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Perform a chi-square goodness-of-fit test.
import chi2gof from 'https://cdn.jsdelivr.net/gh/stdlib-js/stats-chi2gof@esm/index.mjs';
The previous example will load the latest bundled code from the esm branch. Alternatively, you may load a specific version by loading the file from one of the tagged bundles. For example,
import chi2gof from 'https://cdn.jsdelivr.net/gh/stdlib-js/[email protected]/index.mjs';
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
.
// 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
) of expected frequencies, an array-like object (or 1-dimensional ndarray
) of population probabilities summing to one, or a discrete probability distribution name to test against.
// 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.
import Int32Array from 'https://cdn.jsdelivr.net/gh/stdlib-js/array-int32@esm/index.mjs';
import discreteUniform from 'https://cdn.jsdelivr.net/gh/stdlib-js/random-base-discrete-uniform@esm/index.mjs';
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.
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.
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
.
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
.
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
*/
- 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.
<!DOCTYPE html>
<html lang="en">
<body>
<script type="module">
import poisson from 'https://cdn.jsdelivr.net/gh/stdlib-js/random-base-poisson@esm/index.mjs';
import Int32Array from 'https://cdn.jsdelivr.net/gh/stdlib-js/array-int32@esm/index.mjs';
import chi2gof from 'https://cdn.jsdelivr.net/gh/stdlib-js/stats-chi2gof@esm/index.mjs';
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() );
</script>
</body>
</html>
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