Hypothesis Testing
Repeated-Measures Data with
One Sample
Repeated-Measures Testing with
One Sample

One-Sample Z/T-Test:
 Looks
at difference between one sample and
its population at one point in time

What if we wanted to compare a sample
with its population when we had data at
multiple time points (i.e. one subject
provided data for multiple
groups/samples)?
Repeated-Measures Testing with
One Sample

Why would we want to do this?
 If
we were devising a treatment for a particular
disorder, we’d want to know how our subjects fared
post-treatment as a function of their pre-treatment
score


I.e. if we know that Bob has a score of 10/21 on a measure of
anxiety after participating in a treatment for social phobia,
treatment is only successful if Bob scored much less than
this before treatment
Knowing Bob’s score at two time points (before and after
treatment), helps us to make this determination
Repeated-Measures Testing with
One Sample
 Taking
multiple assessments of one subject
over time gives us more data from which to
draw conclusions, allows us to use more
powerful statistical tests, and therefore
requires fewer subjects to be run

“Powerful” = the test can detect a real difference
between two groups, or between two or more time
points with fewer subjects (Remember – sample
size effects the likelihood of obtaining a significant
p-value)
Repeated-Measures Testing with
One Sample

OK, well then why not just pretend the data from Time 1
is one sample, the data from Time 2 is another sample,
and compare the means on the two?

This is a big statistical no-no – one of the assumptions of the
Two-Independent Samples T-Test (covered in the next chapter)
is that the samples are independent/unrelated

If subjects provide data at Time 1(pre-Tx sample) and Time 2 (postTx sample), these two samples are not independent, they are
probably related. If Bob experiences the most anxiety (has the
highest score on our anxiety scale) before treatment, he probably
experiences the most anxiety after treatment (has the highest score
after treatment)
Repeated-Measures Testing with
One Sample

OK, so what do we do then?
 Use
the Two Related Samples/Repeated
Measures/Matched Samples T-Test

How does this test differ from other t-tests?
 It
uses difference/gain scores
 Difference Score = scores representing the difference
between performance on two occasions

i.e. Time 2 – Time 1, or visa-versa
Repeated-Measures Testing with
One Sample

Difference Scores
Time 1
Time 2
Difference Score
2
1
1
5
4
1
8
6
2
6
3
3
12
9
3
15
7
8
23
12
11
14
5
9
11
3
8
9
2
7
Repeated-Measures Testing with
One Sample

Ho for Repeated Measures Tests:
 For a Two-Tailed Test:
 There is no difference in scores from Time 1 to Time 2
 AKA The average of the population of difference scores (μD)
=0
 AKA μD = μ1 (population mean of Time 1 scores) – μ2
(population mean of Time 2 scores) = 0

H1 for Repeated Measures Tests:
 For a Two-Tailed Test:
 There is a difference in scores from Time 1 to Time 2
 μD = μ1– μ2 ≠ 0
Repeated-Measures Testing with
One Sample

Ho for Repeated Measures Tests:
 For a One-Tailed Test:
 If the Time 1 scores are lower than Time 2 (if they’re higher
than you would just use “>” instead of “<“)
 μ1 (population mean of Time 1 scores) < μ2 (population mean
of Time 2 scores)

H1 for Repeated Measures Tests:
 For a Two-Tailed Test:
 The Time 1 scores are equal to or higher than Time 2 (if not,
use “” instead of “”)
 μD = μ1 μ2
Repeated-Measures Testing with
One Sample

How do we calculate a Repeated
Measures T-Test?
 Same
as previous t-tests, just using difference
scores as opposed to raw scores

D0
t
sD
N
Repeated-Measures Testing with
One Sample

D = the mean of the difference scores
s = the standard deviation of difference
scores
 N = the number of difference scores (NOT
the total number of scores, which is twice
the number of difference scores)
 df = N – 1 = number of pairs minus 1

D
Repeated-Measures Testing with
One Sample

Review – When to use Repeated
Measures T-Tests:
 1.
One subject provides data for two time
points
 2. One subjects provides data for two
groups/samples
Repeated-Measures Testing with
One Sample

Example #1:
 The
following data is from an experimental treatment
that I conducted to help prevent depression by taking
pessimistic people (who are at risk for developing
depression) and making them more optimistic. The
data are levels of depression as assessed with the
Beck Depression Inventory-II (BDI-II), both before and
after treatment, for a subset of those in the group
receiving treatment (we also had a group that didn’t
receive treatment).

Why did we have a no treatment group?
Repeated-Measures Testing with
One Sample
BDI@Time 1
BDI@ Time 2
Difference Score
19
19
0
0
1
-1
11
9
2
5
5
0
23
23
0
4
0
4
13
3
10
9
2
7
3
8
-5
3
3
0
Repeated-Measures Testing with
One Sample
Example #1:
 Difference Scores: 0, -1, 2, 0, 0, 4, 10, 7,
-5, 0
 Diff. Scores2: 0, 1, 4, 0, 0, 16, 100, 49, 25,
0
 D = 17/10 = 1.7
2
2
 s =D 2  D  = [195 - (172/10)]/9 = 18.46

D
N
N 1
Repeated-Measures Testing with
One Sample
s = √18.46 = 4.30
 t = (1.7 – 0)/(4.30/√10) = 1.25
 Critical t (df = 9, two-tailed, p < .05) =
2.262
 We would fail to reject Ho

D
Repeated-Measures Testing with
One Sample

Example #2:

Hoaglin, Mosteller, and Tukey (1983) present data on blood
levels of beta-endorphin as a function of stress. They took betaendorphin levels for 19 patients 12 hours before surgery, and
again 10 minutes before surgery. The data are presented in
fmol/ml:







Get into groups of 2 or more
Based on the data, what are the df?
Are you going to use a one- or two-tailed test? Why?
What level of α are you going to use?
What are you df and what is your critical t?
What are the Ho and H1?
What effect does increased stress level have on endorphin levels?
12 Hours Before
10 Minutes Before
10
6.5
6.5
14
8
13.5
12
18
5
14.5
11.5
9
5
18
3.5
42
7.5
7.5
5.8
6
4.7
25
8
12
7
52
17
20
8.8
16
17
15
15
11.5
4.4
2.5
2
2
Repeated-Measures Testing with
One Sample
Example #2:
 D = -7.70
s D= 13.52
 t = -2.48
 df = 18
 For α = .05, One-Tailed Critical t = 1.734
Two-Tailed Critical t = 2.101
We would reject Ho, and conclude that stress
raises beta-endorphin levels.

Repeated-Measures Testing with
One Sample

Why not to use Repeated Measures data?
 1.
If subjects are asked to take the same
measures at two time points, they may
respond similarly to Time 2 as to Time 1 if
they can recall their earlier responses.

Solution: Create two measures that are similar, but
not identical

Although this involves proving that they’re similar
Repeated-Measures Testing with
One Sample

Why not to use Repeated Measures data?
 2.
The questions asked on the test at Time 1
may tip off subjects to the point of the
experiment, and contaminate the results.

Solution: Throw in “filler” items to the measures
that have nothing to do with your experiment and
that you’ll essentially ignore in data analysis to
throw off your wiley subjects