Bios 6649: Clinical Trials - Statistical Design and Monitoringcsph.ucdenver.edu/sites/kittelson/Bios6649-2015/Lctnotes/2015/lctS… · 3.1 On Choosing the Sample Size Addressing inadequate

3. Design offixed-sample trial3.1 On choosing the samplesize

3.2 Frequentist evaluationof a fixed-sample trial

(a) Examples (introduced)

(b) Pre-determinedinformation

(c) Selecting theinformation

* Decision theoretic

* Statistical power

(d) Commonmean-variancerelationships

Bios 6649- pg 1

Bios 6649:Clinical Trials - Statistical Design and Monitoring

Spring Semester 2015

John M. KittelsonDepartment of Biostatistics & Informatics

Colorado School of Public HealthUniversity of Colorado Denver

c©2015 John M. Kittelson, PhD







* Statistical power


Bios 6649- pg 2

3. Fixed-sample Clinical Trial Design

Overview

3.1 On choosing the sample sizeI Importance of adequate informationI Key elements of informationI Evaluation of informationI Addressing inadequate information

3.2 Frequentist evaluation of a fixed-sample trial(a) Examples

I Phase II Iloprost chemoprevention trialI Sepsis trialI Daptomycin trialI PLCO trial

(b) When information is pre-specified(c) Selecting the information(e) Common mean-variance relationships







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3.1 On Choosing the Sample Size

The importance of adequate information

I A trial requires adequate information to answer the designquestions (i.e., to inform science and clinical practice).

I Without adequate information the trial cannot answer therelevant questions.

I A trial that cannot answer the design question is notethical.







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Key elements of information

I Information can be measured by:I Confidence interval widthI PowerI Fisher information: N/σ2.

I Sample size equation contains key elements ofinformation:

N =

(zα + zβθ+ − θ∅

)2

V

I Sample size (N): Number of subjects in the trial.I Variance (V ): Inherent variability (“noise") in the outcome

measure.I Statistical and scientific operating characteristics:

I Specificity (zα): statistical standard for evidence.I Sensitivity (zβ ): power







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Key elements of information (con’t)

I Smallest important difference (θ+ − θ0): Variously definedas:

I Smallest important effect: Smaller differences are notclinically important.

I Most likely effect: Anticipated effect based on past experiencein a disease setting.

I Effect observed previously: Magnitude observed in an earlierstudy.

I Self-designing trials: Alter θ+ − θ0 based on observed effectsin the midst of the current trial.

I Detectable difference: θ+ − θ0 is often called the detectabledifference.







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Bios 6649- pg 6


Evaluation of information

I Frequentist:I PowerI Design alternative hypothesis with power βI Sample size

I Bayesian:I Predicted powerI Sample sizeI (Loss functions in decision-theroetic constructions)

I Scientific:I Confidence interval width (97.5% power point)I Sample size in relation to previous studies.







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Addressing inadequate information

I During trial design (before trial starts)I Reduce variability:

Standardize endpoint measurement proceduresCentral endpoint adjudicationReduce non-compliance with a carefully-selectedstudy population

I Increase sample size:Expand the number of centers participating in the trial.“Large simple trial": fewer measurements on more

people.Time-to-event trials: increase duration to get more events.

I Increase retention:“The best retention plan is a good recruitment plan."Reduce participant burden.Allow treatment termination, but retain for measurement.

I Change endpoints: Use a different endpoint with higherpower.This likely means changing the scientific objectives.







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Bios 6649- pg 8


Addressing inadequate information (con’t)

I During trial conduct (after trial starts)I Routine problems:

I Poor recruitment: Many trials fail due to poor recruitmentI Excessive dropout or loss to follow-up: May also introduce

biasI Excessive non-compliance: May inflate variation

I Solutions that preserve the scientific design:I Expand recruitment effortsI Expand the trial to other centersI Fix problems with retentionI Extend the planned trial duration to allow additional

recruitmentI Solutions that may alter the scientific design:

I Change eligibility criteria: This will alter the study populationand could render the trial uninterpretable.

