signrank

Wilcoxon signed rank test

Syntax

p = signrank(x)

p = signrank(x,y)

p = signrank(x,y,Name,Value)

[p,h] = signrank(___)

[p,h,stats] = signrank(___)

[___] = signrank(x,m)

[___] = signrank(x,m,Name,Value)

Description

p = signrank(x) returns the p-value of a two-sided Wilcoxon signed rank test.

signrank tests the null hypothesis that data in the vector x come from a distribution whose median is zero at the 5% significance level. The test assumes that the data in x come from a continuous distribution symmetric about its median.

p = signrank(x,y) returns the p-value of a paired, two-sided test for the null hypothesis that x – y comes from a distribution with zero median.

p = signrank(x,y,Name,Value) returns the p-value for the sign test with additional options specified by one or more Name,Value pair arguments.

[p,h] = signrank(___) also returns a logical value indicating the test decision. h = 1 indicates a rejection of the null hypothesis, and h = 0 indicates a failure to reject the null hypothesis at the 5% significance level. You can use any of the input arguments in the previous syntaxes.

[p,h,stats] = signrank(___) also returns the structure stats with information about the test statistic.

[___] = signrank(x,m) returns any of the output arguments in the previous syntaxes for the null hypothesis that the data in x are observations from a distribution with median m.

[___] = signrank(x,m,Name,Value) returns any of the output arguments in the previous syntaxes for the signed rank test with additional options specified by one or more Name,Value pair arguments.

Examples

Test for Zero Median of a Single Population

Test the hypothesis of zero median.

Generate the sample data.

rng('default') % for reproducibility
x = randn(1,25) + 1.30;

Test the hypothesis that the data in x has zero median.

[p,h] = signrank(x)

p = 
3.2229e-05

h = logical
   1

At the default 5% significance level, the value h = 1 indicates that the test rejects the null hypothesis of zero median.

Test the Median of Differences of Paired Samples

Test the hypothesis of zero median for the difference between paired samples.

Generate the sample data.

rng('default') % for reproducibility
x = lognrnd(2,.25,10,1);
y = x + trnd(2,10,1);

Test the hypothesis that x – y has zero median.

[p,h] = signrank(x,y)

p = 
0.3223

h = logical
   0

The results indicate that the test fails to reject the null hypothesis of zero median in the difference at the default 5% significance level.

Signed Rank Test for Large Samples

Conduct a -sided test on a large sample using approximation.

Load the sample data.

load('gradespaired.mat');

Test the null hypothesis that the median of the grade differences of students before and after participating in a tutoring program is 0 against the alternate that it is less than 0.

[p,h,stats] = signrank(gradespaired(:,1),...
		gradespaired(:,2),'tail','left')

p = 
0.0047

h = logical
   1

stats = struct with fields:
          zval: -2.5982
    signedrank: 2.0175e+03

Because the sample size is greater than 15, signrank uses an approximate method to calculate the $$p$$-value and also returns the value of the $$z$$-statistic. The value h = 1 indicates that the test rejects the null hypothesis that there is no difference between the grade medians at the 5% significance level. There is enough statistical evidence to conclude that the median grade before the tutoring program is less than the median grade after the tutoring program.

Repeat the test using the exact method.

[p,h,stats] = signrank(gradespaired(:,1),gradespaired(:,2),...
		'tail','left','method','exact')

p = 
0.0045

h = logical
   1

stats = struct with fields:
    signedrank: 2.0175e+03

The results obtained using the approximate method are consistent with the exact method.

Two-Sided Test for the Median of a Single Population

Load the sample data.

load mileage

The data contains the mileages per gallon for three different types of cars in columns 1 to 3.

Test the hypothesis that the median mileage for the type of cars in the second column differs from 33.

[p,h,stats] = signrank(mileage(:,2),33)

p = 
0.0312

h = logical
   1

stats = struct with fields:
    signedrank: 21

At the 5% significance level, the results indicate that the median mileage for the second type of cars differs from 33. Note that signrank uses an exact method to calculate the $$p$$-value for small samples and does not return the $$z$$-statistic.

Right-Sided Test for the Median of a Single Population

Use the name-value pair arguments in signrank.

Load the sample data.

load mileage

The data contains the mileage per gallon for three different types of cars in columns 1 to 3.

Test the hypothesis that the median mileage for the type of cars in the second row are larger than 33.

[p,h,stats] = signrank(mileage(:,2),33,'tail','right')

p = 
0.0156

h = logical
   1

stats = struct with fields:
    signedrank: 21

Repeat the same test at the 1% significance level using the approximate method.

[p,h,stats] = signrank(mileage(:,2),33,'tail','right',...
'alpha',0.01,'method','approximate')

p = 
0.0180

h = logical
   0

stats = struct with fields:
          zval: 2.0966
    signedrank: 21

This result, h = 0, indicates that the null hypothesis cannot be rejected at the 1% significance level.

