Section V

Sample Size and Power

Multiple hypothesis testing

 Sample size (n) based on
estimation precision-CI width

Can plan sample size so that standard

errors (SEs) and the corresponding
confidence intervals are sufficiently
small.
(How small? How small do you need it
to be?)
 Sample size for precision/CIs
π = proportion with TB in the population (prevalence)
p = proportion with TB in a sample of size n,
SE(p) = √[π(1- π)/n]
95% confidence interval for π: p ± 1.96(SE)
Precision: want to estimate true prevalence (π) within ± 6%
Solve for n: 1.96(SE) = 1.96 √[π(1- π)/n] = 0.06
n = 1.962 π(1- π)/(0.06)2 = 3.84 π(1- π)/0.0036
Can estimate π using the observed p or use maximum at π=0.5
If π=0.15, n = 3.84 (0.15)(0.85)/0.0036 = 136
At π=0.50, n = 3.84 (0.50)(0.50)/0.0036 = 267 (worst case)
Rule of thumb: For 95% CI for π, conservative n for precision w
is n = 1/w2
Sample size based on
hypothesis testing
 Hypothesis test decision table
No difference Actual
in population difference
(Null is true) (Null is false)
Test:
Do not reject null 1-α β
(correct) (Type II error)

Test:
Reject null α 1-β = power
hypothesis (Type I error) (correct)
 Determinants of power
Power (1-β) depends on
δ = delta = true difference
σ = sigma = true SD or true variation
α = alpha = significance criteiron
n = sample size
(Or, n depends on δ, σ, α, 1-β )
 Alpha versus Power
The top distribution shows
• the sampling distribution of
a test statistic under the
assumption that delta (δ) is
zero (the null hypothesis is
α true).

-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0

The bottom distribution shows

the true population distribution
1-β (unknown at the time of testing),
1-β with a true population delta=2.5.

-1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0

 Power calculation
Zpower = Zobs – Z1-α/2 = (δ/SE) – Z1-α/2
Treatment n Mean HBA1c SD SE
chg
Liraglutide 5 -1.24 0.99 0.44
Sitaglipin 4 -0.90 0.98 0.49
Difference 0.34=δ √[0.442 + 0.492] = 0.66

Zobs=0.34/0.66 = 0.516 (<1.96, so not statistically

significant, p=0.622)
Zpower = Zobs - Z α = 0.516 – 1.96 = -1.44
From the Gaussian table, Zpower=-1.44 yields power
about 7%.
 Interpretation of power
If test is “statistically” significant (p < α), we have a
“positive” or “significant” outcome (& accept the
false positive probability of α).
If test is not statistically significant (p > α) either
there is no relationship (“negative” outcome) or
sample size is inadequate (inconclusive
outcome).
If power is low for a given δ, results are
inconclusive, not negative.
If power is high, results are affirmatively negative.
(But better to quote Confidence interval after the
study is published)
 Sample size to test difference between 2 means
(this is NOT a universal formula)
Two independent groups, each with sample size
n = 2(Zpower+ Z1-α/2)2 (σ/δ)2

Z0.975 = 1.96 and Zpower = 0.842 (for power of 80%), so

n = 2(0.842 + 1.96)2 (σ/δ)2 = 15.7 (σ/δ)2
or
n per group approximately ≈ (range/δ)2

(since 15.7 ≈16, 16(σ/δ)2 = (4σ/δ)2 and the range ≈4σ )

 Power for increasing delta
δ =2.5

δ=0
δ = 3.5

-3 -2 -1 0 1 2 3 4 5 6 7

Areas under the curves and right of the vertical line are
α for the black curve and power for the other curves.
The power is larger for the red curve than for the blue.
 Power Summary
Power increases as:
• True difference (δ) increases
• Sample size (n) increases
• α increases (less strict significance criterion)
• Patient heterogeneity (σ) decreases

