Web-based Supplementary Materials for
“Sample Size/Power Calculation for Stratified Case-Cohort Design” by
Wenrong Hu, Applied Statistics, Department of Mathematical Sciences, The University of
Memphis; Department of Biostatistics, CSL Behring; Jianwen Cai, Department of Biostatistics,
School of Public Health, University of North Carolina Chapel Hill; Donglin Zeng, Department of
Biostatistics, School of Public Health, University of North Carolina Chapel Hill
Appendix I: Asymptotic variance derivation
From
the
of  l
expression
in
Section
2.3
and
note
that
d l ( t )  
dS l (t )
,
S l (t )
~
~
dN l1 (t )  dN l 2 (t )
Yl1 (t )
Yl 2 (t )
 l (t )  pl  ~
,  l  l1 (t ) S l (t )  ~ , and (1 -  l ) l 2 (t ) S l (t )  ~ ,  can be
~
0
nl
nl
Yl1 (t )  Yl 2 (t )
approximated by

~
~
~
~
~
~
1 L  pˆ l (1  pˆ l )
 (t ) ( w) Yl1 (t )Yl1 ( w)Yl 2 (t  w)  Yl 2 (t )Yl 2 ( w)Yl1 (t  w)
ˆ   nl 
~
~
 Y~l1 (t )  Y~l 2 (t )
n l 1 
n~l
Yl1 ( w)  Yl 2 ( w)

dN l1 ( w)  dN l 2 ( w) dN l1 (t )  dN l 2 (t ) 

~
~
~
~
Yl1 ( w)  Yl 2 ( w)
Yl1 (t )  Yl 2 (t ) 
1 L 
 (t )

  (1  pˆ l )  ~
~
n l 1 
(Yl1 (t )  Yl 2 (t )) 2

~
~
~
t  ( w)Y ( w)Y (t  w)
l1
l2
[dN l1 ( w)  dN l 2 ( w)]
Yl1 (t ) 0 ~
~
(Yl1 ( w)  Yl 2 ( w)) 2

~
~


t  ( w)Y ( w)Y (t  w)
~

l2
l1
 Yl 2 (t ) 
[
d
N
(
w
)

d
N
(
w
)]
[
d
N
(
t
)

d
N
(
t
)]


~
~
l
1
l
2
l
1
l
2
2
0
(Yl1 ( w)  Yl 2 ( w))




1 L 
 (t )

(1  pˆ l )  ~

~
n l 1 
(Yl1 (t )  Yl 2 (t )) 2

 ~
t  ( w) I ( w  t )
~
[dN l1 ( w)  dN l 2 ( w)]
2Yl1 (t )Yl 2 (t ){0 ~
~
Yl1 ( w)  Yl 2 ( w)



 ( w) I ( w  t )

 ~
}[dN l1 ( w)  dN l 2 ( w)][dN l1 (t )  dN l 2 (t )]
~
Yl1 ( w)  Yl 2 ( w)




~ ~
nl 2
 (t )Yl1 (t )Yl 2 (t )  t  ( w) I ( w  t ) nl1
1 L 
ˆ
2
(
1

p
)
[
dN
(
w
)

dN li 2 (w)]
 ~

 li1

~
~
l  ~
n l 1 
(Yl1 (t )  Yl 2 (t )) 2 0 Yl1 ( w)  Yl 2 ( w) i 1
i 1
nl 2
nl 2

 nl 1
 ( w) I (w  t ) nl1
 ~
[
dN
(
w
)

dN
(
w
)]
[
dN
(
t
)

dN li 2 (t )]
  li1



~
li1
li 2
Yl1 ( w)  Yl 2 ( w) i 1

i 1
i 1
 i 1
~
Since Ylj (t ) 
n~l
I(X
i 1
nlj
lij
.
 t ), N lj (t )   N lij (t ), and N lij (t )   lij I ( X lij  t ),
i 1
1
~
~
 2 nlj  lij ( X lij )Yl1 ( X lij )Yl 2 ( X lij ) 2 nlj   lij ( X lij ) I ( X lij  X lij )
1 L 
ˆ   2(1  pˆ l ) 
{ ~
~
~
~
2
n l 1 
(
Y
(
X
)

Y
(
X
))
Yl1 ( X lij )  Yl 2 ( X lij )
j

1
i

1
j 1 i1

l
1
lij
l
2
lij




 ( X lij )
1

}

~
~
2 Yl1 ( X lij )  Yl 2 ( X lij )  

