Chapter 14Nonparametric Tests

Part III: Additional Hypothesis Tests

Renee R. Ha, Ph.D.James C. Ha, Ph.DIntegrative Statistics for the Social & Behavioral Sciences

Chi-square Test

The chi-square test is appropriate when you have nominal independent variable(s) but only group membership (frequencies) as your raw data.

Table 14.1

Democrat Green Independent Republican

40 20 12 40

Chi-square Test

Chi-square Test

Calculation of expected valuesIf the expected frequencies are based purely on chance (random model), then you can calculate them based on how many categories you have in your analysis.

E.g.: in political parties example, the probability of being in each category simply due to chance would be ¼ or 25%.

Chi-square Test

Calculation of expected valuesSecond form of expected frequencies, called a “goodness-of-fit” model.

Calculate expected frequencies based on some prior knowledge of what those values should be that are not necessarily random (the goodness-of-fit model).

Table 14.2

• Chi-square Test

• Expected frequencies of tongue curling based on Mendelian genetics

Can Curl Tongue Cannot Curl Tongue

75% 25%

Chi-square Test

Calculation of the Chi-square (χ2) Test fe = expected frequency with random sampling from the null hypothesis population fo = observed frequency in the sample k = number of cells (or categories)

χ2obtained =

kcell

cell e

eo

f

ff

1

2)(

Chi-square Test

Table of critical values for χ2 (Table G back of text book).

Critical values based on the degrees of freedom (k – 1) and α, where k is the number of categories.

Evaluations of χ2 obtained are always nondirectional.

 

Results of Chi-square test using SPSS

Npar TestsChi-square TestParty

Test Statistics

a. 0 cells (.0%) have expected frequencies less than 5. The minimum expected cell frequency is 28.0

Test of Independence Between Two Variables:

Contingency Table Analysis

Null hypothesis (Ho): Factor A is not related to Factor B.

Alternative hypothesis (HA): Factor A and Factor B are related.

Test of Independence Between Two Variables:

Contingency Table Analysis

Equation for contingency table analysis is the same one as chi-square, except that the fe and the df are calculated differently for this analysis. To calculate fe:

For df: (r-1)(c-1)

where r is # rows and c is # columns

(Row marginal for that cell)(Column marginal for that cell)

Total N

Assumptions for Chi-square and Contingency Table

1. Each subject has only one entry, and the categories are mutually exclusive.

2. In chi-square designs that are larger than 2 × 2, the expected frequency in each cell must be at least five.

Other Non-parametric Tests

Table 14-3 - Non-parametric equivalents of popular parametric tests

Parametric Test Equivalent Nonparametric Test

Correlated (paired) t-test Wilcoxon

Independent t-test Mann-Whitney U

One-Way ANOVA Kruskal-Wallis

Wilcoxon Signed Rank Test (T)

Calculation Steps

1. Calculate the differences between the two conditions (e.g., before vs. after). This is your difference score.

2. Rank the absolute values of the difference scores from the smallest to the largest. If there are ties on the ranked scores, then you should take an average rank, but do not reuse ranks.

Wilcoxon Signed Rank Test (T)

Calculation Steps3. Assign the sign of the original difference score to the rank.

4. Sum all of the positive ranks together and then separately sum the negative ranks. Calculate the absolute value of the summed ranks for each group.

Wilcoxon Signed Rank Test (T)

Calculation Steps5. The smaller of the summed ranks becomes the T-obtained value.

6. Evaluate with the Wilcoxon table.Note: Reject if Tobt < Tcrit .

Results if you use SPSS to calculate the Wilcoxon

Npar TestsWilcoxon Signed Ranks TestRanks

a. POSTTEST<PRETESTb. POSTTEST>PRETESTc. PRETEST=POSTTEST

Test Statistics

a. Based on negative ranksb. Wilcoxon Signed Ranks Test

Mann-Whitney U test

The Mann-Whitney U test is appropriate when you have an independent (or between-groups) design with two groups (e.g., experimental vs. control groups).

This test is designed to evaluate the separation between the groups.

Mann-Whitney U test

Calculation Steps1. Place scores in rank order and rank scores from both samples regardless of experimental condition. Handle tie scores by assigning the average rank as you did with the Wilcoxon test.

2. Sum the ranks separately by experimental group (R1 and R2).

Mann-Whitney U test

Calculation Steps3. Calculate two U-obtained values by using the following equations:

U obtained = 21nn + 111

2

)1(R

nn

U obtained = 21nn + 222

2

)1(R

nn

The smaller value becomes the true U-obtained value while the larger value is referred to as the U′ (prime) obtained value.

Mann-Whitney U test

Calculation Steps4. Evaluate the U obtained value using the Mann-Whitney table and reject the null hypothesis if Uobt < Ucrit.

Assumptions: Mann-Whitney U test

1. The dependent variable must be measured in at least the ordinal scale, but interval or ratio data can be “collapsed” into ordinal data via the ranking.

2. Only appropriate when you have an independent or between groups design and two groups/conditions.

3. Random sampling is required.

Results if you use SPSS to calculate the Mann-Whitney U test

Npar TestsMann-Whitney U testRanks

Test Statistics

a. Not corrected for tiesb. Grouping Variable: GROUP

Kruskal-Wallis Test

The Kruskal-Wallis test is used when data are in three or more groups and is ordinal, and it replaces the one-way (independent) ANOVA when either or both of the assumptions are broken.

Kruskal-Wallis Test

Calculation Steps1. Rank all scores from the smallest to the largest.

2. Sum ranks for each group (Ri).

Kruskal-Wallis Test

Calculation Steps3. Calculate the H-obtained value using the following formula:

Hobt =

)1(

12

NN

i

i

n

R 2)(- 3(N + 1)

Kruskal-Wallis Test

Calculation Steps4. Evaluate using chi-square table, df = k – 1, where k = # of groups, and reject the null hypothesis if and only if Hobt ≥ Hcrit.

5. If the null hypothesis is rejected, pairwise combinations of Mann-Whitney U tests should be used to determine the post hoc pattern of differences among the groups.

Results if you use SPSS to calculate the Kruskal Wallace test

Npar TestsKruskal-Wallace testRanks

Test Statistics

a. Kruskal Wallace testb. Grouping Variable: CLASS

Assumptions for the Kruskal-Wallis Test

1. The dependent variable (data) are measured on an ordinal or better scale.

2. The scores come from the same underlying distribution.

3. You must have at least five scores per group to use the chi-square table.

4. Random sampling

Spearman’s Rank Order Correlation

The Spearman rank correlation test is appropriate when one or both of the variables of interest are on an ordinal scale.

This test replaces Pearson’s correlation test when the data are not collected on an interval or ratio scale but are at least ordinal.

Spearman’s Rank Order Correlation

Calculation Steps1. Independently rank each variable X and Y (strongest response = 1).

2. Give tied ranks an average ranking score, but then don’t reuse the ranks that you averaged.

3. Calculate the “difference in rank” scores and then square those differences.

Spearman’s Rank Order Correlation

Calculation Steps4. Calculate rho (rs) using the following formula:

5. Evaluate rs using Table I in the Appendix.

rs = 1 - NN

Di

3

2 )(6

Spearman’s Rank Order Correlation

Results of the rs correlation if you use SPSS

Correlations

Final Flowchart on Choosing the Appropriate Test