Transcript Bayesian decision-theoretic approach for phase I dose

Medical and Pharmaceutical
Statistics Research Unit
Seminar, Bordeaux School of Public Health
8 June 2011
Combining endpoints in clinical trials to
increase power
John Whitehead
Medical and Pharmaceutical Statistics Research Unit
Tel: +44 1524 592350
Fax: +44 1524 592681
E-mail: [email protected]
Department of Mathematics and Statistics
Fylde College
Lancaster University
Lancaster LA1 4YF, UK
1. Ordinal endpoints in stroke studies

Treatments for acute stroke are administered for a few
days following diagnosis

The primary endpoint is the functional status of the
patient, 90 days after the stroke

Several scoring systems exist, including the Barthel index,
the modified Rankin score and the NIH stroke scale

All are ordinal scales from full recovery to vegetative
state, to which death before 90 days can be added
MPS Research Unit
1
Analysis of an ordinal response
R1 = best response (full recovery)
Rk = worst response (death before day 90)
Response
Control
Experimental
Total
R1
R2
c1
c2
e1
e2
t1
t2

Rk

ck

ek

tk
Total
nC
nE
n
MPS Research Unit
2
Let
Ch = c1 +…+ ch
Ch = the number of controls with response Rh or better
Let
Ch = ch +…+ ck
Ch = the number of controls with response Rh or worse
Similarly define Eh, Eh, Th and Th
MPS Research Unit
3
Let
QCh = P(a control has response Rh or better)
QEh = P(an experimental has response Rh or better)
(then QCk = QEk = 1)
Put
 Q Eh (1  QCh ) 
qh  log 

 QCh (1  Q Eh ) 
h = 1,…, k – 1
qh is the log-odds ratio for response Rh or better, E:C
MPS Research Unit
4
The proportional odds assumption is
q1 = q2 = … = qk–1 = q
The common value, q, is a measure of the advantage of the
experimental treatment
q
>
=
<
MPS Research Unit
0 experimental better
0 no difference
0 control better
5
Under PO, the most efficient test of treatment advantage
 greatest power for any given sample size
is based on the test statistics
and
1 k
Z   a h (Bh 1  Bh 1 )
n h 1
3
k

nCn E 
 t h  
V 
1     
3n  h 1  n  
For large samples and small q, approximately Z ~ N(qV, V)
Z is the score statistic and V is Fisher’s information
MPS Research Unit
6
To test for treatment difference, refer
Z2/V
2

to 1
 This is the Mann-Whitney test
 Also known as the Wilcoxon test
Under the null hypothesis of no treatment effect, PO is true
with q = 0
Thus the hypothesis test and the p-value are valid without
assumptions
Estimates of and confidence intervals for q do rely on
assumptions, as does adjustment for prognostic factors
MPS Research Unit
7
How should investigators choose which scale to use?
An alternative to choosing is to combine more than one stroke
scale in the analysis
Tilley et al. (1996) combined four scales in the trial of rTPA
as a treatment in acute stroke conducted by the National
Institute of Neurological Disorders and Stroke
 the trial was positive and the approach caught on
If the treatment has a beneficial effect on all scales, then
combining them will increase the power to demonstrate the
advantage of the treatment
MPS Research Unit
8
2. Example: The ICTUS trial in stroke
•
Currently ongoing in 60 centres in Europe
•
Patients who have suffered acute stroke
•
Randomised between citicoline and placebo
•
Assessed at 90 days on Barthel index, modified Rankin
score and NIH stroke scale
•
Prognostic factors
 baseline NIHSS
 time from stroke to treatment ( or > 12 hours)
 age ( or > 70 years)
 site of stoke (right or left side)
 use of rTPA (yes or no)
MPS Research Unit
9
The approach used by Tilley et al.
Combine the three analyses using GEE
(based on an independence covariance structure: IEE)
That is, analyse as if the three scores were independent, but
adjust the standard error of the treatment effect estimate using
the sandwich estimator
•
•
•
complicated to understand
no associated sample size formula
failed in test data set of 1000 patients with binary
responses and adjustment for 60 centres
MPS Research Unit
10
An alternative general approach
The log-odds ratio q and the test statistics Z and V, for the
analysis of the ith response will be denoted by qi, Zi and Vi
i = 1 is Barthel index
i = 2 is modified Rankin score
i = 3 is NIH stroke score
W will test H0: q1 = q2 = q3 = 0 (no effect of treatment on any
of the scales) using
Z = Z1 + Z2 + Z3
MPS Research Unit
11
For each scale,
Zi ~ N(qiVi, Vi)
if Vi is large and qi is small
If q1 = q2 = q3 = q, then approximately
Z ~ N  qV, V  C 
where V = V1 + V2 + V3, C = 2(C12 + C23 + C31) and
Cij = cov(Zi, Zj)
MPS Research Unit
12
It follows that, if
ZV
Z* 
VC
V2
and V* 
VC
then
Z* ~ N  qV*, V *
as required for a 2 test and for sample size calculation
What we need to use this is an expression for
Cij = cov(Zi, Zj)
MPS Research Unit
13
The binary case, no covariates
 only one response
Control
Experimental
Total
Success (R1)
c1
e1
t1
Failure (R2)
c2
e2
t2
Total
nC
nE
n
n C e1  n E c1
Z
n
MPS Research Unit
n C n E t1 t 2
and V 
n3
14
The binary case, no covariates
 ith of several responses
Control
Experimental
Total
Success (R1)
ci1
ei1
ti1
Failure (R2)
ci2
ei2
ti2
Total
nC
nE
n
n C ei1  n E ci1
Zi 
n
n C n E t i1t i2
and Vi 
n3
 assuming that each patient provides all responses
MPS Research Unit
15
Covariance between Zi and Zj
For two such statistics, we have
nCn E
Cij  cov  Zi , Z j   3  nt (ij),1  t i1t j1 
n
where ti1 is the number of patients succeeding on the ith scale,
tj1 the number succeeding on the jth scale and t(ij),1 the number
succeeding on both scales (Pocock, Geller and Tsiatis, 1987)
MPS Research Unit
16
The ordinal case, no covariates
 ith of several responses
R1
Control
ci1
Experimental
ei1
Total
ti1
Rk
cik
eik
tik
Total
nC
nE
n
with Cih = ci1 +…+ cih and Cih = cih +…+ cik
MPS Research Unit
17
Covariance between Zi and Zj
For two such statistics, we have
nCn E
Cij  2
n

