Nonparametric Tests 2

Mann-Whitney U test

*Mann-Whitney U test* - Nonparametric alternative to the independent-samples t-test (two Group comparison control group and experimental but one or more of assumptions are violated through a parametric test) 1) Rank the scores if they are not already ordinal 2) Use sample size and direction of test to look up test value in the *MWU table* 3) Separate ranks by groups, calculate sum of ranks, one for each group 4) Plug sample sizes and sums into formulas (Ch. 15) to find sample statistics, one for each group 5) Test smallest of these against critical value in table: You are looking for the test statistic to be *smaller than the cut-off*

Choosing a Rank Order Test

*Parametric Test*: (Nominal Data) 1) Dependent samples t-test 2) Independent samples t-test 3) One-way ANOVA 4) Pearson product moment correlation ( r ) *Nonparametric Counterpart* 1) Wilcoxon test 2) Mann-Whitney U test 3) Kruskal-Wallis H Test 4) Spearman rank-order correlation ( Spearman's rho) ( r s )

Important Aspects of Rank-Order Tests

-- For experimental designs, null hypothesis states that there is no difference between population medians (no means) -- Rank-Ordering creates a predictable arrangement of scores that does not require estimated population parameters (distribution free)

Reporting Results in APA Style

-- Report why the nonparametric method was chosen -- Report statistical results of test, with name of test statistic, degrees of freedom and the significance level ℗ -- State results and general interpretation in sentence form: See SPSS tutorials for examples.

Steps of Rank-Order Testing

1) Rank scores from lowest to highest regardless of group 2) Add ranks together in group that is expected to have the lower total 3) Compare to cut-off from table: if sum of ranks *is less than or equal* to the cut-off, REJECT THE NULL HYPOTHESIS

SPSS APA Results Section Outline for Whitney U Test

General Outline: State what kind of test, what variables include IV or grouping and dependent or test variables (IV) Formal APA statistical statement always includes z and p values.Mean rank of dependent variable for each level of IV (Mean VSMT rank for each age group), Brief interpretation in English of the results in context of the research problem: Significant or not? Mean ranks differed in which Direction? Is the null hypothesis rejected/fail to reject? See Graphic.

Nonparametric tests Ordinal Data

Involve using: -- Rank-ordered values of the data to arrange scores -- Can be used with research designs that are otherwise suited to familiar parametric tests

Parametric Tests

Nonparametric tests Ordinal Data. To use this test: data at scale level, normally distributed. Parametric tests will not be accurate or reliable or any hypothesis tests.

SPSS Test Results Mann Whitney U Test

Report Z Statistic and p value in results section from Test Statistics with Mean Ranking. *z= -1.97, p = .-48* The results of the Manning Whitney U Test indicate that (example) that younger and older women perform differently on average on the VSMT.

SPSS Mann-Whitney U Test: Effect Size Statistics

SPSS does not report an effect size index for this test, but simple indices can be computed to communicate the size of the effect. For example, difference in mean ranks or medians between the two groups can serve as an effect size index.

Spearman rank-order correlation

Spearman's rho -- Cases/participants have scores on 2 variables -- Data is ordinal or is transformed to ordinal level (ranks) -- Interpreted similar to Pearson's correlation coefficient (mod. 5) Computing r s: -- Find a difference between ranks for each participant (rank var1 - rank var 2) -- Square the difference -- Fill values into formula (Ch. 15) for final answer

What if assumptions are not met?

TWO APPROACHES: 1. Data transformation 2. Nonparametric Tests

SPSS Mann-Whitney U test evaluations

To understand how the Mann-Whitney U test evaluates differences in medians, it is useful to first describe what data are analyzed with this test Because analyses for the Mann-Whitney U test are conducted on ranked scores, the distributions of the test variable for the two populations do not have to be of any particular form (e.g., normal). However, these distributions should be continuous and have identical forms.

SPSS APA Written for Whitney U Test

Written Example: A Mann-Whitney U test was conducted to determine whether older or younger women differ in their performance on a visa-spatial memory task (VSMT). The results indicate a significant difference between the mean ranks, z= -.197, p= .048. The average rank of younger women was significantly higher (M= 23.17) than the average rank of older women (M= 15.54). Thus we reject the null hypothesis of no difference between the age groups: younger women scored higher on average than older women on the VASMT.

SPSS Mann-Whitney U Test

A Mann-Whitney U test can analyze data from different types of studies. . Experimental studies Quasi-experimental studies Field studies The Mann-Whitney U test evaluates whether the medians on a test variable differ significantly between two groups. In an SPSS data file, each case must have scores on two variables, the grouping variable and the test variable. The grouping variable divides cases into two groups or categories, and the test variable assesses individuals on a variable with at least an ordinal scale. Although SPSS uses the terms grouping variable and test variable, the grouping variable may also be referred to as the independent or categorical variable, and the test variable may be referred to as the dependent or the quantitative variable.

