Student's t-Test

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

Usage

var ttest = require( '@stdlib/math/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 numdigits option that controls the number of decimal digits displayed for the outputs.

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/math/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
*/