Randomisation in Clinical Trials - Idlir Shkurti

7/17/2019 Randomisation in Clinical Trials - Idlir Shkurti

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Idlir Shkurti Queen Mary University of London (2014)

Randomisation in clinical trials

Idlir Shkurti

December, 2014

1.0 Introduction

In this project I will be observing clinical trials where two or more treatments are being

compared. The most common type of clinical trials is a therapeutic trial where a new treatment is

usually compared to a conventional treatment. These trials are usually referred to as phase III clinical

trials since they consist of the third stage of the conventional four-stage process in which a new

promising treatment is investigated. During the first phase, the toxicity levels of different doses of

the drug are observed. The second phase is a preparatory stage which studies the toxicity and the

efficacy of the drug whilst the fourth stage consists of the follow up of patients symptoms after the

drug has been granted a licence. Rosenberger and Lachin in their Randomization in Clinical Trials

book considered randomisation to be the pivotal point of phase III clinical trials. Randomisation plays

a key role in clinical trials as it tends to reduce the bias in the comparisons between the treatments.

Cornfield (1959, p.245) described the importance of randomisation the following way:

Randomisation controls the probability that the treated and control groups differ more than

a calculable amount in their exposure to disease, in immune history or with respect to any

other variable, known or unknown to the experiment that may have an influence on the

outcome of the trial.

This idea is very widely used during clinical trials because researchers usually want to keep

these variables under control between different treatment groups. The earliest methods of

randomisation have been around since 1929 when R. Fisher introduced the flip a coin mechanism.

This means that given there are two treatments available, the probability of each patient being

assigned to each treatment is 0.5. It is easy to observe that this method is not very ideal, particularly

with small sample sizes, since it is likely that this technique will cause very different number of

patients assigned to each treatment. Also different patient characteristics such as the patients

medical history or other variables which may influence the outcome of the trial will be unbalanced

between treatment groups. It is argued that in trials with several clinics and researchers, random

allocation is not required since the order of arrival of the patients is sufficient to protect against any

bias. However this arises more questions such as how many clinics/clinicians would be minimally

required for this to work? Therefore in this project I will be observing the cases where

randomisation is required and give an insight on different randomisation approaches.

2.0 Different Approaches to Randomisation

We can distinguish between four different types of randomisation procedures: complete

randomisation, restricted randomisation, covariate-adaptive randomisation and response-adaptive

randomisation. Let for i=1,,n be the sequence of treatment assignments when two treatments, Aand B are available. We denote


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1, 0, (1)If we use the flip a coin method which is a complete randomisation procedure, then

~(1 2 )for all i = 1,,n.We can think of the treatment assignments as a random walk

with a probability equal to a half of the random walk going on either side. Assume we move once to

the right if treatment A is assigned to the patient and once to the left if treatment B is assigned.

Assume the first patient is assigned to treatment A hence we move to the right. We would prefer

this imbalance now to get cancelled out by the next patients. If we denote the current position of

the assignments with +1, then the probability of exacerbating this imbalance is relatively not small. If

the probabilities of the random walk moving to either side are equal (exactly like in our case) then

the following formula is used to calculate the probability of the random walk starting at n reaching N

before M, given that M < n < N. In our case we assigned n to +1.

Pr (2)

Using this formula we can find the probability of exacerbating the previously mentioned. n is equal

to +1 for our example and we are interested in the random walk reaching a certain value N greater

than +1 before it reaches N = 0. Hence this probability is:

Pr 0 1 0 0 1If N = 10 then we can see that there is a 10% probability that this imbalance will become 10 before

the two treatment groups are balanced out. These computations give us a slight idea of how this

method could cause numerical imbalances between the cardinality of each treatment group.

However it does take under consideration that there are only n patients available to be assigned to a

treatment but it assumes that the random walk could potentially continue forever. The following

way of working with these treatment assignments is much more convenient for calculations.

2.1 Complete Randomisation

Let = for i=1,,n, be the number of patients assigned to treatment A when i patientshave been assigned a treatment. Likewise, let be the number of patients assignedto treatment B once i patients have been assigned to a treatment group. By the Central Limit

Theorem,

is asymptotically distributed with

~ , 4.

Let

(3) describes the degree of imbalance between the two treatment groups A and B andasymptotically ~0, . We use the distribution of to calculate the probability of theimbalance being larger than a real value R >0 between the two treatment groups.