Generally, we set α = 0.05 & power = 1 – β = 0.80. To

determine n, we need to estimate δ and σ.
Often we use values of δ/σ for the calculation
For time to event outcomes (survival), n also depends on follow up time
since “n” is the number of events. The sample size for comparing two
survival curves is often computed based on comparing the corresponding
two hazards.
 Sample Size Checklist
Effect size (δ) = smallest clinically important difference
(n increases as δ decreases)
Variability = patient heterogeneity = group SD
(n increases as variability increases)
Power = prob of detecting the effect, set at 80% or higher
(n increases with power)
α level = prob of rejecting when δ=0, usually set at 0.05, two-sided
(n decreases with larger α)
** for time to event (survival) outcomes ***
Time of comparison = how long it takes to achieve the effect (n decreases
with time)
Follow up time = time each patient is followed (n reduced if patients followed
longer). In survival “n” is the number who have the outcome.
Also consider
percentage who will agree to participate
patient accrual rate, dropout / loss rate
Sample size for selected δ/σ -2 means
(2 means difference= δ, SD=σ, two-sided α=0.05)

δ/σ 70% power 80% power 90% power

0.10 1,234 1,570 2,102
0.15 549 698 934
0.20 309 392 525
0.25 198 251 336
0.50 49 63 84
0.75 22 28 37
1.00 12 16 21
1.25 8 10 13
1.50 5 7 9
Sample size for comparing two proportions
80% power, alpha=0.05
Difference between P1 and P2
|P1- P2|=δ
Smaller of P1 & P2 0.05 0.10 0.15 0.20
0.05 434 140 71 45

0.10 685 199 99 62

0.15 904 250 120 72

Hypothesis testing
limitations
 Pseudo replication
Most variation is between persons, not within person.
Two blood samples on n=10 is not a sample size of 20.
Observed value = true population mean
+ between person variation (σp)
+ within person variation (σe)
Example: To estimate the mean
1. Compute a mean for each person using her “m”
observations per person.
2. Compute the group mean from the “n” person means.
SEM = √[σp2/n + σe2/nm], usually σe < σp
Statistical vs Medical “significance”
(“A difference, in order to be a difference, must
make a difference”–Gertrude Stein?).
Average drop in weight (kg) after 3 months

Diet Mean Drop p 95% CI

I 0.50 <0.001 (0.45,0.55)

II 10.0 0.16 (-5.0, 25.0)
 p value limitations (ASA)
1. p values do not measure the probability that the
studied hypothesis is true, or the probability that the data
were produced by random chance alone (ignoring the
model).
2. Conclusions should not be based only on whether a p-
value passes a specific threshold.
3. Proper inference requires full reporting &
transparency.
4. A p-value does not measure the size of an effect or
the importance of a result.
5. A p-value alone does not provide a good measure of
evidence regarding a model or hypothesis.
19
 R A Fisher on p values
Statistical Methods and Scientific Inference, Hafner, New York, ed. 1, 1956

“The concept that the scientific worker can regard

himself as an inert item in a vast co-operative
concern working according to accepted rules, is
encouraged by directing attention away from his
duty to form correct scientific conclusions,…and
by stressing his supposed duty to mechanically
make a succession of automatic “decisions”. ...The
idea that this responsibility can be delegated to a
giant computer programmed with Decision
Functions belongs to a phantasy of circles, rather
remote from scientific research”. [pp. 104–105]
Multiple Hypothesis testing
Multiple efficacy endpoints /outcomes

