PSY 307 – Statistics for the Behavioral SciencesChapter 20 – Tests for Ranked Data, Choosing Statistical TestsWhat To Do with Non-normal Distributions Tranformations (pg 382): The shape of the distribution can be changed by applying a math operation to all observations in the data set. Square roots, logs, normalization (standardization). Rank order tests (pg 387): Use a nonparametric statistic that has different assumptions about the shape of the underlying distribution.Pros and Cons Tranformations must be described in the Results section of your manuscript. Effects of transformations on the validity of your t or F statistical tests is unclear. Nonparametric tests may be preferable but make probability of Type II error greater.Nonparametric Tests A parameter is any descriptive measure of a population, such as a mean. Nonparametric tests make no assumptions about the form of the underlying distribution. Nonparametric tests are less sensitive and thus more susceptible to Type II error.When to Use Nonparametric Tests When the distribution is known to be non-normal. When a small sample (n < 10) contains extreme values. When two or more small samples have unequal variances. When the original data consists of ranks instead of values.Mann-Whitney Test (U Test) The nonparametric equivalent of the independent group t-test. Hypotheses: H0: Pop. Dist. 1 = Pop. Dist. 2 H1: Pop. Dist. 1 ≠ Pop. Dist. 2 The nature of the inequality is unspecified (e.g., central tendency, variability, shape).Calculating the U-Test Convert data in both samples to ranks. With ties, rank all values then give all equal values the mean rank. Add the ranks for the two groups. Substitute into the formula for U. U is the smaller of U1and U2. Look up U in the U table.ObservationsRanksTV FavorableTV UnfavorableTV FavorableTV Unfavorable01.501.5132445575757109121014112012421343144915R1= 72R2= 48Calculating U20723656722)18(8)7)(8(2)1(111211RnnnnU36482856482)17(7)7)(8(2)1(222212RnnnnUU = whichever is smaller – U1or U2= 20Testing U H0: Population distribution 1 = population distribution 2H1: Population distribution 1 ≠ population distribution 2 Look up critical values in U Table. Instead of degrees of freedom, use n’s for the two groups to find the cutoff. Since 20 is larger than 10, retainthe null (not reject).Interpretation of U U represents the number of times individual ranks in the lower group exceed those in the higher group. When all values in one group exceed those in the other, U will be 0. Reject the null (equal groups) when U is less than the critical U in the table.Directional U-Test Similar variance is required in order to do a directional U-test. The directional hypothesis states which group will exceed which: H0: Pop Dist 1 ≥ Pop Dist 2 H1: Pop Dist 1 < Pop Dist 2 In addition to calculating U, verify that the differences in mean ranks are in the predicted direction.Wilcoxon T Test Equivalent to paired-sample t-test but used with non-normal distributions and ranked data. Compute difference scores. Rank order the difference scores. Put plus ranks in one group, minus ranks in the other. Sum the ranks. Smallest value is T. Look up in T table. Reject null if < than critical T.Kruskal-Wallis H Test Equivalent to one-way ANOVA for ranked data or non-normal distributions. Hypotheses: H0: Pop A = Pop B = Pop C H1: H0is false. Convert data to ranks and then use the H formula. With n > 4, look up in 2table.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