Wilcoxon Signed Rank Test

One-sample and paired Wilcoxon signed rank test.

Usage

var wilcoxon = require( '@stdlib/stats/wilcoxon' );

wilcoxon( x[, y][, opts] )

Performs a one-sample t-test for the null hypothesis that the data in array or typed array x is drawn from a distribution that is symmetric around zero (i.e., with median zero).

// Differences in plant heights, see Cureton (1967)
var x = [ 6, 8, 14, 16, 23, 24, 28, 29, 41, -48, 49, 56, 60, -67, 75 ];
var out = wilcoxon( x );
/* e.g., returns
    {
        'rejected': true,
        'alpha': 0.05,
        'pValue': 0.04125976562499978,
        'statistic': 96,
        // ...
    }
*/

When array or typed array y is supplied, the function tests whether the paired differences x - y come from a distribution that is symmetric around zero (i.e., with median zero).

// Patient measurements at first (x) and second (y) visit, see Hollander & Wolfe (1973)
var x = [ 1.83, 0.50, 1.62, 2.48, 1.68, 1.88, 1.55, 3.06, 1.30 ];
var y = [ 0.878, 0.647, 0.598, 2.05, 1.06, 1.29, 1.06, 3.14, 1.29 ];

var out = wilcoxon( x, y );
/* e.g., returns
    {
        'rejected': true,
        'alpha': 0.05,
        'pValue': 0.0390625,
        'statistic': 40,
        // ...
    }
*/

The returned object comes with a .print() method which when invoked will print a formatted output of the hypothesis test results. print accepts a digits option that controls the number of decimal digits displayed for the outputs and a decision option, which when set to false will hide the test decision.

console.log( out.print() );
/* e.g., =>
    Paired Wilcoxon signed rank test

    Alternative hypothesis: Median of the difference `x - y` is not equal to 0

        pValue: 0.0391
        statistic: 40

    Test Decision: Reject null in favor of alternative at 5% significance level
*/

The wilcoxon function accepts the following options:

  • alpha: number in the interval [0,1] giving the significance level of the hypothesis test. Default: 0.05.
  • alternative: Either two-sided, less or greater. Indicates whether the alternative hypothesis is that the mean of x is larger than mu (greater), smaller than mu (less), or equal to mu (two-sided). Default: two-sided.
  • correction: continuity correction adjusting the Wilcoxon rank statistic by 0.5 towards the mean when using the normal approximation. Default: true.
  • exact: Determines whether to force use of the exact distribution instead of a normal approximation when there are more than fifty data points. Default: false.
  • mu: number denoting the hypothesized median under the null hypothesis. Default: 0.
  • zeroMethod: Method governing how zero-differences are handled (pratt, wilcox, or zsplit). Default: 'wilcox'.

By default, the hypothesis test is carried out at a significance level of 0.05. To choose a different significance level, set the alpha option.

var table;
var out;
var arr;

arr = [ 2, 4, 3, 1, 0 ];
out = wilcoxon( arr, {
    'alpha': 0.01
});
table = out.print();
/* e.g., returns
    One-Sample Wilcoxon signed rank test

    Alternative hypothesis: Median of `x` is not equal to 0

        pValue: 0.035
        statistic: 21

    Test Decision: Reject null in favor of alternative at 5% significance level
*/

out = wilcoxon( arr, {
    'alpha': 0.1
});
table = out.print();
/* e.g., returns
    One-Sample Wilcoxon signed rank test

    Alternative hypothesis: Median of `x` is not equal to 0

        pValue: 0.035
        statistic: 21

    Test Decision: Fail to reject null in favor of alternative at 1% significance level
*/

To test whether the data comes from a distribution with a median different than zero, set the mu option.

var arr = [ 4, 4, 6, 6, 5 ];
var out = wilcoxon( arr, {
    'mu': 5
});
/* e.g., returns
{
    'rejected': false,
    'pValue': 1,
    'statistic': 0,
    // ...
}
*/

By default, a two-sided test is performed. To perform either of the one-sided tests, set the alternative option to less or greater.

var arr = [ 4, 4, 6, 6, 5 ];
var out = wilcoxon( arr, {
    'alternative': 'less'
});
var table = out.print();
/* e.g., returns
    One-Sample Wilcoxon signed rank test