Pr|| > 2 1 (4)


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Where is the cumulative distribution function of a standard normal distribution. This formula givesthe probability at which complete randomisation could give an imbalance greater than R between

treatment groups within a sample of n patients. The table below gives the percentiles of the

distribution of

|

|for different sample sizes.

n 0.33 0.25 0.10 0.005 0.025

50 6.9 8.1 11.6 13.9 15.9

100 9.7 11.5 16.5 19.6 22.4

200 13.8 16.3 23.3 27.7 31.7

400 19.5 23.0 32.9 39.2 44.8

800 27.6 32.5 46.5 55.4 63.4

This table tells us that for a sample of size 50 there is 33.3% probability of an imbalance of

6.9 or worse, a 25% probability of an imbalance of 8.1 or worse and so on if we randomisecompletely without taking any other factors into account. Unlike the previous method calculated byformula (2), this method takes into consideration that there is a limited number n of patients

available to be assigned.

2.2 Restricted Randomisation

In restricted randomisationprocedures are dependent since the main objective of theseprocedures is to have an equal number of patients assigned to each treatment group. A good

example of a random allocation rule which ensures equal number of patients on both treatment

groups is the following:

Let , , }.() , 2 , , (5)

0.5This method will make sure that there are exactly n/2 patients on each treatment group (assuming

there are two possible treatments to be tested) from an initial group of size n. However there is a

problem with this method as there is usually with most restricted randomisation procedures. Once

n/2 patients are assigned to one treatment group the allocation becomes predictable as all the

remaining patients will have to be assigned to the other treatment. There are methods which try to

balance the treatment groups whilst still keeping the allocations not entirely predictable such as

Weis urn design orEfronsbiased coin design. Weis urn design requires an urn with balls inside.

These balls are divided into two equal groups, call them type A and type B. When a patient is ready

to be assigned to a treatment group, a ball is randomly chosen from the urn (i.e. the probability of

the first patient being assigned to each treatment group is a half) and replaced. If this ball was a type

A ball, then the patient is assigned to treatment A and a pre- decided number of type B balls , are

added to the urn. The procedure is trivial when a type B ball is drawn. This method keeps the patient

assignments unpredictable but increases the probability of balancing the treatment groups. This


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method could similarly be applied to cases when there are more than two treatments to be tested,

by increasing the number of the ball types which were not drawn from the urn.

Efrons biased coin design(1971) on the other hand works the following way:

() 0 < 01 > 0 (6)where p is any value within (0.5, 1] and and are the quantity and the set defined earlier. It isfairly easy to compute that the expectation of , simply by considering both cases when j-1 is evenand when j-1 is odd, regardless of the value of p. The expectation of is equal to .

2.3 Adaptive Design Methods

Adaptive design methods are randomisation procedures which use prior information inclinical trial designs. The adaptations which we will be considering for this project will be covariates

and responses from previous patients. The primary goal of the restricted randomisation methods

was to balance the sample sizes for each treatment group; however researchers are usually

interested in keeping other factors under control too, depending on the type of treatment they are

working with. Covariate-adaptive randomisation techniques aims to balance covariates (whether

they are quantitative or categorical) between the treatment groups. A good example of a covariate-

adaptive randomisation is Franes method (1998), where he aimed to balance the treatment groups

such that the next patient assignment caused the least imbalance among the pre-specified

covariates. The method works as follows:

Step 1: Temporarily assign the new patient to treatment group A.

Step 2: Calculate the PearsonsGoodness of Fit test statistic for the covariate groups towhich the new patient would belong.

Step 3: Identify the maximumtest stat among all covariate groups. Step 4: Remove the patient from group A and repeat steps 1-3 for all the other treatment

groups.

Step 5: Identify the minimum test stat over all the identified test statistics (one fromeach repetition of step 1-3).

Step 6: Assign the new patient to the group for which the minimum

test statistics was

achieved.

A low goodness of fit test statistic means there is more balance between treatment groups.

The following example gives us a good idea of when this method might be appropriate and the most

ideal way to use it. Assume we are comparing a conventional blood pressure drug A with a new

blood pressure drug B. The comparison is made between patients with high blood pressure

considering the following covariates: baseline blood pressure (hypertensive or pre-hypertensive) and

age (greater than or equal to 65 or less than 65 years old). The first 20 patients are allocated the

following way.


