Session 9 : Advances in Sample Size Methods
The 36th Annual Meeting of Society for Clinical Trials
Arlington, Virginia, USA, May 17-20, 2015
Sample size considerations for clinical trials
with two primary time-to-event outcomes
Toshimitsu Hamasaki, PhD, pstat®
National Cerebral and Cardiovascular Center
Tomoyuki Sugimoto, PhD
Hirosaki University School of Sciences and Technology
Koko Asakura, PhD
National Cerebral and Cardiovascular Center
Scott R. Evans, PhD
Harvard School of Public Health
Mayumi Fukuda, MD
National Cerebral and Cardiovascular Center
Haruko Yamamoto, MD, PhD
National Cerebral and Cardiovascular Center
2
Goals and outlines
To discuss sample size determination when using two time-to-event
outcomes for comparing two interventions in superiority clinical
trials as co-primary or primary contrasts
 A use of two primary or co-primary time-to-event endpoints has
become common in clinical trials evaluating interventions in many
disease areas such as infectious disease, oncology, or
cardiovascular disease.
Structure of presentation
1. Design issues in clinical trials with two time-to-event outcomes
2. Study designs, hypothesis testing and powers
3. Time dependency association and censoring scheme
4. Behavior of sample size
5. Summary
3
Examples: clinical trials with two time-to-event outcomes
Disease area
Outcomes
HIV (Kaposi’s
sarcoma in HIVinfected)
 Kaposi’s sarcoma
(KS) progression
 HIV virologic failure
Oncology
 Overall survival (OS)  OS requires long follow-up periods after disease
 Time to progression
progression, which leads to quite long and also
(TTP) or
expensive studies.
Progression-fee
 PFS is often included as a short-term primary
survival (PFS)
endpoint, defined as the time from randomization
until tumor progression or death from any cause,
whichever may occurs earlier than OS.
Cardiovascular
 Major cardiac
adverse event
(MACE)
 Death
 Both events are not fatal and each event is not
censored by other event.
 Subjects who do not experience both events yet
are censored at the same time (e.g., by the end of
the study or patient drop-out) in the end of followup period.
 MACE is a composite endpoints, including multiple
types of clinical events of varying degrees of
relatedness.
 Death is included as a component of MACE and it
is the most important event
4
Design Issues: sample size
(i) Inferential goal for multiple outcomes
 To evaluate a joint statistical significance on BOTH outcomes
“Multiple Co-Primary Endpoints”
 To evaluate a statistical significance on AT LEAST ONE outcome
“Multiple Primary Endpoints”
Sample sizing for two time-to-event outcomes could be more complex compared with other
scale outcomes such as continuous or binary outcomes- the following aspects should
carefully be considered in sample size determination in clinical trials with two time-to-event
outcomes
(ii) Censoring scheme between two outcomes
 Whether an event of interest is FATAL or NON-FATAL
 A non-fatal event could be censored by the fatal event (DEPENDENT censoring)
(iii) Time dependent association between two outcomes
 Whether the association between the two time-to-event outcomes could be changed with
the time
5
Superiority clinical trials with two time-to-event outcomes
∗
Endpoint (EP1):
,
“Positively” correlated
∗
Endpoint (EP2):
,
,
1,2:
1, … ,
Underlying continuous survival time and potential
censoring time for the th outcome for the th subject
0 →→→
:accrual duration →→→
EP1
Total
sample
size
EP2
Test
Hazard ratio
/
1
 Observed bivariate survival data
, , , ,
min ∗ ,
∗
,
∙ is the index function
1 → thsubject to test
2 → thsubject to control
 Hazard function
lim
→
/
Censored
: follow-up duration
∗