dfv dgw  H fg H vw  n C H fg K vw  n E K fg H vw 
f ,g,v,w
where dfv = 1, 0 or 1 if f <, =, > v respectively,
Kfg = tfi tgj/n2, Hfg = t(ij),(fg)/n  Kfg,
t(ij),(fg) is the count of patients who have both response Rf,i on
the ith scale and response Rg,j on the jth scale
MPS Research Unit
18
Adjustment for covariates
The approach can be extended to allow for prognostic factors
via stratification and/or linear modelling of covariates
Stratification: sum Z and V statistics over strata, and assume
that the treatment effect is constant over strata
Covariate adjustment: use proportional hazards regression,
plus binary logistic regression to model the simultaneous
occurrence of particular responses on different scales (such as
complete recovery on Barthel index and partial recovery on
the modified Rankin)
MPS Research Unit
19
3. Sample size calculation for the combined test
For power of 90% to detect a log-odds ratio of qR as significant
at level 0.05 (two-sided), we need
2
 1.960  1.282  10.5
V
  2
qR
qR


for a test based on a single response, and
2
 1.960  1.282  10.5
V*  
  2
qR
qR


for a test based on the combined approach
MPS Research Unit
20
For a single binary (success/fail) response, with an overall
success probability of p,
4
42
n
V
p(1  p)
p(1  p)q2R
For three binary responses, each having an overall success
probability of p, and with the probability of success on any two
responses being g
n

 V*  

4 p(1  p)  2  g  p 2 
42 p(1  p)  2  g  p 2 
3p(1  p)
3p(1  p) q2R
MPS Research Unit
2
2
21
Suppose that g = p2 (independence), then
42p(1  p)
42
n

2 2
2
3p(1

p)
q
3p(1  p) qR
R
 that is one third of the sample size using only one response
For g = p (responses coincide), then
n
42p(1  p)  2p(1  p)
3p(1  p) q2R
2
42

p(1  p)q2R
 that is the same as the sample size using only one response
Otherwise, combining the responses reduces sample size by up
to one third, depending on the correlation between the responses
MPS Research Unit
22
Now suppose that p = 0.2 and that g = 0.1 (correlation = 0.75)
then for one response
n
42
262.5