I Change treatments to encourage recruitment (esp. controltreatment)

I Institute ancillary treatments to improve tolerability andretention.







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Bios 6649- pg 9

3.2 Frequentist evaluation of a fixed-sample trial

Organization

(a) Examples (introduced)(b) Pre-specified information (sample size):

I Inference at the boundaryI Examples (revisited)

(c) Selecting the information (sample size):I Hypotheses discriminated (decision-theoretic approach)I Statistical powerI Examples (revisited)

(d) Common mean-variance relationships







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Bios 6649- pg 10


Phase II Iloprost

I Background: A previous phase II chemoprevention trial of13-cis retinoic acid (vitamin A) did not show effect on lunghistology. There was good biological rationale (and labdata) to suggest that iloprost might be effective. A phase IItrial was designed to evaluate effects on histology.

I Study Population: Former or current smokers with > 20pack years, and sputum atypia.

I Scientific Hypotheses: Iloprost treatment will result inimproved histology relative to placebo.







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Bios 6649- pg 11


Phase II Iloprost

I Outcome Parameterization:I Outcome (Yik ): 6-month change in average histology (also

maximum histology and dysplasia index)I Probability model: Non-parametric: Yi0 ∼ F0(y); Yi1 ∼ F1(y)I Functional: Mean: E(Yi0) = θ0, E(Yi1) = θ1I Contrast: Difference: θ = θ1 − θ0.







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Bios 6649- pg 12


Sepsis trial

I Background:I Critically ill patients often get overwhelming bacterial

infection (sepsis), after which mortality is high.I Gram negative sepsis is often characterized by production

of endotoxin which is thought to be the cause of many of theill effects of gram negative sepsis.

I Hypothesize that administering an antibody to the endotoxinmay decrease morbidity and mortality.

I Two previous randomized clinical trials showed slight benefitwith suggestion of difference in benefit within subgroups.

I There are no safety concerns (based on previous studies).







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Bios 6649- pg 13


Sepsis trial

I Design: Double-blind, placebo controlled RCT.I Study Population: Patients entering ICU with newly proven

gram-negative sepsis.I Scientific Hypotheses: Antibody to the endotoxin will

improve survivalI Outcome Parameterization:

I Outcome (Yik ): 28-day mortalityI Probability model: Bernoulli: Yi0 ∼ B(1, p0); Yi1 ∼ B(1, p1)I Functional: Mean: E(Yi0) = p0 = θ0, E(Yi1) = p1 = θ1I Contrast: Difference: θ = θ1 − θ0.







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Bios 6649- pg 14


Daptomycin Trial

I Background: Standard therapy for S. aureus bacteremiaand endocarditis is not fully successful. Daptomycin isanother antibiotic that might also be effective for thiscondition.

I Clinical Question: Is daptomycin as successful asstandard care in treating S. aureus bacteremia?

I Study Population: Patients with S. aureus bacteremiaI Outcome Parameterization:

I Outcome (Yik ): Success at 42-days after the end oftreatment

I Probability model: Bernoulli: Yi0 ∼ B(1, p0); Yi1 ∼ B(1, p1)I Functional: Mean: E(Yi0) = p0 = θ0, E(Yi1) = p1 = θ1I Contrast: Difference: θ = θ1 − θ0.







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Bios 6649- pg 15


Prostate cancer screening (PLCO)

I Background: PSA screening for prostate cancer iscommon, but the tradeoff between risks and benefits is notknown.

I Public Health Question: Should screening for prostatecancer be part of routine practice?

I Study Population: Men age 55-74







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Bios 6649- pg 16



I Outcome Parameterization:I Outcome Yik = (Tik , δik ): Time to death from prostate

cancer:

Tik = time from study entry to end of follow-up

δik = 1 if died from prostate cancer; 0 otherwise.

