The Paired t-Test (A.K.A. Dependent Samples t-Test, or t-Test for Correlated Groups)

1
The Paired t-Test(A.K.A. Dependent Samples
t-Test, or t-Test for Correlated Groups)

• Advanced Research Methods in Psychology
• - lecture
• Matthew Rockloff

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When to use the paired t-test

• In many research designs, it is helpful to
  measure the same people more than once.
• A common example is testing for performance
  improvements (or decrements) over time.
• However, in any circumstance where multiple
  measurements are made on the same person (or
  experimental unit), it may be useful to observe
  if there are mean differences between these
  measurements.
• The paired t-test will show whether the
  differences observed in the 2 measures will be
  found reliably in repeated samples.

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Example 4.1

• In this example, we will look at the throwing
  distance for junior varsity javelin toss (in
  meters).
• Five players are selected at random from the
  entire league.
• We are interested in the following research
  question
• Do players improve on their distance between the
  pre and post season?
• The average throwing distance, in both pre and
  post season, is recorded in columns 1 and 2
  (see next slide) for each of 5 people (P1-P5)

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Example 4.1 (cont.)
Column 1 Column 2 Column 3
Column 4
X1 Pre-season X2 Post-season
P1 1 2 1 4
P2 2 4.5 0 0.25
P3 2 3 0 1
P4 2 4.5 0 0.25
P5 3 6 1 4
2 4
0.4 1.9
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Example 4.1 (cont.)

• Unlike the independent samples t-test, on each
  row the numbers in columns 2 and 3 come from the
  same people.
• Person 1, for example, threw an average of 1
  meter pre-season, but improved to an average of 2
  meters in the post-season (after all competition
  was completed).
• It appears that this player may have improved
  through practice.
• How can we find if the league has improved
  overall from the pre to the post season?

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Example 4.1 (cont.)

• The paired t-test will allow us to see if the
  improvement that we see in this sample is
  reliable.
• If we selected another 5 players at random from
  the league, would we still see an improvement?
• Without having to go through the trouble and
  expense of repeated sampling (called
  replication), we can estimate whether the
  difference in the 2 means is so large in
  magnitude that we would likely find the same
  result if we chose another 5 persons.

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Example 4.1 (cont.)
, df n-1
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Example 4.1 (cont.)

• This paired t needs a couple more values that
  we have not yet computed.
• First, we need to find the Standard Deviation of
  X1 and X2, called Sx1 and Sx2.
• These are simply the square-root of the variances
  ( and
  ).

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Example 4.1 (cont.)

• Second, we need to find the correlation between
  the pre and post-season distances ( ), or
  likewise columns 2 and 3.
• Another section will illustrate how to compute a
  correlation.
• This computation is somewhat long, so well avoid
  it for now.
• Ill just tell you the correlation is
  rx1x20.9177.
• Any scientific or statistical calculator can get
  you this answer.

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Example 4.1 (cont.)
-4.78, df 4
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Example 4.1 (cont.)

• Finally, this computed t statistic must be
  compared with the critical value of the
  t-distribution.
• The critical value of the t is the highest
  magnitude we should expect to find if there is
  really no difference between the population means
  of X1 and X2, or in other words, no difference
  between performance in the pre and post season in
  the league.
• Since we expect there should be improvement in
  throwing distance, this is a 1-tailed test.

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Example 4.1 (cont.)

• The C.V. t(4), a.05 2.132, therefore we reject
  the null hypothesis because the absolute value of
  our t at 4.78 is greater than the critical
  value.
• This is a 1-tailed t-test, so we must verify this
  conclusion by noting that the mean of the post
  season at 4 meters, is greater than the mean of
  the pre-season throw average of 2 meters.

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Example 4.1 (cont.)

• Our research conclusion states the facts in
  simple terms
• Throwing distances increased significantly from
  the pre-season (M 2) to the post-season (M
  4), t(4) 4.78, p lt .05 (one-tailed).

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Example 4.1 Using SPSS

• First, we must setup the variables in SPSS.
• Although not strictly necessary, it is good
  practice to give a unique code to each
  participant (personid).
• Unlike the independent samples t-test, the paired
  t-test has separate entries for 2 dependent
  variables, rather than an independent and
  dependent
• DependentVariable1 preseas (for Pre-season
  scores)
• DependentVariable2 postseas (for Post-season
  scores)

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Example 4.1 Using SPSS (cont.)

• In our example, the variables are setup as
  follows in the SPSS variable view

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Example 4.1 Using SPSS (cont.)

• It is no longer necessary to provide codes (or
  values) for the independent variable, simply
  because one does not exist! We can proceed to
  typing in the data in the SPSS data view

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Example 4.1 Using SPSS (cont.)

• Notice, this is where the personid variables
  has helped.
• If we had incorrectly tried to analyze this
  problem as an independent samples t-test, then we
  would have coded for 10 people under personid.
• Of course, since we have only 5 people in this
  example, this would have been incorrect.
• The personid variable thus allows a simple check
  for whether we have typed-in the data correctly.
• The number of rows in SPSS should always equal
  the number of subjects (or likewise,
  experimental units).

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Example 4.1 Using SPSS (cont.)

• Next, we need the SPSS syntax to run a paired
  t-test. The code is as follows
• t-test pairs DependentVariable1
  DependentVariable2.
• In our example, the following code is written

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Example 4.1 Using SPSS (cont.)

• After running the syntax, the following appears
  in the SPSS output viewer

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Example 4.1 Using SPSS (cont.)

• You should focus your attention first of the mean
  values for the pre and the post season
  performance.
• As before, the means (Pre-season2 and
  Post-season4) give us our conclusion.
• Namely, we conclude that performance increased
  from the pre to the post season.
• The statistics tell us that our conclusion is
  true not only for this sample of 5 persons, but
  also for other samples of 5 persons in the league.

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Example 4.1 Conclusion

• Our test is 1-tailed, so we must divide the
  2-tailed probability provided by SPSS in half
  (p.009/2 .0045).
• When expressed to 2 significant digits, this
  value will round to .00 and as a result the
  lowest value that can be represented in APA style
  is plt.01.
• In short, we can now write our conclusion as
  follows
• Throwing distances increased significantly from
  the pre-season (M 2) to the post-season (M
  4), t(4) 4.78, p lt .01 (one-tailed).

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The Paired t-Test(A.K.A. Dependent Samples
t-Test, or t-Test for Correlated Groups)
Thus concludes

• Advanced Research Methods in Psychology
• - Week 3 lecture
• Matthew Rockloff