jankowski & Leitenberg(pers. comm.)Jankowski & Leitenberg(pers. comm.) XDoes abuse continue? Do adults who are, and are not, being abused differ in childhood history of abuse?Do adults who are, and are not, being abused differ in childhood history of abuse? One variable = adult abuse (yes or no)One variable = adult abuse (yes or no) Other variable = number of abuse categories (out of 4) suffered as childrenOther variable = number of abuse categories (out of 4) suffered as children XSexual, Physical, Alcohol, or Personal violence

25Chapter 19 Chi-Square

26Chapter 19 Chi-Square Chi-Square Calculation

27Chapter 19 Chi-SquareConclusions > > 7.82 XReject H 0 XConclude that adult abuse is related to childhood abuse XIncreasing levels of childhood abuse are associated with greater levels of adult abuse. e.g. Approximately 10% of nonabused children are later abused as adults.e.g. Approximately 10% of nonabused children are later abused as adults. Cont.

28Chapter 19 Chi-SquareConclusions--cont. Approximately 40% of highly abused children are later abused as adults.Approximately 40% of highly abused children are later abused as adults. These data suggest that childhood abuse doesn’t stop when children grow up.These data suggest that childhood abuse doesn’t stop when children grow up.

29Chapter 19 Chi-Square Tests on Proportions Proportions can be converted to frequencies, and tested using  2.Proportions can be converted to frequencies, and tested using  2. Use a z test directly on the proportions if you have two proportionsUse a z test directly on the proportions if you have two proportions From last exampleFrom last example X10% of nonabused children abused as adults X40% of abused children abused as adults Cont.

30Chapter 19 Chi-SquareProportions--cont. There were 566 nonabused children and 30 heavily abused children.There were 566 nonabused children and 30 heavily abused children. Cont.

31Chapter 19 Chi-SquareProportions--cont. z = 5.17z = 5.17 This is a standard z score.This is a standard z score. XTherefore.05 (2-tailed) cutoff = XReject null hypothesis that the population proportions of abuse in both groups are equal. This is just the square root of the  2 you would have with  2 on those 4 cells.This is just the square root of the  2 you would have with  2 on those 4 cells.

32Chapter 19 Chi-Square Nonindependent Observations We require that observations be independent.We require that observations be independent. XOnly one score from each respondent XSum of frequencies must equal number of respondents If we don’t have independence of observations, test is not valid.If we don’t have independence of observations, test is not valid.

33Chapter 19 Chi-Square Small Expected Frequencies Assume O would be normally distributed around E over many replications of experiment.Assume O would be normally distributed around E over many replications of experiment. This could not happen if E is small.This could not happen if E is small. Rule of thumb: E > 5 in each cellRule of thumb: E > 5 in each cell XNot firm rule XViolated in earlier example, but probably not a problem Cont.

34Chapter 19 Chi-Square Expected Frequencies--cont. More of a problem in tables with few cells.More of a problem in tables with few cells. Never have expected frequency of 0.Never have expected frequency of 0. Collapse adjacent cells if necessary.Collapse adjacent cells if necessary.

35Chapter 19 Chi-Square Effect Size Phi and Cramer’s PhiPhi and Cramer’s Phi XDefine N and k XNot limited to 2X2 tables Odds RatioOdds Ratio Cont.

36Chapter 19 Chi-Square Effect Size—cont. Everitt & cc dataEveritt & cc data Cont.

37Chapter 19 Chi-Square Effect Size—cont. Odds Dep|Lack IntimacyOdds Dep|Lack Intimacy X26/112 =.232 Odds Dep | No LackOdds Dep | No Lack X11/270 =.041 Odds Ratio =.232/.041 = 5.69Odds Ratio =.232/.041 = 5.69 Odds Depressed = 5.69 times great if experiencing lack of intimacy.Odds Depressed = 5.69 times great if experiencing lack of intimacy.

38Chapter 19 Chi-Square Review Questions What are categorical data?What are categorical data? What is the difference between a goodness-of-fit test and a test on contingency table?What is the difference between a goodness-of-fit test and a test on contingency table? What are expected frequencies?What are expected frequencies? What is H 0 in goodness-of-fit test?What is H 0 in goodness-of-fit test? Cont.

39Chapter 19 Chi-Square Review Questions--cont. What is H 0 in test on contingency tables?What is H 0 in test on contingency tables? What is difference between independence of variables and independence of observations?What is difference between independence of variables and independence of observations? Why are small expected frequencies a problem?Why are small expected frequencies a problem? Does the z test on proportions tell you something that  2 does not?Does the z test on proportions tell you something that  2 does not? Cont.

40Chapter 19 Chi-Square Review Questions—cont. What are odds?What are odds? What is an odds ratio?What is an odds ratio?