- Email: [email protected]
On the possibility of incorporating patients from non-randomising centres into a randomised clinical trial
J Chron Dis 1979. Vol. 32. pp. 347 to 353 0 Pergamon Press Ltd. Printed in Great Britain
002 I-968 1 ?9~050143347502
00 0
ON THE POSSIBILITY OF INCORPORATING PATIENTS FROM NON-RANDOMISING CENTRES INTO A RANDOMISED CLINICAL TRIAL DAVID E.O.R.T.C.
Data
MACHIN
Center. lnstitut Jules Bordct. 1000 Brussels. Belgium
(Remred
in revised
form
rue Hcgcr-Border
3 Aup~v
I.
19781
Abstract-The implications of incorporating patients from non-randomising centres into the design of a 2 arm randomised multicentre clinical trial are investigated from the point of view of Bayesian statistics. It is concluded that although the inclusion of such patients may contribute to the reduction in recruitment time to a trial m the special circumstances when any biases introduced by such centres can be assumed very small. this is offset by a ldrgcr number of patients to be followed than in the corresponding fully randomised trial.
INTRODUCTlOK
recognised that one important method of satisfactorily evaluating the relative efficacy of alternative therapies is by means of multicentre randomised clinical trials. The conduct of such trials is not always easy; they may be difficult to organize and it may be difficult to recruit sufficient patients. One barrier to recruitment of patients may be ethical objections in certain centres to one or more of the therapies suggested by the protocol. Some centres may find it difficult to support a protocol in which one arm is an untreated control while others may find this acceptable but are unable to accept, for example, the intensive chemotherapy proposed on another arm. Some of these ‘ethical’ objections may well be imposed by, for example, public opinion, the law pertaining in a certain country, failure to convince a department head. and not directly ethical objections of the physician himself. It is usually the case that if a centre has ‘ethical’ objections to one arm of a trial then that centre would not enter any patients onto a trial even if the other arms were acceptable. Such non-cooperation may well severely limit recruitment to the trial and. hence, at the very least extend its duration. This paper discusses two arm trials in which centres who have ethical objections to one arm nevertheless enter patients for the other. The assumption is that there will be some centres who randomise between the two alternative therapies C (control) and T (treatment), while some centrcs opt only for C and others opt only for 7: The methods used follow very closely the work of Pocock [I] who has discussed the combination of randomised and historical controls in clinical trials. The effect of ethical design considerations on statistical analysis has been considered by Lindley [Z] in a situation in which an optimal experimental design may require a patient to be given a treatment which the attending physician does not consider best for the patient. IT
IS GENERALLY
THE
STATISTICAL
MODEL
As already indicated Pocock [l] shows how historical controls may be combined with randomised controls to improve trial design. He lays down, however, very strict conditions of acceptability of the historical control group. We use Pocock’s approach 347
DAVID MACHIS
348
to study the situation which allows the inclusion into an ongoing trial those centres which can give only one of the two protocol treatments. We shall assume that one is performing a ‘randomised’ clinical trial to compare the effect of a new treatment T with a standard ‘untreated’ control C. Randomisation is used to assign patients to treatments (not necessarily in equal proportions) in some centres, while others opt for either one or other of the arms. It is implicitly assumed that non randomising centres would give their chosen treatment in precisely the same way as the randomising centres. that all eligible patients (or at least a random sample of eligible patients) are entered in the trial by the non randomising centres and that methods of treatment evaluation are the same. Let n,, nc be the total number of patients receiving the new treatment T and the control treatment C respectively in those centres randomising between T and C, while rnr and mc represent the number of patients entered by centres only giving T or C respectively. Let N = nT + nc + inT + mc be the total number of patients on-study. Note the three special cases (i) mc = 0, (ii) mr = 0 and (iii) mC = m7 = 0, the latter corresponding to the usual randomised clinical trial situation. Suppose that the efficacies of the treatments are to be evaluated by means of the patients survival time .F. Pocock [l] argues that it is not too restrictive to assume a normal distribution for some transformation of .7 to a new random variable. We shall label this random variable X if we are discussing centres giving both treatments, and Y for centres giving only one treatment. We shall assume that X and Y have normal distributions whose parameters depend on which of the 4 patient groups is indicated. Thus: (i) (ii) (iii) (iv)
patients patients patients patients
receiving T after receiving C after from centres only from centres only
randomisation assume randomisation assume giving T assume Yr is giving C assume Yr is
X, is a sample from I$,.. CT:-): Xc is a sample from .V‘(pc, a:): a sample from :V(V~, tot); a sample from .,b*(vc, (0:).
