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Within subjects t tests
• Related samples
• Difference scores
• t tests on difference scores
• Advantages and disadvantages
Related Samples
• The same participants give us data on two
measures
 e. g. Before and After treatment
 Usability problems before training on PP and after
training
• With related samples, someone high on one
measure probably high on other(individual
variability).
Cont.
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
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
Effect of training
Before
Mean
St. Dev.
21
24
21
26
32
27
21
25
18
23.84
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
8.17
3.60
Results
• The training decreased the number of problems
with Powerpoint
• Was this enough of a change to be significant?
• Before and After scores are not independent.
 See raw data
 r = .64
Cont.
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 one sample test
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.
t test--cont.
• With 8 df, t.025 = +2.306 (Table E.6)
• We calculated t = 6.85
• Since 6.85 > 2.306, reject H0
• Conclude that the mean number of
problems after training was less than
mean number before training
Advantages of Related
Samples
• Eliminate subject-to-subject variability
• Control for extraneous variables
• Need fewer subjects
Disadvantages of Related
Samples
• Order effects
• Carry-over effects
• Subjects no longer naïve
• Change may just be a function of time
• Sometimes not logically possible
Between subjects t test
• Distribution of differences between
means
• Heterogeneity of Variance
• Nonnormality
Powerpoint training again
• Effect of training on problems using
Powerpoint
 Same study as before --almost
• Now we have two independent groups
 Trained versus untrained users
 We want to compare mean number of
problems between groups
Effect of training
Before
Mean
St. Dev.
21
24
21
26
32
27
21
25
18
23.84
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
8.17
3.60
Differences from within
subjects test
Cannot compute pairwise differences, since we
cannot compare two random people
We want to test differences between the two
sample means (not between a sample and
population)
Analysis
• How are sample means distributed if H0
is true?
• Need sampling distribution of differences
between means
 Same idea as before, except statistic is
(X1 - X2) (mean 1 – mean2)
Sampling Distribution of Mean
Differences
• Mean of sampling distribution = m1 - m2
• Standard deviation of sampling
distribution (standard error of mean
differences) =
sX X
1
2
1
2
2
2
s s


n1 n2
Cont.
Sampling Distribution--cont.
• Distribution approaches normal as n
increases.
• Later we will modify this to “pool”
variances.
Analysis--cont.
• Same basic formula as before, but with
accommodation to 2 groups.
X1  X 2 X1  X 2
t
 2
2
sX X
s1 s2

n1 n2
1
2
• Note parallels with earlier t
Degrees of Freedom
• Each group has 6 subjects.
• Each group has n - 1 = 9 - 1 = 8 df
• Total df = n1 - 1 + n2 - 1 = n1 + n2 - 2
9 + 9 - 2 = 16 df
• t.025(16) = +2.12 (approx.)
Conclusions
• T = 4.13
• Critical t = 2.12
• Since 4.13 > 2.12, reject H0.
• Conclude that those who get training
have less problems than those without
training
Assumptions
• Two major assumptions
 Both groups are sampled from populations
with the same variance
• “homogeneity of variance”
 Both groups are sampled from normal
populations
• Assumption of normality
 Frequently violated with little harm.
Heterogeneous Variances
• Refers to case of unequal population variances.
• We don’t pool the sample variances.
• We adjust df and look t up in tables for
adjusted df.
• Minimum df = smaller n - 1.
 Most software calculates optimal df.