Fundamental Statistics for the Behavioral Sciences, 4th edition Chapter 19 Chi-Square Fundamental Statistics for the Behavioral Sciences, 4th edition David C. Howell ©1999 Brooks/Cole Publishing Company/ITP 

Major Points Categorical variables One-way classification Chapter 19 Chi-Square Major Points Categorical variables One-way classification An example Contingency tables 2X2 tables Larger tables Cont.

Major Points--cont. Tests on proportions Non-independent observations Chapter 19 Chi-Square Major Points--cont. Tests on proportions Non-independent observations Small expected frequencies Review questions

Categorical Variables Chapter 19 Chi-Square 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

One-way Classification Chapter 19 Chi-Square 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.

Chapter 19 Chi-Square 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)

Goodness-of-Fit Chi-square Chapter 19 Chi-Square Goodness-of-Fit Chi-square Fombonne (1989) Season of birth and childhood psychosis Are children born at particular times are 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 Fombonne, E. (1989) Season of birth and childhood psychosis. British Journal of Psychiatry. 155, 655-661.

Chapter 19 Chi-Square Fombonne’s Data

Chi-Square (c2) Compare Observed (O) with Expected (E) Chapter 19 Chi-Square 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.

Chapter 19 Chi-Square Calculation of c2 2.05(11) = 19.68

Chapter 19 Chi-Square

Conclusions Obtained 2= 14.58 df = c - 1, where c = # categories Chapter 19 Chi-Square 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.

Elaboration Degrees of freedom Why formula makes logical sense Chapter 19 Chi-Square Elaboration Degrees of freedom Why formula makes logical sense How to read critical values from table Why reject for only large positive 2 What would “significantly small” mean?

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

Intimacy and Depression Chapter 19 Chi-Square Intimacy and Depression Everitt & Smith (1979) Asked depressed and non-depressed women about intimacy with boyfriend/husband Data on next slide Everitt, B.S., & Smith, A.M.R. (1979) Interactions in contingency tables: A brief discussion of alternative methods. Psychological Medicine, 9, 581-583.

Chapter 19 Chi-Square Data

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

Chi-Square on Contingency Table Chapter 19 Chi-Square Chi-Square on Contingency Table Same formula Expected frequencies E = RT X CT GT RT = Row total, CT = Column total, GT = Grand total

Expected Frequencies E11 = 37*138/419 = 12.19 E12 = 37*281/419 = 24.81 Chapter 19 Chi-Square 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

Observed and Expected Freq. Chapter 19 Chi-Square Observed and Expected Freq.

Chi-Square Calculation Chapter 19 Chi-Square Chi-Square Calculation

Degrees of Freedom For contingency table, df = (R - 1)(C - 1) Chapter 19 Chi-Square 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.

Conclusions Since 25.61 > 3.84, reject H0 Chapter 19 Chi-Square 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.

Larger Contingency Tables Chapter 19 Chi-Square 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

Chapter 19 Chi-Square

Chi-Square Calculation Chapter 19 Chi-Square Chi-Square Calculation

Conclusions 29.62 > 7.82 Reject H0 Chapter 19 Chi-Square 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.

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

Chapter 19 Chi-Square Tests on Proportions Proportions can be converted to frequencies, and tested using c2. Use a z test directly on the proportions if you have two proportions From last example 10% of nonabused children abused as adults 40% of abused children abused as adults Cont.

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

Proportions--cont. z = 5.17 This is a standard z score. Chapter 19 Chi-Square Proportions--cont. z = 5.17 This is a standard z score. Therefore .05 (2-tailed) cutoff = +1.96 Reject null hypothesis that the population proportions of abuse in both groups are equal. This is just the square root of the c 2 you would have with c 2 on those 4 cells.

Nonindependent Observations Chapter 19 Chi-Square 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.

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

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

Review Questions What are categorical data? Chapter 19 Chi-Square Review Questions What are categorical data? What is the difference between a goodness-of-fit test and a test on contingency table? What are expected frequencies? What is H0 in goodness-of-fit test? Cont.

Review Questions--cont. Chapter 19 Chi-Square Review Questions--cont. What is H0 in test on contingency tables? What is difference between independence of variables and independence of observations? Why are small expected frequencies a problem? Does the z test on proportions tell you something that c2 does not?