Thus the unknown means of the four groups are pr, pc, vr and rc with unknown varisample means are .U,. .fc, yr and yc. The ances fs+, a,$. 0:. and u$. The observed principal objective of the trial is taken to be the estimation of pr - pc which measures the effectiveness of the new and standard treatments. When mr = mc = 0, i.e. the usual randomised trial situation, the best estimate of available the ‘ethical’ centres pr - pc is XT - Xc. However, we now have potentially data summarised by jr and jc. Unfortunately, however, neither lr and vr. or pc and vc are necessarily equal and one must allow for the possibility of unknown biases in the ethical groups. We define these biases by:
The possible presence of such biases is one of the arguments used for the conduct of randomised trials. We shall assume, see Pocock [I], that ST and 6, both have normal distributions with mean zero and variances & and ~5 respectively. These Bayesian prior distributions for 8r and 6c express the degree of confidence one has in the ethical centres data. Considering for a moment one arm of the trial only and dropping temporarily the suffices T or C, then one may suppose that prior to commencement of the clinical trial one has complete ignorance regarding the values of p and v; however, after the trial and using the prior distribution of 6. it is possible to show by application of
Incorporating
Bayes theorem with mean.
Patients
from Non-randomising
that the posterior
distribution
Centres mta a Randomised
of p (the parameter
Clinical
Trial
of interest)
349
is normal
and variance
(2) whet-c A = l/q2 + nr/02. WC note the following (i) the fully randomised
special cases: clinical trial i.e. In = 0.
/Ti = s. C’ = a2!n; (ii) total ignorance
of (5 (ignore
ethical
centre data) i.e.. ‘I’-+ IL.
ii = .s. V = (i’!ll: (iii) complete
information ~ =
on 6. implying
?%::a’ + I n.:a2 +
The weights
rn~jo
21
,
’
mis2;
given to .U and y depend b=---_-
~3E 0. i.e. q2 + 0.
V = l/(nia2
+ m;02)
only on n, m, a’ and 02. If further
a2 = to2, then
n.u+ my ; t1 +
m
(iv) II = t-p. m = 2cp where r is the number of randomising centres and c the number of non-randomising ccntres. The c centres entering only one arm enter twice the number of patients on that arm than do the corresponding randomising ccntres.
If wc write
then restoring the suffices T and C. one can test the null hypothesis H,:p, (assuming 0;. r$. c$-. (0: known) by the standardised normal random variable I, =
{(‘;$iy \
In practice.
however.
_ (-yg!L$)} _-. -. (CT +
only estimates SAMPLE
(3)
a
of the variances SIZE
= 11~.
will be available.
IMPLICATIONS
We assume that r centres are randomising between T and C, that t centres give treatment T only and c centres C only. We further assume that centres giving only one treatment have twice the number of patients available for that treatment than does a corresponding centre randomising between the two alternative treatments and that a:. = af = a2 and w2r = 0,; = (U2. Let the total entry for each institute be 2p patients and the randomised patients can then be allocated to treatments T and C in the proportion U:l - O(0 < fI < I)
350
DAVID MACHIN
thus; nT = 20rp.
nc = 2( 1 - @i-p,
mT = 2tp,
mc = 2cp,
and N = nr + n, + mr + mc = 2(r + t + c)p. Choice
ofo.
We choose a value of (1 which minimised vr + V, =
‘1“.
Now
Differentiating I” with respect to 8 and setting the resulting derivative obtains after some algebra:
to zero. one
(5) where 4 = a2,102. and which can be shown to minimise 1,: We note that if: (i) c = 0. i.e. there are no centres contributing control patients only, then 0 < l/2. (ii) t = 0, i.e. there are no centres contributing treatment patients only. then 9 > 112. (iii) c = t = 0 then 0 = lj2. Substituting for 0 in (4) we obtain Yii” = 2 ;i$
+ $[(&)
(6)
+ &)I}.