I Probability model: Incidence rate follows Poissondistribution:

θ̂k =

N∑i=1δik

N∑i=1

Tik

and θ̂k

N∑i=1

Tik ∼ P(λk

N∑i=1

Tik ).

I Functional: Mean rate: θ0 = λ0, θ1 = λ1I Contrast: Ratio: θ = θ1/θ0.







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(b) Pre-specified information (sample size)

I Suppose that the sample size is pre-determined.I How should we evaluate the information?







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Pre-specified information (sample size)

I Recall inference upon trial completion:I Point estimateI Interval estimateI P-valueI Decision

I Can we examine the potential inference upon trialcompletion prior to starting the trial?

I (Reference: Goodman and Berlin, Ann Intern Med. 1994)







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Potential inference upon completion

I Inference at the boundary:I Critical value: Threshold for declaring that the treatment

works.I Hypotheses discriminated (interval estimate):

I What hypothesis will be rejected if the results are significant?I What hypothesis will be rejected if the results are not

significant?I P-value at the boundary?







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Phase II Iloprost

I Sample Size: N = 76 per groupI Variance: From 13 cis-retinoic acid trial:

Observation Retinoic AcidNon-smokers Smokers Non-smokers Smokers

Avg. Histol. 0.20 (0.59) 0.19 (0.73) -0.27 (0.84) -0.47 (1.22)Worst Histol. 0.00 (1.22) -0.50 (1.38) -0.20 (1.32) -0.45 (1.63)Dysp. Index 0.05 (0.25) 0.08 (0.25) -0.04 (0.21) -0.12 (0.27)

(Notice: Potential mean-variance relationship.)







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Bios 6649- pg 21


Phase II Iloprost

I Variance: Preliminary variance estimate:

Spooled =

√14(0.592 + 0.732 + 0.842 + 1.222)

≈ 0.90

SE = Spooled

√1

N0+

1N1

= 0.90

√276

= 0.146







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Bios 6649- pg 22


Phase II Iloprost

I Potential Inference:I Critical value: cv = −1.96× SE = −0.286I Interval estimate at cv: (−0.572, 0)I Possible conclusions:

I If θ̂ > cv then reject θ ≤ −0.572I If θ̂ ≤ cv then reject θ ≥ 0







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Bios 6649- pg 23

(b) Pre-determined sample size

Phase II Iloprost

I RCTdesign implementation:I Primary function for defining any design in RCTdesign isseqDesign()

> iloII <- seqDesign(prob.model="normal",arms=2,test.type="lessr",+ variance=0.9^2,sample.size=152)> iloIICall:seqDesign(prob.model = "normal", arms = 2, variance = 0.9^2,

sample.size = 152, test.type = "less")

PROBABILITY MODEL and HYPOTHESES:Theta is difference in means (Treatment - Comparison)One-sided hypothesis test of a greater alternative:

Null hypothesis : Theta >= 0.0000 (size = 0.025)Alternative hypothesis : Theta <= -0.5723 (power = 0.975)(Fixed sample test)

STOPPING BOUNDARIES: Sample Mean scaleFutility Efficacy

Time 1 (N= 152) -0.2862 -0.2862







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Bios 6649- pg 24


Sepsis trial

I Sample Size: N = 850 per groupI Variance: Assume that placebo mortality rate is p0 = 0.3.

