An Adaptive Design For Discrete Responses Of Patients In Clinical Trials

An Adaptive Design For Discrete Responses Of Patients In Clinical Trials

An Adaptive Design For Discrete Responses Of Patients In Clinical Trials

V Lakshmi Naryana1,

S.P. Kishore2, K. Santosh Reddy3

Vardhaman College of Engineering, Hyderabad, India

ABSTRACT

Adaptive design is a trial design that allows modifications to one or more specified aspects of the study design and hypothesis based on the analysis of data after its initiation without undermining the validity and integrity of the trial.. It has become very attractive to the pharmaceutical industries. In this paper, discrete adaptive designs for clinical trials survival responses are studied using computer simulations. The greatest interest in adaptive design clinical trials has been in the adequate and well controlled study setting intended to support marketing a drug. The discrete adaptive design (DAD) described here decides the immediate future allocation of treatment to patients based on the analysis of data .At each phase of analysis the decision to terminate or continue the trial is based on the expected loss function .

Keywords: Adaptive design, robustness, decision making, allocation pattern, loss function.

1. INTRODUCTION

   The objective of clinical trials is saving or curing the maximum number of patients and of several alternative therapies, treatment plans or drug formulations the physician has to decide on adopting one that would yield the maximum cure rate or minimum mortality. The statistician has a vital role to play in these sorts of testing plans.

   Take for instance a clinical trial where in subjects (human patients) arrive sequentially desiring to receive, adaptively, the better of two competing treatments A and B in which, A is a drug that is already administered widely, and B is the new introduction. Those subjected to such a trial are human beings and are proposed in this paper as a DAD that would allocate less efficient treatment therapy to fewer numbers of patients. This scheme of allocation does not just dichotomize the response of the patients, but used for the next allocation, all the information gathered from previous discrete responses.

   Various data-dependent adaptive allocation rules are available for a sequential chain of patients undergoing clinical trials. There is the concept of play-the- winner (PW) rule introduced by Zelen (1969). Subsequently Wei and Durham (1978) and Wei (1979) proposed the randomized PW rule. Durham and Yu (1990) designed a two arm urn model that ignores failure and in which the urn composition changes only as and when a success occurs. The contributions of Ware (1989) and Rout et al. (1993), Anderson et al. (1994) Faries et al. (1995), Bandyopadhyay and Biswas (1996, and 1997), and Durham et al.(1998) have made available other adaptive designs. However over the past two decades, the contributed number of real life response adaptive designs, is not very large. Statisticians can actively influence the design and analysis of clinical trials since recent advancement in computer technology could make clinical trials more adaptive in all respects. A new type of urn design called a birth and death urn of Ivan ova et al. (2000) has enabled them to derive the maximum likelihood estimators of success probabilities and their limiting distribution. All designs listed above are urn designs where responses of the patients are assumed

   to be dichotomous (success or failure).

2. The DAD, now being described, decides the immediate future allocation of treatment to patients based on the quantum of updated information. The population of responses under treatment A is represented by X and that of

   B by Y. The real valued random variables X and Y are assumed to be of the discrete type. Let Xi or Yi denote the response of treatment A or B by the ith patient depending on the one that has been allocated to him/her. Consider

   i

   to be the indicator of assignment that takes the value 1 (or 0) if the ith patient is allocated to treatment A (B).

   The first patient is allocated to treatment A and the second to treatment B and their responses are measured

   as 1=1, X1,Y1=0 , and 2 = 1, X2 , Y2 = 0respectively. Starting from the third up to the nth allocation,

   incoming patients are allocated to either treatment A or B according to an adaptive allocation scheme (AAS), which is detailed below. When n is the maximum number of patients participating in the trial, this AAS used the

   statistics of observed sample based on the discrete responses of triplets i , Xi, Yi ; i =1, 2,..kof the previous

   k < n patients to decide the value of k + 1 without dichotomizing them.

   The stage at which the number of patients NAK and NBk treated by treatment A and B respectively just after the entry of kth patient is

   NAK =

   k

   i

   i

   i=1

   and NBK =

   k

   i

   i

   (1- ) i=1

   (1)

   The observed sample means for the two populations X and Y can respectively be denoted by

   K

   iXi AK = i=1

   NAK

   K

   (1-i)Yi BK = i=1

   NBK

   (2)

   The steps leading to the decision rule for allocating k + 1th

   patient to either treatment A or B as per AAS or

   equivalently for the determination of k + 1th value are as follows:

   Step 1: A symmetric and continuous cumulative distribution function G (.) is chosen i.e G(0)=0.5, G (-x) =1-G(x).