Sample sizr for fl yicen l’iin. Suppose we wish to conduct a two allowing the inclusion of ‘ethical’ centres and we require it to be a way that if ‘ethical’ centres were excluded the corresponding fully would involve r centres with 2P patients from each centre, giving Vi f ,I” = 2d,bP we obtain:
Rearranging
arm clinical trial designed in such randomised trial = 2djrP. Setting
we obtain the following cubic in p:
4rctp3 + 2(rrc + /3rt + 4rrt
+ 4/k?
- 2rctP)p’
+ (rap + 4a/?c + &x/It - 2rrcP
which can be solved for p to obtain the required Special cases :
- 2/?rtP)p - rrj3P = 0
(7)
number of patients per institute.
(i) if c = t = 0 then (7) reduces to rafip - ra/.?P = 0 or p = P; (ii) if qt. 115--) T_ i.e., no prior information on bT and 6c, then (7) reduces to 4rctp*(p - P) or p = P: (iii) if qc. $ -+ 0 which implies ST = 6, E 0 or no bias in ethical centres. (7) becomes (r + cd + tb)p - rP = 0
Incorporating
Patients
from Non-randomising
Centres
into a Randomised
Clinical
351
Trial
or (rP/a’)
(8)
’ = (r/a2 + c/w2 + f/&’
It is easy to see from (8) that p < P and this situation, i.e. no bias in the ethical ccntres. corresponds to the minimum value possible for p. If a2 = (,~2 then (8) becomes p = rP!(r + c + t). (iv) if c = 0. I f 0. then (7) reduces
to the quadratic
2rtp2 + (w + c#W - 2rtP)p - rrP = 0; (v) if c # 0, t = 0 then (7) reduces
(9)
to
2rcp2 + (jr + C#J~C - 2rcP)p - @P = 0. Some numerical in the following
solutions section.
of equations
(7) and
NUMERICAL
(9) are given
(IO) for some
special
cases
RESULTS
In principle for given r, c, t. r. /A C#Iand P equations (7). (9) or (IO) can be solved to obtain p. Extensive tabulation of the solutions however is not practicable. Table I gives for a fully randomised trial with N = 2rP = 200. the comparative ‘ethical’ trial sizes assuming r = 8. C#J= 1 and c = t (note that these conditions imply 0 = +,). It is clear from Table 1 that, if 2 is large i.e., r7: and r$ are small. then there can be a considerable reduction in the numbers of patients recruited by each centre. for example for 2P = 20. r = 10. c = t = 10. 3: = 100 gives 2p = 10.0 so that the duration of entry onto trial could well be halved (2p/2P = IO/20 = 0.5). The total number of patients involved on the trial would be N’ = nr + n, + mT + mc = 50 + 50 + 100 + 100 = 300 rather than the N = 200 of the corresponding fully randomised trial. In practice the situation corresponding to equation (9) may be more common in which centres only ‘object’ to the one of the arms, which implies c = 0 and t z 0 or c # 0 and I = 0. Table 2 shows some solutions of equations (9) for given r. t and IX, assuming 4 = I and N = 2rP = 200. Thus for 2P = 20. r = IO, c = 0, t = IO. r = 100, 2p = 14.14 so that the duration of the trial would be reduced to 70% (2p/ZP = 14.14!20) of the correTABLE
I.
CFNTRFS.
NUMRER
OF
PATlENTS
CORRESPOSDING
TO
REQUIRED A
FULLY
PER CENTRE RANDOMISED
($) TRIAL
FOR A CLINICAL WITH
r
CENlRtS
TRlAl
ALLOWING
NON-RANDOMlSlN(;
CONTRIBL’TING
A
IOIAL
OF
?o()
PATIENTS
Number of patients per centre in a Number of fully ranrandomising dnmised trial centres
?P IO
r 20
Number of control arm nonrandomising centres
Number of treatment arm nonrandomising centres
c
I
0
40
10
5
r+r+c
Ratlo of observation variances z = (J/q2
0
5
IO
and bias
loo
I
0 5 IO 20
20 30 40 60
IO 10 IO IO
IO 11.9 9.3 915
IO x.2 8.7 5.1
IO 6.6 5.1 5.0
IO 6.7 5.0 3.1
5 10 20
0 5 IO 20
IO 20 30 50
20 20 20 20
20 18.6 19.0 19.3
20 17.4 IX.1 18.5
20 II.4 10.0 9.2
20 10.0
0 5 10 20
0 5 IO 20
5 I5 25 45
40 40 40 40
40 38.1 38.5 3X.8
40 36.2 37.1 37.5
40 20.0 18.3 17.9
40 13.3 X.0 4.4
5 IO 20 20
Total number of centres
0
6.1 4.0
352
DAVID MACHIN
TARLL 2. Nuuet~ OF PATIENTS RLQI:IRtl) PFR C‘LlriTRt (2~) FOR A CLINICAL ‘TRIAL ALLOWING SON-RANUOMISING CtNTRtS WHO OLUhCT TO THI ‘COKTROL‘ ARM. CORRtSPOPiUING TO A FIJLLY RAHIXNISFD TRIAL WITH r CCSTRIS CONTRILIL’TING A TOTAL ot 200 PATIENTS Number of patients per centre in a fully randomised trial 2P
Number of randomising centres r
IO
Number of control arm nonrandomising centres ‘
Number of trealmenl arm nonrandomising centrcs I
Total number of ccntres r+/+c.