~
~
2 nlj    ( X )Y ( X )Y ( X )
2 nlj   ( X

1 L
 lij
lij l1
lij l 2
lij
lij
lij ) I ( X lij  X lij ) 
  2(1  pˆ l )  ~


~
~
~

2
n l 1
(
Y
(
X
)

Y
(
X
))
Y
(
X
)

Y
(
X
)



j 1 i 1 
j

1
i

1




l
1
lij
l
2
lij
l
1
l
i
j
l
2
l
i
j

~
~
2 n   ( X ) 2 Y ( X )Y ( X )
1 L
lij
lij
l1
lij
l2
lij
.
  (1  pˆ l )
~
~
3
n l 1
(Yl1 ( X lij )  Yl 2 ( X lij ))
j 1 i 1
lj
Appendix II: Simulation Results
The following procedures/parameters are set up for the simulation:
1) there are 4 strata in the full cohort with size n = 2,000, 4,000, or 10,000, with the stratum
proportions of 0.1, 0.2, 0.3, and 0.4;
2) all subjects are assigned to one of the two groups and the group 1 proportion  l (0.3 or 0.5)
is the same over 4 strata in a full cohort ;
3) the event proportion p Dl (1%, 5%, or 10%) and sub-cohort sampling proportion p l (0.1,
0.2, 0.01, or 0.02) are the same over 4 strata;
4) the event time is generated from the exponential distribution with l 1 = l 2 at each stratum
with the values of 0.1, 0.2, 0.3, and 0.5 for stratum 1, 2, 3, and 4, respectively;
5) the censoring time is generated from a uniform distribution between [0,  ], where  is
varied with different censoring proportions in strata based on the given event proportions;
6) the proposed stratified log-rank test for SCC is programmed in SAS. SAS procedure
PROC LIFETEST for the stratified log-rank test is used for the full cohort and the sub-cohort
data analysis; and
7) each simulation is repeated 2,000 times.
For n = 2,000, we considered the situation with the disease rate to be 5% or 10%. From the
results in the first block of Appendix Table A, we note that the empirical type I error rates in
SCC are fairly close to the nominal 0.05 level. For n = 4,000, we considered smaller disease rates
(1%). In a couple of cases with small sub-cohort sampling proportion (1%), the empirical type I
error rates are higher than the nominal level. For example, the event proportion p Dl = 1%, the
group 1 proportion  l = 0.3, and sub-cohort sampling fraction p l = 1% in all strata and shows that
the empirical type I error rates for SCC= 0.069 meanwhile for full cohort = 0.056 and for subcohort= 0.040. The results occur in these cases because some simulated SCC samples contain no
event in at least one of the strata. The numbers of such cases are large when the full cohort size is
small and the event rate and sampling fraction are low. However, the empirical type I error rate
for SCC is improved to 0.050 after the sample fraction is increased to 2% from 1%. Also the
empirical type I error rate for SCC is improved to 0.050 when the full cohort size is increased
2
from 4,000 to 10,000. When n is increased to 10,000, the empirical type I error rates are fairly
close to the nominal level.
The simulated samples in Table B are generated similarly to Table A, with the exception that
the event time is generated from the exponential distribution with l 1 = 1.5 l 2 at each stratum
with l1 values of 0.15, 0.3, 0.45, and 0.75 for stratum 1, 2, 3, and 4, respectively, while the l 2
remains the same as before.
Table A. Empirical Type I Error of Stratified Case-Cohort Design
n
2,000
l
0.3
pl
pD
10%
5%
0.5
10%
5%
4,000
10,000
0.3
1%
0.5
1%
0.3
1%
0.5
1%
Full
10%
20%
10%
20%
10%
20%
10%
20%
1%
2%
1%
2%
1%
2%
1%
2%
0.058
0.058
0.059
0.059
0.059
0.059
0.049
0.049
0.056
0.056
0.051
0.051
0.051
0.051
0.050
0.050
SCC
0.057
0.054
0.051
0.050
0.059
0.043
0.055
0.048
0.069
0.050
0.070
0.051
0.050
0.050
0.049
0.055
Sub
0.049
0.046
0.045
0.052
0.056
0.045
0.049
0.054
0.040
0.044
0.018
0.009
0.036
0.043
0.005
0.018
Footnote: n =full cohort size, p D =mean event proportion,  l =group 1 proportion, p l =sub-cohort
sampling fraction in stratum l. SCC=empirical type I error of SCC, Full=empirical type I error of
full cohort, and Sub=empirical type I error of sub-cohort. Significant level   0.05 .
3
Table B. Simulated Testing Power of Stratified Case-Cohort Design
n
l
pD
pl
TFull TSCC
TSub
PSCC
TSCC
TFull
nSCC
n
0.441 0.160 0.469
0.55 19.0%
0.579 0.265 0.597
0.72 28.0%
5%
0.312 0.112 0.336
0.61 14.5%
0.366 0.158 0.395
0.71 24.0%
0.5
10%
0.484 0.147 0.538
0.56 19.0%
0.647 0.276 0.672
0.75 28.0%
5%
0.350 0.083 0.389
0.63 14.5%
0.439 0.154 0.455
0.79 24.0%
2.0%
4,000
0.3
1%
0.145 0.048 0.130
0.56
3.0%
0.142 0.051 0.159
0.55
2.0%
0.5
1%
0.143 0.016 0.146
0.53
3.0%
0.155 0.014 0.181
0.58
2.0%
10.000
0.3
1%
0.233 0.062 0.260
0.46
3.0%
0.301 0.071 0.330
0.60
2.0%
0.5
1%
0.251 0.014 0.300
0.46
3.0%
0.352 0.021 0.382
0.64
Footnote: n =full cohort size, p D =mean event proportion,  l =group 1 proportion, p l =sub-cohort
2,000
0.3
10%
10%
20%
10%
20%
10%
20%
10%
20%
1%
2%
1%
2%
1%
2%
1%
2%
0.804
0.804
0.514
0.514
0.861
0.861
0.557
0.557
0.257
0.257
0.269
0.269
0.504
0.504
0.549
0.549
sampling fraction in stratum l. TSCC = simulated testing power of SCC,
TFull =simulated testing
TSub = simulated testing power of sub-cohort, PSCC =theoretical power
of SCC, nSCC =SCC sample size, Significant level   0.05 .
power of full cohort, and
Appendix III: Analytic comparison between stratified design and unstratified design
In this section we compare the power of the proposed optimal stratified design with stratified test
against the unstratified design with stratified or unstratified test as follows.
Comparison with unstratified-design-stratified-test
For an unstratified-design-stratified-test, p l is the same across strata. Thus, this design is
equivalent to the proportional design that we discussed in the paper. Our analytic results and the
empirical results have already shown that the power loss can be significant as compared to the
optimal stratified-design with stratified-test when the event rates are heterogeneous and the subcohort sampling rate is low.
Comparison with unstratified-design-unstratified-test
Note the power formula in (4) for a stratified-design-stratified test is
4



