Sample size estimation in a single randomised clinical trial with or without interimanalyses

1
Sample size estimation in a single randomised
clinical trialwith or withoutinterim-analyses

• Jørn Wetterslev M.D., Ph.D.
• Copenhagen Trial Unit
• Centre for Clinical Intervention Research
• Rigshospitalet

2
Presentation

• The risk of type I and type II errors
• Sample size continuous outcome measure
• Sample size binary outcome measure
• Practical approach
• Expansion of sample size in multicentre trials
  with interim analyses and heterogeneity

3
Type I error risk

• The probability that you find a difference of a
  given size (or more) given the null hypothesis,
  that there is no difference, is true a

4
Type II error risk

• The probability that you discard a difference of
  a given size (or more) even though the
  alternative hypothesis, that there is a
  difference, is true ß

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Sample size in a randomised clinical trials

• The number of participants to be included in
  the trial to detect or reject an anticipated
  intervention effect µ with the chosen error risks
• Type 1 error ?
• Type 2 error ? (power 1- ?)

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t-test statistic

• Continuous variable X N(X, SD2)

µ
X1 - X2
t
( )
?
SD
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Sample size in a RCT with a continuous outcome
measure

• The number of participants N to be included in
  the trial to detect or reject an anticipated
  intervention effect µ with a variance of ? with
  the chosen ? and power 1- ?
• N 2 ?

(Z2aZß)2 ? ?
µ2
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Sample size in a RCT with a continuous outcome
measure

• Equal group size and sample size N
• with ? 0.05 and ? 0.20
• N 32 ? 32 ?

2
2
SD
Noise
µ
Signal
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Sample size (SS) randomised clinical trial with
binary outcome equal group sizes

• SS 4 (Z?/2 Z?)2 ? / ?2
• ? PC - PE intervention effect
• ? P (1- P) the variance P (PCPE) / 2
• PC and PE
• event rates in control and intervention group

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Sample size in RCT with equal group size Type I
error risk 0.05 and power0.10
16(RR1)pCont(RR21) pCont (1RR)2
N
With RR pIntervent / pCont
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Sample size in a RCT with a binary outcome
measure and equal group sizes

• The number of participants N to be included in
  the trial to detect or reject an estimated
  intervention effect µ with an stimated variance
  of ? with a chosen
• ? 0.05 and power 1- ? 1- 0.20
• N 4 ? 64 ?

16 ? ? 2
2
?
µ
µ2
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Sample size in a RCT with a binary outcome measure

• Sample size N
• with ? 0.05 and ? 0.20
• N 64 ? 64 ?

2
2
?
Noise
µ
Signal
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Practical approach

• It is easy to calculate N if µ and ? are known
  (true effect and true variance), - but they never
  are !!!
• Estimating or guessing a µ for µ and ? for ? is
  difficult and may in the first trial be
  impossible
• If a trial is preceeded by numerous trials a
  meta-analysis is a prerequisite to make estimates
  of a realistic interventon effect µ and variance
  ?

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Practical approach

• Power and sample size calculater at
• http//biostat.mc.vanderbilt.edu/twiki/bin/view/Ma
  in/PowerSampleSize

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Sample size in randomised trials with interim
analyses

• Multiple tests on accumulating data
• Heterogeneity in multicentre trials

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Expansion of fixed sample size by multiple
looks in interim analyses

• Multiple analyses of accumulating data may
  overinterpret results of interim analyses
• The total type I error risk doing repetitive
• significance testing on accumulating data with
• a 5

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Group sequential boundaries same group size
Z 1.96
Z - 1.96
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Sample size adjustment in sequential designs
a 0.05 1-ß 0.80

• Calculate the adjusted sample size by
  multiplying of fixed sample size with a factor
  from a relevant table (choice of a-spending)
• Pococks design

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Group sequential boundaries same group size
Z 1.96
Z - 1.96
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Adjusting sample sizefor multiple
looksOBrien-Flemming
Sample size adjustment factor Mk a 0.05
1-ß0.80

• Calculate the adjusted maximum sample size by
  multiplying fixed sample size with the adjusting
  factor from appropriate tables (choice of
  a-spending)

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Expected proportion of maximumfixed sample size
OBrien-Flemming
Percentage of fixed sample size a 0.05 1-ß
0.80

• The percentage of the maximum sample size
  expected on average in trials with the
• OBrien- Flemming design

22
Adjusting sample sizefor multiple
looksOBrien-Flemming
Sample size adjustment factor Mk a 0.05
1-ß 0.80

• Calculate the adjusted maximum sample size by
  multiplying fixed sample size with the adjusting
  factor from appropriate tables (choice of
  a-spending)

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Adjusted multiple looks sample size (M SS)

• Calculated setting µ and ? to relevant values
  possibly found in evidence based on other
  interventions in that area

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Sample size in a multicentre trial

• If you do not account for variability in the
  number of recruited participants and the estimate
  of the intervention effect between the sites in a
  multicenter trial it may increase type I error
  risk and reduce power
• Valerij Fedorov og Byron Jones (GSK Pfizer)
• Statistical Methods in Medical Research
  200514205-48

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A priori heterogeneity- adjusted multiple looks
sample size (APHM SS)

• Calculated setting µ and ? to a clinically
  relevant value possibly found in evidence based
  on other interventions in that area

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A priori heterogeneity adjusted multiple sample
size (APHM SS)

• Calculated setting µ and ? to
• clinical relevant values or realistic estimates

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Conclusions

• Estimate or calculate relevant or anticipated
  intervention effect
• Estimate or calculate relevant variance
• Calculate fixed sample size (without interim
  looks)
• Adjust for multiple looks when interim analyses
  are planned
• Adjust for anticipated heterogeneity between
  sites in multicenter trials

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Type 1 error risk and power

• Type 1 error risk Pr?t?gt c ?
• when the effect of the intervention 0
• c ?-1(1 - ?/2) ( 1.96)
• Power Pr ?t?gt c 1 - ?
• when the effect of the intervention ? 0
• and ? Type 2 error risk