20
0 0 0 0
0 5 IO 20
20
IO
0 0 0 0
40
5
0 0 0 0
Ratlo of observation variances z = (I/ tq
and bias
0
5
IO
100
-,
20 25 30 40
IO 10 IO IO
10 9.7 9.8 9.8
IO 9.6 9.6 9.5
IO x.5 7.8 7.1
IO 8 6.7 5
0 5 IO 20
IO 15 20 30
20 20 20 20
20 19.5 19.5 19.4
20 19.1 19.1 19.0
20 15.6 14.1 12.8
20 13.3 IO 6.7
0 5 IO 20
5 IO I5 ‘5
40 40 40 40
40 39.0 39.0 39.0
40 38.1 38.1 38.0
40 28.3 25.6 23.5
40 20 13.3 X
sponding fully randomised clinical trial. The total number of patients involved on the trial would be N’ = nr + nc + rnr + mc = 70.7 + 70.7 + 141.4 + 0 = 282.8 rather than the 200 of the corresponding fully randomised trial.
DISCUSSION
One of the major difficulties when it comes to launching a multicentre randomised clinical trial is to recruit sufficient patients within a relatively short period so that the questions posed by the trial can themselves be answered as soon as possible. It is tempting in such circumstances to recruit patients at all costs and perhaps to allow certain institutions (who would not otherwise enter the trial) to enter patients for only one of the trial arms. It is widely recognised that if recruitment of patients from such institutes is permitted they may introduce unknown biases into the overall treatment comparison if they are merely pooled with the remaining patients for analysis. However. regression model methods of analysis for example allow the exploration of data in greater detail and may suggest whether or not the pooling of data from such centres is justified. In planning a cooperative trial there is usually more than one end point of interest, for example the different responses to a given treatment occurring in different institutes or countries-or the confirmation or otherwise of the value of some accepted prognostic variable, for which the non-randomising institutes may be willing to provide patient data. Tables 1 and 2 suggest that if patients from non-randomising institutes are included. then for this to make any real reduction in the overall duration of the trial r = 02;~~. and /I = o’/qi must be large, which implies both & and $ would need to be small and that the biases 6, and Sc are effectively very close to zero. It is necessary therefore before allowing such centres to influence the design of a fully randomised trial to be reasonably certain that any biases will be small. These biases would almost certainly not be negligible if the non-randomising centres only entered a selected (non randomised) proportion of their eligible patients. Even if these assumptions could be made and such centres do include patients, then more patients would have to enter the trial in total than the corresponding fully randomised trial thus increasing the cost of the trial which must be balanced against earlier termination of the study.
Incorporating
Patients
from Non-randomising
Centres
into a Randomised
Chnical
Trial
353
In short, although the inclusion of patients from non-randomising centres appears superficially attractive, there is little that can be gained by their inclusion, particularly as the possibility of entering one arm only of a trial may be used as an excuse to opt out of randomising by a centre which otherwise may have (albeit reluctantly) agreed to randomise between the alternative arms. There may be more scope for inclusion of such centrez in 3 or more arm trials where a minimum participation of randomisation to 2 arms is required by the protocol but these have not yet been investigated. A~,knowlenyements--I am grateful to M. J. Staquet paper. This investigation was supported by Grand Cancer Institute. DHEW.
and R. J. Sylvester for comments on a draft of this Number 5R10 CA 11488-08 awarded by the National
REFERENCES I. 3.
