time-to-botec

README.md (5821B)

---

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 # bartlettTest
 
 > Compute Bartlett’s test for equal variances.
 
 <section class="intro">
 
 Bartlett's test is used to test the null hypothesis that the variances of k groups are equal against the alternative that at least two of them are different.
 
 For `k` groups each with `n_i` observations, the test statistic is
 
 <!-- <equation class="equation" label="eq:bartlett-test-statistic" align="center" raw="\chi^2 = \frac{N\ln(S^2) - \sum_{i=0}^{k-1} n_i \ln(S_i^2)}{1 + \frac{1}{3(k-1)}\left(\sum_{i=0}^{k-1} \frac{1}{n_i} - \frac{1}{N}\right)}" alt="Equation for Bartlett's test statistic."> -->
 
 <div class="equation" align="center" data-raw-text="\chi^2 = \frac{N\ln(S^2) - \sum_{i=0}^{k-1} n_i \ln(S_i^2)}{1 + \frac{1}{3(k-1)}\left(\sum_{i=0}^{k-1} \frac{1}{n_i} - \frac{1}{N}\right)}" data-equation="eq:bartlett-test-statistic">
     <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@4b1db4ebd815eb54bf53a3fa132b992604743d9c/lib/node_modules/@stdlib/stats/bartlett-test/docs/img/equation_bartlett-test-statistic.svg" alt="Equation for Bartlett's test statistic.">
     <br>
 </div>
 
 <!-- </equation> -->
 
 where `N` is the total number of observations, `S_i` are the biased group-level variances and `S^2` is a (biased) pooled estimate for the variance. Under the null hypothesis, the test statistic follows a _chi-square_ distribution with `df = k - 1` degrees of freedom.
 
 </section>
 
 <!-- /.intro -->
 
 <section class="usage">
 
 ## Usage
 
 ```javascript
 var bartlettTest = require( '@stdlib/stats/bartlett-test' );
 ```
 
 #### bartlettTest( a\[,b,...,k]\[, opts] )
 
 For input arrays `a`, `b`, ... holding numeric observations, this function calculates Bartlett’s test, which tests the null hypothesis that the variances in all `k` groups are the same. 
 
 ```javascript
 // Data from Hollander & Wolfe (1973), p. 116:
 var x = [ 2.9, 3.0, 2.5, 2.6, 3.2 ];
 var y = [ 3.8, 2.7, 4.0, 2.4 ];
 var z = [ 2.8, 3.4, 3.7, 2.2, 2.0 ];
 
 var out = bartlettTest( x, y, z );
 /* returns
     {
         'rejected': false,
         'alpha': 0.05,
         'df': 2,
         'pValue': ~0.573,
         'statistic': ~1.112,
         ...
     }
 */
 ```
 
 The function accepts the following `options`:
 
 -   **alpha**: `number` in the interval `[0,1]` giving the significance level of the hypothesis test. Default: `0.05`.
 -   **groups**: an `array` of group indicators. If set, the function assumes that only a single numeric array is provided holding all observations.
 
 By default, the test is carried out at a significance level of `0.05`. To choose a custom significance level, set the `alpha` option.
 
 ```javascript
 var x = [ 2.9, 3.0, 2.5, 2.6, 3.2 ];
 var y = [ 3.8, 2.7, 4.0, 2.4 ];
 var z = [ 2.8, 3.4, 3.7, 2.2, 2.0 ];
 
 var out = bartlettTest( x, y, z, {
     'alpha': 0.01
 });
 /* returns
     {
         'rejected': false,
         'alpha': 0.01,
         'df': 2,
         'pValue': ~0.573,
         'statistic': ~1.112,
         ...
     }
 */
 ```
 
 The function provides an alternate interface by supplying an array of group indicators to the `groups` option. In this case, it is assumed that only a single numeric array holding all observations is provided to the function.
 
 <!-- eslint-disable array-element-newline -->
 
 ```javascript
 var arr = [
     2.9, 3.0, 2.5, 2.6, 3.2,
     3.8, 2.7, 4.0, 2.4,
     2.8, 3.4, 3.7, 2.2, 2.0
 ];
 var groups = [
     'a', 'a', 'a', 'a', 'a',
     'b', 'b', 'b', 'b',
     'c', 'c', 'c', 'c', 'c'
 ];
 var out = bartlettTest( arr, {
     'groups': groups
 });
 ```
 
 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.
 
 ```javascript
 var x = [ 2.9, 3.0, 2.5, 2.6, 3.2 ];
 var y = [ 3.8, 2.7, 4.0, 2.4 ];
 var z = [ 2.8, 3.4, 3.7, 2.2, 2.0 ];
 
 var out = bartlettTest( x, y, z );
 console.log( out.print() );
 /* =>
     Bartlett's test of equal variances
 
     Null hypothesis: The variances in all groups are the same.
 
         pValue: 0.5735
         statistic: 1.1122
         df: 2
 
     Test Decision: Fail to reject null in favor of alternative at 5% significance level
 */
 ```
 
 </section>
 
 <!-- /.usage -->
 
 <section class="examples">
 
 ## Examples
 
 <!-- eslint no-undef: "error" -->
 
 ```javascript
 var bartlettTest = require( '@stdlib/stats/bartlett-test' );
 
 // Data from Hollander & Wolfe (1973), p. 116:
 var x = [ 2.9, 3.0, 2.5, 2.6, 3.2 ];
 var y = [ 3.8, 2.7, 4.0, 2.4 ];
 var z = [ 2.8, 3.4, 3.7, 2.2, 2.0 ];
 
 var out = bartlettTest( x, y, z );
 /* returns
     {
         'rejected': false,
         'alpha': 0.05,
         'df': 2,
         'pValue': ~0.573,
         'statistic': ~1.112,
         ...
     }
 */
 
 var table = out.print();
 /* returns
     Bartlett's test of equal variances
 
     Null hypothesis: The variances in all groups are the same.
 
         pValue: 0.5735
         statistic: 1.1122
         df: 2
 
     Test Decision: Fail to reject null in favor of alternative at 5% significance level
 */
 ```
 
 </section>
 
 <!-- /.examples -->
 
 <section class="references">
 
 </section>
 
 <!-- /.references -->
 
 <section class="links">
 
 </section>
 
 <!-- /.links -->