e-Statistics
Introduction
  • Advanced User Guide
    Assignment & Quiz
  • User Account
  • Guide to Worksheet 1-2
  • Guide to Worksheet 3-4
  • Guide to Worksheet 5-6
  • Guide to Worksheet 7-8
    Data in Statistical Studies
  • Overview
  • Data from SMDA
  • Data and Story Library
    Exploratory Graphics
  • Overview
  • Histogram
  • Stem and Leaf Plot
  • Boxplot
  • Scatter Plot
  • QQ plot
  • Bar Graph
    Descriptive Statistics
  • Overview
  • Summary Statistics
  • Grouped Data
  • Frequency Table
    Discrete Distributions
  • Frequency Function
  • Binomial Distribution
  • Normal Approximation
    Normal Distribution
  • Normal Density Function
  • Critical Point
  • Z-score
  • Central Limit Theorem
    Inference on Mean
  • t-Distribution
  • Confidence Interval
  • Hypothesis Test
  • Power of Test
  • One-sided CI
    Comparison of Two Groups
  • Overview
  • t-Test
  • Nonparametric Test
  • Test for Paired Data
    Inference on Variances
  • Chi-square Distribution
  • Testing a Standard Deviation
  • F-Distribution
  • F-Test
    Explore Statistics in Drug Safety

    • Nonparametric Test

    The sample data from ``Group 1'' and ``Group 2'' are either (a) shown separately in two different columns, or (b) stored in one column with their status of group made available by another column.

    Group 1: in factor level

    Group 2: in factor level

    In case of (b) all the measurements are shown in a single variable, and grouped by another variable. The name of this variable for factor levels must be specified here.

    Here 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. The Wilcoxon rank sum 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. Recall that the validity of t-test is based on the ``normality of population distributions,'' and that 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.

    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 estimate ``estimate.shift'' and the confidence interval ``(lower.bound,upper.bound)'' 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 the estimate ``estimate.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.