Power and Non-Inferiority

Richard L. Amdur, Ph.D.Chief, Biostatistics & Data Management Core, DC VAMC

Assistant Professor, Depts. of Psychiatry & SurgeryGeorgetown University Medical Center

Power and Non-Inferiorityin Clinical Trials

Richard L. Amdur, Ph.D.Chief, Biostatistics & Data Management Core, DC VAMC

Assistant Professor, Depts. of Psychiatry & SurgeryGeorgetown University Medical Center

If you can not reject the null hypothesis of ‘no effect’, this does not ‘prove’ there is no

effect

Why?

Frequency Distribution for One VariableSubject Score

1 92 103 104 115 116 117 128 129 12

10 1211 1312 1313 1314 1315 1316 1417 1418 1419 1420 1521 1522 1523 1624 1625 1626 1727 1728 1829 1830 1931 2032 21

   

ScoreSubjectCount

% of total

9 1 0.5010 2 1.0011 3 1.5012 4 2.0013 5 2.5014 4 2.0015 3 1.5016 3 1.5017 2 1.0018 2 1.0019 1 0.5020 1 0.5021 1 0.50

total 32  

Frequency Table

9 10 11 12 13 14 15 16 17 18 19 20 210

1

2

3

4

5

6

Score

Co

un

t

mean

sd

Compare the outcomes of treatment vs. control groups

Effect Size = (meanTX – meanCON) / SDCON

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 2000

2

4

6

8

10

12

Control Test Tx

Outcome Score (Level of Functioning)

nu

mb

er

of s

ub

ject

s

^CON mean

^TX mean

Mean difference=1

If SD=3, ES = 1/3= 0.33

Mean difference=2

If SD=3, ES = 2/3 = 0.67

Mean difference=3

If SD=3, ES = 3/3 = 1.0

Mean difference=4

If SD=3, ES = 4/3 = 1.33

Mean difference=4 SD = 3 ES = 1.33

Mean difference=4 SD = 1.94 ES = 2.1

Mean difference=4 SD = 1.1 ES = 3.6

Type-I and Type-II Errors

Different 1- (Power)

The Same 1-

In fact, TX & Placebo are:

Different The Same

In your experiment, you observe thatTX & Placeboare:

a the rate of false positives, Type I error rateβ the rate of false negatives, Type II error ratePower = 1 – β, the rate of true positives

Plot of Score Distribution under the Null and Alternative Hypotheses

0

0.1

0.2

0.3

-3 -2 -1 0 1 2 3 4 5

critical t = 1.97897

α2β

Using 2-tailed independent-groups t-test with alpha=.05, and power = .80

H0 H1

N needed per group is 64

t

p

Plot of Score Distribution under the Null and Alternative Hypotheses

Using 2-tailed independent-groups t-test with alpha=.05, and power = .95

0

0.1

0.2

0.3

-2 0 2 4 6

critical t = 1.97143

α2

β

N needed per group is 105

t

p

Total sample size

Pow

er

(1-β

err

pro

b)

t tests - Means: Difference between two independent means (two groups)Tail(s) = Two, α err prob = 0.05, Allocation ratio N2/N1 = 1, Effect size d = 0.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

20 40 60 80 100 120 140 160 180 200

Total sample size

Pow

er

(1-β

err

pro

b)

t tests - Means: Difference between two independent means (two groups)Tail(s) = Two, α err prob = 0.05, Allocation ratio N2/N1 = 1, Effect size d = 1

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

20 40 60 80 100 120 140 160 180 200

Power is reduced by:• Measurement Error

- This will tend to ‘muddy’ the outcome scores, making tx effect harder to distinguish – i.e., it increases the SD of both the CON & TX groups, reducing the ES.

• Intent-to-treat analysis- If subjects drop out because they see no progress.

- S’s rarely drop out because they get cured early, but if they did, then completer-analysis would reduce power.

• Low disease severity- Less room for improvement

If you can not reject the null hypothesis of ‘no effect’, this does not ‘prove’ there is no

effectWhy?

