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Student's t-Test

One-sample and paired Student's t-Test.

Installation

npm install @stdlib/stats-ttest

Alternatively,

• To load the package in a website via a script tag without installation and bundlers, use the ES Module available on the esm branch (see README).
• If you are using Deno, visit the deno branch (see README for usage intructions).
• For use in Observable, or in browser/node environments, use the Universal Module Definition (UMD) build available on the umd branch (see README).

The branches.md file summarizes the available branches and displays a diagram illustrating their relationships.

To view installation and usage instructions specific to each branch build, be sure to explicitly navigate to the respective README files on each branch, as linked to above.

Usage

var ttest = require( '@stdlib/stats-ttest' );

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

The function performs a one-sample t-test for the null hypothesis that the data in array or typed array x is drawn from a normal distribution with mean zero and unknown variance.

var normal = require( '@stdlib/random-base-normal' ).factory;

var rnorm;
var arr;
var out;
var i;

rnorm = normal( 0.0, 2.0, {
    'seed': 5776
});
arr = new Array( 100 );
for ( i = 0; i < arr.length; i++ ) {
    arr[ i ] = rnorm();
}
out = ttest( arr );
/* e.g., returns
    {
        'rejected': false,
        'pValue': ~0.722,
        'statistic': ~0.357,
        'ci': [~-0.333,~0.479],
        // ...
    }
*/

When array or typed array y is supplied, the function tests whether the differences x - y come from a normal distribution with mean zero and unknown variance via the paired t-test.

var normal = require( '@stdlib/random-base-normal' ).factory;

var rnorm;
var out;
var i;
var x;
var y;

rnorm = normal( 1.0, 2.0, {
    'seed': 786
});
x = new Array( 100 );
y = new Array( 100 );
for ( i = 0; i < x.length; i++ ) {
    x[ i ] = rnorm();
    y[ i ] = rnorm();
}
out = ttest( x, y );
/* e.g., returns
    {
        'rejected': false,
        'pValue': ~0.191,
        'statistic': ~1.315,
        'ci': [ ~-0.196, ~0.964 ],
        // ...
    }
*/

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 t-test

    Alternative hypothesis: True difference in means is not equal to 0

        pValue: 0.1916
        statistic: 1.3148
        df: 99
        95% confidence interval: [-0.1955,0.9635]

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

The ttest 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.
• mu: number denoting the hypothesized true mean under the null hypothesis. Default: 0.

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 = ttest( arr, {
    'alpha': 0.01
});
table = out.print();
/* e.g., returns
    One-sample t-test

    Alternative hypothesis: True mean is not equal to 0

        pValue: 0.0474
        statistic: 2.8284
        df: 4
        99% confidence interval: [-1.2556,5.2556]

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

out = ttest( arr, {
    'alpha': 0.1
});
table = out.print();
/* e.g., returns
    One-sample t-test

    Alternative hypothesis: True mean is not equal to 0

        pValue: 0.0474
        statistic: 2.8284
        df: 4
        90% confidence interval: [0.4926,3.5074]

    Test Decision: Reject null in favor of alternative at 10% significance level
*/

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

var out;
var arr;

arr = [ 4, 4, 6, 6, 5 ];

out = ttest( arr, {
    'mu': 5
});
/* e.g., returns
{
    'rejected': false,
    'pValue': 1,
    'statistic': 0,
    'ci': [ ~3.758, ~6.242 ],
    // ...
}
*/

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 table;
var out;
var arr;

arr = [ 4, 4, 6, 6, 5 ];

out = ttest( arr, {
    'alternative': 'less'
});
table = out.print();
/* e.g., returns
    One-sample t-test

    Alternative hypothesis: True mean is less than 0

        pValue: 0.9998
        statistic: 11.1803
        df: 4
        95% confidence interval: [-Infinity,5.9534]

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

out = ttest( arr, {
    'alternative': 'greater'
});
table = out.print();
/* e.g., returns
    One-sample t-test

    Alternative hypothesis: True mean is greater than 0

        pValue: 0.0002
        statistic: 11.1803
        df: 4
        95% confidence interval: [4.0466,Infinity]

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

Examples

var normal = require( '@stdlib/random-base-normal' ).factory;
var ttest = require( '@stdlib/stats-ttest' );

var rnorm;
var arr;
var out;
var i;

rnorm = normal( 5.0, 4.0, {
    'seed': 37827
});
arr = new Array( 100 );
for ( i = 0; i < arr.length; i++ ) {
    arr[ i ] = rnorm();
}

// Test whether true mean is equal to zero:
out = ttest( arr );
console.log( out.print() );
/* e.g., =>
    One-sample t-test

    Alternative hypothesis: True mean is not equal to 0

        pValue: 0
        statistic: 15.0513
        df: 99
        95% confidence interval: [4.6997,6.127]

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

// Test whether true mean is equal to five:
out = ttest( arr, {
    'mu': 5.0
});
console.log( out.print() );
/* e.g., =>
    One-sample t-test

    Alternative hypothesis: True mean is not equal to 5

        pValue: 0.2532
        statistic: 1.1494
        df: 99
        95% confidence interval: [4.6997,6.127]

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

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See Also

• @stdlib/stats-ttest2: two-sample Student's t-Test.

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Notice

This package is part of 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, see the main project repository.

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License

See LICENSE.

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