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Drug A 12 Drug B 8Baseline Blood Pressure Hypertensive

Pre-Hypertensive

8

4

3

5

Age 65 years


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3.1Zelens rule

Zelens rule (1974) uses a pre-assigned randomisation sequence ignoring strata. Assume for

simplicity that we are considering a trial with two treatment groups. The method works by first

defining

and

. Let

be the number of patients on stratum i on treatment k = 1, 2

after n patients have been assigned a treatment. For a certain stratum i, we define to be the treatment imbalance within this stratum. When the (n+1)th patientbelonging to this stratum is ready to be assigned to a treatment we use the definition of torandomise them. We define an integer constant c, such that if the absolute value of 1issmaller than c we randomise the patient according to the schedule. If | 1| c then weassign the patient to the opposite treatment. This idea of controlling the treatment imbalances

within each stratum was also used by Pocock and Simon in their 1975 paper as described below.

3.2 Minimisation

In certain clinical trials the slight imbalances between treatment groups, which restrictedrandomisation methods try to fix, are not as influential on the outcomes of the experiment as other

factors. Minimisation is an adaptive randomisation method first introduced by Taves in 1974. This

approach aims to reduce treatment imbalances between treatment groups over prognostic factors.

Pocock and Simon in their 1975 paper developed this minimisation method which separates the

patients into different covariates according to factors which are believed to be of prognostic

importance. These factors are believed to have an impact on the outcome of the experiment. A good

example would be the previous example which illustrated Franes method. In this example we

divided the patients according to two prognostic factors: baseline blood pressure (hypertensive and

pre-hypertensive) and age (greater than or equal to 65 or less than 65). Both of these factors are

believed to have an impact on the outcome of the drug which aims to reduce blood pressure.

Another example could be grouping the patients into a block with two levels, smoker and non-

smoker, if we are testing a treatment which aims to cure lung cancer. Minimisation is usually more

efficient and better at minimising treatment imbalances than stratification. The Pocock and Simon

approach to minimisation is simply a broader and more developed procedure to the Taves method,

hence next we shall be observing the general minimisation procedure as proposed by Pocock and

Simon. We shall compare minimisation with other adaptive methods and also discuss some issues

and further areas of development with his method.

The usual way to compute this method is to divide each prognostic factor into different

levels. For example if we are trying to observe the outcome of a certain treatment, we might assume

that the clinic at which the patient was treated and the sex of the patient might be relevant factors.

Assume we have four different clinics at which the patients were treated hence the clinic factor is

divided into four levels: clinic 1, 2, 3 or 4. Also gender is clearly divided into two levels. However one

must be careful when dividing the factor blocks into levels since the number of strata increases

rapidly as the number of levels increases. In the clinic-gender example we would have 4 x 2 = 8

strata as shown by the table below:


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Clinic 1 Clinic 2 Clinic 3 Clinic 4

Male

Clinic 1 & Male Clinic 2 & Male Clinic 3 & Male Clinic 4 & Male

Female

Clinic 1 & Female Clinic 2 & Female Clinic 3 & Female Clinic 4 & Female

In the previous blood pressure drug example we have two different prognostic factors, each

divided into two levels which means that we have 2 x 2 = 4 different strata. These previous examples

have had a reasonable number of strata but if for example we have four prognostic factors, each

separated into three levels. This means that the total number of strata equals 3 x 3 x 3 x 3 = 81.

A large number of strata are problematic in a clinical trial. Assume for the previous

mentioned example with 81 strata we have 100 patients available to be assigned to a treatment

group. This means that we cannot even obtain 2 patients per strata; which means that we would

have to compromise some strata so that they contain no patients. Hence the whole clinical trialwould fail its objective.

3.2.1 Pocock and Simon procedure

Pocock and Simon in their 1975 paper described a new method which aimed to find a

solution to procedures which failed to balance the effects of prognostic factors across all treatments.

In this paper they introduce the use of sequential treatment assignments in order to balance

prognostic factors across treatments. They describe this method to be a generalisation of Efrons

biased coin design for more than two treatments and several prognostic factors. The two-treatment

case of this procedure works as follows. Assume we are interested in a clinical trial where patients

enter sequentially waiting for a treatment assignment. Suppose we have two treatments available

and M prognostic factors for which treatment balance is required. Hence we define to bethe number of patients assigned to stratum j = 1, ., of covariate i = 1, .,M on treatment k = 1, 2(1 for treatment A and 2 for treatment B), once n patients have been assigned to a treatment group.

