com.imsl.stat
Class WilcoxonRankSum

java.lang.Object
  com.imsl.stat.WilcoxonRankSum

All Implemented Interfaces:

Serializable, Cloneable

public class WilcoxonRankSum
extends Object
implements Serializable, Cloneable

Performs a Wilcoxon rank sum test.

Class WilcoxonRankSum performs the Wilcoxon rank sum test for identical population distribution functions. The Wilcoxon test is a linear transformation of the Mann-Whitney U test. If the difference between the two populations can be attributed solely to a difference in location, then the Wilcoxon test becomes a test of equality of the population means (or medians) and is the nonparametric equivalent of the two-sample t-test. Class WilcoxonRankSum obtains ranks in the combined sample after first eliminating missing values from the data. The rank sum statistic is then computed as the sum of the ranks in the x sample. Three methods for handling ties are used. (A tie is counted when two observations are within fuzz of each other.) Method 1 uses the largest possible rank for tied observations in the smallest sample, while Method 2 uses the smallest possible rank for these observations. Thus, the range of possible rank sums is obtained.

Method 3 for handling tied observations between samples uses the average rank of the tied observations. Asymptotic standard normal scores are computed for the W score (based on a variance that has been adjusted for ties) when average ranks are used (see Conover 1980, p. 217), and the probability associated with the two-sided alternative is computed.

Hypothesis Tests

In each of the following tests, the first line gives the hypothesis (and its alternative) under the assumptions 1 to 3 below, while the second line gives the hypothesis when assumption 4 is also true. The rejection region is the same for both hypotheses and is given in terms of Method 3 for handling ties. If another method for handling ties is desired, another output statistic, stat[0] or stat[3], should be used, where stat is the array containing the statistics returned from the getStatistics method.

Test	Null Hypothesis	Alternative Hypothesis	Action
1			Reject if stat[9] is less than the significance level of the test. Alternatively, reject the null hypothesis if stat[6] is too large or too small.
2			Reject if stat[6] is too small
3			Reject if stat[6] is too large

Assumptions

1. x and y contain random samples from their respective populations.
2. All observations are mutually independent.
3. The measurement scale is at least ordinal (i.e., an ordering less than, greater than, or equal to exists among the observations).
4. If f(x) and g(y) are the distribution functions of x and y, then g(y) = f(x + c) for some constant c(i.e., the distribution of y is, at worst, a translation of the distribution of x).

The p-value is calculated using the large-sample normal approximation. This approximate calculation is only valid when the size of one or both samples is greater than 50. For smaller samples, see the exact tables for the Wilcoxon Rank Sum Test.

See Also:

Example 1, Example 2, Serialized Form

Constructor Summary

WilcoxonRankSum(double[] x, double[] y)
Constructor for WilcoxonRankSum.

Method Summary

double	compute() Performs a Wilcoxon rank sum test.
double[]	getStatistics() Returns the statistics.
void	setFuzz(double fuzz) Sets the nonnegative constant used to determine ties in computing ranks in the combined samples.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

WilcoxonRankSum

public WilcoxonRankSum(double[] x,
                       double[] y)

Constructor for WilcoxonRankSum.

Parameters:

x - A double array containing the first sample.

y - A double array containing the second sample.

Method Detail

compute

public final double compute()

Performs a Wilcoxon rank sum test.

Returns:

A double scalar containing the two-sided p-value for the Wilcoxon rank sum statistic that is computed with average ranks used in the case of ties.

getStatistics

public double[] getStatistics()

Returns the statistics. Note that the compute method must be invoked first before invoking this method. Otherwise, the method throws a NullPointerException exception.

Returns:

A double array of length 10 containing the following statistics:

Row	Statistics
0	Wilcoxon W statistic (the sum of the ranks of the x observations) adjusted for ties in such a manner that W is as small as possible
1	2 x E(W) - W, where E(W) is the expected value of W
2	probability of obtaining a statistic less than or equal to min{W, 2 x E(W) - W}
3	W statistic adjusted for ties in such a manner that W is as large as possible
4	2 x E(W) - W, where E(W) is the expected value of W, adjusted for ties in such a manner that W is as large as possible
5	probability of obtaining a statistic less than or equal to min{W, 2 x E(W) - W}, adjusted for ties in such a manner that W is as large as possible
6	W statistic with average ranks used in case of ties
7	estimated standard error of Row 6 under the null hypothesis of no difference
8	standard normal score associated with Row 6
9	two-sided p-value associated with Row 8

setFuzz

public void setFuzz(double fuzz)

Sets the nonnegative constant used to determine ties in computing ranks in the combined samples.

Parameters:

fuzz - A double scalar containing the nonnegative constant used to determine ties in computing ranks in the combined samples. A tie is declared when two observations in the combined sample are within fuzz of each other. Default: