PSY 307 – Statistics for the Behavioral SciencesChi-Square (c2) TestTwo TestsFrequenciesCalculating Chi-Square (c2)Blood Type ExampleCalculating c2Chi-Square DistributionChi Square TableAbout c2Two-Way c2Returned Letter ExampleCalculating Expected FrequenciesCalculating Two-Way c2Slide 15Effect Size for c2PrecautionsA Repertoire of Hypothesis TestsNull and Alternative HypothesesDeciding Which Test to UseSummary of t-testsSummary of ANOVA TestsSummary of Nonparametric TestsSummary of Qualitative TestsPSY 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 2 ratio.Compare against the 2 table with appropriate degrees of freedom.Blood Type ExampleBlood TypeFrequency O A B AB TotalObserved (fo) 38 38 20 4 100Expected (fe) 44 41 10 5 100H0: PO = .44, PA = .41, PB = .10, PAB = .05H1: H0 is falseeeofff22)(Calculating 2eeofff22)(24.1120.00.1022.82.511010041944365)1(10)10(41)3(44)6(5)54(10)1020(41)4138(44)4438(22222222df = 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 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.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: H0 is false.Returned Letter ExampleNeighborhoodReturned LettersDowntown Suburbia Campus TotalYes 41 32 47 120No 19 38 23 80Total 60 70 70 200H0: Type of neighborhood and return rate of lost letters are independent.H1: H0 is false.Calculating Expected FrequenciesNeighborhoodReturned LettersDowntown Suburbia Campus TotalYes fo41 32 47 120 fe36 42 42No fo19 38 23 80 fe24 28 28Total 60 70 70 200 ) )( (totalgrandtotalrowtotalcolumnfe362007200200)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 2 the same as in a one-way test.Calculating 2eeofff22)(17.989.057.304.1060.38.269.028)2823(28)2838(24)2419(42)4247(42)4232(36)3641(222222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.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(22kncwhere 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, 2, 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)