Pocock SJ: The combination of randomized and historical controls in clinical trials. J Chron 175-188. 1976 Lindley DV: The effect of ethical design considerations on statistical analysis. J Royal Statist 218-228. 1975
Dis 29: Sot 24:
002 I-968 1 ?9~050143347502
00 0
ON THE POSSIBILITY OF INCORPORATING PATIENTS FROM NON-RANDOMISING CENTRES INTO A RANDOMISED CLINICAL TRIAL DAVID E.O.R.T.C.
Data
MACHIN
Center. lnstitut Jules Bordct. 1000 Brussels. Belgium
(Remred
in revised
form
rue Hcgcr-Border
3 Aup~v
I.
19781
Abstract-The implications of incorporating patients from non-randomising centres into the design of a 2 arm randomised multicentre clinical trial are investigated from the point of view of Bayesian statistics. It is concluded that although the inclusion of such patients may contribute to the reduction in recruitment time to a trial m the special circumstances when any biases introduced by such centres can be assumed very small. this is offset by a ldrgcr number of patients to be followed than in the corresponding fully randomised trial.
INTRODUCTlOK
recognised that one important method of satisfactorily evaluating the relative efficacy of alternative therapies is by means of multicentre randomised clinical trials. The conduct of such trials is not always easy; they may be difficult to organize and it may be difficult to recruit sufficient patients. One barrier to recruitment of patients may be ethical objections in certain centres to one or more of the therapies suggested by the protocol. Some centres may find it difficult to support a protocol in which one arm is an untreated control while others may find this acceptable but are unable to accept, for example, the intensive chemotherapy proposed on another arm. Some of these ‘ethical’ objections may well be imposed by, for example, public opinion, the law pertaining in a certain country, failure to convince a department head. and not directly ethical objections of the physician himself. It is usually the case that if a centre has ‘ethical’ objections to one arm of a trial then that centre would not enter any patients onto a trial even if the other arms were acceptable. Such non-cooperation may well severely limit recruitment to the trial and. hence, at the very least extend its duration. This paper discusses two arm trials in which centres who have ethical objections to one arm nevertheless enter patients for the other. The assumption is that there will be some centres who randomise between the two alternative therapies C (control) and T (treatment), while some centrcs opt only for C and others opt only for 7: The methods used follow very closely the work of Pocock [I] who has discussed the combination of randomised and historical controls in clinical trials. The effect of ethical design considerations on statistical analysis has been considered by Lindley [Z] in a situation in which an optimal experimental design may require a patient to be given a treatment which the attending physician does not consider best for the patient. IT
IS GENERALLY
THE
STATISTICAL
MODEL
As already indicated Pocock [l] shows how historical controls may be combined with randomised controls to improve trial design. He lays down, however, very strict conditions of acceptability of the historical control group. We use Pocock’s approach 347
DAVID MACHIS
348
to study the situation which allows the inclusion into an ongoing trial those centres which can give only one of the two protocol treatments. We shall assume that one is performing a ‘randomised’ clinical trial to compare the effect of a new treatment T with a standard ‘untreated’ control C. Randomisation is used to assign patients to treatments (not necessarily in equal proportions) in some centres, while others opt for either one or other of the arms. It is implicitly assumed that non randomising centres would give their chosen treatment in precisely the same way as the randomising centres. that all eligible patients (or at least a random sample of eligible patients) are entered in the trial by the non randomising centres and that methods of treatment evaluation are the same. Let n,, nc be the total number of patients receiving the new treatment T and the control treatment C respectively in those centres randomising between T and C, while rnr and mc represent the number of patients entered by centres only giving T or C respectively. Let N = nT + nc + inT + mc be the total number of patients on-study. Note the three special cases (i) mc = 0, (ii) mr = 0 and (iii) mC = m7 = 0, the latter corresponding to the usual randomised clinical trial situation. Suppose that the efficacies of the treatments are to be evaluated by means of the patients survival time .F. Pocock [l] argues that it is not too restrictive to assume a normal distribution for some transformation of .7 to a new random variable. We shall label this random variable X if we are discussing centres giving both treatments, and Y for centres giving only one treatment. We shall assume that X and Y have normal distributions whose parameters depend on which of the 4 patient groups is indicated. Thus: (i) (ii) (iii) (iv)
patients patients patients patients
receiving T after receiving C after from centres only from centres only
randomisation assume randomisation assume giving T assume Yr is giving C assume Yr is
X, is a sample from I$,.. CT:-): Xc is a sample from .V‘(pc, a:): a sample from :V(V~, tot); a sample from .,b*(vc, (0:).
