273 lines
7.2 KiB
Markdown
273 lines
7.2 KiB
Markdown
<!--
|
|
|
|
@license Apache-2.0
|
|
|
|
Copyright (c) 2018 The Stdlib Authors.
|
|
|
|
Licensed under the Apache License, Version 2.0 (the "License");
|
|
you may not use this file except in compliance with the License.
|
|
You may obtain a copy of the License at
|
|
|
|
http://www.apache.org/licenses/LICENSE-2.0
|
|
|
|
Unless required by applicable law or agreed to in writing, software
|
|
distributed under the License is distributed on an "AS IS" BASIS,
|
|
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
|
|
See the License for the specific language governing permissions and
|
|
limitations under the License.
|
|
|
|
-->
|
|
|
|
# Student's t-Test
|
|
|
|
> Two-sample Student's t-Test.
|
|
|
|
<section class="usage">
|
|
|
|
## Usage
|
|
|
|
```javascript
|
|
var ttest2 = require( '@stdlib/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][mdn-array] or [typed arrays][mdn-typed-array] `x` and `y` is independently drawn from normal distributions with _equal_ means.
|
|
|
|
```javascript
|
|
// 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. `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.
|
|
|
|
<!-- run-disable -->
|
|
|
|
```javascript
|
|
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.
|
|
|
|
```javascript
|
|
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`.
|
|
|
|
```javascript
|
|
// 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`.
|
|
|
|
```javascript
|
|
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.
|
|
|
|
```javascript
|
|
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
|
|
*/
|
|
```
|
|
|
|
</section>
|
|
|
|
<!-- /.usage -->
|
|
|
|
<section class="examples">
|
|
|
|
## Examples
|
|
|
|
<!-- eslint no-undef: "error" -->
|
|
|
|
```javascript
|
|
var incrspace = require( '@stdlib/array/incrspace' );
|
|
var ttest2 = require( '@stdlib/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
|
|
*/
|
|
```
|
|
|
|
</section>
|
|
|
|
<!-- /.examples -->
|
|
|
|
<section class="links">
|
|
|
|
[mdn-array]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Array
|
|
|
|
[mdn-typed-array]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Typed_arrays
|
|
|
|
</section>
|
|
|
|
<!-- /.links -->
|