   Step2: The observed sample means AK and BK are determined as in (2) based on the data

   (i, Xi,Yi ), i = 1, 2,……kof the previous k < n patients.

   Step 3: The k < n patient is allocated to treatment A with probability G (.) and to treatment B With probability 1-G (.), where c is some appropriate scaling constant.

   This kind of an allocation will always be biased towards the treatment that has led on average to larger number of favorable responses on average in the past, and that is what the AAs aims to achieve. (See Biswas and Basu (2001)).

3. The decision-making problem addressed in (3) below declares that the sample evidence supports one of the following three hypothesis (H):

   H0 : µA =µB , H1 : µA >µB , H2 : µA <µB

   (3)

   If the observed difference d = An – Bn of sample means obtained on the basis of n sample observations

   i , Xi, Yi ; i =1, 2,..n and sample means An and Bn , is greater than a pre-specified cut-off point v or less than -v then we accept H1 or H2 otherwise we tend to accept H0

   There is the opportunity for identifying a few outlying observations. Each outlier can exert undue influence on the values of the sample means An and Bn . Special attention has been paid in recent years to the statistical

   property called robustness which is an indicative of the extent to the estimation procedures that are adversely affected by violations of underlying assumptions.

   Making use of appropriate robustness considerations as Biswas and Basu (2001) have in the estimates

   An and Bn , the influence of outlier may be minimized. Accommodation methods such as trimming data and Winsorization are other commonly used procedures for dealing with outliers. The computer simulations alone can solve most of the questions of robustness that are difficult to answer.

   In most of the applications of adaptive designs available in literature, the responses are either assumed to be dichotomous (success or failure) or intentionally dichotomized (if continuous only) setting appropriate threshold value. But the desgn being considered, measured the responses of patients in discrete units after the allocation of treatment to patients.

   For our numerical study, responses X and Y are assumed to follow binomial distributions Bx; m, p0 and

   By; m, p1 respectively so that mean µA = mean of X = mp0 , µB = mean of Y = mp1 ,

   2A = variance of

   X = mp0 1- p0 , and 2B = variance of Y = mp1 1- p1,.

   Normal distribution can provide a close approximation of the binomial

   Bx; m, p

   distribution when the

   number of trials m, is very large and p, the probability of success on an individual trial is close to 0.5. However, normal distribution is often applied to approximate binomial probabilities even when m is fairly small. A good rule of thumb is to use this approximation only when mp and m1- p values are greater than 5. Algorithm of drawing a

   sample from binomial distribution Bx; m, p is outlined below subject to the condition mp > 5 and

   m1- p > 5 :

   Step 1: A point u at random is drawn from the uniform population U0,1.

   Step2: The value z1 of Z =

   X – mp mp(1 – p)

   ~ N0,1is computed such that P Z z1 u

   Where N 0,1 stands for the standard normal distribution.

   Step3: Computed is X=number of successes = z1( mp(1- p)) + mp

   Step 4: Steps 1 to 3 are repeated until the sample desired is selected from the binomial population

   X ~ Bx; m, p

   Detailed below is data on i , Xi, Yi obtained for a fixed number n of patients i.e. i = 1, 2,.n. The first

   patient is allocated to treatment A and second to treatment B and random observation X1 from B(x;m,p0) and observation Y2from B(y;m,p1) populations are drawn so that 1 = 1, X1, Y1 = 0, and 2 = 0, X2 = 0, Y1 are

   completely known. For each of successive allocations, the current estimate

   Ak and Bk can be computed and

   the allocation determined according to the algorithm of AAS. And so the next allocation will either be to treatment

   Ak – Bk

   A i.e. k+1 = 1with probability G c or treatment B i.e. k+1 = 0 with probability

   Ak – Bk

   1 – G c .