Multiple safety endpoints/outcomes

Multiple treatment arms and/or doses

Multiple interim analyses

Multiple patient subgroups

Multiple analyses
 Exploratory vs confirmatory
protein example
750 proteins are compared between two groups – 12 are significant at p < 0.05
Protein name Atril fib Atherosclerosis p value
RAS guanyl-releasing protein 2 33.3% 0.0% 0.0000
Glutathione S-transferase P 38.9% 100.0% 0.0000
Selenium-binding protein 1 22.2% 0.0% 0.0000
Nucleosome assembly protein 1-like 4 16.7% 0.0% 0.0000
Integrin beta;Integrin beta-2 11.1% 50.0% 0.0000
Spectrin alpha chain, non-erythrocytic 1 11.1% 0.0% 0.0000
Pituitary tumor-transforming gene 1 protein-interacting 11.1% 0.0% 0.0000
WW domain-binding protein 2 16.7% 50.0% 0.0000
Syntaxin-4 5.6% 0.0% 0.0006
CD9 antigen 27.8% 50.0% 0.0013
ATP synthase-coupling factor 6, mitochondrial 27.8% 50.0% 0.0013
Flotillin-1 77.8% 100.0% 0.0037
Aconitate hydratase, mitochondrial 38.9% 50.0% 0.1142
Fructose-bisphosphate aldolase C 94.4% 100.0% 0.4402
Alpha-adducin 50.0% 50.0% 1.0000
40S ribosomal protein SA 1.0% 1.0% 1.0000
Abl interactor 1 1.0% 1.0% 1.0000
Bone marrow proteoglycan;Eosinophil granule major basic 1.0% 1.0% 1.0000
Tubulin alpha-4A chain 100.0% 100.0% 1.0000
… (750 proteins total)
Exploratory vs confirmatory
Who killed Tweety Bird?
Did Sylvester do it?
 Motivation (class discussion)
Tweety Bird is murdered by a cat who left a DNA
sample. The particular DNA profile found in the
sample is known to occur in one of every one
million cats. There is also about a 0.1% false
positive rate for this test.
Is the level of evidence (guilt) equal in these two
scenarios?
1. Sylvester only is tested and is a match.
2. A DNA database on 100,000 cats (but not all
cats), including Sylvester, is searched and
Sylvester is a match, although not necessarily
the only match. No prior belief that Sylvester
is guilty.
 Motivation (class discussion)
The “disease score” ranges from 2 (good) to
12 (worst).
Scenario A: Due to prior suspicion (prior
information), only patients 19 and 47 are
measured and both have scores of 12. We
report that they are “significantly” ill.
Scenario B: The score is measured on 72
patients. Only patients 19 and 47 have
scores of 12. We report that they are
“significantly” ill.
Is the amount of “evidence” or “belief” that
patients 19 and 47 “really” are very ill (have
“true” score of 12) the same in both
scenarios? The data for patients 19 and 47
are the same in both scenarios.
Most would agree that, if both patients were
retested (confirmation step), and came out
with lower scores, this would decrease the
belief that there “true” score is 12. If they
came out with 12 again, this would increase
the belief that the true score is 12.
 Multiple testing
“If you torture the data long enough, it will eventually
confess”
Two different situations for new arthritis treatment compared
to aspirin.
A. Only pain (0-10) and swelling (0-10) are measured. Both
are significantly better at p < 0.05 on the new treatment
compared to aspirin.
B. Ten different outcomes measured: pain, swelling, activities
of daily living, quality of life, sleep, walking, bending, lifting,
grinding, climbing. Only the two that are significant are
reported after all 10 are evaluated. (fraud?)

Confirmatory studies specify outcomes in advance. Misleading

to report only statistically significant results.
How to really lie with stats for fun and profit

1. Bet on the horse after you know who

won (Movie -> The “Sting”)

2. Send financial advice after you know how

the market did (example in class)
 Multiple Testing
Assume all results are reported, not just the significant ones.
Bonferroni: out of m independent tests at 0.05, num significant by
chance alone, even when all null hypotheses are true (assumes
independence).
# tests=m Probability reject at least one

1 0.0500
2 0.0975
3 0.1426
4 0.1855
5 0.2262
10 0.4013
20 0.6415
25 0.7226
50 0.9231
 Multiple testing-What to do?
Option 1: Use nominal alpha level for
significance. Creates too many false
positives.
Option 2: Use Bonferroni criterion –Declare
significance if p < α/m if “m” tests are
made. Has too many false negatives.
Option 3: Use Holm/Hochberg criterion – a
compromise
 Holm/Hochberg criterion
Rule for m (not necessarily independent) significance
tests. Keeps overall false positive rate at α.

