e-Statistics
e-Statistics
For Students
For Contributors
Data in Statistical Studies
Textbook Data Sets
Worksheet Data Sets
Data and Story Library
Exploratory Graphics
Stem and Leaf Plot
Histogram
Boxplot
Scatter Plot
QQ plot
Bar Graph
Descriptive Statistics
Summary Statistics
Frequency Table
Grouped Data
Transformed Data
Normal Distribution
Normal Density Function
Z-score
Critical Value
Central Limit Theorem
Discrete Distributions
Binomial Distribution
Frequency Function
Normal Approximation
Inference on Mean
t-Distribution
Confidence Interval
Hypothesis Test
Power of Test
One-sided CI
Comparison of Two Groups
Test for Independent Groups
Nonparametric Test
Test for Paired Data
Simple Linear Regression
Regression Line
R Square
Residual Plot
Inference on Parameters
Predicting Responses
One-Way ANOVA Model
F-Distribution
Analysis of Variance
Strip Chart
Multiple Comparisons
Multiple Linear Regression
Model Estimate and Residuals
Model Selection and F-test
Two-Way Additive Model
Test of Effect
Analysis of Variance
Inference on Variances
Testing a Standard Deviation
Test for Two Groups
Test for Homogeneity
Inference on Proportions
Testing a Proportion
Comparison of Two Proportions
Chi-square Test
Chi-square Distribution
Goodness of Fit
Test for Homogeneity
Logistic Regression
Inference on parameters
Adjusted log OR

Nonparametric Test

Wilcoxon rank sum test (also called Mann–Whitney–Wilcoxon test) is based on ranks—the rank 1 is assigned to the smallest measurement in both groups, and 2 to the second smallest, and so on. The validity of t-test is based on the “normality of population distributions,” and it requires that either sample distributions are approximately normal, or the sample sizes are appropriately large ( $ n,m \ge 30$). Since a normal distribution is “parametric,” the rank sum test procedure is referred as “nonparametric,” free from the assumption of normal distribution.

Data must be arranged either (i) in two columns each of which contains data for the respective groups, or (ii) in a form of grouped data. Here the first column specifies (i) Group 1 data or (ii) the whole data, and the second column specifies (i) Group 2 data or (ii) the column of two categorical values by which Group 1 and 2 are identified.

Here we identify Group 1 and Group 2 as and . It is assumed that the distribution of Group 1 and that of Group 2 share the same shape with possible shift, but not necessarily normally distributed. Then the null hypothesis is stated as “the distributions from two groups are identical,” and the test will determine whether it is rejected in favor of the following alternative hypothesis.

$ H_A:$ The distribution of Group 1 that of Group 2

It produces the estimated “Shift” and the confidence interval “(Lower, Upper)” for the shift (the difference of locations) with confidence interval. The value p_value indicates the significance of the test: If p-value < $ \alpha$, it suggest evidence against the null hypothesis in favor of the alternative hypothesis.

Here “Shift” is calculated from the sample median of the difference  $ (X_i - Y_j)$ between $ X_1,\ldots,X_n$ in Group 1 and $ Y_1,\ldots,Y_m$ in Group 2. It is called the Hodges-Lehmann estimator.

© TTU Mathematics