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S.F.Kelsey/class2181/lecture 4-sample size 1
DESIGN OF CLINICAL TRIALSEPIDEMIOLOGY 2181
“SAMPLE SIZE DETERMINATION AND STATISTICAL POWER IN CLINICAL TRIALS”
October 30, 2008Lecture 4
SHERYL F. KELSEY, PhD
Department of Epidemiology
S.F.Kelsey/class2181/lecture 4-sample size 2
QUIZAssume 90% Power, a = 0.05 two-sided
(x) more with A(y) more with B(z) the same
1. Mortality 20% vs 10% 40% vs 30%
2. Mortality 20% vs 10% 20% vs 15%
3. Diastolic 80 vs 85 mmHg 90 vs 95 mmHg BP
4. Diastolic 80 vs 85 mmHg 80 vs 85 mmHg BP
A B
(x) more with A(y) more with B(z) the same
(x) more with A(y) more with B(z) the same
(x) more with A(y) more with B(z) the same
(St Dev 10) (St Dev 10)
(St Dev 10) (St Dev 8)
How manysubjects?
S.F.Kelsey/class2181/lecture 4-sample size 3
1. More with B
2. More with B
3. The same
4. More with A
ANSWERSVariance of the binomialbigger 50% smaller 0% 100%
Small difference need more subjects
Only standard deviation matters
Bigger standard deviation more subjects
S.F.Kelsey/class2181/lecture 4-sample size 4
SHALL WE COUNT THE LIVING OR THE DEAD?
40% vs 20% 20% 50% “reduction” in mortality lower mortality
20% vs 10% 10% 50% “reduction” in mortality lower mortality
10% vs 5% 05% 50% “reduction” in mortality lower mortality
60% vs 80% 20% 33% “improvement” in survival higher mortality
80% vs 90% 10% 2.5% “improvement” in survival higher mortality
90% vs 95% 05% 5.6% “improvement” in survival higher mortality
Absolute Relative
S.F.Kelsey/class2181/lecture 4-sample size 5
Even more confusing with continuous variables
Blood pressure (St Dev 10)
5.9% “reduction” 80 vs 85 mmHg
5.3% “reduction” 90 vs 95 mmHg
S.F.Kelsey/class2181/lecture 4-sample size 6
Intervention A has “increased” survival
No: Intervention A has longer, better, greater survival
“Increase” should be used for changes over time
S.F.Kelsey/class2181/lecture 4-sample size 7
PERCENTS Absolute difference
Relative difference
Percents as continuous measures
Proceed with caution
S.F.Kelsey/class2181/lecture 4-sample size 8
Legal null hypothesis: innocent until proven guilty
Scientific null hypothesis: no difference in response between treatment groups
Inno
cent
Gui
lty
Innocent Guilty
Truth
Dec
isio
n of
Ju
dge/
Jury
ok
ok
guilty goes freetype IIerror ()
hang theinnocenttype Ierror ()
Treatment Different
TreatmentSame
Truth
Sam
eD
iffer
ent
Obs
erve
d D
ata
ok
ok
miss goodtreatmenttype IIerror ()
promoteworthlessTx type Ierror ()
S.F.Kelsey/class2181/lecture 4-sample size 9
FUNDAMENTAL POINT
Clinical trials should have sufficient statistical power to detect differences between groups considered to be of clinical interest. Therefore, calculation of sample size with provision for adequate levels of significance and power is an essential part of planning.
S.F.Kelsey/class2181/lecture 4-sample size 10
THE RAW INGREDIENTS
What is your question, precisely?
What is your outcome, precisely?
Who will be measured?
Type 1 and type 2 error rates
Variability
S.F.Kelsey/class2181/lecture 4-sample size 11
PRIMARY COMPARISONS
Dichotomous Response Variables
The event rates in the intervention group (Pi) and the control group (Pc) are compared
Continuous Response Variables
The true but unknown mean value in the intervention group is compared with the true, but unknown
mean value in the control group.
Survival Data
A hazard rate is often compared for the two study groups or at least is used for sample size
estimation.
S.F.Kelsey/class2181/lecture 4-sample size 12
SAMPLE SIZE ISSUES
Choice of outcome – primary endpoint
Change from baseline only if correlation > .5 when in doubt don’t
The difference____________ to detect you want you believe is clinically meaningful you believe is biologically credible you can afford to
S.F.Kelsey/class2181/lecture 4-sample size 13
ASSESSMENTS OF EVENT RATE IN THE CONTROL AND INTERVENTION GROUPS
The estimate for the control group event rate is usually obtained from a previous study of similar people. Good data base desirable
The investigator must choose the difference in event rate based on preliminary evidence of the potential effectiveness of the intervention or be willing to specify some minimum difference.