0.16q2R
qR2
and for three responses
n
420.16  2  0.10  0.04 
30.16 q2R
2
153.1
 2
qR
58% of the sample size using a single response
MPS Research Unit
23
If the success rate on control is 18%, and the trial is to be
powered to detect an improvement to 22%, then the log-odds
ratio is
qR
 0.22(1  0.18) 
 log 
  0.25
 0.18(1  0.22) 
so that, for one response
n = 4200
and for three responses
n = 2450
MPS Research Unit
24
ICTUS trial
Fixed sample size using only
Barthel:
modified Rankin:
NIH stroke scale:
2590
3584
5494
Combined test:
2421
This is for dichotomised responses, based on the previous data
available
ICTUS is using a sequential design
MPS Research Unit
25
Ordinal scales
For sample size calculation for combining several ordinal
responses, probabilities of every pair of responses on every pair
of responses must be anticipated
 Databases from previous trials can be used
 A mid-trial sample size review can be used
MPS Research Unit
26
4. Evaluation of the combined approach
The first of a series of interim analyses of the ICTUS trial takes
place when data from 1000 patients are available
A dataset from four previous studies comparing citicoline with
placebo is available (Davalos et al., 2002) comprising 1,372
patients
First, one dataset of 1,000 was extracted and analysed using the
combined test and the GEE approach
Then 10,000 datasets of size 200, 500 or 1,000 were randomly
selected, the treatment code was removed and randomly
reassigned
 in some runs an artificial treatment effect of known
magnitude was introduced
MPS Research Unit
27
Analyses of a synthetic stroke dataset, n = 1000
sd qˆ
68.19
0.2280
0.1211
0.0597
15.57
68.94
0.2259
0.1204
0.0607
GEE
17.27
60.09
0.2874
0.1290
0.0259
comb
17.84
62.85
0.2839
0.1261
0.0244
Method Z*
no factors
GEE
15.55
comb
all factors
binary

q̂
Adjusting
V*
p
all factors + GEE
Failed to converge
centre
comb
19.64
58.23
0.3373
0.1311
0.0100
no factors
GEE
9.02
89.85
0.1004
0.1055
0.3413
comb
7.84
83.92
0.0935
0.1092
0.3918
GEE
17.15
89.17
0.1923
0.1059
0.0695
comb
15.18
82.41
0.1842
0.1102
0.0945
all factors + GEE
20.96
79.86
0.2624
0.1119
0.0190
centre
18.49
80.33
0.2302
0.1116
0.0391
all factors
ordinal
MPS Research Unit
comb
28
Results from 10,000-fold simulations of the
combined score test and the GEE approach
sample hypot
size
hesis
200
500
1000
true q
# rejections according to
q̂ from
comb
GEE
both
comb
GEE
H0
0
232
268
228
0.002
0.002
H1
0.781
9170
9186
9122
0.795
0.808
H0
0
251
255
239
0.001
0.001
H1
0.494
8950
8942
8894
0.480
0.472
H0
0
227
226
211
0.000
0.000
H1
0.349
9010
8995
8956
0.345
0.334
MPS Research Unit
29
5. Conclusions
Use of the combined approach can reduce sample size,
provided that the treatment effect is apparent on all responses
being combined
The score approach used here matches the GEE approach, and
is more reliable in small samples
The approach can combine quantitative responses and
survival responses, it can also be used to combine different
types of response
MPS Research Unit
30
References
Bolland, K., Whitehead, J., Cobo, E. and Secades, J. J. (2009). Evaluation of a sequential global
test of improved recovery following stroke as applied to the ICTUS trial of citicoline.
Pharmaceutical Statistics 8, 136-149.
Dávalos A, Castillo J, Álvarez-Sabin J, Secades JJ, Mercadal J, López S, Cobo E, Warach S,
Sherman D, Clark WM, Lozano R. (2002). Oral citicoline in acute ischemic stroke.
Stroke 33, 2850-2857.
Dávalos A. (2007). Protocol 06PRT/3005: ICTUS study: International Citicoline Trial on acUte
Stroke (NCT00331890) Oral citicoline in acute ischemic stroke. Lancet Protocol Reviews.
Pocock, S.J., Geller, N. L. and Tsiatis, A. A. (1987). The analysis of multiple endpoints in
clinical trials. Biometrics 43, 487-498.
Tilley, P. C., Marler, J., Geller, N. L., Lu, M., Legler, J., Brott, T., Lyden, P. and Grotta, J. for
the National Institute of Neurological Disorders and Stroke (NINDS) rt-PA Stroke Trial
Study Group. (1996). Use of a global test for multiple outcomes in stroke trials with
application to the National Institute of Neurological Disorders and t-PA Stroke Trial.
Stroke 27, 2136-2142.
Whitehead, J., Branson, M. and Todd, S. (2010). A combined score test for binary and ordinal
endpoints from clinical trials. Statistics in Medicine 29, 521-532.
MPS Research Unit
31