  Z  / 2  n1 / 2 







l 1 ( l (1   l ) pDl vl )
.

L 
1  pl
l 1 ( l (1   l ) pDl vl )(1  1  p / 2  p pDl )  
Dl
l


For ease of comparison, we assume  l   for l = 1, …, L and assume pDl to be small in the strata.
L
Let pD  l 1 pDl vl . Then the above power function can be approximated by
L


B1    Z  / 2  n1/ 2 




.
L
2
p D  l 1 (1 / pl  1) p Dl
vl 

 (1   ) p D
For an unstratified-design-unstratified-test, the above power formula is still applicable but with L
= 1. That is, the power is

B2   Z / 2  n1/ 2 



,
2 
pD  (1 / ps  1) pD 
 (1   ) pD
where ps  l 1 pl vl representing the overall sampling fraction if we do not use the stratified deL
2
2
sign. Therefore, it suffices to compare l 1 (1 / pl  1) pDl
vl with (1 / p s  1) p D in these two power
L
expressions.
In the optimal stratified design, p l is approximately proportional to pDl when event rate is
small, i.e., pl  cp Dl for some constant c. Thus, ps  cp D . Then

L
l 1
2
2
2
(1 / pl  1) pDl
vl  c 1 l 1 pDl vl  l 1 pDl
vl  pD2 / ps  l 1 pDl
vl .
L
L
By Cauchy-Schwarz inequality,
l 1 pDl2 vl 
L

L
p v
l 1 Dl l
L
 p .
2
2
D
2
We obtain l 1 (1 / pl  1) pDl
vl  (1 / ps  1) pD2 . Therefore, B1  B2 .
L
Conclusions
Our conclusion is that when disease rate is low, the power of the optimal stratified-design
with the stratified-test is always larger than or equal to the power of the unstratified design with
stratified or unstratified test.
5