Thus the unknown means of the four groups are pr, pc, vr and rc with unknown varisample means are .U,. .fc, yr and yc. The ances fs+, a,$. 0:. and u$. The observed principal objective of the trial is taken to be the estimation of pr - pc which measures the effectiveness of the new and standard treatments. When mr = mc = 0, i.e. the usual randomised trial situation, the best estimate of available the ‘ethical’ centres pr - pc is XT - Xc. However, we now have potentially data summarised by jr and jc. Unfortunately, however, neither lr and vr. or pc and vc are necessarily equal and one must allow for the possibility of unknown biases in the ethical groups. We define these biases by:
The possible presence of such biases is one of the arguments used for the conduct of randomised trials. We shall assume, see Pocock [I], that ST and 6, both have normal distributions with mean zero and variances & and ~5 respectively. These Bayesian prior distributions for 8r and 6c express the degree of confidence one has in the ethical centres data. Considering for a moment one arm of the trial only and dropping temporarily the suffices T or C, then one may suppose that prior to commencement of the clinical trial one has complete ignorance regarding the values of p and v; however, after the trial and using the prior distribution of 6. it is possible to show by application of
Incorporating
Bayes theorem with mean.
Patients
from Non-randomising
that the posterior
distribution
Centres mta a Randomised
of p (the parameter
Clinical
Trial
of interest)
349
is normal
and variance
(2) whet-c A = l/q2 + nr/02. WC note the following (i) the fully randomised
special cases: clinical trial i.e. In = 0.
/Ti = s. C’ = a2!n; (ii) total ignorance
of (5 (ignore
ethical
centre data) i.e.. ‘I’-+ IL.
ii = .s. V = (i’!ll: (iii) complete
information ~ =
on 6. implying
?%::a’ + I n.:a2 +
The weights
rn~jo
21
,
’
mis2;
given to .U and y depend b=---_-
~3E 0. i.e. q2 + 0.
V = l/(nia2
+ m;02)
only on n, m, a’ and 02. If further
a2 = to2, then
n.u+ my ; t1 +
m
(iv) II = t-p. m = 2cp where r is the number of randomising centres and c the number of non-randomising ccntres. The c centres entering only one arm enter twice the number of patients on that arm than do the corresponding randomising ccntres.
If wc write
then restoring the suffices T and C. one can test the null hypothesis H,:p, (assuming 0;. r$. c$-. (0: known) by the standardised normal random variable I, =
{(‘;$iy \
In practice.
however.
_ (-yg!L$)} _-. -. (CT +
only estimates SAMPLE
(3)
a
of the variances SIZE
= 11~.
will be available.
IMPLICATIONS
We assume that r centres are randomising between T and C, that t centres give treatment T only and c centres C only. We further assume that centres giving only one treatment have twice the number of patients available for that treatment than does a corresponding centre randomising between the two alternative treatments and that a:. = af = a2 and w2r = 0,; = (U2. Let the total entry for each institute be 2p patients and the randomised patients can then be allocated to treatments T and C in the proportion U:l - O(0 < fI < I)
350
DAVID MACHIN
thus; nT = 20rp.
nc = 2( 1 - @i-p,
mT = 2tp,
mc = 2cp,
and N = nr + n, + mr + mc = 2(r + t + c)p. Choice
ofo.
We choose a value of (1 which minimised vr + V, =
‘1“.
Now
Differentiating I” with respect to 8 and setting the resulting derivative obtains after some algebra:
to zero. one
(5) where 4 = a2,102. and which can be shown to minimise 1,: We note that if: (i) c = 0. i.e. there are no centres contributing control patients only, then 0 < l/2. (ii) t = 0, i.e. there are no centres contributing treatment patients only. then 9 > 112. (iii) c = t = 0 then 0 = lj2. Substituting for 0 in (4) we obtain Yii” = 2 ;i$
+ $[(&)
(6)
+ &)I}.
Sample sizr for fl yicen l’iin. Suppose we wish to conduct a two allowing the inclusion of ‘ethical’ centres and we require it to be a way that if ‘ethical’ centres were excluded the corresponding fully would involve r centres with 2P patients from each centre, giving Vi f ,I” = 2d,bP we obtain:
Rearranging
arm clinical trial designed in such randomised trial = 2djrP. Setting
we obtain the following cubic in p:
4rctp3 + 2(rrc + /3rt + 4rrt
+ 4/k?
