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: H 0 : Pop. Dist. 1 = Pop. Dist. 2 H 1 : 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 U 1 and U 2.  Look up U in the U table.

ObservationsRanks TV FavorableTV UnfavorableTV FavorableTV Unfavorable R 1 = 72R 2 = 48

Calculating U U = whichever is smaller – U 1 or U 2 = 20

Testing U  H 0 : Population distribution 1 = population distribution 2 H 1 : 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: H 0 : Pop Dist 1 ≥ Pop Dist 2 H 1 : 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: H 0 : Pop A = Pop B = Pop C H 1 : H 0 is false.  Convert data to ranks and then use the H formula.  With n > 4, look up in  2 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,  , 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 variables.