S =√

p(1− p)

≈√

0.7× 0.3= 0.458

SE = S

√1

N0+

1N1

= 0.458

√2

850= 0.0222







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Bios 6649- pg 25


Sepsis trial

I Potential Inference:I Critical value: cv = 0− 1.96× SE = −0.0436I Interval estimate at cv: (−0.087, 0)I Possible conclusions:

I If θ̂ ≥ cv then reject θ ≤ −0.087I If θ̂ < cv then reject θ ≥ 0







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Bios 6649- pg 26


Sepsis trial

I RCTdesign implementation:

> sepsisFixed <- seqDesign(prob.model="proportions",arms=2,+ test.type="less", null.hypo=0.3,variance="null",sample.size=1700)> sepsisFixedCall:seqDesign(prob.model = "proportions", arms = 2, null.hypothesis = 0.3,

variance = "null", sample.size = 1700, test.type = "less")

PROBABILITY MODEL and HYPOTHESES:Theta is difference in probabilities (Treatment - Comparison)One-sided hypothesis test of a lesser alternative:


STOPPING BOUNDARIES: Sample Mean scaleEfficacy Futility

Time 1 (N= 1700) -0.0436 -0.0436







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Bios 6649- pg 27


Daptomycin Trial

I Sample Size: N = 90 per groupI Variance: Assume that placebo success rate is p0 = 0.65.

S =√

p(1− p)

≈√

0.65× 0.35= 0.477

SE = S

√1

N0+

1N1

= 0.477

√2

90= 0.071







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Bios 6649- pg 28


Daptomycin Trial

I Potential Inference (superiority):I Critical value: cv = 0 + 1.96× SE = 0.139I Interval estimate at cv: (0, 279)I Possible conclusions:

I If θ̂ ≤ cv then reject θ ≥ 0.279I If θ̂ > cv then reject θ ≤ 0

I Potential Inference (non-inferiority):I Critical value: cv = 0I Interval estimate at cv: (−0.139, 0.139)I Possible conclusions:

I If θ̂ ≥ cv then reject θ ≤ −0.139I If θ̂ < cv then reject θ ≥ 0.139







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Bios 6649- pg 29


Daptomycin trial


> daptoSup <- seqDesign(prob.model="proportions",arms=2,+ test.type="greater",null.hypo=0.65,variance="null",+ sample.size=180)

> daptoEquiv <- seqDesign(prob.model="proportions",arms=2,+ test.type="equivalence",null.hypo=0.65,variance="null",+ sample.size=180)







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Bios 6649- pg 30



I Sample Size:N ≈ 38,350 per group; person-years ≈ 250,000 per group

I Variance:I Assuming prostate cancer death rate ≈ 2/10,000 py.I Assuming that there will be about 50 deaths per group.

SE(

log(θ1

θ0)

)=

√1

θ0P1+

1θ0P0

=

√150

+150

= 0.2







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Bios 6649- pg 31



I Potential Inference:I Critical value: cv = e0−1.96×0.2 = 0.676I Interval estimate at cv: (e−0.784, e0) = (0.457, 1.0)I Possible conclusions:

I If θ̂ ≥ cv then reject θ ≤ 0.457I If θ̂ < cv then reject θ ≥ 1.0







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Bios 6649- pg 32




> PLCOsup.1 <- seqDesign(prob.model="poisson",arms=2,+ test.type="less",null.hypo=2/10000,variance="null",+ sample.size=500000)>> PLCOsup.2 <- seqDesign(prob.model="poisson",arms=2,+ test.type="less",null.hypo=1,variance="null",+ sample.size=100)







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Bios 6649- pg 33

3.2 Frequentist evaluation of a fixed-sample trial(c) Selecting the information

Decision-theoretic approach

I Consider design hypotheses:

H0 : θ ≤ θ∅ harmH+ : θ ≥ θ+ important benefit

I Decision-theoretic approach:I Decision for benefit:

I Exclude all harmI Lower limit of 95% confidence interval is above θ∅

I Decision against benefit:I Exclude any important benefitsI Upper limit of 95% confidence interval is below θ+

I Avoid inconclusive result:I Confidence should not include both θ∅ and θ+







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Bios 6649- pg 34



I Confidence interval width:

W = 2× 1.96

√VN

where N is the sample size per group and V is anappropriate expression of the variation.