It is worth mentioning here that the total number of strata in this case would be = . Somefactors might be assumed to have a greater effect on the outcome of the test than others, therefore

we assign different weights for i = 1,,M for each prognostic factor. Assume the (n+1)th patientis waiting to be assigned to a treatment. Assume this patient is a member of the strata , , , of covariates 1, .,M. We define D to be the balance difference like before such that

(7)We use this formula now to define D(n).

= (8)Where are the covariate weights which we have defined above. Pocock and Simon use the valueof D(n) to decide which treatment group should have a higher probability of assignment for the

(n+1)th patient. If D(n) is less than a half then treatment B has been favoured so far. Hence the

(n+1)th patient with the corresponding strata set

, , , should be assigned to treatment A

with a higher probability. The opposite reasoning applies if D(n) is greater than a half. Thisprobability is calculated using a biased coin design with probability:


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+3 (9)If we define p(A) to be the probability of assigning the patient to treatment A, then following rule is

applied when assigning treatments:

< 1 > (10)

And we can see that when is equal to 1, this method is quite similar to the Efrons biased coindesign.

In most clinical trials, we are interested in comparing more than two treatments/drugs and Pocock

and Simon generalise their procedure for more than two treatments. Assume we have now N

treatments to be considered and compared such that k = 1,2,,N. Again we assume that the (n+1)th

patient is part of the , , , levels of factors 1, .,M. We define now , for k = 1,,N to bethe treatment assignment imbalance of the 1, which are the patients within level offactor . That is, the imbalance across this particular strata given that the (n+1)th patient has beenassigned to treatment k.

Let , , , , (11)where G is a function . gives us the total imbalance across all levels , , , of factors1, ., M, given that the (n + 1)th patient has been assigned to treatment k. We find

for all

treatments k = 1,,N and we use them to rank the treatments the following way {(1), (2), , (N)},where (1) is the treatment with the lowest value of , (2) is the treatment with the second lowestvalue and so on.

Ordering the treatments based on their corresponding values of will help us determinethe probability of assigning the (n +1)th patient to treatment T. Assigning this patient to treatment T

is determined by the following probabilities :Pr [T k] (12)where

and

= 1. Hence we can see that the smallest the value of

for

a certain treatment k = 1,,N; the higher the probability of assigning the patient to that specific

treatment. The values of could be functions of such that a slightly greater or smaller value ofwould decrease or increase the value of respectively. However these probabilities could alsobe fixed. The entire procedure is then sequentially repeated when the next patient is ready to be

assigned to a treatment group.

However a few questions come up at this stage regarding the choice of and the functionG. The function G could be defined in several different ways and has been since Pocock and Simon

published their paper. The most general form of the function G is the following:

(, , , , ) ,= (13)


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However, some prognostic factors could be deemed to be more significant towards the

outcome of the clinical trial than others. Hence, just like in formula (8), we can assign certain

constants to these factors and calculate the following way:

( , , , ,) ,= (14)

where is now a weighted sum of , . Hence the more important the factor i, the greater thevalue of .

The choice of however is more complicated. The two main ways to obtain theseprobabilities, which Pocock and Simon mention in their paper are below.

1) Let and for all remaining probabilities, i.e. k = 2,,N. The necessaryrequirement here is that p = , 1. If p is equal to 1/N then we would get the sameprobabilities of assignment for all treatments. Also if p = 1, the procedure would be

predictable.

2) This is the equivalent method to formula (9) and it simplifies to (9) when N = 2. The

probabilities are calculated the following way:

+ . ; for k = 1,,N. (15)Where is a constant between 1/N and 2/(N 1) and the larger the value of the morebias is used in the treatment assignment.

When Efron introduced his biased coin design, he suggested a value of 2/3 for p. However

even when we have a clinical trial with only two treatments available, a 3might be too small ifthe patient sample is small but with many different prognostic factors. It is imperative here that

predictability is not possible for any patient thus the value of cannot be equal to 1 even thoughthe chance of treatment imbalance is minimised when this happens. On the other hand if 1/then the predictability is at its minimum but the chance of treatment imbalance is quite high. No

certain rules have been determined on the choice of , however many factors are considered whendeciding this other than minimising the imbalances, such as accidental biasor correlations between

covariates.

