MPS/MSc in StatisticsAdaptive & Bayesian - Lect 61 Lecture 6 Response adaptive designs 6.1Unequal...

MPS/MSc in Statistics Adaptive & Bayesian - Lect 6 1

Lecture 6

Response adaptive designs

6.1 Unequal treatment allocation

6.2 Varying the allocation ratio

6.3 Play-the-winner rules

6.4 Applications

6.5 Block response-adaptive randomization


6.1 Unequal treatment allocation

Suppose that treatment allocation is planned to be unequal,in an R:1 ratio to E:C then

nE = RnC so that n = nC + nE = nC (R + 1)

Hence

For unequal sized samples, the statistic Z of Lecture 2.5 is generalised to:

C Eandn nR

n nR 1 R 1

E CE C E C

n n1 RnZ y y y y

n R 1


Hence

so that

For equal sized samples, we had:

2 2

E C 22E C

Rn RnZ ~ N ,

R 1 n nR 1

RnZ ~ N 0,1

R 1

Z n 2 ~ N 0,1


Thus, to allow for unequal sample sizes, the sample size for equal samples should be multiplied by

This is also true for binary data, as Lecture 3.9 gives

and when sample sizes are large

2R 1

4R

E C3

n n SFV

n

E C2

n n p 1 p Rnp 1 pV

n R 1


Example:

Normally distributed data

= 0.025 (one-sided) z1 = 1.961 = 0.9 z1 = 1.282 = 1.8R= 0.5

For (1:1) allocation

Recruit 273 patients per treatment group

2

1 12

R

z zn 4 544.87


For other allocation ratios:

R (R+1)2/4R n

1 1 544.87

2 1.125 612.98

3 1.333 721.95

4 1.563 858.17

5 1.800 980.76

The total number of patients increases as the allocation ratio increases

The optimum value of R is 1


Motivation for unequal treatment allocation

If C is a potentially inferior treatment

If C is a standard treatment so that much is already known about it, but we need to gain more experience about E, including safety data

If E is particularly expensive or difficult to produce


In group sequential trials, the test statistics B and V arechosen so that

B ~ N(V, V)

increments (Bi – Bi1) between interims are independent

Essentially, we condition on the ancillary statistics V1, V2, ...

0 V1 V2 V3 V4 ...

then B1, B2, ... will form a “Brownian motion” with drift

6.2 Varying the allocation ratio


Varying the allocation ratio nE:nC will effect the values of the Vi

but not the Brownian motion property, nor its drift, as (Bi Bi1 ) ~ N((Vi Vi1), Vi Vi1) whatever the allocation ratio

For example, in the normal case

Hence, the allocation ratio can be chosen in any way you like

In particular, it can depend on the Bi

Robbins and Siegmund (1974) – normal caseRobbins (1974) – binary case

Ei Ci Ei Cii i Ei Ci2 2

i i

n n n nV B y

nand y

n


Response adaptive designs seek to minimise the number of patients receiving the inferior treatment

The motivation is generally ethical

They proceed by progressively biasing allocation in favour of the more effective treatment

The total sample size is increased, but the number on the inferior treatment is reduced


Robbins (1952), Zelen (1969)

Context: Comparison of E with C Patients give binary responses, parameters pE, pC

Patients treated taken one at a time Responses are immediate

Patient 1: Allocate E with probability ½

Patient n: If Patient (n – 1) succeeded, allocate the same treatment

If Patient (n – 1) failed, allocate the opposite treatment

6.3 Play-the-winner rules


Example:

Patient Treatment Outcome

1 E S

2 E F

3 C F

4 E S


Randomised play-the-winner Wei and Durham (1978)

Context: Comparison of E with C Patients give binary responses, parameters pE,

pC

Patients treated taken one at a time Responses are immediate

Use an urn: u red balls; u blue balls;


Choose a ball at random:

Red – treat next patient with EBlue – treat next patient with C

Replace the ball

If patient SUCCEEDS add balls of same colour and balls of opposite colour

If patient FAILS add balls of same colour and balls of opposite colour

Repeat ...


