Matched-Pair t-Test Task

A Matched-Pair t-Test Task is a statistical hypothesis testing task used to describe an matched-pair t-test.

• AKA: Matched-Pairs t-Test, Dependent t-Test, Repeated Measures t-Test, Related Samples t-Test.
• Context:
  • It can be solved by an Matched-Pair t-Test System (that implements an matched-pair t-test algorithm).
  • Task Input:
    • Input Data :
      • X =[math]\displaystyle{ \{x_1,x_2,\cdots,x_n\} }[/math] , first sample dataset drawn from a statistical population of interest.
      • Y =[math]\displaystyle{ \{y_1,y_2,\cdots,y_n\} }[/math], second sample dataset drawn from the same statistical population of interest
      • Data set requirements: [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] are related (i.e. dependent datasets) and can be paired [math]\displaystyle{ (X,Y) =\{(x_1,y_1),(x_2,y_2),\cdots,(x_n,y_n)\} }[/math]
    • Input Parameters:
      • [math]\displaystyle{ \mu_d=\mu_0 }[/math], a hypothesized difference between population means,
      • [math]\displaystyle{ \alpha_0 }[/math] = a significance level value or a confidence level (a percentage).
  • output:
    • matched-pair t-test statistic value,
    • P-value or Region of Acceptance
    • Region of Rejection (optional)
    • Decision Errors (optional).
  • Task Requirements
    • Verification of Test Requirement(s) (Optional):
      • Population Variances are equal. This may include a Bartlett's Test or a Levene's Test for the homogeneity of variance.
      • Sampling distribution can be approximated to a normal distribution.
    • Hypotheses Statement: Uses the input parameter [math]\displaystyle{ \mu_0 }[/math] to state a null hypothesis and an alternative hypothesis according to one-tailed matched-pair t-test or two-tailed matched-pair t-test.
    • Test Statistic Calculation: It is solved by matched-pair t-statistic calculating system.
    • P-value and/or Region of acceptance computation: these require a t-distribution calculator or t-table.
    • Decision Rule: Null hypothesis is reject if P-value is less than [math]\displaystyle{ \alpha_0 }[/math] or if the t-test statistic value follows outside region of acceptance.
• Example(s):
  • Let's consider paired datasets corresponding Full Scale Intelligent Quotient (FSIQ) and Performance Intelligent Quotient (PIQ) measured and retrieved from http://www.scipy-lectures.org/_downloads/brain_size.csv and the Paired Samples t-Test System based on the scipy statistical function stats.ttest_rel()[1] then:
    • Hypothesis statement:
      • Null Hypothesis: Mean Full Scale IQ (FSIQ) and Mean Performance IQ (PIQ), measured on the same individuals, are equal. [math]\displaystyle{ H_0:\; \mu_{FSIQ}=\mu_{PIQ} }[/math]
      • Alternative Hypothesis: Mean Full Scale IQ (FSIQ) and Mean Performance IQ (PIQ), measured on the same individuals, are unequal. [math]\displaystyle{ H_A:\; \mu_{FSIQ}\neq\mu_{PIQ} }[/math]
    • Task input: sample datasets = data['FSIQ'] and data['PIQ'].
    • Task output: t-statistic = 1.78420194059 , p-value=0.0821726381836
    • For [math]\displaystyle{ \alpha_0=0.001,0.01,0.025, 0.05 }[/math] the test fails to reject null hypothesis is rejected as p-value is greater than significance level. We can concluded the FSIQ and PIQ means are very similar.
  • …
• Counter-Example(s):
  • Welch's t-Test Task.
  • Linear Regression t-Test.
  • Independent Two-Sample t-Test Task.
  • One-Sample t-Test Task.
• See: Independent Two-Sample t-Test System, Statistical Significance, Sample Average, Sample Variance.

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References

2017a

• (Wikipedia, 2017) ⇒ http://en.wikipedia.org/wiki/Student%27s_t-test#
  • This test is used when the samples are dependent; that is, when there is only one sample that has been tested twice (repeated measures) or when there are two samples that have been matched or "paired". This is an example of a paired difference test.

[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 (X_D) and standard deviation (s_D) 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

• (Stattrek, 2017) ⇒ http://stattrek.com/hypothesis-test/paired-means.aspx?Tutorial=AP
  • This lesson explains how to conduct a hypothesis test for the difference between paired means. The test procedure, called the matched-pairs t-test, is appropriate when the following conditions are met:

*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.

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