Entry
Reader's guide
Entries A-Z
Subject index
Wilcoxon Rank Sum Test
The Wilcoxon rank sum test is a nonparametric test that may be used to assess whether the distributions of observations obtained between two separate groups on a dependent variable are systematically different from one another. Developed by Frank Wilcoxon in 1945, the test replaces the obtained values of a dependent variable with a rank score, and inferentially tests whether the sum of the ranks within each group is likely to be obtained by chance from the same population. As the test does not use group means and standard deviations as estimations of population parameters, it is not beholden to the assumption of normality. It is therefore considered an excellent nonparametric alternative to the independent t test in cases where populations might not be normally distributed and the sample sizes of each group are small. Although the calculation of the Wilcoxon rank sum statistic is different from that employed by the Mann-Whitney U test, the two tests are fundamentally similar and linearly related, such that many texts refer to these rank-based statistics in unison as the Wilcoxon −Mann −Whitney (or Mann-Whitney −Wilcoxon) test for independent groups.
Rationale and Calculation
Much as in parametric testing, the Wilcoxon rank sum test makes its inferences by determining the probability that two independently obtained groups are sampled from the same population. If two groups are sampled from a population that does not vary across the levels of the independent variable, then there should be an equal probability that any score obtained on the dependent variable will fall into either experimental group. If group identity is conserved, and the obtained values are ranked in order from smallest to largest, each rank should have an equal likelihood of being in either of the experimental groups.
Take, for example, the distribution of a hypothetical data set of diastolic blood pressure measures from two groups of patients, one receiving a drug and another a placebo. If there were no effect of the drug treatment on blood pressure, the obtained measures might look as follows:
Blood pressure—drug (Group A): 70, 100, 78, 55, 67
Blood pressure—placebo (Group B): 80, 60, 65, 90, 75
To rank these scores, all observations are placed in ascending order, and group identification is maintained, as shown in Table 1.
Descriptively, the values obtained from each group are randomly distributed among the ten ranks, providing little indication that the drug has an effect. If, on the other hand, the treatment had a favorable effect on blood pressure (causing it to be reduced in a group of hypertensive patients), the data might look like the following:
Blood pressure—drug (Group A): 55, 60, 65, 67, 75
Blood pressure—placebo (Group B): 70, 78, 80, 90, 100
Again, the groups are combined and the data rank-ordered (see Table 2).
| Table 1 Wilcoxon Rank Sum Test Example #1 | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Rank: | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Score (BP): | 55 | 60 | 65 | 67 | 70 | 75 | 78 | 80 | 90 | 100 |
| Group: | A | B | B | A | A | B | A | B | B | A |
| Table 2 Wilcoxon Rank Sum Test Example #2 | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Rank: | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Score (BP): | 55 | 60 | 65 | 67 | 70 | 75 | 78 | 80 | 90 | 100 |
| Group: | A | A | A | A | B | A | B | B | B | B |
In this case, the distribution of scores from Group A (the drug treatment) appears shifted to the left of the distribution of the scores from Group B (the placebo group). The Wilcoxon rank sum test determines the probability at which the assigned rankings for two groups would occur because of random sampling from the same population. It does this by summing all the rank scores in one (or both) experimental groups and by determining whether this sum of the ranks is likely to have been obtained from the same population by chance. As in parametric tests, if this probability is sufficiently low, then the null hypothesis is rejected.
...
