1. Chapter 15 The Chi-Square Statistic: Tests for Goodness of Fit and Independence
PowerPoint Lecture Slides Essentials of Statistics for the Behavioral Sciences Eighth Edition by Frederick J. Gravetter and Larry B. Wallnau

2. Chapter 15 Learning Outcomes
Explain when chi-square test is appropriate 2
Test hypothesis about shape of distribution using chi-square goodness of fit 3
Test hypothesis about relationship of variables using chi-square test of independence 4
Evaluate effect size using phi coefficient or Cramer’s V

3. Tools You Will Need Proportions (math review, Appendix A)
Frequency distributions (Chapter 2)

4. 15.1 Parametric and Nonparametric Statistical Tests
Hypothesis tests used thus far tested hypotheses about population parameters
Parametric tests share several assumptions
Normal distribution in the population
Homogeneity of variance in the population
Numerical score for each individual
Nonparametric tests are needed if research situation does not meet all these assumptions

5. Chi-Square and Other Nonparametric Tests
Do not state the hypotheses in terms of a specific population parameter
Make few assumptions about the population distribution (“distribution-free” tests)
Participants usually classified into categories
Nominal or ordinal scales are used
Nonparametric test data may be frequencies

6. Circumstances Leading to Use of Nonparametric Tests
Sometimes it is not easy or possible to obtain interval or ratio scale measurements
Scores that violate parametric test assumptions
High variance in the original scores
Undetermined or infinite scores cannot be measured—but can be categorized

7. 15.2 Chi-Square Test for “Goodness of Fit”
Uses sample data to test hypotheses about the shape or proportions of a population distribution
Tests the fit of the proportions in the obtained sample with the hypothesized proportions of the population

8. Figure 15.1 Grade Distribution for a Sample of n = 40 Students
FIGURE A distribution of grades for a sample of n = 40 students. The same frequency distribution is shown as a bar graph, as a table, and with the frequencies written in a series of boxes.

9. Goodness of Fit Null Hypothesis
Specifies the proportion (or percentage) of the population in each category
Rationale for null hypotheses:
No preference (equal proportions) among categories, OR
No difference in specified population from the proportions in another known population

10. Goodness of Fit Alternate Hypothesis
Could be as terse as “Not H0”
Often equivalent to “…population proportions are not equal to the values specified in the null hypothesis…”

11. Goodness of Fit Test Data
Individuals are classified (counted) in each category, e.g., grades; exercise frequency; etc.
Observed Frequency is tabulated for each measurement category (classification)
Each individual is counted in one and only one category (classification)

12. Expected Frequencies in the Goodness of Fit Test
Goodness of Fit test compares the Observed Frequencies from the data with the Expected Frequencies predicted by null hypothesis
Construct Expected Frequencies that are in perfect agreement with the null hypothesis
Expected Frequency is the frequency value that is predicted from H0 and the sample size; it represents an idealized sample distribution

13. Chi-Square Statistic Notation Chi-Square Statistic
χ2 is the lower-case Greek letter Chi
fo is the Observed Frequency
fe is the Expected Frequency
Chi-Square Statistic

14. Chi-Square Distribution
Null hypothesis should
Not be rejected when the discrepancy between the Observed and Expected values is small
Rejected when the discrepancy between the Observed and Expected values is large
Chi-Square distribution includes values for all possible random samples when H0 is true
All chi-square values ≥ 0.
When H0 is true, sample χ2 values should be small

15. Chi-Square Degrees of Freedom
Chi-square distribution is positively skewed
Chi-square is a family of distributions
Distributions determined by degrees of freedom
Slightly different shape for each value of df
Degrees of freedom for Goodness of Fit Test
df = C – 1
C is the number of categories

16. Figure 15.2 The Chi-Square Distribution Critical Region
FIGURE Chi-square distributions are positively skewed. The critical region is placed in the extreme tail, which reflects large chi-square values.

17. Figure 15.3 Chi-square Distributions with Different df
FIGURE The shape of the chi-square distribution for different values of df. As the number of categories increases, the peak (mode) of the distribution has a larger chi-square value.

18. Locating the Chi-Square Distribution Critical Region
Determine significance level (alpha)
Locate critical value of chi-square in a table of critical values according to
Value for degrees of freedom (df)
Significance level chosen

19. Figure 15.4 Critical Region for Example 15.1
FIGURE For Example 15.1 , the critical region begins at a chi-square value of 7.81.

