Matched-Pair t-Test

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A Matched-Pair t-Test is a student's t-test based on paired samples used for examining the differences between two paired population means

References

2017a

[math]\displaystyle{ t = \frac{\overline{X}_D - \mu_0}{\frac{s_D}{\sqrt{n}}} }[/math]
For this equation, the differences between all pairs must be calculated. The pairs are either one person's pre-test and post-test scores or between pairs of persons matched into meaningful groups (for instance drawn from the same family or age group: see table). The average (XD) and standard deviation (sD) of those differences are used in the equation. The constant μ0 is non-zero if you want to test whether the average of the difference is significantly different from μ0. The degree of freedom used is n − 1, where n represents the number of pairs.

2017b

*The sampling method for each sample is simple random sampling.
*The test is conducted on paired data. (As a result, the data sets are not independent.)
*The sampling distribution is approximately normal, which is generally true if any of the following conditions apply.
* The population distribution is normal.
* The population data are symmetric, unimodal, without outliers, and the sample size is 15 or less.
* The population data are slightly skewed, unimodal, without outliers, and the sample size is 16 to 40.
* The sample size is greater than 40, without outliers.

2017c

  • (SPSS, 2017) ⇒ http://libguides.library.kent.edu/SPSS/PairedSamplestTest
  • The Paired Samples t Test compares two means that are from the same individual, object, or related units. The two means typically represent two different times (e.g., pre-test and post-test with an intervention between the two time points) or two different but related conditions or units (e.g., left and right ears, twins). The purpose of the test is to determine whether there is statistical evidence that the mean difference between paired observations on a particular outcome is significantly different from zero. The Paired Samples t Test is a parametric test.
This test is also known as:
* Dependent t Test
* Paired t Test
* Repeated Measures t Test
*The variable used in this test is known as:
Dependent variable, or test variable (continuous), measured at two different times or for two related conditions or units
(...) Note: The Paired Samples t Test can only compare the means for two (and only two) related (paired) units on a continuous outcome that is normally distributed. The Paired Samples t Test is not appropriate for analyses involving the following: 1) unpaired data; 2) comparisons between more than two units/groups; 3) a continuous outcome that is not normally distributed; and 4) an ordinal/ranked outcome.

2016

For example, suppose managers at a fitness facility want to determine whether their weight-loss program is effective. Because the "before" and "after" samples measure the same subjects, a paired t-test is the most appropriate analysis.
The paired t-test calculates the difference within each before-and-after pair of measurements, determines the mean of these changes, and reports whether this mean of the differences is statistically significant.
A paired t-test can be more powerful than a 2-sample t-test because the latter includes additional variation occurring from the independence of the observations. A paired t-test is not subject to this variation because the paired observations are dependent. Also, a paired t-test does not require both samples to have equal variance. Therefore, if you can logically address your research question with a paired design, it may be advantageous to do so, in conjunction with a paired t-test, to get more statistical power.

2014

  • (McDonald, 2014) ⇒ McDonald, J.H. 2014. Handbook of Biological Statistics (3rd ed.). Sparky House Publishing, Baltimore, Maryland. Pages 180-185: http://www.biostathandbook.com/pairedttest.html
    • Summary: Use the paired t–test when you have one measurement variable and two nominal variables, one of the nominal variables has only two values, and you only have one observation for each combination of the nominal variables; in other words, you have multiple pairs of observations. It tests whether the mean difference in the pairs is different from 0.
When to use it: Use the paired t–test when there is one measurement variable and two nominal variables. One of the nominal variables has only two values, so that you have multiple pairs of observations. The most common design is that one nominal variable represents individual organisms, while the other is "before" and "after" some treatment. Sometimes the pairs are spatial rather than temporal, such as left vs. right, injured limb vs. uninjured limb, etc. You can use the paired t–test for other pairs of observations; for example, you might sample an ecological measurement variable above and below a source of pollution in several streams.
As an example, volunteers count the number of breeding horseshoe crabs on beaches on Delaware Bay every year; here are data from 2011 and 2012. The measurement variable is number of horseshoe crabs, one nominal variable is 2011 vs. 2012, and the other nominal variable is the name of the beach. Each beach has one pair of observations of the measurement variable, one from 2011 and one from 2012. The biological question is whether the number of horseshoe crabs has gone up or down between 2011 and 2012.


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