1. PSY 307 – Statistics for the Behavioral Sciences Chapter 19 – Chi-Square Test for Qualitative Data Chapter 21 – Deciding Which Test to Use

2. Chi-Square (  2 ) Test  For qualitative data  Tests whether observed frequencies are closely similar to hypothesized expected frequencies.  Expected frequencies can be probabilities determined by chance or other values based on theory.

3. Two Tests  One-way (one variable) chi-square: Tests observed frequencies against a null hypothesis of equal or specified proportions.  Two-way (two variable) chi-square: Tests observed frequencies against specified proportions across all cells of two cross-classified variables. Another way of saying this is that it tests for an interaction.

4. Frequencies  Observed frequencies – the obtained frequency for each category in a study.  Expected frequencies – the hypothesized frequency for each category given a true null hypothesis.

5. Calculating Chi-Square (  2 )  Determine the expected frequencies.  Are the differences between the expected and the observed frequencies large enough to qualify as a rare outcome?  Calculate the  2 ratio.  Compare against the  2 table with appropriate degrees of freedom.

6. Blood Type Example Blood Type FrequencyOABABTotal Observed (f o )38 204100 Expected (f e )4441105100 H 0 : P O =.44, P A =.41, P B =.10, P AB =.05 H 1 : H 0 is false

7. Calculating  2 df = categories (c) - 1

8. Chi-Square Distribution

9. Chi Square Table Look up the critical value for our df (c-1) and significance level (e.g., p <.05). Is 11.24 greater than 7.81? If yes, reject the null hypothesis. Conclude blood types are not distributed as in the general population. Reject H 0

10. About  2  Because differences from expected values are squared, the value of  2 cannot be negative.  Because differences are squared, the  2 test is nondirectional.  A significant  2 is not necessarily due to big differences, small ones can add up.

11. Two-Way  2  When observations are cross- classified according to two variables, a two-way test is used.  The two-way test examines the relationship between two variables. It is a test of independence between them.  Null hypothesis: independence.  Alternative hypothesis: H 0 is false.

12. Returned Letter Example Neighborhood Returned Letters DowntownSuburbiaCampusTotal Yes413247120 No19382380 Total6070 200 H 0 : Type of neighborhood and return rate of lost letters are independent. H 1 : H 0 is false.

13. Calculating Expected Frequencies Neighborhood Returned Letters DowntownSuburbiaCampusTotal Yes f o 413247120 f e 3642 No f o 19382380 f e 2428 Total6070 200

14. Calculating Two-Way  2  Expected frequencies are based on the proportions found in the column and row totals.  Degrees of freedom are limited by the column and row totals.  Once expected frequencies and df have been found, calculate  2 the same as in a one-way test.

15. Calculating  2 df = (columns – 1)(rows – 1) df = (3-1)(2-1) = 2 From the Chi Square Table, critical value is 5.99. Our value of 9.17 exceeds 5.99 so reject the null. There is a relationship between neighborhood and letter return rate.

16. Effect Size for  2  Cramer’s Phi Coefficient ( )  Roughly estimates the proportion of explained variance (predictability) between two qualitative variables..01 = small effect.09 = medium effect.25 = large effect where k is the smaller of the number of rows or columns

17. Precautions  Observations must be independent of each other. One observation per subject.  Avoid small expected frequencies – must be 5 or more.  Avoid small sample sizes – increases danger of Type II error (retaining a false null hypothesis).  Avoid very large sample sizes.

18. A Repertoire of Hypothesis Tests  z-test – for use with normal distributions when σ is known.  t-test – for use with one or two groups, when σ is unknown.  F-test (ANOVA) – for comparing means for multiple groups.  Chi-square test – for use with qualitative data.

19. Null and Alternative Hypotheses  How you write the null and alternative hypothesis varies with the design of the study – so does the type of statistic.  Which table you use to find the critical value depends on the test statistic (t, F,  , U, T, H).  t and z tests can be directional.

20. Deciding Which Test to Use  Is data qualitative or quantitative? If qualitative use Chi-square.  How many groups are there? If two, use t-tests, if more use ANOVA  Is the design within or between subjects?  How many independent variables (IVs or factors) are there?

21. Summary of t-tests  Single group t-test for one sample compared to a population mean.  Independent sample t-test – for comparing two groups in a between-subject design.  Paired (matched) sample t-test – for comparing two groups in a within-subject design.

22. Summary of ANOVA Tests  One-way ANOVA – for one IV, independent samples  Repeated Measures ANOVA – for one or more IVs where samples are repeated, matched or paired.  Two-way (factorial) ANOVA – for two or more IVs, independent samples.  Mixed ANOVA – for two or more IVs, between and within subjects.

23. Summary of Nonparametric Tests  Two samples, independent groups – Mann-Whitney (U). Like an independent sample t-test.  Two samples, paired, matched or repeated measures – Wilcoxon (T). Like a paired sample t-test.  Three or more samples, independent groups – Kruskal-Wallis (H). Like a one-way ANOVA.

24. Summary of Qualitative Tests  Chi Square (  2 ) – one variable. Tests whether frequencies are equally distributed across the possible categories.  Two-way Chi Square – two variables. Tests whether there is an interaction (relationship) between the two variables.