Calculation of several sample sizes based on a range of estimates helps one to assess how sensitive the sample size is to these estimates.
S.F.Kelsey/class2181/lecture 4-sample size 14
To Plan with continuous endpoints: Clinical difference worth detecting
1–Power Probability of obtaining a significant result if is true difference
Significance level, must specify one or two-tailed test
(Z Z )2 Multiplier which depends on level of significance and Power 1-
n Sample size for each of two groups
For continuous measures:
Standard deviation
2
222)(
ZZ
n
With a little algebra
Z=1.96 for =.05, two-sided
n
ZZ 222)(
Zn
z 2
(solve for power)
(Solve for difference)
S.F.Kelsey/class2181/lecture 4-sample size 15
For two proportions P1 vs P2, = P1 - P2 With a little algebra
Z = 1.64 for .05, one-sided
Z = 1.96 for .05, two-sided 221
2
arcsinarcsin2
)(
PP
ZZ
221
2
arcsinarcsin2 PP
ZZn
ZPPnZ 21 arcsinarcsin2
2
1
2
2 arcsin2
)(sin
P
n
ZZP
22
1 2
)(arcsinsin
n
ZZP
212
22112 11
PP
PPPPZZn
S.F.Kelsey/class2181/lecture 4-sample size 16
TABLE(Z + Z)2
Needed to determine the size of each sample
(Z22.32 1.645 1.28)
Desired Two-Tailed Tests One-Tailed Tests Power Level Level
Z P 0.01 0.05 0.10 0.01 0.05 0.10
Two groups of unequal size: Calculate the harmonic mean
}2/11
/{1
21
nn
n
This n is what is needed for 2 groups of equal size. Note that equal sized groups are the most efficient, that is the harmonic mean is less than the arithmetic mean.References: Snedecor and Cochran, 7th Edition Statistical Methods, 1980, pp 102-1- 5, 120, 130.
Fleiss, JL. Statistical Methods for Rates and Proportions, 1981, Chapter 3 & Tables. Schlesselman, JJ. Case Control Studies, 1981, Chapter 6 & Tables.
(Z 2.576 1.96 1.645)
0.84 0.80 11.7 7.9 6.2 10.0 6.2 4.5
1.28 0.90 14.9 10.5 8.6 13.0 8.6 6.6
1.645 0.95 17.8 13.0 10.8 15.8 10.8 8.6
S.F.Kelsey/class2181/lecture 4-sample size 17
Example: Compare .10 vs .05
= .05 one sidedPower 80%
arcsin
arcsin
2255.05.
3218.10.
221
2
)arcsin(arcsin2
)(
PP
ZZn
334
)0963(.2
2.62
n
So total study: 334 x 2 = 668
.10 vs .05 with 200 patients in each group
2002x
Power = 61%
with 100 patients Z = .28 39% power
50 patients Z = .68 25% power
nZ 2 | 21 arcsin PP | .0963| - 1.64 = .286| — Z arcsin
S.F.Kelsey/class2181/lecture 4-sample size 18
FURTHER SAMPLE SIZE CONSIDERATIONS
outcome is survival time discount for noncompliance/dropout projections subgroups more than 2 treatment groups unequal group sizes - less efficient - can get more information on new
treatment computer packages (PASS, SAS, MINITAB) ASSUMPTIONS ABOUT EVENT RATES PARAMOUNT
sensitivity analysis
S.F.Kelsey/class2181/lecture 4-sample size 19
ONE-SIDED VERSUS TWO-SIDED TESTS
I Drug A side effects/expensiveDrug B no side effects/cheap
A more efficaciousA&B the sameB more efficacious
II X Nutrition Intervention Strategy-Group sessionsY Nutrition Intervention Strategy-Individual program
X reduce sodium intake moreX&Y the sameY reduce sodium intake more
S.F.Kelsey/class2181/lecture 4-sample size 20
SAMPLE SIZE FOR TESTING “EQUIVALENCY”
OF INTERVENTIONS
The problem in designing positive control studies is that there is no statistical method to demonstrate complete equivalence.
Computing a sample size assuming no difference results in an infinite sample size.
One approach is to specify a value for difference in response such that interventions
with differences that are less than this might be considered equally effective or equivalent.
