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# Chi-square goodness-of-fit test [![NPM version][npm-image]][npm-url] [![Build Status][test-image]][test-url] [![Coverage Status][coverage-image]][coverage-url] > Perform a chi-square goodness-of-fit test.
## Installation ```bash npm install @stdlib/stats-chi2gof ```
## Usage ```javascript var chi2gof = require( '@stdlib/stats-chi2gof' ); ``` #### chi2gof( x, y\[, ...args]\[, options] ) Computes a chi-square goodness-of-fit test for the **null hypothesis** that the values of `x` come from the discrete probability distribution specified by `y`. ```javascript // Observed counts: var x = [ 30, 20, 23, 27 ]; // Expected counts: var y = [ 25, 25, 25, 25 ]; var res = chi2gof( x, y ); var o = res.toJSON(); /* returns { 'rejected': false, 'alpha': 0.05, 'pValue': ~0.5087, 'df': 3, 'statistic': ~2.32, ... } */ ``` The second argument can either be an array-like object (or 1-dimensional [`ndarray`][@stdlib/ndarray/array]) of expected frequencies, an array-like object (or 1-dimensional [`ndarray`][@stdlib/ndarray/array]) of population probabilities summing to one, or a discrete probability distribution name to test against. ```javascript // Observed counts: var x = [ 89, 37, 30, 28, 2 ]; // Expected probabilities: var y = [ 0.40, 0.20, 0.20, 0.15, 0.05 ]; var res = chi2gof( x, y ); var o = res.toJSON(); /* returns { 'rejected': true, 'alpha': 0.05, 'pValue': ~0.0187, 'df': 3, 'statistic': ~9.9901, ... } */ ``` When specifying a discrete probability distribution name, distribution parameters **must** be provided as additional arguments. ```javascript var Int32Array = require( '@stdlib/array-int32' ); var discreteUniform = require( '@stdlib/random-base-discrete-uniform' ); var res; var x; var v; var i; // Simulate expected counts... x = new Int32Array( 100 ); for ( i = 0; i < x.length; i++ ) { v = discreteUniform( 0, 99 ); x[ v ] += 1; } res = chi2gof( x, 'discrete-uniform', 0, 99 ); // returns {...} ``` The function accepts the following `options`: - **alpha**: significance level of the hypothesis test. Must be on the interval `[0,1]`. Default: `0.05`. - **ddof**: "delta degrees of freedom" adjustment. Must be a nonnegative integer. Default: `0`. - **simulate**: `boolean` indicating whether to calculate p-values by Monte Carlo simulation. Default: `false`. - **iterations**: number of Monte Carlo iterations. Default: `500`. By default, the test is performed at a significance level of `0.05`. To adjust the significance level, set the `alpha` option. ```javascript var x = [ 89, 37, 30, 28, 2 ]; var p = [ 0.40, 0.20, 0.20, 0.15, 0.05 ]; var res = chi2gof( x, p ); var table = res.toString(); /* e.g., returns Chi-square goodness-of-fit test Null hypothesis: population probabilities are equal to those in p pValue: 0.0186 statistic: 9.9901 degrees of freedom: 3 Test Decision: Reject null in favor of alternative at 5% significance level */ res = chi2gof( x, p, { 'alpha': 0.01 }); table = res.toString(); /* e.g., returns Chi-square goodness-of-fit test Null hypothesis: population probabilities are equal to those in p pValue: 0.0186 statistic: 9.9901 degrees of freedom: 3 Test Decision: Fail to reject null in favor of alternative at 1% significance level */ ``` By default, the p-value is computed using a chi-square distribution with `k-1` degrees of freedom, where `k` is the length of `x`. If provided distribution arguments are estimated (e.g., via maximum likelihood estimation), the degrees of freedom **should** be corrected. Set the `ddof` option to use `k-1-n` degrees of freedom, where `n` is the degrees of freedom adjustment. ```javascript var x = [ 89, 37, 30, 28, 2 ]; var p = [ 0.40, 0.20, 0.20, 0.15, 0.05 ]; var res = chi2gof( x, p, { 'ddof': 1 }); var o = res.toJSON(); // returns { 'pValue': ~0.0186, 'statistic': ~9.9901, 'df': 3, ... } ``` Instead of relying on chi-square approximation to calculate the p-value, one can use Monte Carlo simulation. When the `simulate` option is `true`, the simulation is performed by re-sampling from the discrete probability distribution specified by `y`. ```javascript var x = [ 89, 37, 30, 28, 2 ]; var p = [ 0.40, 0.20, 0.20, 0.15, 0.05 ]; var res = chi2gof( x, p, { 'simulate': true, 'iterations': 1000 // explicitly set the number of Monte Carlo simulations }); // returns {...} ``` The function returns a results `object` having the following properties: - **alpha**: significance level. - **rejected**: `boolean` indicating the test decision. - **pValue**: test p-value. - **statistic**: test statistic. - **df**: degrees of freedom. - **method**: test name. - **toString**: serializes results as formatted test output. - **toJSON**: serializes results as a JSON object. To print formatted test output, invoke the `toString` method. The method accepts the following options: - **digits**: number of displayed decimal digits. Default: `4`. - **decision**: `boolean` indicating whether to show the test decision. Default: `true`. ```javascript var x = [ 89, 37, 30, 28, 2 ]; var p = [ 0.40, 0.20, 0.20, 0.15, 0.05 ]; var res = chi2gof( x, p ); var table = res.toString({ 'decision': false }); /* e.g., returns Chi-square goodness-of-fit test Null hypothesis: population probabilities are equal to those in p pValue: 0.0186 statistic: 9.9901 degrees of freedom: 3 */ ```
## Notes - The chi-square approximation may be incorrect if the observed or expected frequencies in each category are too small. Common practice is to require frequencies **greater than** five.
## Examples ```javascript var poisson = require( '@stdlib/random-base-poisson' ); var Int32Array = require( '@stdlib/array-int32' ); var chi2gof = require( '@stdlib/stats-chi2gof' ); var N = 400; var lambda = 3.0; var rpois = poisson.factory( lambda ); // Draw samples from a Poisson distribution: var x = []; var i; for ( i = 0; i < N; i++ ) { x.push( rpois() ); } // Generate a frequency table: var freqs = new Int32Array( N ); for ( i = 0; i < N; i++ ) { freqs[ x[ i ] ] += 1; } // Assess whether the simulated values come from a Poisson distribution: var out = chi2gof( freqs, 'poisson', lambda ); // returns {...} console.log( out.toString() ); ```
* * * ## Notice This package is part of [stdlib][stdlib], a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more. For more information on the project, filing bug reports and feature requests, and guidance on how to develop [stdlib][stdlib], see the main project [repository][stdlib]. #### Community [![Chat][chat-image]][chat-url] --- ## License See [LICENSE][stdlib-license]. ## Copyright Copyright © 2016-2026. The Stdlib [Authors][stdlib-authors].
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