PSY 307 – Statistics for the Behavioral SciencesChapter 19 – Chi-Square Test for Qualitative DataChapter 21 – Deciding Which Test to UseChi-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.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.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.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 2ratio. Compare against the 2table with appropriate degrees of freedom.Blood Type ExampleBlood TypeFrequencyOABABTotalObserved (fo)3838204100Expected (fe)4441105100H0: PO= .44, PA= .41, PB= .10, PAB= .05H1: H0is falseeeofff22)(Calculating 2eeofff22)(24.1120.00.1022.82.511010041944365)1(10)10(41)3(44)6(5)54(10)1020(41)4138(44)4438(22222222df = categories (c) - 1Chi-Square DistributionChi Square TableLook 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 H0About 2 Because differences from expected values are squared, the value of 2cannot be negative. Because differences are squared, the 2 test is nondirectional. A significant 2is not necessarily due to big differences, small ones can add up.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: H0is false.Returned Letter ExampleNeighborhoodReturned LettersDowntownSuburbiaCampusTotalYes413247120No19382380Total607070200H0: Type of neighborhood and return rate of lost letters are independent.H1: H0is false.Calculating Expected FrequenciesNeighborhoodReturned LettersDowntownSuburbiaCampusTotalYes fo413247120fe364242No fo19382380fe242828Total607070200 ) )( (totalgrandtotalrowtotalcolumnfe362007200200)120)(60(ef422008400200)120)(70(efCalculating 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 2the same as in a one-way test.Calculating 2eeofff22)(17.989.057.304.1060.38.269.028)2823(28)2838(24)2419(42)4247(42)4232(36)3641(222222df = (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.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 effect2c)1(22kncwhere k is the smaller of the number of rows or columnsPrecautions 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.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.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.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?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.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.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.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