- Descriptive Statistics
- Distributions
- Graphical Displays of Data
- Hypothesis Testing
- Alternative Hypotheses
- Beta
- Critical Value
- Decision Rule
- Hypothesis
- Nondirectional Hypotheses
- Nonsignificance
- Null Hypothesis
- One-Tailed Test
- p Value
- Power
- Power Analysis
- Significance Level, Concept of
- Significance Level, Interpretation and Construction
- Significance, Statistical
- Two-Tailed Test
- Type I Error
- Type II Error
- Type III Error
- Important Publications
- “Coefficient Alpha and the Internal Structure of Tests”
- “Convergent and Discriminant Validation by the Multitrait-Multimethod Matrix”
- “Meta-Analysis of Psychotherapy Outcome Studies”
- “On the Theory of Scales of Measurement”
- “Probable Error of a Mean, The”
- “Psychometric Experiments”
- “Sequential Tests of Statistical Hypotheses”
- “Technique for the Measurement of Attitudes, A”
- “Validity”
- Aptitudes and Instructional Methods
- Doctrine of Chances, The
- Logic of Scientific Discovery, The
- Nonparametric Statistics for the Behavioral Sciences
- Probabilistic Models for Some Intelligence and Attainment Tests
- Statistical Power Analysis for the Behavioral Sciences
- Teoria Statistica Delle Classi e Calcolo Delle Probabilità
- Inferential Statistics
- Association, Measures of
- Coefficient of Concordance
- Coefficient of Variation
- Coefficients of Correlation, Alienation, and Determination
- Confidence Intervals
- Margin of Error
- Nonparametric Statistics
- Odds Ratio
- Parameters
- Parametric Statistics
- Partial Correlation
- Pearson Product-Moment Correlation Coefficient
- Polychoric Correlation Coefficient
- Q-Statistic
- R2
- Randomization Tests
- Regression Coefficient
- Semipartial Correlation Coefficient
- Spearman Rank Order Correlation
- Standard Error of Estimate
- Standard Error of the Mean
- Student's t Test
- Unbiased Estimator
- Weights
- Item Response Theory
- Mathematical Concepts
- Measurement Concepts
- Organizations
- Publishing
- Qualitative Research
- Reliability of Scores
- Research Design Concepts
- Aptitude-Treatment Interaction
- Cause and Effect
- Concomitant Variable
- Confounding
- Control Group
- Interaction
- Internet-Based Research Method
- Intervention
- Matching
- Natural Experiments
- Network Analysis
- Placebo
- Replication
- Research
- Research Design Principles
- Treatment(s)
- Triangulation
- Unit of Analysis
- Yoked Control Procedure
- Research Designs
- A Priori Monte Carlo Simulation
- Action Research
- Adaptive Designs in Clinical Trials
- Applied Research
- Behavior Analysis Design
- Block Design
- Case-Only Design
- Causal-Comparative Design
- Cohort Design
- Completely Randomized Design
- Cross-Sectional Design
- Crossover Design
- Double-Blind Procedure
- Ex Post Facto Study
- Experimental Design
- Factorial Design
- Field Study
- Group-Sequential Designs in Clinical Trials
- Laboratory Experiments
- Latin Square Design
- Longitudinal Design
- Meta-Analysis
- Mixed Methods Design
- Mixed Model Design
- Monte Carlo Simulation
- Nested Factor Design
- Nonexperimental Design
- Observational Research
- Panel Design
- Partially Randomized Preference Trial Design
- Pilot Study
- Pragmatic Study
- Pre-Experimental Designs
- Pretest-Posttest Design
- Prospective Study
- Quantitative Research
- Quasi-Experimental Design
- Randomized Block Design
- Repeated Measures Design
- Response Surface Design
- Retrospective Study
- Sequential Design
- Single-Blind Study
- Single-Subject Design
- Split-Plot Factorial Design
- Thought Experiments
- Time Studies
- Time-Lag Study
- Time-Series Study
- Triple-Blind Study
- True Experimental Design
- Wennberg Design
- Within-Subjects Design
- Zelen's Randomized Consent Design
- Research Ethics
- Research Process
- Clinical Significance
- Clinical Trial
- Cross-Validation
- Data Cleaning
- Delphi Technique
- Evidence-Based Decision Making
- Exploratory Data Analysis
- Follow-Up
- Inference: Deductive and Inductive
- Last Observation Carried Forward
- Planning Research
- Primary Data Source
- Protocol
- Q Methodology
- Research Hypothesis
- Research Question
- Scientific Method
- Secondary Data Source
- Standardization
- Statistical Control
- Type III Error
- Wave
- Research Validity Issues
- Bias
- Critical Thinking
- Ecological Validity
- Experimenter Expectancy Effect
- External Validity
- File Drawer Problem
- Hawthorne Effect
- Heisenberg Effect
