Two Sample T-Test

The Two Sample T-Test of the null hypothesis that data in the samples are random samples from independent normal distributions with equal means, against the alternative that the means are different.

This topic contains the following sections:

Two-Sample T-Test Specification
Implementation
Code Sample
See Also

Two-Sample T-Test Specification

For one sample T-test we consider two time series {X₁} with sample size N₁ and sample variance s₁ and {X₂} with sample size N₂ and sample variance s₂.

Tail.Both must be set if in the alternative hypothesis mean of the first distribution is not equal to mean of the second distribution.

Tail.Left must be set if in the alternative hypothesis mean of the first distribution is less than mean of the second distribution.

Tail.Right must be set if in the alternative hypothesis mean of the first distribution is greater than mean of the second distribution.

Equal Variances

In the case of equal variances the t-statistic is calculated as follows:

where S_X1,X2 is an estimator of the common standard deviation of the two samples:

where S_X1, S_X2 are standard deviations of the samples.

It is defined in this way so that its square is an unbiased estimator of the common variance whether or not the population means are the same. N₁ + N₂ − 2 is the total number of degrees of freedom.

Unequal Variances

In the case of unequal variances the t-statistic is calculated as follows:

where s₁² and s₂² are the unbiased estimators of the variance of the two samples. Note that in this case, it is not a pooled variance.

Welch–Satterthwaite equation is used for approximation of degrees of freedom:

Implementation

The following constructors create an instance of TTestTwoSample class.

Constructor	Description	Performance
two-tailed T-test, default significance level	Constructor without parameters. Creates TTestTwoSample instance with default significance level for two-tailed test. TTestTwoSample
default significance level	Creates TTestTwoSample instance with default significance level and user defined tail. TTestTwoSample(Tail)
two-tailed test, user defined significance level	Creates TTestTwoSample instance for two-tailed test and user-defined significance level. TTestTwoSample(Double)
user defined tail and significance level	Creates TTestTwoSample instance with user defined significance level and tail. TTestTwoSample(Double, Tail)

The class provides the following methods:

Method	Description	Performance
update	Updates test statistic using provided samples and supposition about variances equality. Double array series: Update Vector series: Update
update	Updates test statistic using provided samples which are subsets of double arrays. User specifies the first element to use and the number of elements to use in each of two arrays. Update(Double, Int32, Int32, Double, Int32, Int32)

Method

Description

Performance

update

Updates test statistic using provided samples and supposition about variances equality.

Double array series:

method Update

Vector series:

method Update

update

Updates test statistic using provided samples which are subsets of double arrays. User specifies the first element to use and the number of elements to use in each of two arrays.

method Update(Double, Int32, Int32, Double, Int32, Int32)

The class provides the following properties:

Property	Description	Performance
region of acceptance	We fail to reject null hypothesis if test statistics is between left and right borders of region of acceptance. Region of acceptance left border: AcceptanceRegionLeft Region of acceptance right border: AcceptanceRegionRight
p-value	The probability of obtaining a test statistic at least as extreme as the one that was actually observed, assuming that the null hypothesis is true. PValue

Property

Description

Performance

region of acceptance

We fail to reject null hypothesis if test statistics is between left and right borders of region of acceptance.

Region of acceptance left border:

Property AcceptanceRegionLeft

Region of acceptance right border:

Property AcceptanceRegionRight

p-value

The probability of obtaining a test statistic at least as extreme as the one that was actually observed, assuming that the null hypothesis is true.

Property PValue

Code Sample

The example of TTestTwoSample class usage:

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 1using System;
 2using FinMath.LinearAlgebra;
 3using FinMath.Statistics.HypothesisTesting;
 4using FinMath.Statistics.Distributions;
 5
 6namespace FinMath.Samples
 7{
 8    class TTestTwoSamplesSample
 9    {
10        static void Main()
11        {
12            //Create instances of normal distribution.            
13            Normal distr1 = new Normal(0, 5);
14            Normal distr2 = new Normal(3, 5);
15
16            // Generate random series.
17            Vector series1 = Vector.Random(100, distr1);
18            Vector series2 = Vector.Random(150, distr2);
19
20            // Create an instance of TTestTwoSample
21            TTestTwoSample test = new TTestTwoSample(0.05, Tail.Both);
22            test.Update(series1, series2);
23
24            Console.WriteLine("Test Result:");
25            // Test decision
26            Console.WriteLine($"  The null hypothesis failed to be rejected: {test.Decision}");
27            // The statistic of TTestTwoSample test.
28            Console.WriteLine($"  Statistics = {test.Statistics:0.000}");
29            // The p-value of the test statistic.
30            Console.WriteLine($"  P-Value = {test.PValue:0.000}");
31
32        }
33    }
34}

Reference

FinMath.Statistics.HypothesisTesting