> ≥ 0, so that success is rewarded and failure penalised

Wei and Durham show that, if pE ≥ pC

For example, if = 0,

C CE

C E E

p 1 pn1

n p 1 p

CE

C E

1 pn

n 1 p


The RPW can be used with a fixed sample or a sequentialdesign in order to reduce the sample size on the inferiortreatment

Wei and Durham suggest continuing until either

SE + FC = r select E

or

SC + FE = r select C

and explore the probability of correct selection

The RPW can also be used when responses are delayed: thedrawing of balls occurs when treatment assignment is to bemade, the adding of balls when results are received


(a) ECMO

The extracorporeal membrane oxygenation study Bartlett et al. (1985), see also Begg (1990)

ECMO is a treatment for newborns with respiratory failure

E = ECMO, C = conventional therapy, SUCCESS = survival

Historical data suggest that pC 0.2

Design was RPW with u = 1, = 0, = 1

6.4 Applications


Baby treatment outcome #red #blue

0 1 1

1 E survived 2 1

2 C died 3 1

3 E survived 4 1

4 E survived 5 1

5 E survived 6 1

6 E survived 7 1

7 E survived 8 1

8 E survived 9 1

9 E survived 10 1

10 E survived 11 1

11 E survived 12 1

12 E survived 13 1

Results:


An exact analysis gave p = 0.051 (one-sided)

This result was not generally accepted by clinicians

ECMO was studied again in the UK Collaborative ECMO Trialusing a more conventional design

63 out of 93 babies on ECMO survived, 38 out of 92 babies on conventional care survived, p = 0.0005 – trial was stopped early by the DSMB

Elbourne (1994), UK Collaborative Trial Group (1996)

Ep̂ 0.68Cp̂ 0.41


(b) A trial in depressive disorder

Tamura et al. (1994)

Treatments: fluoxetine (E) vs placebo (C)Response: SUCCESS = 50% reduction in HAMD17 in two consecutive visits

Strata: time to REM after sleep onset > or 65 mins

Design: 3 patients in each stratum allocated at random to E and 3 to C, then RPW with u = 1, = 0, = 1


Month 65 mins > 65 mins

#red #blue mE mC #red #blue mE mC

1 1 1 3 3 1 1 3 4

2 4 4 1 3 2 2 2 1

3 4 4 1 0 3 3 1 5

4 6 4 2 6 8 4 2 1

5 8 4 4 2 10 5 6 4

6 10 6 6 5 11 6 4 2

7 16 6 0 1 17 11 3 2

8 21 10 6 2 19 16 2 2

Total 23 22 23 21

Urn composition at the middle of each month and monthly numbers of recruits:


Month 65 mins > 65 mins

#red #blue mE mC #red #blue mE mC

1 1 1 3 3 1 1 3 4

2 4 4 1 3 2 2 2 1

3 4 4 1 0 3 3 1 5

4 6 4 2 6 8 4 2 1

5 8 4 4 2 10 5 6 4

6 10 6 6 5 11 6 4 2

7 16 6 0 1 17 11 3 2

8 21 10 6 2 19 16 2 2

#red > #blue as E is doing better, but the biased allocation failed to be reflected in the actual treatment choices


• For REM < 65 mins

For REM > 65 mins

• Tamura et al. conducted simulations which showed that the failure of actual allocations to reflect the bias in the urn was not unusual

• The trial had various other novel features, including the use of a surrogate response (SUCCESS) for allocation and a delayed response for analysis, and a Bayesian interpretation

E Cˆ ˆp 0.60, p 0.33

E Cˆ ˆp 0.62, p 0.48


Discussion

Applications of response adaptive designs have so far beendisappointing

One problem is that so much is left to chance Instead of biasing the chance of allocation to E to be R/(R + 1),one could use a group sequential approach in which the nextgroup of R + 1 patients must have R on E and 1 on C

See Hu and Rosenberger (2006)

Magirr (2010)

• The RPW rule is myopic – only one patient is randomized at a time

• Alternative: Use random permuted blocks

EEEEEEEEE CCC

EEEEEEEE CCCC

EEEEEE CCCCCC

EEEE CCCCCCCC

EEE CCCCCCCCC


• 12 is a ‘nice’ block size• Total imbalance can be no more extreme than 3:1• Other block sizes/ratios are possible

6.5 Block response-adaptive randomization

Modified RPWR

1. Initial urn composition: 3:3 E:C

2. First 16 patients are allocated equally between E and C

3. When a subsequent block of 12 patients enter study:

i. Fraction of E balls in urn is rounded to closest of 1/4, 1/3, 1/2, 2/3 or 3/4

ii. This fraction of block receive E

4. When a response is observed:

i. Success on E or Failure on C → E ball added to urn

ii. Failure on E or Success on C → C ball added to urn


Decreasing probability of failure


Example: pC = 0.6 pE = 0.9 α = 0.025 1 β = 0.95

Sample size E(failures)

Equal 100 25.0

RPW 107 20.8

Block RPW 100 20.6

MPS/MSc in StatisticsAdaptive & Bayesian - Lect 61 Lecture 6 Response adaptive designs 6.1Unequal...

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