SPSS Mann-Whitney U test Assumptions

Assumption 1: The Continuous Distributions for the Test Variable Are Exactly the Same (Except Their Medians) for the Two Populations If population distributions differ on characteristics other than their medians, the Mann-Whitney U test does not evaluate whether the medians differ between populations, but whether the distributions themselves differ. In addition, difficulties can arise with the Mann-Whitney U test if there are not a variety of scores on the dependent variable and ties are produced in the ranked scores. SPSS employs a correction for such ties. *Assumption 2*: The Cases Represent Random Samples from the Two Populations, and the Scores on the Test Variable Are Independent of Each Other The Mann-Whitney U test will yield inaccurate p values if the independence assumption is violated. *Assumption 3*: The z-Approximation Test for the Mann-Whitney U Test Requires a Large Sample Size An exact test is computed by SPSS if the number of cases is relatively small: 40 cases or fewer if the number of cases per group is equal and somewhat greater than 40 to the extent that the number of cases per group differs. If an exact test is not reported, the sample size is probably sufficiently large for the z-approximation test to yield relatively accurate p values. The z-approximation test includes a correction for ties but does not include a continuity correction. The results for the Mann-Whitney U test obtained from SPSS may differ from results in other statistical computer packages depending on whether the software uses the same correction procedures.

SPSS Mann-Whitney Graphs to Display the Results

BOXPLOT Because the computations for the Mann-Whitney U test are on ranked scores of a test variable, a reader of a Results section reporting this test would have very little knowledge of what the original, raw scores would look like. Consequently, a graph presenting distributions of raw scores for the two groups would be very informative. Boxplots offer the reader more information than the other graphical options within SPSS. We present a boxplot in Figure 258 showing the distributions for the visual spatial memory scores for the two age groups.

SPSS Whitney U Test Graph

Boxplot: Graphs>>Legacy Dialogs>> Boxplot>>Define Simple>> Summaries of groups of cases DV goes into variable box, age goes to the category box Final box plot each box represents age group and the scores are on the Vertical.

SPSS Data Entry Mann Whitney U Test

Create grouping (independent) Variable and Dependent Variable Columns. Remember to define values of grouping variable under Variable View - Values. Example: age: Grouping Variable (IV) Second Column Raw Scores (DV) Analysis Step 1 Nonparametric Tests>>Legacy Dialogs>> Independent Samples The independent samples option will lead to a dialogue box with several choices of test Analysis Step 2 Select Mann-Whitney U Test - Define groups before continuing

SPSS APA Results for Mann-Whitney

EXAMPLE : A Mann-Whitney U test was conducted to evaluate the hypothesis that older women would score lower, on the average, than younger women on a visual spatial memory task. The results of the test were in the expected direction and significant, z = −1.97, p < .05. Older women had an average rank of 15.54, whereas younger women had an average rank of 23.17. Figure 258 shows the distributions of the scores on the VSMT measure for the two age groups. Alternative Analyses: If a study has two independent samples, and individuals in the samples are assessed on a dependent variable measured on an ordinal scale, then the data from the study should be analyzed with a Mann-Whitney U test. However, if the dependent variable is measured on an interval scale (or ratio scale), then the data from the study can be analyzed with a Mann-Whitney U test or an independent-samples t test. The decision about which test to use with an interval-scaled dependent variable is complex and depends on whether the assumptions underlying these tests are met. If the normality assumption for the independent-samples t test is met, it is the more powerful than, and preferable to, the Mann-Whitney U test. The independent-samples t test yields relatively accurate p values under some conditions when the normality assumption is violated (see Lesson 24), but may not be as powerful as the Mann-Whitney U test for a number of different types of nonnormal distributions, including those with thick tails. See Wilcox (2001) for more details about choices between these two tests and other methods.

SPSS Conducting a Mann-Whitney U Test

To conduct a Mann-Whitney U Test, follow these steps: Click Analyze, click Nonparametric Tests, click Legacy Dialogs, and then click 2 Independent Samples. You will see the Two-Independent-Samples Tests dialog box as shown in Figure 256. Click vsmt, then click to move it to the Test Variable List box. Click age, then click to move it to the Grouping Variable box. Click Define Groups. Type 1 in the Group 1 box to indicate that Group 1 is the younger group. Type 2 in the Group 2 box indicating that Group 2 is the older group. Click Continue. Be sure that Mann-Whitney U is checked in the Test Type area. Click OK.