   For the present numerical illustration we use the symmetric distributions G1 (.) and G2 (.). G1 (.) has been defined as symmetric over the real valued r to r as follows:

   x x + r

   1

   1

   G (x) = g(x)dx =

   -r 2r

   (4)

   So that g(x) = 1 / (2r) ) for

   r x r

   where r is a positive and finite real number. Quantities 0 andr2 / 3 and

   will respectively be the mean and variance values of G1(x).

   The performance of G1 (.) as allocation probability in the DAD is then compared with another cumulative distribution function (CDF) say G2 (.) that has been selected as the CDF (.) of the standard normal distribution N0,1.

   The estimates µAk and µBk are simple means of k observations and for each method, the final

   estimates µAn and µBn and the number of allocations TA and T B to the treatment groups A and B respectively (TA+TB=n) have been calculated.

   The functions listed below will help estimate loss consequent on problem (3) of the final decision-making.

   1. The loss is zero for choosing Hi when Hi is true for i=0,1 and 2.

   2. The loss is 1 for choosing H0 when H1 or H2 is true.

   3. The loss L is greater than zero for choosing H1 when H2 is true or vice versa.

   The reader may refer to Ferguson (1967) for details on a loss function.

   The above binomial

   Bm, p trials have been carried out with n=20 over a large number of samples say

   N=200. The following sample estimates have been made from 200replications of the sampling distribution: (i)E(TA)=Expected value of TA, (ii)V(TA)=Variance of TA ,(iii) Prob1=estimate of the probability of H1 of (3) and Prob2= estimate of the probability of H2 of (3) and (iv)E(p0)and E(p1).

   Numerical calculations carried out for different choices of input parameters such as m, p0, p1,r, v and c of the above DAD, and the corresponding results presented in tables 1 through 4 will help gain a better understanding of the performance of the design.

   TABLE 1: Simulation results for Binomial model. Data (responses) from populations B(m=17, p0=0.45)i.e. A and B (m=17, p1=0.45) i.e. B are generated for n=20 trails which is replicated N=200times with cut off v=2.

   CDF	c	E TA	V TA	Prob1	Prob2	E( p0 )	E( p1 )
   G1(.)=U(-r,r) r=1.7321	5 10 20	10.070 10.135 10.110	7.3651 5.4768 5.2579	0.010 0.005 0.000	0.985 0.995 1.000	0.4480 0.4496 0.4504	0.4493 0.4510 0.4528
   G1(.)=(.)	5 10 20	9.9750 10.125 10.110	9.8642 6.1494 5.4418	0.015 0.005 0.000	0.975 0.995 1.000	0.4453 0.4494 0.4502	0.4465 0.4500 0.4517

   TABLE 2: : Simulation results for Binomial model. Data (responses) from populations B(m=17, p0=0.41)i.e. A and B (m=17, p1=0.52) i.e. B are generated for n=20 trails which is replicated N=200 times with cut off v=2

   CDF	c	E TA	V TA	Prob1	Prob2	E( p0 )	E( p1 )
   G1 (.)=U(-r,r)	5	8.1050	7.2140	0.260	0.740	0.4062	0.5198
   r=1.7321	10	9.1550	5.0510	0.265	0.735	0.4106	0.5222
   	20	9.6350	4.8918	0.260	0.740	0.4116	0.5231
   G2 (.)= (.)	5 10 20	7.4600 8.8450 9.4200	8.4584 5.3710 4.8236	0.280 0.245 0.255	0.72 0.775 0.745	0.4018 0.4096 0.4120	0.5198 0.5215 0.5228

   It could be noted that when r =3=1.7321, variance of G1 (.) equals one so that both G1(.) and G2(.) have the same mean (i.e. zero) and variance(i.e. unity ). A scrutiny of the results presented in the above two tables, will show that both G1(.) and G2(.) produce close sample values but V(TA) of G1(.) is smaller. Further, if the probability of success of finding a positive symptom p for the two treatments are same as in Table 1 or differs moderately by less than

   0.1 unit between p0 and p1 as in Table 2, any one real value between c= 5and c=20, can be preferred as scale value in the allocation process.

   Secondly, Prob2 (which should fall very close to 1 if p0 = p1 values and close to zero if p0 p1) value under G1 (.) estimation is greater than or equal to the corresponding Prob2 value under G2 (.). This pattern of calculation has been repeated in two more combinations with vast differences between the proportions of p0 and p1 and the corresponding results have been tabulated and presented in Table3 below for further verification.

