2006, Tianjin, China 071201sf.ppt - Faragalli 1 Statistical Hypotheses and Methods in Clinical...

071201sf.ppt - Faragalli 1

2006, Tianjin, China2006, Tianjin, China

Statistical Hypotheses and Methods in Clinical Trials with Active Control Non-inferiority Design

Statistical Hypotheses and Methods in Clinical Trials with Active Control Non-inferiority Design

Yong-Cheng Wang

Centocor, Inc., Johnson & Johnson Pharmaceutical Companies, PA, USA

2006 International Conference on Design of Experiments and Its Applications

July 9-13, 2006, Tianjin, P. R. China

Yong-Cheng Wang

Centocor, Inc., Johnson & Johnson Pharmaceutical Companies, PA, USA

2006 International Conference on Design of Experiments and Its Applications

July 9-13, 2006, Tianjin, P. R. China



OutlineOutline

Introduction– Notations & Definitions– Objectives of non-inferiority (NI) trials

Non-inferiority Hypotheses – Fixed margin– Fraction retention

Non-inferiority Analysis Methods– Fixed margin– Fraction retention– Two CI procedure– Bayesian

Summary

Introduction– Notations & Definitions– Objectives of non-inferiority (NI) trials

Non-inferiority Hypotheses – Fixed margin– Fraction retention

Non-inferiority Analysis Methods– Fixed margin– Fraction retention– Two CI procedure– Bayesian

Summary

Yong-Cheng Wang

IntroductionIntroduction

Notations, Definitions, NI objectivesNotations, Definitions, NI objectives



Notations & Definitions Notations & Definitions

Endpoint: time to event (e.g., survival, TTP)

Hazard ratios: HR(T/C) and HR(P/C)

Treatment effect: 1 = HR(T/C) -1

Control effect: 2 = HR(P/C) -1

Fraction retention of control effect: = 1 – {1 / 2}, or

Fraction loss of control effect: 1 - = 1 / 2,

where, T, C and P are treatment, control and placebo, respectively, and are under the constancy assumption:

C = C1 (active control) = C2 (historical control)

Endpoint: time to event (e.g., survival, TTP)

Hazard ratios: HR(T/C) and HR(P/C)

Treatment effect: 1 = HR(T/C) -1

Control effect: 2 = HR(P/C) -1

Fraction retention of control effect: = 1 – {1 / 2}, or

Fraction loss of control effect: 1 - = 1 / 2,

where, T, C and P are treatment, control and placebo, respectively, and are under the constancy assumption:

C = C1 (active control) = C2 (historical control)



Objectives of NI TrialsObjectives of NI Trials

To demonstrate new treatment effect by showing:

Not worse than a given _margin:Test Drug > Active Control - _margin

Retain a fraction of control effect: to test if new treatment preserves a fraction of control effect:

Test Drug > % Active Control

To demonstrate new treatment effect by showing:

Not worse than a given _margin:Test Drug > Active Control - _margin

Retain a fraction of control effect: to test if new treatment preserves a fraction of control effect:

Test Drug > % Active Control

Yong-Cheng Wang

NI HypothesesNI Hypotheses

Fraction retention / Fixed marginFraction retention / Fixed margin



NI Hypotheses – Fraction RetentionNI Hypotheses – Fraction Retention

Fraction retention NI hypotheses:

H0: ≤ 0 vs. Ha: > 0

or, if 2 = HR(P/C) - 1 > 0,

H0: 1 (1- 0 ) 2 vs. Ha: 1 < (1 - 0 ) 2

Fraction retention NI hypotheses:

H0: ≤ 0 vs. Ha: > 0

or, if 2 = HR(P/C) - 1 > 0,

H0: 1 (1- 0 ) 2 vs. Ha: 1 < (1 - 0 ) 2



Selection of Fraction RetentionSelection of Fraction Retention

The selection of depends on several factors:

objective of active control trial– claim non-inferiority or equivalence– claim efficacy

clinical judgment

statistical judgment– distributional properties of the ratio of treatment effect vs.

