Paired-Samples t-test
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Independence
• Knowing one variable tells nothing about
another
– Probabilities: p(X) unchanged by value of Y
– Measures from unrelated people or events
• Critical question for inferential statistics
– Affects sampling distributions
– Example: Household income
• Sample 100 unrelated people
• Sample 50 couples (100 people total)
Independent vs. Paired Samples
• Independent-sample t-test assumes no relation
between Sample A and Sample B
– Unrelated subjects, randomly assigned
– Necessary for standard error to be correct
• Sometimes samples are paired
– Each score in Sample A goes with a score in Sample B
– Before vs. after, husband vs. wife, matched controls
– Paired-samples t-test
Paired-samples t-test
• Data are pairs of scores, (XA, XB)
– Form two samples, XA and XB
– Samples are not independent
• Same null hypothesis as with independent samples
– mA = mB
– Equivalent to mean(XA – XB) = 0
• Approach
– Compute difference scores, Xdiff = XA – XB
– One-sample t-test on difference scores, with m0 = 0
Example
• Breath holding underwater vs. on land
– 8 subjects
– Water:
XA = [54, 98, 67, 143, 82, 91, 129, 112]
– Land:
XB = [52, 94, 69, 139, 79, 86, 130, 110]
• Difference:
Xdiff = [2, 4, -2, 4, 3, 5, -1, 2]
å Xdiff = 17 = 2.13
Mean: M diff =
n
Standard Error:
s M diff =
8
sdiff
6.13
=
= .88
n
8
• Critical value
> qt(.025,7,lower.tail=FALSE)
[1] 2.364624
•
Reliably longer underwater
Mean Square:
Test Statistic:
2
sdiff
t=
=
å( Xdiff - M diff )
2
n -1
M diff 2.13
=
= 2.43
s M diff .88
= 6.13
Comparison of t-tests
Samples
One
2-Indep.
2-Paired
Data
t
Standard Error
X
M -m 0
sM
s
1
= MS
n
n
XA, XB
Xdiff = XA - XB
MA - MB
s M A -M B
M diff
s M diff
MS
(
1
nA
+ n1B
Mean Square
s =
2
)
å( X - M )
df
2
n-1
df
å( X A - M A )
2
+å ( X B - M B )
2
nA + nB – 2
df
sdiff
1
= MS
n
n
2
sdiff
=
å ( Xdiff - M diff )
df
2
n-1