Input Arguments

x — Sample data

vector

Sample data, specified as a vector.

Data Types: single | double

y — Sample data

vector

Sample data, specified as a vector. y must be the same length as x.

Data Types: single | double

m — Hypothesized value of the median

scalar

Hypothesized value of the median, specified as a scalar.

Example: signrank(x,10)

Data Types: single | double

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: 'alpha',0.01,'method','approximate','tail','right' specifies a right-tailed signed rank test with 1% significance level, which returns the approximate p-value.

alpha — Significance level

0.05 (default) | scalar value in the range 0 to 1

Significance level of the decision of a hypothesis test, specified as the comma-separated pair consisting of 'alpha' and a scalar value in the range 0 to 1. Significance level of h is 100 * alpha%.

Example: 'alpha', 0.01

Data Types: double | single

method — Computation method of p

'exact' | 'approximate'

Computation method of p, specified as the comma-separated pair consisting of 'method' and one of the following.

'exact'	Exact computation of the p-value, p. Default value for 15 or fewer observations in x, x – m, or x – y when method is unspecified.
'approximate'	Normal approximation while computing the p-value, p. Default value for more than 15 observations in x, x – m, or x – y when 'method' is unspecified because the exact method can be slow on large samples.

Example: 'method','exact'

tail — Type of test

'both' (default) | 'right' | 'left'

Type of test, specified as the comma-separated pair consisting of 'tail' and one of the following:

'both'	Two-sided hypothesis test, which is the default test type. For a one-sample test, the alternate hypothesis states that the data in x come from a continuous distribution with median different than 0 or m. For a two-sample test, the alternate hypothesis states that the data in x – y come from a distribution with median different than 0.
'right'	Right-tailed hypothesis test. For a one-sample test, the alternate hypothesis states that the data in x come from a continuous distribution with median greater than 0 or m. For a two-sample test, the alternate hypothesis states the data in x – y come from a distribution with median greater than 0.
'left'	Left-tailed hypothesis test. For a one-sample test, the alternate hypothesis states that the data in x come from a continuous distribution with median less than 0 or m. For a two-sample test, the alternate hypothesis states the data in x – y come from a distribution with median less than 0.

Example: 'tail','left'

Output Arguments

p — p-value of the test

nonnegative scalar

p-value of the test, returned as a nonnegative scalar from 0 to 1. p is the probability of observing a test statistic as or more extreme than the observed value under the null hypothesis. signrank computes the two-sided p-value by doubling the most significant one-sided value.

h — Result of the hypothesis test

1 | 0

Result of the hypothesis test, returned as a logical value.

• If h = 1, this indicates the rejection of the null hypothesis at the 100 * alpha% significance level.

• If h = 0, this indicates a failure to reject the null hypothesis at the 100 * alpha% significance level.

stats — Test statistics

structure

Test statistics, returned as a structure. The test statistics stored in stats are:

• signrank: Value of the sign rank test statistic.

• zval: Value of the z- statistic (computed when 'method' is 'approximate').

More About

Wilcoxon Signed Rank Test

The Wilcoxon signed rank test is a nonparametric test for two populations when the observations are paired. In this case, the test statistic, W, is the sum of the ranks of positive differences between the observations in the two samples (that is, x – y). When you use the test for one sample, then W is the sum of the ranks of positive differences between the observations and the hypothesized median value M₀ (which is 0 when you use signrank(x) and m when you use signrank(x,m)).

z-Statistic

For large samples, or when method is approximate, the signrank function calculates the p-value using the z-statistic, given by

$$z=\frac{\left(W-n\left(n+1\right)/4\right)}{\sqrt{\frac{n\left(n+1\right)\left(2n+1\right)-tieadj}{24}}},$$

where n is the sample size of the difference x – y or x – m. For the two-sample case, signrank uses [tie_rank,tieadj] = tiedrank(abs(diffxy),0,0,epsdiff) to obtain the tie adjustment value tieadj.

Algorithms

signrank treats NaNs in x and y as missing values and ignores them.

For the two-sample case, signrank uses a tolerance based on the values epsdiff = eps(x) + eps(y). signrank computes the absolute values of the differences (abs(d(i)) where d(i) = x(i) – y(i)) and compares them to epsdiff. Values with an absolute value less than epsdiff (abs(d(i)) < epsdiff(i)) are treated as ties.

References

[1] Gibbons, J. D., and S. Chakraborti. Nonparametric Statistical Inference, 5th Ed., Boca Raton, FL: Chapman & Hall/CRC Press, Taylor & Francis Group, 2011.

[2] Hollander, M., and D. A. Wolfe. Nonparametric Statistical Methods. Hoboken, NJ: John Wiley & Sons, Inc., 1999.

Version History

Introduced before R2006a