active control effect– mean effect size of active control– variability of active control effect

The selection of depends on several factors:

objective of active control trial– claim non-inferiority or equivalence– claim efficacy

clinical judgment

statistical judgment– distributional properties of the ratio of treatment effect vs.

active control effect– mean effect size of active control– variability of active control effect



NI Hypotheses – Fixed Margin NI Hypotheses – Fixed Margin

If fix control effect 2 = M1 > 0, and define margin M = 0* M1, where 0 is a fixed level of fraction retention, then NI hypotheses become:

H0: 1/M1 0 vs. Ha: 1/M1 < 0 , or

H0: HR(T/C) 1+M vs. Ha: HR(T/C) < 1+M

If fix control effect 2 = M1 > 0, and define margin M = 0* M1, where 0 is a fixed level of fraction retention, then NI hypotheses become:

H0: 1/M1 0 vs. Ha: 1/M1 < 0 , or

H0: HR(T/C) 1+M vs. Ha: HR(T/C) < 1+M



Selection of Fixed MarginSelection of Fixed Margin

Arbitrary margin: questionable

Margin based on control effect ~ two CI method: Based on the lower limit (LL) of % CI for HR(P/C), i.e.

Margin = 0*[LL of % CI for HR(P/C) -1]

e.g., 0 = 0.5, LL of % CI = 1.2, then margin = 0.1

If the 95% CI for HR(T/C) lies entirely beneath 1 + margin (NI cutoff), “non-inferiority” is concluded.

Arbitrary margin: questionable

Margin based on control effect ~ two CI method: Based on the lower limit (LL) of % CI for HR(P/C), i.e.

Margin = 0*[LL of % CI for HR(P/C) -1]

e.g., 0 = 0.5, LL of % CI = 1.2, then margin = 0.1

If the 95% CI for HR(T/C) lies entirely beneath 1 + margin (NI cutoff), “non-inferiority” is concluded.

Yong-Cheng Wang

NI Analysis MethodsNI Analysis Methods



NI Statistical TestsNI Statistical Tests

Linearization method (Mark Rothmann)

Ratio test (Yong-Cheng Wang)

Two 90% (or 95%) CI procedure (CBER/FDA/US)

CI for the ratio (Hasselblad & Kong)

Bayesian (Richard Simon)

Linearization method (Mark Rothmann)

Ratio test (Yong-Cheng Wang)

Two 90% (or 95%) CI procedure (CBER/FDA/US)

CI for the ratio (Hasselblad & Kong)

Bayesian (Richard Simon)



NI Tests – Linearization Method (Rothmann)NI Tests – Linearization Method (Rothmann)

NI hypotheses: Assuming HR(P/C) > 1

H0(1): logHR(T/C) (1-0)logHR(P/C) vs.

Ha(1): logHR(T/C) < (1-0)logHR(P/C)

Test statistic for H0(1) vs. Ha

(1):

NI hypotheses: Assuming HR(P/C) > 1

H0(1): logHR(T/C) (1-0)logHR(P/C) vs.

Ha(1): logHR(T/C) < (1-0)logHR(P/C)

Test statistic for H0(1) vs. Ha

(1):

0(1)

2 2 20

ˆ ˆlogHR(T/C)-(1-δ )logHR(P/C)

ˆ ˆse (logHR(T/C))+(1-δ ) se (logHR(P/C))Z



NI Tests – Ratio Test (Wang)NI Tests – Ratio Test (Wang)

Estimate of :

where and are estimates of hazard ratios.

Estimate of :

where and are estimates of hazard ratios.ˆHR(P/C)

ˆ ˆ ˆ[HR(P/C)-1]-[HR(T/C)-1] HR(T/C)-1ˆ 1ˆ ˆHR(P/C)-1 HR(P/C)-1

ˆHR(T/C)




NI hypothesis:

H0: 0 vs. Ha: > 0

Test statistic:

Concern: Is Z* normal?