I Sample size required to achieve a width of W :

N =

(2× 1.96

W

)2

V







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Bios 6649- pg 35



I Selecting sample size so that CI cannot include both θ∅and θ+:

N =

(2× 1.96θ+ − θ∅

)2

V

I This sample size allows 2 possible conclusions uponcompletion:

1. Treatment is beneficial:I Rule out harmI Reject θ ≤ θ∅

2. Treatment lacks important benefit:I Rule out any important benefitI Reject θ ≥ θ+.

I Inclusive results (θ ≤ θ∅ and simultaneously θ ≥ θ+) areimpossible.







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Bios 6649- pg 36

(c) Selecting the information: decision-theoretic approach

Phase II Iloprost

I Design hypotheses:I θk = 6-month change in average histology with treatment k

(θk < 0 indicates improvement)I θ = θ1 − θ0 (θ < 0 indicates Iloprost is better).

H0 : θ ≥ 0 harm

H− : θ ≤ −0.47 important benefit







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Bios 6649- pg 37


Phase II Iloprost

I Variance:

var(θ̂1 − θ̂0) =

√σ2

0

N+σ2

1N

From previous lecture σ20 = σ2

1 ≈ 0.92 so that V = 2× 0.92.I Sample size:

N =

(2× 1.96θ− − θ∅

)2

V

=

(2× 1.96−0.47

)2

(2× 0.92)

= 113

i.e., 113 per group are required to discriminate the abovehypotheses.







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Bios 6649- pg 38


Phase II Iloprost

I Potential conclusions:1. Iloprost results in a significantly greater improvement in

histology (θ < 0)2. The effects of iloprost are significantly smaller than any

important difference (θ > −0.47).

RCTdesign implementation:

> iloII.975 <- seqDesign(prob.model="mean",arms=2,test.type="less",+ variance=0.9^2,null.hypo=0,alt.hypo=-0.47,power=0.975)> iloII.975Call:seqDesign(prob.model = "mean", arms = 2, null.hypothesis = 0,

alt.hypothesis = -0.47, variance = 0.9^2, test.type = "less",power = 0.975)

PROBABILITY MODEL and HYPOTHESES:Theta is difference in means (Treatment - Comparison)One-sided hypothesis test of a lesser alternative:


STOPPING BOUNDARIES: Sample Mean scaleEfficacy Futility

Time 1 (N= 225.37) -0.235 -0.235







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Bios 6649- pg 39


Sepsis trial

I Design hypotheses:I θk = 28-day mortality with treatment kI θ = θ1 − θ0 (θ < 0 indicates antibody is better).

H0 : θ ≥ 0 harm

H+ : θ ≤ −0.07 important benefit







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Bios 6649- pg 40


Sepsis trial

I Variance:

var(θ̂1 − θ̂0) =

√σ2

0

N+σ2

1N

where σ20 ≈ 0.3× 0.7 = 0.21; σ2

1 ≈ 0.23× 0.77 = 0.177I Sample size:

N =

(2× 1.96θ+ − θ∅

)2

V

=

(2× 1.96−0.07

)2

(0.21 + 0.177)

= 1214








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Bios 6649- pg 41


Sepsis trial

I Potential conclusions:1. Antibody injection results in a significantly greater survival

(θ < 0)2. Antibody injection does not result in any important in

survival (θ > −0.070)


> sepsis.975 <- seqDesign(prob.model="proportions",arms=2,+ test.type="less", null.hypo=c(0.3,0.3),alt.hypo=c(0.23,0.3),+ variance="alt",power=0.975)







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Bios 6649- pg 42


Daptomycin Trial

I Design hypotheses:I θk = 42-day success probability with treatment kI θ = θ1 − θ0 (θ > 0 indicates daptomycin is better).