20. In the Literature
Report should describe whether there were significant differences between category preferences
Report should include
χ2 df, sample size (n) and test statistic value
Significance level
E.g., χ2 (3, n = 50) = 8.08, p < .05

21. “Goodness of Fit” Test and the One Sample t Test
Nonparametric versus parametric test
Both tests use data from one sample to test a hypothesis about a single population
Level of measurement determines test:
Numerical scores (interval / ratio scale) make it appropriate to compute a mean and use a t test
Classification in non-numerical categories (ordinal or nominal scale) make it appropriate to compute proportions or percentages to do a chi-square test

22. Learning Check
The expected frequencies in a chi-square test _____________________________________ A
are always whole numbers B
can contain fractions or decimal values C
can contain both positive and negative values D
can contain fractions and negative numbers

23. Learning Check - Answer
The expected frequencies in a chi-square test _____________________________________ A
are always whole numbers B
can contain fractions or decimal values C
can contain both positive and negative values D
can contain fractions and negative numbers

24. Learning Check
Decide if each of the following statements is True or False
T/F
In a Chi-Square Test, the Observed Frequencies are always whole numbers
A large value for Chi-square will tend to retain the null hypothesis

25. Learning Check - Answers
True
Observed frequencies are just frequency counts, so there can be no fractional values
False
Large values of chi-square indicate observed frequencies differ a lot from null hypothesis predictions

26. 15.3 Chi-Square Test for Independence
Chi-Square Statistic can test for evidence of a relationship between two variables
Each individual jointly classified on each variable
Counts are presented in the cells of a matrix
Design may be experimental or non-experimental
Frequency data from a sample is used to test the evidence of a relationship between the two variables in the population using a two-dimensional frequency distribution matrix

27. Null Hypothesis for Test of Independence
Null hypothesis: the two variables are independent (no relationship exists)
Two versions
Single population: No relationship between two variables in this population.
Two separate populations: No difference between distribution of variable in the two populations
Variables are independent if there is no consistent predictable relationship
Note 1: under the second version of the null hypothesis, the null hypothesis does NOT say that the two distributions are identical; instead it says they have the same proportions.
Note 2: stating that there is no relationship between variables (version 1) is equivalent to stating that the distributions have equal proportions.

28. Observed and Expected Frequencies
Frequencies in the sample are the Observed frequencies for the test
Expected frequencies are based on the null hypothesis prediction of the same proportions in each category (population)
Expected frequency of any cell is jointly determined by its column proportion and its row proportion

29. Computing Expected Frequencies
Frequencies computed by same method for each cell in the frequency distribution table
fc is frequency total for the column
fr is frequency total for the row

30. Computing Chi-Square Statistic for Test of Independence
Same equation as the Chi-Square Test of Goodness of Fit
Chi-Square Statistic
Degrees of freedom (df) = (R-1)(C-1)
R is the number of rows
C is the number of columns

31. Chi-Square Compared to Other Statistical Procedures
Hypotheses may be stated in terms of relationships between variables (version 1) or differences between groups (version 2)
Chi-square test for independence and Pearson correlation both evaluate the relationships between two variables
Depending on the level of measurement, Chi-square, t test or ANOVA could be used to evaluate differences between various groups

32. Chi-Square Compared to Other Statistical Procedures (cont.)
Choice of statistical procedure determined primarily by the level of measurement
Could choose to test the significance of the relationship
Chi-square
t test
ANOVA
Could choose to evaluate the strength of the relationship with r2

33. 15.4 Measuring Effect Size for Chi-Square
A significant Chi-square hypothesis test shows that the difference did not occur by chance
Does not indicate the size of the effect
For a 2x2 matrix, the phi coefficient (Φ) measures the strength of the relationship
So Φ2 would provide proportion of variance accounted for just like r2

34. Effect size in a larger matrix
For a larger matrix, a modification of the phi-coefficient is used: Cramer’s V
df* is the smaller of (R-1) or (C-1)

35. Interpreting Cramer’s V
Small Effect
Medium Effect
Large Effect
For df* = 1
0.10
0.30
0.50
For df* = 2
0.07
0.21
0.35
For df* = 3
0.06
0.17
0.29

36. 15.5 Chi-Square Test Assumptions and Restrictions
Independence of observations
Each observed frequency is generated by a different individual
Size of expected frequencies
Chi-square test should not be performed when the expected frequency of any cell is less than 5

37. Learning Check
A basic assumption for a chi-square hypothesis test is ______________________ A
the population distribution(s) must be normal B
the scores must come from an interval or ratio scale C
the observations must be independent D
None of the other choices are assumptions for chi-square

38. Learning Check - Answer
A basic assumption for a chi-square hypothesis test is ______________________ A
the population distribution(s) must be normal B
the scores must come from an interval or ratio scale C
the observations must be independent D
None of the other choices are assumptions for chi-square

39. Learning Check
Decide if each of the following statements is True or False
T/F
The value of df for a chi-square test does not depend on the sample size (n)
A positive value for the chi-square statistic indicates a positive correlation between the two variables

40. Learning Check - Answers
True
The value of df for a chi-square test depends only on the number of rows and columns in the observation matrix
False
Chi-square cannot be a negative number, so it cannot accurately show the type of correlation between the two variables

41. Figure 15.5: Example 15.2 SPSS Output—Chi-square Test for Independence
FIGURE The SPSS output for the chi-square test for independence in Example 15.2

42. Any Questions?
Concepts?
Equations?