NI hypothesis:

H0: 0 vs. Ha: > 0

Test statistic:

Concern: Is Z* normal?

0ˆ-δˆs.e.( )

Z



NI Tests – Ratio Test (Wang)Asymptotic Normality of Z*





Interim statistic:

Zk* is approximately normally distributed, and

Interim statistic:

Zk* is approximately normally distributed, and

2 20

2

ˆlog( ) -log(δ +k)ˆs.e.(log( ) )

kk

Zk

* * N(0, 1) (when )kZ Z k




Normality of Z* (Xeloda trials, simulation runs=100,000)Normality of Z* (Xeloda trials, simulation runs=100,000)

Number of Events

600 800 1000 1200 1400 1600 1800

p 68.2% 80.9% 88.9% 93.8% 96.6% 98.2% 99.1%

where p = proportion of simulation runs passed Shapiro-Wilk test.



NI Tests – Two 95% CI Method (FDA/US)NI Tests – Two 95% CI Method (FDA/US)

Two 95% CI method:

Define the non-inferiority cutoff (1+margin) as

+ 0.5*LL of 95% CI for HR(P/C) - 1)

If the 95% CI for HR(T/C) lies entirely beneath this cutoff, non-inferiority is concluded.

Two 95% CI method:

Define the non-inferiority cutoff (1+margin) as

+ 0.5*LL of 95% CI for HR(P/C) - 1)

If the 95% CI for HR(T/C) lies entirely beneath this cutoff, non-inferiority is concluded.



NI Tests – CI of Ratio Method (H & K)NI Tests – CI of Ratio Method (H & K)

A “95%” confidence interval is calculated using a normal distribution with standard error

A “95%” confidence interval is calculated using a normal distribution with standard error

1 1 2

1 1

ˆ ˆlog ( / ) log ( / )ˆˆlog ( / )

HR P C HR T C

HR P C

2

2 2 1 12 2

1 1 2 1 1

ˆ ˆ ˆlog ( / ) (log ( / )) (log ( / ))ˆ ˆ ˆlog ( / ) log ( / ) log ( / )

HR T C Var HR T C Var HR P C

HR P C HR T C HR P C



NI Tests – Bayesian Method (Simon)NI Tests – Bayesian Method (Simon)

The posterior density for 1=logHR(T/C) is normal with mean (1+2) and variance (1+ 2).

1: log HR(T/C) 2: logHR(C/P)

1: var(log HR(T/C)) 2 : var(logHR(C/P))

The posterior prob (T is superior to C):

P(<0)=1-[(1+2)/sqrt(1+ 2)]

The prob (1-k)100% of the effect of C to P is not lost with T is

Pr(-k<0, <0).

The posterior density for 1=logHR(T/C) is normal with mean (1+2) and variance (1+ 2).

1: log HR(T/C) 2: logHR(C/P)

1: var(log HR(T/C)) 2 : var(logHR(C/P))

The posterior prob (T is superior to C):

P(<0)=1-[(1+2)/sqrt(1+ 2)]

The prob (1-k)100% of the effect of C to P is not lost with T is

Pr(-k<0, <0).

Yong-Cheng Wang

SummarySummary



Summary of Active Control NI TrialsSummary of Active Control NI Trials

If control effect is small, active control trial should be a “superiority” trial, not a “non-inferiority” trial.

Appropriate assessment of the control effect based on historical data may be difficult when

– few trials, especially only one trial– changing the population– changing the standard care

Selection of the fraction retention should be based on both clinical and statistical judgments.

Interpretation of results needs to be with caution.

If control effect is small, active control trial should be a “superiority” trial, not a “non-inferiority” trial.

Appropriate assessment of the control effect based on historical data may be difficult when

– few trials, especially only one trial– changing the population– changing the standard care

Selection of the fraction retention should be based on both clinical and statistical judgments.

Interpretation of results needs to be with caution.

Yong-Cheng Wang

ENDEND

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2006, Tianjin, China 071201sf.ppt - Faragalli 1 Statistical Hypotheses and Methods in Clinical...

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