- Internal Validity
- John Henry Effect
- Mortality
- Multiple Treatment Interference
- Multivalued Treatment Effects
- Nonclassical Experimenter Effects
- Order Effects
- Placebo Effect
- Pretest Sensitization
- Random Assignment
- Reactive Arrangements
- Regression to the Mean
- Selection
- Sequence Effects
- Threats to Validity
- Validity of Research Conclusions
- Volunteer Bias
- White Noise
- Sampling
- Cluster Sampling
- Convenience Sampling
- Demographics
- Error
- Exclusion Criteria
- Experience Sampling Method
- Nonprobability Sampling
- Population
- Probability Sampling
- Proportional Sampling
- Quota Sampling
- Random Sampling
- Random Selection
- Sample
- Sample Size
- Sample Size Planning
- Sampling
- Sampling and Retention of Underrepresented Groups
- Sampling Error
- Stratified Sampling
- Systematic Sampling
- Scaling
- Software Applications
- Statistical Assumptions
- Statistical Concepts
- Autocorrelation
- Biased Estimator
- Cohen's Kappa
- Collinearity
- Correlation
- Criterion Problem
- Critical Difference
- Data Mining
- Data Snooping
- Degrees of Freedom
- Directional Hypothesis
- Disturbance Terms
- Error Rates
- Expected Value
- Fixed-Effects Models
- Inclusion Criteria
- Influence Statistics
- Influential Data Points
- Intraclass Correlation
- Latent Variable
- Likelihood Ratio Statistic
- Loglinear Models
- Main Effects
- Markov Chains
- Method Variance
- Mixed- and Random-Effects Models
- Models
- Multilevel Modeling
- Odds
- Omega Squared
- Orthogonal Comparisons
- Outlier
- Overfitting
- Pooled Variance
- Precision
- Quality Effects Model
- Random-Effects Models
- Regression Artifacts
- Regression Discontinuity
- Residuals
- Restriction of Range
- Robust
- Root Mean Square Error
- Rosenthal Effect
- Serial Correlation
- Shrinkage
- Simple Main Effects
- Simpson's Paradox
- Sums of Squares
- Statistical Procedures
- Accuracy in Parameter Estimation
- Analysis of Covariance (ANCOVA)
- Analysis of Variance (ANOVA)
- Barycentric Discriminant Analysis
- Bivariate Regression
- Bonferroni Procedure
- Bootstrapping
- Canonical Correlation Analysis
- Categorical Data Analysis
- Confirmatory Factor Analysis
- Contrast Analysis
- Descriptive Discriminant Analysis
- Discriminant Analysis
- Dummy Coding
- Effect Coding
- Estimation
- Exploratory Factor Analysis
- Greenhouse-Geisser Correction
- Hierarchical Linear Modeling
- Holm's Sequential Bonferroni Procedure
- Jackknife
- Latent Growth Modeling
- Least Squares, Methods of
- Logistic Regression
- Mean Comparisons
- Missing Data, Imputation of
- Multiple Regression
- Multivariate Analysis of Variance (MANOVA)
- Pairwise Comparisons
- Path Analysis
- Post Hoc Analysis
- Post Hoc Comparisons
- Principal Components Analysis
- Propensity Score Analysis
- Sequential Analysis
- Stepwise Regression
- Structural Equation Modeling
- Survival Analysis
- Trend Analysis
- Yates's Correction
- Statistical Tests
- Bartlett's Test
- Behrens-Fisher t′ Statistic
- Chi-Square Test
- Duncan's Multiple Range Test
- Dunnett's Test
- F Test
- Fisher's Least Significant Difference Test
- Friedman Test
- Honestly Significant Difference (HSD) Test
- Kolmogorov-Smirnov Test
- Kruskal-Wallis Test
- Mann-Whitney U Test
- Mauchly Test
- McNemar's Test
- Multiple Comparison Tests
- Newman-Keuls Test and Tukey Test
- Omnibus Tests
- Scheffé Test
- Sign Test
- t Test, Independent Samples
- t Test, One Sample
- t Test, Paired Samples
- Tukey's Honestly Significant Difference (HSD)
- Welch's t Test
- Wilcoxon Rank Sum Test
- z Test
- Theories, Laws, and Principles
- Bayes's Theorem
- Central Limit Theorem
- Classical Test Theory
- Correspondence Principle
- Critical Theory
- Falsifiability
- Game Theory
- Gauss-Markov Theorem
- Generalizability Theory
- Grounded Theory
- Item Response Theory
- Occam's Razor
- Paradigm
- Positivism
- Probability, Laws of
- Theory
- Theory of Attitude Measurement
- Weber-Fechner Law
- Types of Variables
- Validity of Scores
- Loading...
Get a 30 day FREE TRIAL
-
Watch videos from a variety of sources bringing classroom topics to life -
Read modern, diverse business cases
-
Explore hundreds of books and reference titles
Read next
More like this
Sage Recommends
We found other relevant content for you on other Sage platforms.
Have you created a personal profile? Login or create a profile so that you can save clips, playlists and searches