Title: The Paired t-Test (A.K.A. Dependent Samples t-Test, or t-Test for Correlated Groups)
1The Paired t-Test(A.K.A. Dependent Samples
t-Test, or t-Test for Correlated Groups)
- Advanced Research Methods in Psychology
- - lecture
- Matthew Rockloff
2When 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.
3Example 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)
4Example 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
5Example 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?
6Example 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.
7Example 4.1 (cont.)
, df n-1
8Example 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
).
9Example 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.
10Example 4.1 (cont.)
-4.78, df 4
11Example 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.
12Example 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.
13Example 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).
14Example 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)
15Example 4.1 Using SPSS (cont.)
- In our example, the variables are setup as
follows in the SPSS variable view
16Example 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
17Example 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).
18Example 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
19Example 4.1 Using SPSS (cont.)
- After running the syntax, the following appears
in the SPSS output viewer
20Example 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.
21Example 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).
22The 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