- 2rctP)p’
+ (rap + 4a/?c + &x/It - 2rrcP
which can be solved for p to obtain the required Special cases :
- 2/?rtP)p - rrj3P = 0
(7)
number of patients per institute.
(i) if c = t = 0 then (7) reduces to rafip - ra/.?P = 0 or p = P; (ii) if qt. 115--) T_ i.e., no prior information on bT and 6c, then (7) reduces to 4rctp*(p - P) or p = P: (iii) if qc. $ -+ 0 which implies ST = 6, E 0 or no bias in ethical centres. (7) becomes (r + cd + tb)p - rP = 0
Incorporating
Patients
from Non-randomising
Centres
into a Randomised
Clinical
351
Trial
or (rP/a’)
(8)
’ = (r/a2 + c/w2 + f/&’
It is easy to see from (8) that p < P and this situation, i.e. no bias in the ethical ccntres. corresponds to the minimum value possible for p. If a2 = (,~2 then (8) becomes p = rP!(r + c + t). (iv) if c = 0. I f 0. then (7) reduces
to the quadratic
2rtp2 + (w + c#W - 2rtP)p - rrP = 0; (v) if c # 0, t = 0 then (7) reduces
(9)
to
2rcp2 + (jr + C#J~C - 2rcP)p - @P = 0. Some numerical in the following
solutions section.
of equations
(7) and
NUMERICAL
(9) are given
(IO) for some
special
cases
RESULTS
In principle for given r, c, t. r. /A C#Iand P equations (7). (9) or (IO) can be solved to obtain p. Extensive tabulation of the solutions however is not practicable. Table I gives for a fully randomised trial with N = 2rP = 200. the comparative ‘ethical’ trial sizes assuming r = 8. C#J= 1 and c = t (note that these conditions imply 0 = +,). It is clear from Table 1 that, if 2 is large i.e., r7: and r$ are small. then there can be a considerable reduction in the numbers of patients recruited by each centre. for example for 2P = 20. r = 10. c = t = 10. 3: = 100 gives 2p = 10.0 so that the duration of entry onto trial could well be halved (2p/2P = IO/20 = 0.5). The total number of patients involved on the trial would be N’ = nr + n, + mT + mc = 50 + 50 + 100 + 100 = 300 rather than the N = 200 of the corresponding fully randomised trial. In practice the situation corresponding to equation (9) may be more common in which centres only ‘object’ to the one of the arms, which implies c = 0 and t z 0 or c # 0 and I = 0. Table 2 shows some solutions of equations (9) for given r. t and IX, assuming 4 = I and N = 2rP = 200. Thus for 2P = 20. r = IO, c = 0, t = IO. r = 100, 2p = 14.14 so that the duration of the trial would be reduced to 70% (2p/ZP = 14.14!20) of the correTABLE
I.
CFNTRFS.
NUMRER
OF
PATlENTS
CORRESPOSDING
TO
REQUIRED A
FULLY
PER CENTRE RANDOMISED
($) TRIAL
FOR A CLINICAL WITH
r
CENlRtS
TRlAl
ALLOWING
NON-RANDOMlSlN(;
CONTRIBL’TING
A
IOIAL
OF
?o()
PATIENTS
Number of patients per centre in a Number of fully ranrandomising dnmised trial centres
?P IO
r 20
Number of control arm nonrandomising centres
Number of treatment arm nonrandomising centres
c
I
0
40
10
5
r+r+c
Ratlo of observation variances z = (J/q2
0
5
IO
and bias
loo
I
0 5 IO 20
20 30 40 60
IO 10 IO IO
IO 11.9 9.3 915
IO x.2 8.7 5.1
IO 6.6 5.1 5.0
IO 6.7 5.0 3.1
5 10 20
0 5 IO 20
IO 20 30 50
20 20 20 20
20 18.6 19.0 19.3
20 17.4 IX.1 18.5
20 II.4 10.0 9.2
20 10.0
0 5 10 20
0 5 IO 20
5 I5 25 45
40 40 40 40
40 38.1 38.5 3X.8
40 36.2 37.1 37.5
40 20.0 18.3 17.9
40 13.3 X.0 4.4
5 IO 20 20
Total number of centres
0
6.1 4.0
352
DAVID MACHIN
TARLL 2. Nuuet~ OF PATIENTS RLQI:IRtl) PFR C‘LlriTRt (2~) FOR A CLINICAL ‘TRIAL ALLOWING SON-RANUOMISING CtNTRtS WHO OLUhCT TO THI ‘COKTROL‘ ARM. CORRtSPOPiUING TO A FIJLLY RAHIXNISFD TRIAL WITH r CCSTRIS CONTRILIL’TING A TOTAL ot 200 PATIENTS Number of patients per centre in a fully randomised trial 2P
Number of randomising centres r
IO
Number of control arm nonrandomising centres ‘
Number of trealmenl arm nonrandomising centrcs I
Total number of ccntres r+/+c.
