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Chapter 13
Hypothesis Tests: Two
Related Samples
Fundamental Statistics for the
Behavioral Sciences, 5th edition
David C. Howell
©2003 Brooks/Cole Publishing Company/ITP
Chapter 13 Hypothesis Tests: Two
Related Samples
Major Points
• Related samples?
• Difference scores?
• An example
• t tests on difference scores
• Advantages and disadvantages
• Effect sizes
• Review questions
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Chapter 13 Hypothesis Tests: Two
Related Samples
3
Related Samples
• The same participants give us data on
two measures
 e. g. Before and After treatment
 Aggressive responses before video and
aggressive responses after
• With related samples, someone high on
one measure is probably high on other.
Cont.
Chapter 13 Hypothesis Tests: Two
Related Samples
Related Samples--cont.
• Correlation between before and after
scores
 Causes a change in the statistic we can use
• Sometimes called matched samples or
repeated measures
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Chapter 13 Hypothesis Tests: Two
Related Samples
Difference Scores
• Calculate difference between first and
second score
 e. g. Difference = Before - After
• Base subsequent analysis on difference
scores
 Ignoring Before and After data
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Chapter 13 Hypothesis Tests: Two
Related Samples
An Example
• Therapy for rape victims
 Foa, Rothbaum, Riggs, & Murdock (1991)
• One group received Supportive
Counseling
• Measured post-traumatic stress disorder
symptoms before and after therapy
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Chapter 13 Hypothesis Tests: Two
Related Samples
Therapy for PTSD
Before
Mean
St. Dev.
21
24
21
26
32
27
21
25
18
23.89
4.20
After
15
15
17
20
17
20
8
19
10
15.67
4.24
Diff.
6
9
4
6
15
7
13
6
8
8.22
3.60
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Chapter 13 Hypothesis Tests: Two
Related Samples
Results
• The Supportive Counseling group
decreased number of symptoms
• Was this enough of a change to be
significant?
• Before and After scores are not
independent.
 See raw data
 r = .64
Cont.
Chapter 13 Hypothesis Tests: Two
Related Samples
Results--cont.
• If no change, mean of differences should
be zero
 So, test the obtained mean of difference
scores against m = 0.
 Use same test as in Chapter 12.
• We don’t know s, so use s and solve for t
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Chapter 13 Hypothesis Tests: Two
Related Samples
t test
D and sD = mean and standard deviation of differences.
D  m 8.22 8.22
t


 6.85
sD
3.6
1.2
n
9
df = n - 1 = 9 - 1 = 8
Cont.
Chapter 13 Hypothesis Tests: Two
Related Samples
t test--cont.
• With 8 df, t.025 = +2.306
• We calculated t = 6.85
• Since 6.85 > 2.306, reject H0
• Conclude that the mean number of
symptoms after therapy was less than
mean number before therapy.
• Supportive counseling seems to work.
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Chapter 13 Hypothesis Tests: Two
Related Samples
Advantages of Related
Samples
• Eliminate subject-to-subject variability
• Control for extraneous variables
• Need fewer subjects
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Chapter 13 Hypothesis Tests: Two
Related Samples
Disadvantages of Related
Samples
• Order effects
• Carry-over effects
• Subjects no longer naive
• Change may just be a function of time
• Sometimes not logically possible
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Chapter 13 Hypothesis Tests: Two
Related Samples
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Effect Size Again
• We could simply report the difference in
means.
 Diff = 8.22
 But the units of measurement have no
particular meaning to us—Is 8.22 large?
• We could “scale” the difference by the
size of the standard deviation.
Cont.
Chapter 13 Hypothesis Tests: Two
Related Samples
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Effect Size, cont.
m1  m2 m Before  m After
d

s
s Before
23.89  15.67 8.22


 1.96
4.20
4.20
Cont.
Chapter 13 Hypothesis Tests: Two
Related Samples
Effect Size, cont.
• The difference is approximately 2
standard deviations, which is very large.
• Why use standard deviation of Before
scores?
• Notice that we substituted statistics for
parameters.
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Chapter 13 Hypothesis Tests: Two
Related Samples
SPSS
• Next slide shows SPSS Printout
 Similar printout from other software
 Results match ours
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Chapter 13 Hypothesis Tests: Two
Related Samples
Paired Samples Statistics
Mean
Pair
1
Std.
Deviation
N
Std. Error
Mean
POST
15.6667
9
4.2426
1.4142
PRE
23.8889
9
4.1966
1.3989
Paired Samples Correlations
N
Pair 1
POST & PRE
Correlation
9
.637
Sig.
.065
Paired Samples Test
Paired Differences
POST - PRE
Mean
Std.
Deviation
Std. Error
Mean
-8.2222
3.5978
1.1993
95% Confidence
Interval of the
Difference
Lower
Upper
t
-10.99
-5.46
-6.86
df
Sig.
(2-tailed)
8
.000
Chapter 13 Hypothesis Tests: Two
Related Samples
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Review Questions
• Why do we say that the two sets of
measures are not independent?
• What are other names for “related
samples?”
• How do we calculate difference scores?
 What happens if we subtract before from
after instead of after from before?
Cont.
Chapter 13 Hypothesis Tests: Two
Related Samples
Review Questions--cont.
• Why do we usually test H0: mD = 0?
• Why do we have 8 df in our sample when we
have 18 observations?
• What are the advantages and disadvantages of
related samples?
• What do effect sizes tell you in this case?
• How would you calculate the confidence
interval that SPSS produced?
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