H0 : θ ≤ −0.1 important harm

H+ : θ ≥ 0.1 important benefit







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Bios 6649- pg 43


Daptomycin Trial

I Variance:

var(θ̂1 − θ̂0) =

√σ2

0

N+σ2

1N

where σ20 = σ2

1 ≈ 0.65× 0.35 = 0.2275I Sample size:

N =

(2× 1.96θ+ − θ∅

)2

V

=

(2× 1.96

0.1− (−0.1)

)2

(2× 0.2275)

= 175








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Bios 6649- pg 44


Daptomycin Trial

I Potential conclusions:1. The daptomycin success rate is no worse than standard

care (−0.10 < θ).2. The daptomycin success rate is worse than standard care

(θ < 0.10).


> dapto.975 <- seqDesign(prob.model="proportions",arms=2,+ test.type="equivalence", null.hypo=c(0.65,0.65),alt.hypo=c(0.65,0.55),+ variance="null",power=0.975)







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Bios 6649- pg 45


PLCO Trial

I Design hypotheses:I θk = cause-specific death rate with treatment kI θ = θ1/θ0 (θ < 1.0 indicates antibody is better).

H0 : θ ≥ 1.0 harm

H+ : θ ≤ 0.5 important benefit







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Bios 6649- pg 46


PLCO Trial

I Variance:

var(log(θ̂1)− log(θ̂0)) =

√1D

+1D

where D is the number of deaths from prostate cancer ineach arm.

I Sample size:

D =

(2× 1.96θ+ − θ∅

)2

V =

(2× 1.96log(0.5)

)2

2

= 64

i.e., 128 deaths (total) are required to discriminate theabove hypotheses, and the total sample size must bechosen to get 128 deaths.







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Bios 6649- pg 47


PLCO Trial

I Potential conclusions:1. Screening does not reduce the prostate cancer death rate

(0.5 < θ).2. Screening reduces risk of death from prostate cancer

(θ < 1.0).


> PLCO.975 <- seqDesign(prob.model="poisson",arms=2,+ test.type="less",null.hypo=c(1,1),alt.hypo=c(0.5,1),+ variance="null",power=0.975)







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Bios 6649- pg 48

3.2 Frequentist evaluation of a fixed-sample trialSelecting the information

Statistical Power

I Recall the definition of power:I Power = probability that the null hypothesis is rejected upon

trial completion.I Power = 1− probability of type II error.I Power = βI Power = sensitivity of the design

I Power is a function of θ







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Bios 6649- pg 49

(c) Selecting the information: statistical power

Statistical Power

I Setting:I Treatment effect parameters:

θ0 = Outcome with control treatment

θ1 = Outcome with active treatment

θ = θ1 − θ0

I Variance:I Suppose there is unequal randomization with N1

N0= r , and let

N0 = N so that N1 = rN.I Then:

var(θ̂) =V0

N+

V1

rN[Notational note: often V0 = σ2

0 = var(Yi0) andV1 = σ2

1 = var(Yi1).]







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Bios 6649- pg 50


Statistical Power

I Define a standardized scale:

δ =θ − θ∅√V0N + V1

rN

=θ − θ∅√V0 +

1r V1

√N

I Note: in a fixed-sample trial δ̂ ∼̇ N (δ, 1) by CLT.







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Bios 6649- pg 51


Statistical Power

I Hypotheses:

H0 : θ ≤ θ∅H+ : θ ≥ θ+

I Standardized hypotheses:

H0 : δ ≤ 0H+ : δ ≥ δ+

whereδ+ =

θ+ − θ∅√V0 +

1r V1

√N







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Bios 6649- pg 52


Statistical Power

I Sample size equation is a map between the standardizedscale and the scale of the problem:

δ+ =θ+ − θ∅√V0 +

1r V1

√N

N =

(δ+

θ+ − θ∅

)2

(V0 +1r

V1)

I Calculating sample size:I Choose θ+ − θ∅ and V = V0 + V1

1r (per the specific clinical

setting).I Find δ+ in the standardized scale.I Calculate N as above.







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Bios 6649- pg 53


Statistical Power

I What is δ+?Distance between null hypothesis δ = 0 and designalternative δ+ with the desired power.