20
0 0 0 0
0 5 IO 20
20
IO
0 0 0 0
40
5
0 0 0 0
Ratlo of observation variances z = (I/ tq
and bias
0
5
IO
100
-,
20 25 30 40
IO 10 IO IO
10 9.7 9.8 9.8
IO 9.6 9.6 9.5
IO x.5 7.8 7.1
IO 8 6.7 5
0 5 IO 20
IO 15 20 30
20 20 20 20
20 19.5 19.5 19.4
20 19.1 19.1 19.0
20 15.6 14.1 12.8
20 13.3 IO 6.7
0 5 IO 20
5 IO I5 ‘5
40 40 40 40
40 39.0 39.0 39.0
40 38.1 38.1 38.0
40 28.3 25.6 23.5
40 20 13.3 X
sponding fully randomised clinical trial. The total number of patients involved on the trial would be N’ = nr + nc + rnr + mc = 70.7 + 70.7 + 141.4 + 0 = 282.8 rather than the 200 of the corresponding fully randomised trial.
DISCUSSION
One of the major difficulties when it comes to launching a multicentre randomised clinical trial is to recruit sufficient patients within a relatively short period so that the questions posed by the trial can themselves be answered as soon as possible. It is tempting in such circumstances to recruit patients at all costs and perhaps to allow certain institutions (who would not otherwise enter the trial) to enter patients for only one of the trial arms. It is widely recognised that if recruitment of patients from such institutes is permitted they may introduce unknown biases into the overall treatment comparison if they are merely pooled with the remaining patients for analysis. However. regression model methods of analysis for example allow the exploration of data in greater detail and may suggest whether or not the pooling of data from such centres is justified. In planning a cooperative trial there is usually more than one end point of interest, for example the different responses to a given treatment occurring in different institutes or countries-or the confirmation or otherwise of the value of some accepted prognostic variable, for which the non-randomising institutes may be willing to provide patient data. Tables 1 and 2 suggest that if patients from non-randomising institutes are included. then for this to make any real reduction in the overall duration of the trial r = 02;~~. and /I = o’/qi must be large, which implies both & and $ would need to be small and that the biases 6, and Sc are effectively very close to zero. It is necessary therefore before allowing such centres to influence the design of a fully randomised trial to be reasonably certain that any biases will be small. These biases would almost certainly not be negligible if the non-randomising centres only entered a selected (non randomised) proportion of their eligible patients. Even if these assumptions could be made and such centres do include patients, then more patients would have to enter the trial in total than the corresponding fully randomised trial thus increasing the cost of the trial which must be balanced against earlier termination of the study.
Incorporating
Patients
from Non-randomising
Centres
into a Randomised
Chnical
Trial
353
In short, although the inclusion of patients from non-randomising centres appears superficially attractive, there is little that can be gained by their inclusion, particularly as the possibility of entering one arm only of a trial may be used as an excuse to opt out of randomising by a centre which otherwise may have (albeit reluctantly) agreed to randomise between the alternative arms. There may be more scope for inclusion of such centrez in 3 or more arm trials where a minimum participation of randomisation to 2 arms is required by the protocol but these have not yet been investigated. A~,knowlenyements--I am grateful to M. J. Staquet paper. This investigation was supported by Grand Cancer Institute. DHEW.
and R. J. Sylvester for comments on a draft of this Number 5R10 CA 11488-08 awarded by the National
REFERENCES I. 3.
Pocock SJ: The combination of randomized and historical controls in clinical trials. J Chron 175-188. 1976 Lindley DV: The effect of ethical design considerations on statistical analysis. J Royal Statist 218-228. 1975
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