3.2.2 Begg and Iglewicz method

Begg and Iglewicz in their 1980 paper, introduced a new way of calculating D. They do not

agree with Pocock and Simons choice for D since it does not aim directly at increasing the statistical

efficiency of the trial. They aim to minimise a function which is an approximation to the variance of

the treatment effects in a linear model. In this paper, for simplicity I shall demonstrate their choice

of D by assuming only two treatments, A and B, are available.

Let be a factor with two levels and let us denote those levels 1 and 2. Let , , , be the treatment totals on both levels of factor

, such that

represents the total amount of

patients within the first level of factor receiving treatment A. Pocock and Simon aimed to minimisethe imbalances within every strata. This means that their choice of D was used to minimise either


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or , depending on whether the new patient fell on level 1 or 2 of factor .However Begg and Iglewiczs approach aimed to minimise an approximation to the variance of the

treatment effect. Their choice of D aimed to minimise - for each prognosticfactor

.

Begg and Iglewicz claim that this method is superior to the Pocock and Simon method. This

is because Pocock and Simon do not take under consideration in their calculations, the variance of

the treatment effects in a linear model relating to the outcome variable to chosen prognostic factors

and interactions. This method outperforms the Pocock and Simon method when comparing

efficiency loss. This means that the variance of the treatment effects is minimised. This outcome is

predictable since this is the very aim of this procedure. However it is outperformed when comparing

treatment balance. The imbalance difference was however very small. This small imbalance is

usually compromised in order to obtain a higher statistical efficiency.

3.2.2Atkinsons Optimal Design

Similar to Begg and Iglewicz, Atkinson (1982) believes that during a clinical trial one should

be concerned about minimising the variance of the estimated treatment effect, assuming we have a

number of prognostic factors present. The following is the randomisation procedure presented in

Atkinsons (1982) paper. Atkinson presents a linear model linking these factors and the treatment

effects. First we define the classical regression model

1 , , (16)

Where

are the independent responses with variance

, and

includes a treatment indicator

and selected covariates of interest. We estimate by such that the variance of this is() Where is the dispersion matrix of size p x p of n observations. In order to construct optimaldesigns we need to find an optimal sequence of n treatment assignments. We denote this n-point

design by and its dispersion matrix is /. Instead of working with the variance }we work with the standardised variance

, (17)

In his 1982 paper Atkinson compares this -optimality to standard D-optimality which maximisesthe log determinant of . D-optimality is relevant when all parameters in a model are ofinterest. However in clinical trials, researchers are typically interested in contrasts between

treatments hence we use thevector, where Ais an s x p matrix of contrasts such that s is strictlyless than p. Since Ais a matrix of contrasts then the elements ofare s linear combinations of thetreatment effect contrasts. Atkinsons optimal design model aims to minimise the variance of .There are many ways to do this, one of those methods being the analogue to D-optimality

introduced by Sibson (1974), called -optimality which maximises|| (18)


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The optimal design which minimises (18) must satisfy the following equations:, , (19)

,

(20)

where , is simply the analogue of variance (17) and is defined as follows, (21)and is the design region of the formulated problem in terms of . consists of as many points asthe number of treatments available (say K in our case), and the i-th point in this design region

corresponds to assigning treatment i to the next patient. Its corresponding value of , isdenoted by , . This is used to assign patients to treatments sequentially. The optimum designaims to assign the (n + 1)th patient to the treatment for which , is a maximum. This optimumdesign aims to assign a fraction

of the total number of patients to treatment i such that all

, are equal. If they are not equal then some of the treatments might beunder/overrepresented. The randomisation proposed for this procedure is a biased coin design,where the probability of assigning a patient to treatment is

, , (22)When and (20) is satisfied, then . If the design is not the optimum design then thetreatment assignments will help bring the design closer to the optimum design. The analogue D-

optimum design aims to assign the same number of patients to each treatment hence

should be

equal to 1/K for each i = 1,,K. Then (22) will simplify to , , (23)So now when and (20) is satisfied, the probability of allocating a patient to each treatment isequal.

4.0 Conclusions

This project deals with some different approaches to randomisation, in particular covariate adaptive

randomisation. If patients in a clinical trial are assigned to completely random treatment group then

the outcome of the trial which determines which treatment is more effective might not be very

accurate. That is because there might be certain prognostic factors which we might be ignoring.

