1. 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 

2. 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.

3. 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

4. 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

5. 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.

6. 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)

7. 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,

8. Chapter 19 Chi-Square
Fombonne’s Data

9. 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.

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

11. Chapter 19 Chi-Square

12. 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 > 14.58, do not reject H0
Conclude that birth month distribution of children with psychoses doesn’t differ from normal.

13. 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?

14. 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

15. 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,

16. Chapter 19 Chi-Square
Data

17. 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

18. 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

19. 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 =
E22 = 382*281/419 =
Enter on following table

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

21. Chi-Square Calculation
Chapter 19 Chi-Square
Chi-Square Calculation

22. 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.

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

24. 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

25. Chapter 19 Chi-Square

26. Chi-Square Calculation
Chapter 19 Chi-Square
Chi-Square Calculation

27. 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.

28. 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.

29. 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.

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

31. 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.

32. 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.

33. 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.

34. 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.

35. 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.

36. 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?