Press n or j to go to the next uncovered block, b, p or k for the previous block.
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* @license Apache-2.0
*
* Copyright (c) 2018 The Stdlib Authors.
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
'use strict';
// MODULES //
var setReadOnly = require( '@stdlib/utils/define-read-only-property' );
var isNumberArray = require( '@stdlib/assert/is-number-array' );
var format = require( '@stdlib/string/format' );
var sqrt = require( '@stdlib/math/base/special/sqrt' );
var abs = require( '@stdlib/math/base/special/abs' );
var normalCDF = require( '@stdlib/stats/base/dists/normal/cdf' );
var min = require( '@stdlib/math/base/special/min' );
var validate = require( './validate.js' );
// eslint-disable-next-line stdlib/no-redeclare
var print = require( './print.js' );
// FUNCTIONS //
/**
* Comparator for ascending sort of value-index pairs.
*
* @private
* @param {Array} a - first [value, index] pair
* @param {Array} b - second [value, index] pair
* @returns {number} comparison result
*/
function sortPairs( a, b ) {
return a[ 0 ] - b[ 0 ];
}
// MAIN //
/**
* Computes the Mann–Whitney U rank-sum test for two independent samples.
*
* @param {NumericArray} x - first sample
* @param {NumericArray} y - second sample
* @param {Options} [options] - function options
* @param {number} [options.alpha=0.05] - significance level
* @param {string} [options.alternative='two-sided'] - alternative hypothesis (`two-sided`, `less`, `greater`)
* @throws {TypeError} must provide two numeric arrays
* @throws {RangeError} alpha option must be a number in `[0,1]`
* @throws {TypeError} alternative option must be a string
* @throws {Error} alternative option must be `two-sided`, `less` or `greater`
* @returns {Object} test results
*/
function mannWhitneyU() {
var values;
var tieSum;
var pValue;
var alpha;
var ranks;
var pairs;
var opts;
var mean;
var alt;
var avg;
var err;
var out;
var R1;
var U1;
var U2;
var sd;
var n1;
var n2;
var N;
var x;
var y;
var k;
var t;
var U;
var z;
var i;
var j;
opts = {};
x = arguments[ 0 ];
y = arguments[ 1 ];
if ( !isNumberArray( x ) || !isNumberArray( y ) ) {
throw new TypeError( format( 'invalid argument. Must provide two numeric arrays.' ) );
}
n1 = x.length;
n2 = y.length;
N = n1 + n2;
if ( arguments[ 2 ] ) {
err = validate( opts, arguments[ 2 ] );
}
if ( err ) {
throw err;
}
alpha = ( opts.alpha === void 0 ) ? 0.05 : opts.alpha;
alt = opts.alternative || 'two-sided';
values = x.concat( y );
pairs = [];
for ( i = 0; i < N; i++ ) {
pairs.push( [ values[ i ], i ] );
}
pairs.sort( sortPairs );
ranks = [];
tieSum = 0.0;
i = 0;
while ( i < N ) {
j = i + 1;
while ( j < N && pairs[ j ][ 0 ] === pairs[ i ][ 0 ] ) {
j += 1;
}
t = j - i;
avg = ( i + j + 1 ) / 2.0;
for ( k = i; k < j; k++ ) {
ranks[ pairs[ k ][ 1 ] ] = avg;
}
if ( t > 1 ) {
tieSum += ( t * t * t ) - t;
}
i = j;
}
R1 = 0.0;
for ( i = 0; i < n1; i++ ) {
R1 += ranks[ i ];
}
U1 = R1 - ( n1 * ( n1 + 1 ) * 0.5 );
U2 = ( n1 * n2 ) - U1;
if ( alt === 'two-sided' ) {
U = min( U1, U2 );
} else {
U = U1;
}
mean = n1 * n2 * 0.5;
if ( tieSum === 0.0 ) {
sd = sqrt( ( n1 * n2 * ( N + 1 ) ) / 12.0 );
} else {
// eslint-disable-next-line max-len
sd = sqrt( ( n1 * n2 / 12.0 ) * ( ( N + 1 ) - ( tieSum / ( N * ( N - 1 ) ) ) ) );
}
z = ( U - mean ) / sd;
if ( alt === 'two-sided' ) {
pValue = 2.0 * ( 1.0 - normalCDF( abs( z ), 0.0, 1.0 ) );
} else if ( alt === 'greater' ) {
pValue = 1.0 - normalCDF( z, 0.0, 1.0 );
} else if ( alt === 'less' ) {
pValue = normalCDF( z, 0.0, 1.0 );
} else {
throw new Error(format('invalid option. `%s` option must be one of `two-sided`, `less`, `greater`. Option: `%s`.', 'alternative', alt));
}
out = {};
setReadOnly( out, 'U', U );
setReadOnly( out, 'U1', U1 );
setReadOnly( out, 'U2', U2 );
setReadOnly( out, 'pValue', pValue );
setReadOnly( out, 'rejected', pValue <= alpha );
setReadOnly( out, 'alpha', alpha );
setReadOnly( out, 'alternative', alt );
setReadOnly( out, 'method', 'Mann–Whitney U test' );
setReadOnly( out, 'print', print );
return out;
}
// EXPORTS //
module.exports = mannWhitneyU;
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