1)Sort the “m” p values from lowest to highest.

2)Declare the ith ordered p significant if it is less

than α/(m+1-i). If p > α/(m+1-i), this & all
larger p values are declared non significant.
Overall type I error rate (FWER) is ≤ α.
FWER = family wise error rate
Holm/Hochberg Example for m=5, α=0.05

i p value α/(6-i) 0.05/(6-i)

1 p1-smallest α/5 0.0100
2 p2 α/4 0.0125
3 p3 α/3 0.0167
4 p4 α/2 0.0250
5 p5-largest α 0.0500
No adjustment vs Hochberg vs Bonferroni
0.06
m=5, α=0.05

0.05
significance criterion

0.04 no adjustment
Bonferroni
Hochberg
0.03
p value

0.02

0.01

0
1 2 3 4 5
i
 m=5, alpha=0.05

i no adjustment Bonferroni Hochberg p value

1 0.05 0.01 0.0100 0.007
2 0.05 0.01 0.0125 0.011
3 0.05 0.01 0.0167 0.014
4 0.05 0.01 0.0250 0.044
5 0.05 0.01 0.0500 0.049
 FWER vs FDR
If a “family” of “m” hypothesis tests are carried
out, the family wise error rate (FWER) is the
chance of any “false positive” type I error
assuming that the null is true for all m tests.
Rather than control the FWER, it may be
preferable to control the number of “positive”
tests (not all tests) that are false positives. This
is called controlling the false discovery rate
(FDR), a less stringent criterion.
For FDR, the ith ordered p value must be less
than (i/m)α which is larger than α/(m+1-i) for
FWER.
 FDR vs FWER
errors committed when testing “m” null hypotheses
Declared non sig Declared sig Total

Truth-Null true U V m0

Truth-Null false T S m-m0

total m-R R m

FWER= V/m (prob V ≥ 1)

FDR = V/R (average V/R)
FDR is more liberal
FWER vs FDR significance criteria
m=5 hypothesis, 5 p values
α=0.05
p value FDR criteria FEWR criteria
p1-smallest (1/5) α=0.01 α/5=0.01
p2 (2/5) α=0.02 α/4=0.0125
p3 (3/5) α=0.03 α/3=0.0167
p4 (4/5) α=0.04 α/2=0.025
p5-largest α=0.05 α=0.05
 Multiple testing & primary outcomes
As “m’, the number of outcomes, increases, individual αi
for each outcome must be smaller so n must be larger if
overall α is to stay constant (ie at α=0.05).
But not all outcomes are equally important. Designate
important outcomes “primary” & the rest secondary so m
is only the number of primary outcomes. Assumes less
concern if there is a false positive finding among
secondary outcomes.
Must designate primary vs secondary outcomes in
advance, before study results are known. It is not fair to
declare which outcomes are primary and which are
secondary based on their p values.
 Statistical Analysis Plan
Statistical models and methods to answer study questions
Conclusions = data + models (assumptions)
Each specific aim needs a stat analysis section.

Sample size and power follows the analysis plan.

Outline:
•Outcomes: denote primary & secondary
•Primary predictors or comparison groups
•Covariates/confounders/effect modifiers
•Methods for missing data, dropouts
•Interim analyses (for efficacy, for safety)
 Common Methods
Univariate analysis
Continuous outcome: Means, SDs, medians
Time to event: Survival curves
Discrete: Proportions
Multivariate analysis
Continuous outcome: Linear regression,correlation
Positive integers: Poisson regression
Binary (yes/no): Logistic regression
Time to event: Proportional hazard regression

ANOVA and t-test are special cases of linear regression