    Alternative hypothesis: Median of `x` is less than 0

        pValue: 0.9853
        statistic: 15

    Test Decision: Fail to reject null in favor of alternative at 5% significance level
*/

out = wilcoxon( arr, {
    'alternative': 'greater'
});
table = out.print();
/* e.g., returns
    One-Sample Wilcoxon signed rank test

    Alternative hypothesis: Median of `x` is greater than 0

        pValue: 0.0284
        statistic: 15

    Test Decision: Reject null in favor of alternative at 5% significance level
*/

By default, all zero-differences are discarded before calculating the ranks. Set zeroMethod to pratt when you wish differences of zero to be used in the rank calculation but then drop them or to zsplit when differences of zero are shall be used in the ranking procedure and the ranks then split between positive and negative ones.

var arr = [ 0, 2, 3, -1, -4, 0, 0, 8, 9 ];
var out = wilcoxon( arr, {
    'zeroMethod': 'pratt'
});
/* e.g., returns
    {
        'rejected': false,
        'alpha': 0.05,
        'pValue': ~0.331,
        'statistic': 28,
        // ...
    }
*/

out = wilcoxon( arr, {
    'zeroMethod': 'zsplit'
});
/* e.g., returns
    {
        'rejected': false,
        'alpha': 0.05,
        'pValue': ~0.342,
        'statistic': 31,
        // ...
    }
*/

By default, the test uses the exact distribution of the rank statistic to calculate the critical values for the test in case of no ties and no zero-differences. Since it is more computationally efficient, starting with fifty observations a normal approximation is employed. If you would like the test to use the correct distribution even for larger samples, set the exact option to true.

var normal = require( '@stdlib/random/base/normal' ).factory;
var rnorm;
var arr;
var out;
var i;

rnorm = normal( 0.0, 4.0, {
    'seed': 100
});
arr = new Array( 100 );
for ( i = 0; i < arr.length; i++ ) {
    arr[ i ] = rnorm();
}

out = wilcoxon( arr, {
    'exact': false
});
/* e.g., returns
    {
        'rejected': false,
        'alpha': 0.05,
        'pValue': ~0.422,
        'statistic': 2291,
        // ...
    }
*/

out = wilcoxon( arr, {
    'exact': true
});
/* e.g., returns
    {
        'rejected': false,
        'alpha': 0.05,
        'pValue': ~0.424,
        'statistic': 2291,
        // ...
    }
*/

By default, when using the normal approximation, the test uses a continuity correction, which adjusts the Wilcoxon rank statistic by 0.5 towards the mean. To disable this correction, set correction to false.

var normal = require( '@stdlib/random/base/normal' ).factory;
var rnorm;
var arr;
var out;
var i;

rnorm = normal( 0.0, 4.0, {
    'seed': 100
});
arr = new Array( 100 );
for ( i = 0; i < arr.length; i++ ) {
    arr[ i ] = rnorm();
}

out = wilcoxon( arr, {
    'correction': false
});
/* e.g., returns
    {
        'rejected': false,
        'alpha': 0.05,
        'pValue': ~0.421,
        'statistic': 2291,
        // ...
    }
*/

out = wilcoxon( arr, {
    'correction': true
});
/* e.g., returns
    {
        'rejected': false,
        'alpha': 0.05,
        'pValue': ~0.422,
        'statistic': 2291,
        // ...
    }
*/

Examples

var uniform = require( '@stdlib/random/base/discrete-uniform' ).factory;
var wilcoxon = require( '@stdlib/stats/wilcoxon' );

var table;
var runif;
var arr;
var out;
var i;

runif = uniform( -50.0, 50.0, {
    'seed': 37827
});
arr = new Array( 100 );
for ( i = 0; i < arr.length; i++ ) {
    arr[ i ] = runif();
}

// Test whether distribution is symmetric around zero:
out = wilcoxon( arr );
table = out.print();
/* e.g., returns
    One-Sample Wilcoxon signed rank test

    Alternative hypothesis: Median of `x` is not equal to 0

        pValue: 0.7714
        statistic: 2438.5

    Test Decision: Fail to reject null in favor of alternative at 5% significance level
*/

// Test whether distribution has median of five:
out = wilcoxon( arr, {
    'mu': 5.0
});
table = out.print();
/* e.g, returns
    One-Sample Wilcoxon signed rank test

    Alternative hypothesis: Median of `x` is not equal to 5

        pValue: 0.0529
        statistic: 1961.5

    Test Decision: Fail to reject null in favor of alternative at 5% significance level
*/