HW5. Effect Size, Power and Sample Size Estimations
Problem 1. Our research hypothesis is that patients with A-T will show
decreased (from 20% to 30%) brain glucose metabolism in the cerebellar
cortex due to decreased inhibitory modulation by Purkinje cells in the
cerebellum when compared with their homozygous non affected siblings.
For this hypothesis, a paired samples t-test will be performed to examine whether A-T
patients have decreased brain glucose metabolism in cerebellar cortex in comparison
to their homozygous non-affected siblings. The standard deviations for the two groups
(A-T patients and their unaffected siblings) are estimated to be equal and is about 10
~ 15% of the unaffected sibling mean brain glucose metabolism in the cerebellar
cortex. Ten A-T patients and 10 of their unaffected siblings will be recruited for the
study. Please answer the following questions.
1. Given a conservative estimation of a correlation of 0.5 between sibling pairs,
please prove that the estimated standard deviation of the paired differences is
also 10 to 15% of the non-affected sibling mean.
2. Given the hypothesized decrease of 20-30%, please show that the estimated
effect size (=expected mean difference divided by the standard deviation of the
paired differences) is 1.3 to 3.0.
3. Please derive the general formula for the power of the paired samples t-test when
the effect size is E, and the significance level is α. Please derive the formula for
the one-sided tests and the two-sided test separately.
4. At the significance level of 0.05 (one-sided), please show that the estimated
power of the paired samples t-test is between 98% to 99.999%.
5. Confirm the above analysis results using G*Power.
Solution:
1. σ2d = var(X − Y) = σ12 − 2ρσ1 σ2 + σ22 = σ2 − 2(0.5)σ ∗ σ + σ2 =
σ2
Thus σd = σ = 0.1μX ~0.15μX
2. μd = μX − μY = 0.2μX ~0.3μX
EFF =
3.
μd
0.2μX ~0.3μX
0.2 0.3
=
=
~
= 1.3~3.0
σd 0.1μX ~0.15μX 0.15 0.1
(3a)
H 0 :  d  0

H a :  d    0
2nd scenario, Normal population,  d2 is unknown.
Test statistic : T0 
D0
Sd
n
H0
~t
n 1
1    P(T0  t n 1, |  d  )
 P(T  

Sd
n
 t n 1, |  d  )
where T 
,
D
Sd
n
Ha
~t
n 1
 P(T  Eff * n  t n 1, |  d  )
Because Eff  
(3b)

d


, where   0
Sd
H 0 :  d  0

H a :  d    0
2
2nd scenario, Normal population,  d is unknown.
Test statistic : T0 
D0
Sd
n
H0
~t
n 1
1    P(T0  t n 1, |  d  )
 P(T  

Sd
n
 t n 1, |  d  )
,
where T 
 P(T   Eff * n  t n 1, |  d  )
Because Eff 
(3c)

d


, where   0
Sd
H 0 :  d  0

H a :  d    0
2
2nd scenario, Normal population,  d is unknown.
Test statistic : T0 
D0
Sd
n
H0
~t
n 1
D
Sd
n
Ha
~t
n 1
1    P( T0  t n 1, / 2 |  d  )
 P(T0  t n 1, / 2 |  d  )  P(T0  t n 1, / 2 |  d  )
 P(T  
Here T 

Sd
n
D
Sd
n
 t n 1, / 2 |  d  )  P(T  

Sd
n
 t n 1, / 2 |  d  )
Ha
~t
n 1
Without loss of generality, let us assume that   0 , then Eff 
1    P(T  

Sd
n
 t n 1, / 2 |  d  )  P(T  

d

Sd
n


,
Sd
 t n 1, / 2 |  d  )
 P(T   Eff * n  t n 1, / 2 |  d  )  P(T   Eff * n  t n 1, / 2 |  d  )
H 0 :  d  0
H a :  d    0
4. 
|Eff |= 1.333 ~ 3.0
1    P(T   Eff * n  t n 1, |  d  )
 P(T   Eff * 10  t 9,0.05 |  d  )
 P(T   Eff * 3.162  1.833 |  d  )
When |Eff |= 1.333
1    P(T  1.333 * 3.162  1.833 |  d  )  0.979
When |Eff |= 3.0
1    P(T  3 * 3.162  1.833 |  d  )  0.99998
5. We have shown how to obtain the G*Power values and
plots in class – including both the paired samples t-test and
the independent samples (Problem 2) t-test scenarios.
Problem 2. Please reproduce the two figures using G*Power.
Statistical Analysis: This pilot study will be the first attempt to use a certain
chemical to assess the effects of alcohol intoxication to regional brain metabolism in
human brain in thirty male subjects (n=30). Half of the subjects will be low alcohol
users and the other half will be heavy alcohol users. The two major hypotheses
(Specific Aims 1 and 2) are based on a PET study where each subject will receive two
brain scans, one after drinking a placebo drink (no alcohol) and an alcohol drink.
Paired samples t-test will be conducted to estimate changes in brain metabolism from
baseline to alcohol intoxication (Specific Aim 1). For Specific Aim 2, independent
samples t-test will be conducted to compare the changes in brain metabolism between
heavy and light drinkers.
Power Analysis: Our power analysis is based on the two specific aims. For Specific
Aim 1, we will perform the analysis for each group separately. With 15 subjects in each
group, at the significance level of 0.05 (1-sided), we will have a power between 97.6%
to 99.9% via the paired samples t-test to compare changes in brain metabolism due to
alcohol intoxication in a range of effect size between 1 to 1.5 (moderate to large; Figure
1a). For Specific Aim 2, at the significance level of 0.05 (1-sided), with 15 subjects per
group (that is, 30 subjects in total at the ratio of 1:1), we will have a power between
84.8% to 98.9% via the independent samples t-test to compare changes in brain
metabolism due to alcohol intoxication between light and heavy drinkers in a range of
effect size between 1 to 1.5 (moderate to large; Figure 1b). In the unexpected case of
non-normality, the Wilcoxon signed rank test (for paired samples) and the Wilcoxon
rank-sum test (for independent samples) will be used with an expected power loss of 5%
resulting in 92.7% to 94.9% and 80.6% to 94.1% power for Specific Aims 1 and 2
respectively.
a
b
Figure 1. Power of the (a) paired and (b) independent samples t-tests for Specific Aims
1 and 2 respectively, over a range of effect sizes from 1 to 1.5, moderate to large.