Control
0
1
Allocation ratio
Study end
∗
1(test); 2 (control)
|
∗
,
6
Logrank test statistics for each endpoint
Hypothesis for each endpoint
H :
H
:
1,
1,
logrank test statistics
forall
atsome
For large sample, each
is
approximately normally distributed
∼ N 0,1 under H
d
d
⋯Bivariate (weighted) logrank statistics process
: Cumulative hazard function
: Nelson-Aalen estimator of Λ
: “At risk” process in group j
⋯Well-known conditional variance of
∑
,
)
7
Superiority hypothesis testing for both or at least one endpoint
Co-Primary
H :
H
:
1or
1and
Primary
1, forall
1, atsome
H :
H
:
Intersection-union test
Z2
1, forall
1, atsome
Union intersection test
Weighted Bonferroni adjustment
Rejection region of H
é{Z > z } Ç {Z > z }ù
êë 1
a
2
a úû
Z k  N(0,1)
1or
1and
Rejection region of H
Z2
é{Z > z } È {Z > z }ù
êë 1
g1a
2
g2 a ú
û
Z k  N(0,1)
12
Z
corr[Z 1, Z 2 ] = rZ12
corr[Z 1, Z 2 ] = r
z
z
Z1
⋯ significant level for hypothesis testing
⋯a upper th percent point of the standard normal
distribution
z 2
z1
Z1
⋯ weight
1
⋯a upper
th percent point of the standard
normal distribution
8
Power for detecting the effect on both or at least one endpoint
Co-Primary
1
Primary
Pr
1
|H
Conjunctive power
Pr
|H
Disjunctive power
Weighted Bonferroni adjustment
 The power is calculated by the cumulative distribution function of bivariate standardized
normal distribution with correlation

 
2
1  b  Pr  j 1 Z k  z a  


bivariate normal
density function

z a  n m1
(
V11


z a  n m2
z k* = U k - n mk
∙,∙;
)
V22
Vkk
f (z1* , z 2* ; rZ )dz1dz 2
æ1
ç
rZ = çç 21
ççèrZ
is the bivariate normal density function with zero mean vector and correlation matrix
rZ12 ö÷÷
÷
1 ÷÷ø
9
Censoring scheme: dependent or independent censoring
Ti 1*
 Dependent or independent censoring
 One event is subject to censoring by other event?
Ti*2
1. Both non-fatal
2. One fatal
min 3. Both fatal
E1:
E2:
Censored
Sugimoto et al (2013)
HIV trial
⋯ Time to virologic failure
⋯ Discontinuation due to toxicity
E1:
E2:
E1:
E2:
Study
end
Semi-competing risk (Fine et al, 2001)
Cardiovascular Trial/ Oncology
, ) ⋯ Hospitalization
TTS
⋯ Death
OS
Oncology Trial
min , ) ⋯ Disease-specific mortality
min , ) ⋯ Other cause- specific mortality
4. Composite
E1’:
E2’:
min or
,
)
Composite and its component
 MACE and Death
 PFS and OS
Fine JPH et al. Biometrika 2001; 88:907-919: Sugimoto et al. Biostatistics 2013; 14:409-421
10
Modeling time-dependent association
Ti 1*
corr
*
i2
T
∗
,
∗
?
Joint survival Function
Marginal survival function
,
∗
Pr
Pr
∗
,
∗
|
Study
end
|
Censored
 Correlation between the two cumulative hazard variates (Hsu, Prentice, 1996)
corr
∗
∗
,
,
1
0
In absence of censoring,
can be estimated replacing functions Λ
with Nelsoncorr ∗ , ∗
Aalen estimators . If each marginal is exponential distribution,
 Correlation between the two test statistics
V12 =
t
ò ò
0
t
0
ìï dL(1) (t, s )
dL(2) (t, s ) üïï
ï
H 1 (t )H 2 (s )C (t, s )G (t, s ) í (1) (1)
+
ý
ïïa y (t )y (1) (s ) a (2)y (2) (t )y (2) (s )ïï
1
2
1
2
îï
þï
ì
ï dL(1) (t )
t
dL(2)
(t ) üïï
2ï
k
k
Vkk = ò H k (t ) í (1) (1) + (2) (2) ý
0
ï
ï
ï
ïa yk (t ) a yk (t )ïþï
î
L( j ) (t, s ) =
t
ò ò
0
a (k ) = n k / N
s
0
l ( j ) (x , y )dydx
yi( j ) (t ) = E[Yi( j ) ] / n ( j )
11
Modeling time-dependent association by copulas
Ti 1*
S ( j ) (t, s ) = C (S1( j ) (t ), S 2( j ) (s ) : q( j ) )    Marginal - Exponential
Ti*2
Early-time dependency
No-time dependency
Less
correlated
Positive Stable (PS)
Late-time dependency
More
correlated
More
correlated
Less
correlated
Frank (FR)
Study
end
Clayton (CL)
be a function which generates the joint survival functions
,
and
with association parameter
the two marginal
from
Clayton DG.