I Critical value in standardize scale:

Reject δ ≤ 0 when δ̂ > zα (usually zα = 1.96)

I For power = β, δ+ must be zβ units above the critical value:

δ+ = zα + zβ

I In group sequential designs we find δ+ by computer searchusing the standardized sequential sampling density.







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Bios 6649- pg 54


Statistical Power

Statistical Power: δ+ = 1.96 + 1.28P

roba

bilit

y de

nsity

0.0

0.1

0.2

0.3

0.4

0.5

−1.96 0.00 1.96 3.24 5.20δδ0 δδ+

NullHypothesis

AlternativeHypothesis

Red area = 0.025

Blue area = 0.90







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Bios 6649- pg 55


Statistical Power

I Calculating the power given a pre-determined samplesize:

I Sample size equation:

δ+ =θ+ − θ∅√V0 +

1r V1

√N

or:

zβ =θ+ − θ∅√V0 +

1r V1

√N − zα

and β is then found using this z-score.I The power for each decision should be calculated

separately.E.g., in a 2-sided test you should report power for:

I Finding A better than BI Finding B better than AI Finding equivalence







* Statistical power


Bios 6649- pg 56


Statistical Power: Superiority trial

Hypotheses: H0 : δ ≤ 0

H+ : δ ≥ δ+

−4 −2 0 2 4

0.0

0.2

0.4

0.6

0.8

1.0

δδ

P(r

ejec

t nul

l)

Superiority decisionNon−superiority decision







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Bios 6649- pg 57


Statistical Power: Approximate equivalence trial

Hypotheses: H0 : δ ≤ −δ+H+ : δ ≥ δ+

−4 −2 0 2 4

0.0

0.2

0.4

0.6

0.8

1.0

δδ

P(r

ejec

t nul

l)

EquivalentInferior







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Bios 6649- pg 58


Statistical Power: Non-inferiority trial

Hypotheses: H0 : δ ≤ −δ+H+ : δ ≥ 0

−4 −2 0 2 4

0.0

0.2

0.4

0.6

0.8

1.0

δδ

P(r

ejec

t nul

l)

Non−inferiorInferior







* Statistical power


Bios 6649- pg 59


Statistical Power: Equivalence trial (2-sided test)

Hypotheses: H− : δ ≤ −δ+H0 : δ = 0

H+ : δ ≥ δ+

−4 −2 0 2 4

0.0

0.2

0.4

0.6

0.8

1.0

δδ

P(r

ejec

t nul

l)

A better than BEquivalentB better than A







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Bios 6649- pg 60


Phase II Iloprost

Sample size with 90% power for θ+ = −0.47:

N =

(1.96 + 1.28

−0.47

)2(2 × 0.92)

= 77 Per group

−0.5 0.0 0.5

0.0

0.2

0.4

0.6

0.8

1.0

θθ

P(r

ejec

t nul

l)

SuperiorNot Superior







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Bios 6649- pg 61


Sepsis trial

Sample size with 90% power for θ+ = −0.07:

N =

(1.96 + 1.28

−0.07

)2(0.21 + 0.177)

= 830 Per group

−0.10 −0.05 0.00 0.05 0.10

0.0

0.2

0.4

0.6

0.8

1.0

θθ

P(r

ejec

t nul

l)








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Bios 6649- pg 62


Daptomycin trial

Sample size with 90% power for θ+ − θ∅ = 0.2:

N =

(1.96 + 1.28

0.2

)2(2 × 0.2275)

= 120 Per group

−0.3 −0.2 −0.1 0.0 0.1 0.2 0.3

0.0

0.2

0.4

0.6

0.8

1.0

θθ

P(r

ejec

t nul

l)