Because your power to detect an effect might have been low.

Faul, F., Erdfelder, E., Buchner, A., & Lang, A.-G. (2009). Statistical power analyses using G*Power 3.1: Tests for correlation and regression analyses. Behavior Research Methods, 41, 1149-1160.    

Faul, F., Erdfelder, E., Lang, A.-G., & Buchner, A. (2007). G*Power 3: A flexible statistical power analysis program for the social, behavioral, and biomedical sciences. Behavior Research Methods, 39, 175-191. 

http://www.psycho.uni-duesseldorf.de/abteilungen/aap/gpower3/download-and-register

     

Equivalence & Non-inferiority trials

How do you show that a new treatment is not inferior to a standard treatment?

Quality of the evidence baseThere should be several 2-arm trials of ‘old’ tx vs. placebo, in order to get a range of effect sizes, and response rates (% who improve).

Ideally, there is at least one 3-arm double-blind placebo-controlled comparison (‘old’ tx vs. new tx vs. placebo)

New Tx Beats Placebo

• Effect size vs. placebo is clinically significant.

•Mean difference on the primary outcome is statistically significant

• Response rate (% responders) is higher than placebo [how much higher is determined by prior studies]

New Tx not substantially worse than established tx

• New tx mean on primary outcome is closer to the est. tx mean than the placebo mean.• New tx is not significantly different from the established tx. • New tx responder rate is not much lower than that of the established tx. [should be just within the range seen in prior studies]

• Lower bound of the 95% confidence interval for primary outcome falls above ∆.

How to select ∆

• It is lower than the range of outcome differences seen in prior to studies of established tx vs. placebo. • The smallest value that could be considered a clinically meaningful effect (vs. placebo).• The mean difference that corresponds to a x% difference in responder rates. [x is determined by prior studies of the established tx vs. placebo]

Other Criteria• Dosing & duration of each tx are within the range of known efficacy.• No confounds (despite randomization)• Sample size provides adequate power to detect a clinically significant difference.• Subjects have moderate disease severity.• ‘Per protocol’ set of subjects may be best (most conservative).

Other Criteria• Tx compliance should be similar in both groups.• Low measurement error. If this is an interview or ratings, there is careful training & inter-rater reliability testing. If using a survey, the test is psychometrically sound._________________________________________________________________________________________________________________________

These threats all create bias in favor of finding equivalence, unlike a superiority trail, where they bias the study against finding an effect.

Summary

• Evidence base adequate (for established tx).• New Tx beats placebo.• New Tx not substantially worse than established tx.• Study design features do not bias the results toward equivalence.

Hypothetical Example

study mean diff

Mean placebo sx

scoreMean Est. Tx

sx score placebo sd ESPlacebo % responder

Est tx % responder

1 10 20 10 6.0 1.7 0.25 0.4

2 12 22 10 5.8 2.1 0.2 0.48

3 14 30 16 6.0 2.3 0.15 0.49

4 16 25 9 6.4 2.5 0.22 0.6

avg 13 24.25 11.25 6.05 2.14 20.5% 49.3%

1.2 1.4 1.6 1.8 2 2.2 2.4 2.60.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

f(x) = 0.209287575839711 x + 0.0441554948606184

Effect size

% T

rea

tme

nt

Re

sp

on

de

rs

Mean Difference v. placebo

Mean sx score

ES vs. placebo

ES vs. Est. tx

Estimated New tx % responder

8 16 1.33 -1.00 32.3%

9 15 1.50 -0.80 35.8%

10 14 1.67 -0.60 39.3%

Previous studies

Effect size x % respondersPossible values for ∆

If ∆ = 9, the lower bound of the 95% CI for the new tx primary outcome score must be < 15in order to claim non-inferiority. This is equivalent to a 36% responder rate and ES of 1.5 vs. placebo and -0.8 vs. established tx.

Hypothetical ResultsA) New tx mean = 16 B) New tx mean = 17

New tx is not inferior to EST. Tx New tx is inferior to EST. tx