# bartlettTest
> Compute Bartlett’s test for equal variances.
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
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.
## 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.
```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
*/
```
## Examples
```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
*/
```
