**5.4****
****Two Sample Paired T-Test** (.NET, C#, CSharp, VB, Visual Basic, F#)

Class **TwoSamplePairedTTest**
tests the null hypothesis that the population mean of the *paired* differences of two samples is zero.
Pairing involves matching up individuals in two samples so as to minimize
their dissimilarity except in the factor under study. Paired samples
often occur in pre-test/post-test studies in which subjects are measured
before and after an intervention. They also occur in matched-pairs (for
example, matching on age and sex), cross-over trials, and sequential
observational samples. Paired samples are also called *matched*
samples and *dependent* samples.

**NOTE—****TwoSamplePairedTTest
is equivalent to performing a OneSampleTTest on the paired differences
(see ****Section ****5.3****).
**

For example, suppose we measure the thickness of plaque (mm) in the carotid artery of 10 randomly selected patients with mild atherosclerotic disease. Two measurements are taken: before treatment with Vitamin E (baseline), and after two years of taking Vitamin E daily. The mean difference between paired measurements is 0.045 with a standard deviation of 0.0264.

As described Section 5.1,
all hypothesis test classes provide two paths for constructing instances
of that type: a parameter-based method and a data-based method. Thus,
you can construct a **TwoSamplePairedTTest**
object by explicitly specifying the mean difference between paired observations
(), the standard
deviation of the differences (), and the sample size (), like so:

Code Example – C# paired t-test

double xbar = 0.045; double s = 0.0264; int n = 10; var test = new TwoSamplePairedTTest( xbar, s, n );

Alternatively, you can supply two sets of sample data.
For instance, this code adds data to a **DataFrame**
(Chapter 2):

Code Example – C# paired t-test

var df = new DataFrame(); df.AddColumn( new DFNumericColumn( "Baseline" ) ); df.AddColumn( new DFNumericColumn( "Vit E" ) ); df.AddRow( 1, 0.66, 0.60 ); df.AddRow( 2, 0.72, 0.65 ); df.AddRow( 3, 0.85, 0.79 ); df.AddRow( 4, 0.62, 0.63 ); df.AddRow( 5, 0.59, 0.54 ); df.AddRow( 6, 0.63, 0.55 ); df.AddRow( 7, 0.64, 0.62 ); df.AddRow( 8, 0.70, 0.67 ); df.AddRow( 9, 0.73, 0.68 ); df.AddRow( 10, 0.68, 0.64 );

And this code constructs a **TwoSamplePairedTTest**
from the two columns of data:

Code Example – C# paired t-test

TwoSamplePairedTTest test = new TwoSamplePairedTTest( df[ "Baseline" ], df[ "Vit E" ] );

The mean difference between paired measurements, the standard deviation, and the sample size are calculated from the given data.

In addition to the properties common to all hypothesis
test objects (Section 5.1),
a **TwoSamplePairedTTest** object provides
the following read-only properties:

● Xbar gets the mean of the differences between paired observations.

● S gets the standard deviation of the differences between paired observations.

● N gets the number of pairs.

● DegreesOfFreedom gets the degrees of freedom.

By default, a **TwoSamplePairedTTest**
object performs a two-sided hypothesis test () with . Non-default test parameters
can be specified at the time of construction using constructor overloads,
or after construction using the provided Type
and Alpha properties.

Once you've constructed and configured a **TwoSamplePairedTTest** object, you can
access the various test results using the provided properties, as described
in Section 5.1:

Code Example – C# paired t-test

Console.WriteLine( "t-statistic = " + test.Statistic ); Console.WriteLine( "deg of freedom = " + test.DegreesOfFreedom ); Console.WriteLine( "p-value = " + test.P ); Console.WriteLine( "reject the null hypothesis? " + test.Reject);

The output is:

t-statistic = 5.4 deg of freedom = 9 p-value = 0.000433006432003502 reject the null hypothesis? True

This indicates that we can reject the null hypotheses (). We can conclude that the true mean thickness of plaque after two years treatment with Vitamin E is significantly different than before treatment.

Finally, remember that the ToString() method returns a formatted string representation of the complete test results:

Two Sample t Test (Paired) -------------------------- Mean of differences between pairs = 0.045 Standard deviation of differences between pairs = 0.0263523138347365 Sample size (number of pairs) = 10 Computed t statistic: 5.4, df = 9 Hypothesis type: two-sided Null hypothesis: true mean of differences between pairs = 0 Alt hypothesis: true mean of differences between pairs != 0 P-value: 0.000433006432003502 REJECT the null hypothesis for alpha = 0.01 0.99 confidence interval: 0.0179180371533991 0.0720819628466008