Student's t-Test

Two-sample Student's t-Test.

Usage

var ttest2 = require( '@stdlib/math/stats/ttest2' );

ttest2( x, y[, opts] )

By default, the function performs a two-sample t-test for the null hypothesis that the data in arrays or typed arrays x and y is independently drawn from normal distributions with equal means.

// Student's sleep data:
var x = [ 0.7, -1.6, -0.2, -1.2, -0.1, 3.4, 3.7, 0.8, 0.0, 2.0 ];
var y = [ 1.9, 0.8, 1.1, 0.1, -0.1, 4.4, 5.5, 1.6, 4.6, 3.4 ];

var out = ttest2( x, y );
/* e.g., returns
    {
        rejected: false,
        pValue: ~0.079,
        statistic: ~-1.861,
        ci: [ ~-3.365, ~0.205 ],
        // ...
    }
*/

The returned object comes with a .print() method which when invoked will print a formatted output of the results of the hypothesis test.

console.log( out.print() );
/* e.g., =>
    Welch two-sample t-test

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

        pValue: 0.0794
        statistic: -1.8608
        95% confidence interval: [-3.3655,0.2055]

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

The 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 x has a larger mean than y (greater), x has a smaller mean than y (less) or the means are the same (two-sided). Default: two-sided.
  • difference: number denoting the difference in means under the null hypothesis. Default: 0.
  • variance: string indicating if the test should be conducted under the assumption that the unknown variances of the normal distributions are equal or unequal. Default: unequal.

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 x = [ 0.7, -1.6, -0.2, -1.2, -0.1, 3.4, 3.7, 0.8, 0.0, 2.0 ];
var y = [ 1.9, 0.8, 1.1, 0.1, -0.1, 4.4, 5.5, 1.6, 4.6, 3.4 ];

var out = ttest2( x, y, {
    'alpha': 0.1
});
var table = out.print();
/* e.g., returns
    Welch two-sample t-test

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

        pValue: 0.0794
        statistic: -1.8608
        90% confidence interval: [-3.0534,-0.1066]

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

By default, a two-sided test is performed. To perform either of the one-sided tests, set the alternative option to less or greater.

// Student's sleep data:
var x = [ 0.7, -1.6, -0.2, -1.2, -0.1, 3.4, 3.7, 0.8, 0.0, 2.0 ];
var y = [ 1.9, 0.8, 1.1, 0.1, -0.1, 4.4, 5.5, 1.6, 4.6, 3.4 ];

var out = ttest2( x, y, {
    'alternative': 'less'
});
var table = out.print();
/* e.g., returns
    Welch two-sample t-test

    Alternative hypothesis: True difference in means is less than 0

        pValue: 0.0397
        statistic: -1.8608
        df: 17.7765
        95% confidence interval: [-Infinity,-0.1066]

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

out = ttest2( x, y, {
    'alternative': 'greater'
});
table = out.print();
/* e.g., returns
    Welch two-sample t-test

    Alternative hypothesis: True difference in means is greater than 0

        pValue: 0.9603
        statistic: -1.8608
        df: 17.7765
        95% confidence interval: [-3.0534,Infinity]

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

As a default choice, the ttest2 function carries out the Welch test (using the Satterthwaite approximation for the degrees of freedom), which does not have the requirement that the variances of the underlying distributions are equal. If the equal variances assumption seems warranted, set the variance option to equal.

var x = [ 2, 3, 1, 4 ];
var y = [ 1, 2, 3, 1, 2, 5, 3, 4 ];

var out = ttest2( x, y, {
    'variance': 'equal'
});
var table = out.print();
/* e.g., returns
    Two-sample t-test

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

        pValue: 0.8848
        statistic: -0.1486
        df: 10
        95% confidence interval: [-1.9996,1.7496]

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

To test whether the difference in the population means is equal to some other value than 0, set the difference option.

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

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

rnorm = normal({
    'seed': 372
});

x = new Array( 100 );
for ( i = 0; i < x.length; i++ ) {
    x[ i ] = rnorm( 2.0, 3.0 );
}
y = new Array( 100 );
for ( i = 0; i < x.length; i++ ) {
    y[ i ] = rnorm( 1.0, 3.0 );
}

out = ttest2( x, y, {
    'difference': 1.0,
    'variance': 'equal'
});
/* e.g., returns
    {
        rejected: false,
        pValue: ~0.642,
        statistic: ~-0.466,
        ci: [ ~-0.0455, ~1.646 ],
        // ...
    }
*/

table = out.print();
/* e.g., returns
    Two-sample t-test

    Alternative hypothesis: True difference in means is not equal to 1

        pValue: 0.6419
        statistic: -0.4657
        df: 198
        95% confidence interval: [-0.0455,1.646]

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

Examples

var incrspace = require( '@stdlib/math/utils/incrspace' );
var ttest2 = require( '@stdlib/math/stats/ttest2' );

var table;
var out;
var a;
var b;

a = incrspace( 1, 11, 1 );
b = incrspace( 7, 21, 1 );

out = ttest2( a, b );
table = out.print();
/* e.g., returns
    Welch two-sample t-test

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

        pValue: 0
        statistic: -5.4349
        95% confidence interval: [-11.0528,-4.9472]

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