PSY 307 – Statistics for the Behavioral Sciences Chapter 20 – Tests for Ranked Data, Choosing Statistical Tests
What 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 U1 and U2. • Look up U in the U table.
Calculating U U = whichever is smaller – U1 or U2 = 20
Testing U • H0: Population distribution 1 = population distribution 2 H1: 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, retain the 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: H0 is false. • Convert data to ranks and then use the H formula. • With n > 4, look up in c2 table.
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, c2, 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 (c2) – 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.