   TABLE 3: Simulation results for Binomial model. Data (responses) from populations B(m=17, p0=0.42)i.e. A and B (m=17, p1=0.67) i.e. B are generated for n=20 trails which is replicated N=200 times with cut off v=2.

   CDF	c	E TA	V A	Prob1	Prob2	E( p0 )	E( p1 )
   G1 (.)=U(-r,r) r=1.7321	5 10	5.450 7.985	6.2775 5.3548	0.955 0.965	0.045 0.035	0.4007 0.4203	0.6698 0.6706
   	20	9.09	4.7719	0.955	0.045	0.4213	0.6713
   G2 (.)= (.)	5 10 20	4.500 7.150 8.740	4.6600 5.4975 4.8324	0.960 0.945 0.955	0.040 0.055 0.045	0.3975 0.4186 0.4210	0.6709 0.6703 0.6717

   TABLE4: Simulation results for Binomial model. Data (responses) from populations B(m=25, p0=0.42)i.e. A and B (m=25, p1=0.67) i.e. B are generated for n=20 trails which is replicated N=200 times with cut off v=2.

   CDF	c	E TA	V TA	Prob1	Prob2	E( p0 )	E( p1 )
   G1 (.)=U(-r,r) r=1.7321	5 10	3.175 6.775	4.3144 5.4664	1.000 1.000	0.000 0.000	0.3817 0.4179	0.6706 0.6702
   	20	8.600	4.8100	1.000	0.000	0.4203	0.6712
   G2 (.)= (.)	5 10 20	2.990 5.790 7.985	3.1299 4.8159 5.0548	0.995 1.000 1.000	0.005 0.000 0.000	0.3917 0.4129 0.4197	0.6713 0.6698 0.6700

   It could again be noticed from a study of the results presented in Table3 and Table 4 that each expected value E(.) from the simulated samples are found to be so close under G1(.) and G2(.) corresponding to each scaling factor c= 5,10, and 20. This ensures that allocation between treatments A and B according to both probabilities G1(.) and G2(.) as specified by DAD, not only produces similar results but also favors the treatment that performs better with reference to the response of the patients. Secondly, the estimate of Prob1 (which should be very close to 1 if

   p p ) value under G1 (.) is greater than or equal to that of the corresponding Prob1 value under G2 (.).

   0 1

4. It could be seen that (i) the scaling factor should increase with an increase in each value of m and (ii) any one of the allocation processes from the family {G1 (.) , G2(.)} may be selected for allocating patients between the competing treatments A and B when the number of successes out of m symptoms from patient follow the binomial

   Bm, p law. For each treatment, p the proportion of success is such that mp > 5 and m1- p > 5 . The value of

   r can be selected from the neighborhoods of 1.7321, in the allocation process under G1 (.) =U(-r, r ), depending upon the input values of DAD.The proposed DAD will perform as expected if there is no delayed response. But the allocation type of next incoming patient to the type of treatment in adaptive trails is decided by the summary of responses up to the current trials. Better estimates may be obtained on the basis of available responses before the next entry of a patient if the number of delayed responses is very few.

   The allocation pattern of patients is governed by the scaling factor c and the value of r of the distribution G1 (.) =U (-r, r) and they are allocated to either treatment, according to the state at any stage of the data on responses. Small values of c and r, for example, will have sensitive impacts on outliers. On the other hand, large values tend to influence information and say the allocation ratio of patients to treatments A and B eventually towards a 50:50 pattern. The choice of c and r and the standard deviation of the estimated difference in mean responses of samples selected from A and B populations are factors that influence the decision of allocation of patients to the better of the two lines of treatments and hence they should always be viewed in conjunction.

   The allocation probabilities will be different for different choices of G (.), but this difference can be minimized by choosing c depending on G(.). in general the selection of c is a more difficult task than that of the choice of G(.).

   In most of the applications of adaptive designs, the responses are either assumed to be dichotomous or intentionally dichotomized setting appropriate threshold value, but the design being considered here measured the responses of patients in discrete units after allocation of treatments to patients. Adaptive design can increase the number of responses in a trial and provide more benefits to the patient in comparison to the classic design.

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