1. Repeated-Measures Data with One Sample
Hypothesis Testing
Repeated-Measures Data with One Sample

2. 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)?

3. 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

4. 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)

5. 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 (post-Tx 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)

6. 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

7. Repeated-Measures Testing with One Sample
Difference Scores
Time 1
Time 2
Difference Score 2 1 5 4 8 6 3 12 9 15 7 23 11 14

8. 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:
There is a difference in scores from Time 1 to Time 2
μD = μ1– μ2 ≠ 0

9. 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

10. 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

11. Repeated-Measures Testing with One Sample
= 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

12. 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

13. 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?

14. Repeated-Measures Testing with One Sample1
Time 2
Difference Score 19 1 -1 11 9 2 5 23 4 13 3 10 7 8 -5

15. 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
= 17/10 = 1.7
s2 = = [195 - (172/10)]/9 = 18.46

16. Repeated-Measures Testing with One Sample
Critical t (df = 9, two-tailed, p < .05) = 2.262
We would fail to reject Ho

17. 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 beta-endorphin 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?

18. 12 Hours Before
10 Minutes Before 10
6.5 14 8
13.5 12 18 5
14.5
11.5 9
3.5 42
7.5
5.8 6
4.7 25 7 52 17 20
8.8 16 15
4.4
2.5 2

19. Repeated-Measures Testing with One Sample
Example #2:
= s = 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.

20. 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

21. 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