Biometrika
1978; 65:14-151.
Hougaard P.
Biometrika
1984; 71:75-83.
Frank DJ.
Aequationes
Mathematicae
1974; 19:194226.
12
Total sample size for “co-primary” endpoints: “late-time” dependency
One fatal
One fatal composite
0.6
Correlation
0.8
1.0
0.667
0.2
0.4
0.6
Correlation
0.8
1.0
475
500
0.025
1
0.80
0.5
450
0.76
0.4
0.5
0.58
0.3
0.5
0.43
0.2
0.5
425
Total Sample Size Required
600
550
500
0.0
400
0.4
650
700
0.2
450
Total Sample Size Required
0.0
400
475
450
425
400
Total Sample Size Required
500
Both non-fatal
0.0
0.2
0.4
0.6
0.8
1.0
Correlation
 For both nonfatal, the sample size decreases as correlation goes toward one, maximum
is given when the correaltion is zero
 For one fatal and one fatal composite, the sample size increases until some point and
then decreases as correaltion goes toward one, maximum depends on hazard ratios- For
is significant effect of censoring by other event on sample size behavior.
13
Total sample size for “co-primary” endpoints: “early-time” dependency
One fatal
0.4
0.6
Correlation
0.8
1.0
0.667
0.2
0.4
0.6
Correlation
0.8
1.0
475
500
0.025
1
0.80
0.5
450
0.76
0.4
0.5
0.58
0.3
0.5
0.43
0.2
0.5
425
Total Sample Size Required
650
600
550
500
0.0
400
0.2
450
Total Sample Size Required
0.0
400
500
475
450
425
Total Sample Size Required
400
One fatal composite
700
Both non-fatal
0.0
0.2
0.4
0.6
0.8
1.0
Correlation
 For both nonfatal, one fatal and one fatal composite, the sample size decreases as
correaltion goes toward one, maximum is given when the correaltion is zero
 For one fatal and one fatal composite, there is no significant effect of censoring by other
event on sample size behavior.
14
Total sample size for “primary” endpoints: un-weighted Bonferroni
Both non-fatal
0.6
Correlation
1.0
450
1.0
0.2
0.4
0.6
Correlation
400
350
300
0.0
0.2
1.0
0.6
0.8
1.0
0.8
1.0
Correlation
350
400
450
0.8
0.4
300
Total Sample Size Required
400
350
0.0
0.5
250
Total Sample Size Required
0.8
0.667
0.0125
0.80
1
250
0.6
Correlation
450
0.8
0.4
200
350
0.2
200
0.4
0.0
300
Total Sample Size Required
400
0.2
400
450
1.0
350
0.0
300
Total Sample Size Required
200
0.8
250
0.6
Correlation
300
Total Sample Size Required
0.4
200
0.2
250
200
Early-time
One fatal composite
250
400
350
300
Total Sample Size Required
250
0.0
450
200
Late-time
450
Non-fatal
0.0
0.2
0.4
0.6
Correlation
0.76
0.4
0.5
0.58
0.3
0.5
0.43
0.2
0.5
15
Summary
Focus
 Methods for power and sample size determination for comparing the effect of two
interventions in superiority clinical trials with two time-to-event outcomes, when the aim is
(i) to evaluate a joint effect on both outcomes, or (ii) to evaluate an effect on at least one
outcome
Findings
 Sample sizing in clinical trials with two time-to-event outcome is more complex compared
with other scale endpoints such as continuous or binary outcomes- many aspects to be
considered in sample size determination in clinical trials.
 Co-primary: Assuming zero correlation is not conservative when one event is fatal
and the association between the two time-to-event is late-time dependency
 Primary: Assuming one correlation is not conservative when one event is fatal and
the association between the two time-to-event is late-time dependency
 The relationship between two time-to-event outcomes including censoring scheme and
time dependency associations should be carefully evaluated when sample size is
determined
Thank you for your kind attention
If you have any questions, please e-mail to
[email protected]