These factors could affect the outcome of the trial. A good example was mentioned earlier in this

paper where the fact that if a patient is a smoker or non-smoker could affect a lung cancer

treatment. Covariate adaptive randomisation takes these prognostic factors under consideration.

These procedures divide the patients into covariates and each covariate into different levels such

that we have patients assigned to strata. Then from this point, covariate adaptive procedures aim to

balance the overall treatment allocations as well as the treatment allocations within each stratum.


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This is because we want to balance the number of patients on each treatment who get affected by

these prognostic factors similarly.

A simple example of a covariate adaptive randomisation procedure introduced by Frane (1998) was

used as an example to show how patients could be allocated to a treatment considering which

stratum they are part of. The rest of the project focuses on minimisation, the procedure, its

applications and how it has been developed during the years. Minimisation was first introduced by

Taves (1974). The most imperative aim of this procedure is to minimise the treatment allocation

imbalances within each level of prognostic factors. However this aim should be achieved without

fully compromising the predictability of the trial. Hence the requirement should not be that the

number of patients allocated to each trial is equal because the procedure would eventually become

predictable. Efrons biased coin design is used to achieve this.

The Pocock and Simon procedure is a development to the Taves (1974) and it aims to minimise the

imbalances over known covariates. This procedure does not involve complicated computations and

hence the theory behind it is widely used and has been developed further by other statisticians.

However there are a few problematic areas with this procedure. One of them is the choice of ,which is a constant between 1/N and 2/(N 1) used to calculate the probabilities . There are afew recommendations in this paper about the choice of however no specific rules. Also, aspointed out by Begg and Iglewicz (1980), this procedure does not take into account the variance of

the estimated treatment effect in the presence of covariates. This means that the Pocock and Simon

procedure is not as statistically efficient as it could be.

Begg and Iglewicz were the first to introduce the idea of minimising the variance of the estimated

treatment effect in the presence of covariates. This procedure, when compared to the Pocock and

Simon procedure in a sample of 100 patients was much more statistically efficient. However it did

slightly compromise the treatment imbalances which the Pocock and Simon procedure specifically

guarded against. Is this increase worth compromising? Also this procedure involves more

complicated computations than the Pocock and Simon procedure hence it might not be preferred.

Atkinsons optimal design approach is even more difficult to compute and might easily result in

calculation errors. Also this method relies on a specific linear model which relates the factors to the

treatment effects, which might not be appropriate for some trials. Hence this method cannot be

applied to every trial. Also similarly to the Begg and Iglewicz procedure, the goal of this optimal

design is to minimise the variance of the treatment effect estimator in the presence of covariates.

This means that some treatment imbalances over prognostic factors might be compromised to

achieve this.

Covariate adaptive randomisation is a good way to assign patients to treatment groups over known

covariates. However there are other methods such as response adaptive randomisation techniques

which take into account the responses from previous patients. Some treatments might be obviously

superior to other treatments, hence it would be appropriate to assign more patients to these

treatments. Covariate adaptive randomisation does not consider this. Using solely response adaptive

randomisation would not be advisable because prognostic factors are ignored. However covariate-

adaptive response-adjusted randomisation is an interesting area of study which juxtaposes these

two techniques with each other so that they can be used simultaneously in order to balance


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treatments over covariates, as well as having a much more efficient trial. These procedures are

relatively new when compared to covariate adaptive or response adjusted techniques, thus not a lot

of work has been done. However there are already a few problematic areas with these procedures

as the statistical computations used are usually quite complicated. Due to the time limit of this

project I will not be analysing these points into further depth.

References

William F. Rosenberger, John M. Lachin (2004) Randomization in clinical trials: Theory and Practice

B. Efron (1971) Forcing a sequential experiment to be balanced

Stuart J. Pocock, Richard Simon (1975) Sequential treatment assignment with balancing for

prognostic factors in the controlled clinical trial

A. C. Atkinson (1982) Optimum biased coin designs for sequential clinical trials with prognostic

factors

Colin B. Begg, Boris Iglewicz (1980) A treatment allocation procedure for sequential clinical trial

Frane J.W. (1998) A method of biased coin randomization, its implementation and its validation

Jenna Colavincenzo (2013) Doctoring your clinical trial with adaptive randomisation

K. Fernandez (2005) A review of minimisation: Methods and practical concerns

Randomisation in Clinical Trials - Idlir Shkurti

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