Paired t-test: tD
I.
Introduction To The Repeated
Measures Design:What is a
repeated measure?
II. Finding an Experimental Effect In
a Single Group: Before vs. After
III. Creating a new distribution tD.
IV. Reduces Sampling Error: It’s a
more powerful test
V. Limited Applicability
Pre-Measure
Manipulation
Before-After
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Post-Measure
2
It doesn’t have to be “Before-After”
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3
Matched Subject Design
For a given study the two groups of subjects
could be closely matched
1. Age
2. IQ
3. Blood Pressure
4. Income
5. Education Level
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4
The Basic Idea
• Standard t-test
n
x1
x2
2
6
13
17
24
28
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5
The Basic Idea
• Standard t-test
n
average
x1
x2
2
6
13
17
24
28
13
17
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The Basic Idea
• Standard t-test
n
x1
x2
2
6
13
17
24
28
average 13
17
• Is 13 different than 17? Or 13-17 different than 0?
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The Basic Idea
• Repeated Measures t-test
n
x1
x2
A
2
6
B
13
17
C
24
28
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The Basic Idea
• Repeated Measures t-test
n
x1
x2
D
A
2
6
4
B
13
17
4
C
24
28
4
• Create A New Variable, D
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The Basic Idea
• Repeated Measures t-test
subject
x1
x2
D
A
2
6
4
B
13
17
4
C
24
28
4
average
4
• Is 4 different than 0?
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The Basic Idea
The fundamental advantage?
• The error term in the within subjects design is smaller
• In the simplified example, the standard error terms
would be higher in the two sample version versus the
difference test (in this case the sMD is zero)
• The advantage is that individual differences
(2, 13, 24, and 5, 16, 27) are not part of the error in tD
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The Basic Idea
Are there limitations?
•
The repeated measure design (before – after) must be
used cautiously used in many experimental designs
1. Memory
Subjects learn
2. Medicine and Clinical Psych Substantial time passes
3. Social Psych
Minor deceptions
•
Loss of half the degrees freedom
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Distribution of the Paired
t-Statistic
Suppose x is a variable on each of two populations whose
members can be paired. Further suppose that the paired-difference
variable D is normally distributed. Then, for paired samples of size
n, the variable
t
M D  M D
sD / n

M D  M D
sM D
has the t-distribution with df = n – 1.
The normal null hypothesis is that μD = 0
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The paired t-test for two
population means (Slide 1 of 3)
Step 1 The null hypothesis is H0: D = 0; the alternative
hypothesis is one of the following:
Ha: D  0
Ha: D < 0
Ha: D > 0
(Two Tailed) (Left Tailed) (Right Tailed)
Step 2 Decide on the significance level, 
Step 3 The critical values are
±t/2
-t
+t
(Two Tailed) (Left Tailed) (Right Tailed)
with df = n - 1.
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The paired t-test for two
population means (Slide 2 of 3)
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The paired t-test for two
population means(Slide 3 of 3)
Step 4 Compute the value of the test statistic
M D  D M D  D
t

sM D
sD / n
where normally D  0
Step 5 If the value of the test statistic falls in the
rejection region, reject H0, otherwise do not
reject H0.
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The number of doses of medication needed for
asthma attacks before and after relaxation training.
SS
14.8

 1.92
n 1
4
M
 3.2
t D 
 3.72
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sM D 1.92 5
sD 
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A Direct Comparison
xx1
1
xx2
2
25.7
20
28.4
13.7
18.8
12.5
28.4
8.1
23.1
10.4
24.9
18.8
27.7
13
17.8
11.3
27.8
8.2
23.1
9.9
t -test: Two-Sample
Mean
Variance
Observations
Pooled Variance
df
Standard Error
t stat
t critical two-tail
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Variable 1 Variable 2
18.91
18.25
55.83211 55.03833
10
10
55.43522
18
1.754917
0.198215
2.100924
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A Direct Comparison
xx1
1
25.7
20
28.4
13.7
18.8
12.5
28.4
8.1
23.1
10.4
x2
x2
24.9
18.8
27.7
13
17.8
11.3
27.8
8.2
23.1
9.9
D
0.8
1.2
0.7
0.7
1
1.2
0.6
-0.1
0
0.5
t -test: Paired
Mean
Variance
Observations
df
Standard Error of D
t stat
t crit two-tail
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Variable 1 Variable 2
18.91
18.25
55.83211 55.03833
10
10
9
0.14
4.714286
2.262159
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