S.F.Kelsey/class2181/lecture 4-sample size 21
T = Innovative Therapy
S = Standard Therapy
“Superiority”H0: death rate T = death rate S
Halt:death rate T < death rate S
Equivalence H0: death rate T death rate S +
Halt:death rate T < death rate S +
In general equivalence studies require more patients
S.F.Kelsey/class2181/lecture 4-sample size 22
Patients: Acute MI
Treatment: Double bolus vs accelerated Alteplace
Outcome: 30 day mortality
COBALT Equivalence
Death rate within 0.4%
GUSTO III SuperiorityDouble bolus reduce mortality by 20%
WARE AND ANTMAN EDITORIAL
S.F.Kelsey/class2181/lecture 4-sample size 23
MORTALITY RESULTS
COBALT GUSTO III
N 7169 15059
Double bolus 7.98% 7.47%
Accelerated 7.53% 7.24%
Difference 0.45% 0.23%
95% CI Approx. (-.85%, 1.66%) (-.66%, 1.10%)
Action reject equivalence accept null not significantly different from zero
S.F.Kelsey/class2181/lecture 4-sample size 24
DESIGN OF CLINICAL TRIALSEPIDEMIOLOGY 2181
RANDOMIZATION IN CLINICAL TRIALS
SHERYL F. KELSEY, PH.D
S.F.Kelsey/class2181/lecture 4-sample size 25
WHY RANDOMIZE? Best way to assure comparability
In the long run balance of factors known unknown
Statistical hypothesis test based on random assignment
Selection is impartial: “dice not trying to prove a point” - must convince others of validity of comparison
S.F.Kelsey/class2181/lecture 4-sample size 26
RANDOMIZATIONFIXED ALLOCATION: Assigns with pre-specified probability (not necessarily, though usually, equal)
ADAPTIVE: Changes probabilities during study Baseline adaptive:
• on basis of number per group• on basis of variables
Responsive adaptive:• depends on prior outcome• assumes:
• rapid response• stable population source
S.F.Kelsey/class2181/lecture 4-sample size 27
STEPS IN THE RANDOMIZATION OF A PATIENT
Check eligibility
Informed consent
Formal identification
RANDOMIZE
Confirmation of patient entry
S.F.Kelsey/class2181/lecture 4-sample size 28
HOW RANDOM TREATMENT ASSIGNMENTS ARE MADE
Model: Slips in a hat or flipping a coin
Masked drugs numbered and given in order: pharmacy, drug manufacturer
Envelopes
Telephone to central unit Microcomputer at the site
Central computer – internet access
S.F.Kelsey/class2181/lecture 4-sample size 29
HOW TO DO THE SCHEME
Simple randomization
Biased coin, urn models
Example:
Start with 2 balls, one black and one white
Draw-replace and add one of opposite color
Prevents imbalance with high probability early on
Random permuted block
Balance at the end of block
Could predict with unmasked trial
S.F.Kelsey/class2181/lecture 4-sample size 30
BLOCKS OF SIZE 4
4
24
2 2
4 3 2 1
2 1 2 16
!
! !
* * *
* * *1) 1100
2) 1010
3) 1001
4) 0110
5) 0101
6) 0011
S.F.Kelsey/class2181/lecture 4-sample size 31
HOW TO USE BLOCKS WHEN TREATMENT IS NOT MASKED
Choose the block sizes at random, too
Example: 2 treatment, equal allocation
Block sizes 4, 6, and 8
Balance in each block
S.F.Kelsey/class2181/lecture 4-sample size 32
SHOULD YOU STRATIFY?
Clinical sites - generally yes
Prognostic variables - generally no
Size
Practical considerations
Often governed by custom rather than statistical justification
Stratified ANALYSIS is generally preferable
S.F.Kelsey/class2181/lecture 4-sample size 33
MINIMIZATION
Advantages: Balance several prognostic factors Balance marginal treatment totals Good for small trials (<100 patients) Computer makes this fairly easily
Disadvantages: Can’t prepare treatment assignment Scheme in advance Need up-to-date record Not really random - could predict but can introduce
random element by using say 3/4, 1/4
S.F.Kelsey/class2181/lecture 4-sample size 34
TABLE 5.7. - TREATMENT ASSIGNMENTS BY THE FOUR PATIENT FACTORS FOR 80 PATIENTS IN AN ADVANCED BREAST CANCER TRIAL
Factor Level No. on each Next treatment patient A B
Performance status Ambulatory 30 31 Non-ambulatory 10 9
Age <50 18 17 50 22 23
Disease-free interval <2 years 31 32 2 years 9 8
Dominant metastatic lesion Visceral 19 21 Osseous 8 7 Soft tissue 13 12
Pocock S. Clinical Trials: A Practical Approach. John Wiley & Sons, Chichester, England, 1991, p. 85.
Thus, for A this sum = 30 + 18 + 9 + 19 = 76while for B this sum = 31 + 17 + 8 + 21 = 77
S.F.Kelsey/class2181/lecture 4-sample size 35
INTERNAL VALIDITY
compare treatmentsExternal Validity/ Generalizability
extrapolate to other patients
Not realistic to find a random sample of patients for recruitment (at the very least they have to consent)
More important to establish efficacy of treatment before deciding if it can be broadly applied
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