EquivalentNot Equivalent







* Statistical power


Bios 6649- pg 63


Daptomycin trial

I Note that using N = 120 per group requires shifting thecritical value in order to maintain the desired α/2 = 0.025type I error rate at θ∅ = −0.1:

cv = −0.1 + 1.96× SE

= −0.1 + 1.96√

2× 0.65× 0.35/120

= 0.0207







* Statistical power


Bios 6649- pg 64


PLCO trial

Sample size with 90% power for θ+ = log(0.5):

D =

(1.96 + 1.28

log(0.5)

)2× 4

= 88 total deaths (both groups)

0.4 0.6 0.8 1.0 1.2

0.0

0.2

0.4

0.6

0.8

1.0

θθ

P(r

ejec

t nul

l)








* Statistical power


Bios 6649- pg 65



I Common mean-variance relationships used whencalculating sample size:

I Two-sample trials:I Difference of meansI Difference of proportionsI Ratio of proportionsI Ratio of oddsI Ratio of hazards

I Single-sample trials







* Statistical power


Bios 6649- pg 66


Difference of means

I Estimates: θ̂0 = Y 0, θ̂1 = Y 1

I Sample size:

N =


)2

V

I Variance:V = σ2

0 +1rσ2

1

I Under Null: σ21 = σ2

0I Under Alternative: σ2

1 6= σ20

I Example: Phase II iloprost (see above)







* Statistical power


Bios 6649- pg 67


Difference of proportions


I Sample size:

N =


)2

V

I Variance:V = θ0(1− θ0) +

1rθ1(1− θ1)

I Under Null: θ1 = θ0 + θ∅I Under Alternative: θ1 = θ0 + θ+

I Example: DaptomycinI Under Null: θ0 = 0.65, θ1 = 0.65− 0.1 = 0.55 (90% power:

N = 125)I Under Alt: θ0 = 0.65, θ1 = 0.65 + 0.1 = 0.75 (90% power:

N = 109)







* Statistical power


Bios 6649- pg 68


Ratio of proportions


I Sample size:

N =

(zα + zβ

log(θ+)− log(θ∅)

)2

V

I Variance:V =

1− θ0

θ0+

1− θ1

rθ1

I Under Null: θ1 = θ0 × θ∅I Under Alternative: θ1 = θ0 × θ+

I Example: Daptomycin (assume θ∅ = 0.550.65 =, θ+ = 0.75

0.65 )I Under Null: θ0 = 0.65, θ1 = 0.55 (90% power: N = 149)I Under Alt: θ0 = 0.65, θ1 = 0.75 (90% power: N = 96)







* Statistical power


Bios 6649- pg 69


Ratio of odds

I Estimates: θ̂0 = Y 0

1−Y 0, θ̂1 = Y 1

1−Y 1

I Sample size:

N =

(zα + zβ


)2

V

I Variance (p0 = θ01+θ0

; p1 = θ11+θ1

):

V =1

p0(1− p0)+

1rp1(1− p1)

I Under Null: θ1 = θ0 × θ∅I Under Alternative: θ1 = θ0 × θ+

I Example: Daptomycin

(assume θ∅ =0.55/0.450.65/0.35 = 0.658,

θ+ = 0.75/0.250.65/0.35 = 1.615)

I Under Null: p0 = 0.65, p1 = 0.55 (90% power: N = 110)I Under Alt: p0 = 0.65, p1 = 0.75 (90% power: N = 127)







* Statistical power


Bios 6649- pg 70


Ratio of hazards

I Estimates: From score equationsI Sample size (D = total number of deaths in both groups):

D =

(zα + zβ


)2

V

I Variance: V = 4 under null hypothesisI Example: PLCO (assume θ∅ = 1, θ+ = 0.5)

I Under Null: (90% power: D = 88 total deaths)

Bios 6649: Clinical Trials - Statistical Design and Monitoringcsph.ucdenver.edu/sites/kittelson/Bios6649-2015/Lctnotes/2015/lctS… · 3.1 On Choosing the Sample Size Addressing inadequate

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