Binomial Test

Exact test for the success probability in a Bernoulli experiment.

Usage

var binomialTest = require( '@stdlib/stats/binomial-test' );

binomialTest( x[, n][, opts] )

When supplied nonnegative integers x (number of successes in a Bernoulli experiment) and n (total number of trials), the function computes an exact test for the success probability in a Bernoulli experiment. Alternatively, x may be a two-element array containing the number of successes and failures, respectively.

var out = binomialTest( 550, 1000 );
/* returns
    {
        'rejected': true,
        'pValue': ~0.001,
        'statistic': 0.55,
        'ci': [ ~0.519, ~0.581 ],
        // ...
    }
*/

out = binomialTest( [ 550, 450 ] );
/* returns
    {
        'rejected': true,
        'pValue': ~0.001,
        'statistic': 0.55,
        'ci': [ ~0.519, ~0.581 ],
        // ...
    }
*/

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

    Alternative hypothesis: True correlation coefficient is not equal to 0.5

        pValue: 0.0017
        statistic: 0.55
        95% confidence interval: [0.5186,0.5811]

    Test Decision: 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 the true ratio of variances is greater than one (greater), smaller than one (less), or that the variances are the same (two-sided). Default: two-sided.
  • p: success probability under the null hypothesis. Default: 0.5.

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 out = binomialTest( 59, 100, {
    'alpha': 0.1
});
/* returns
    {
        'rejected': true,
        'pValue': ~0.089,
        'statistic': 0.59,
        'ci': [ ~0.487, ~0.687 ],
        // ...
    }
*/

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

out = binomialTest( 550, 1000, {
    'alternative': 'greater'
});
table = out.print();
/** e.g., returns
    Exact binomial test

    Alternative hypothesis: True correlation coefficient is greater than 0.5

        pValue: 0.0009
        statistic: 0.55
        95% confidence interval: [0.5235,1]

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

out = binomialTest( 550, 1000, {
    'alternative': 'less'
});
table = out.print();
/* e.g., returns
    Exact binomial test

    Alternative hypothesis: True correlation coefficient is less than 0.5

        pValue: 0.9993
        statistic: 0.55
        95% confidence interval: [0,0.5762]

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

To test whether the success probability in the population is equal to some other value than 0.5, set the p option.

var out = binomialTest( 23, 100, {
    'p': 0.2
});
/* returns
    {
        'rejected': false,
        'pValue': ~0.453,
        'statistic': 0.23,
        'ci': [ ~0.152, ~0.325 ],
        // ...
    }
*/

var table = out.print();
/* e.g., returns
    Exact binomial test

    Alternative hypothesis: True correlation coefficient is not equal to 0.2

        pValue: 0.4534
        statistic: 0.23
        95% confidence interval: [0.1517,0.3249]

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

Examples

var binomialTest = require( '@stdlib/stats/binomial-test' );

var out = binomialTest( 682, 925 );
/* returns
    {
        'rejected': true,
        'pValue': ~3.544e-49,
        'statistic': 0.737,
        'ci': [ ~0.708, ~0.765 ],
        // ...
    }
*/

out = binomialTest( [ 682, 925 - 682 ] );
/* returns
    {
        'rejected': true,
        'pValue': ~3.544e-49,
        'statistic': 0.737,
        'ci': [ ~0.708, ~0.765 ],
        // ...
    }
*/

out = binomialTest( 682, 925, {
    'p': 0.75,
    'alpha': 0.05
});
/* returns
    {
        'rejected': false,
        'pValue': ~0.382
        'statistic': 0.737,
        'ci': [ ~0.708, ~0.765 ],
        // ...
    }
*/

out = binomialTest( 21, 40, {
    'p': 0.4,
    'alternative': 'greater'
});
/* returns
    {
        'rejected': false,
        'pValue': ~0.382,
        'statistic': 0.737,
        'ci': [ ~0.385, 1.0 ],
        // ...
    }
*/