1. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 61 Lecture 6 Response adaptive designs 6.1Unequal treatment allocation 6.2Varying the allocation ratio 6.3Play-the-winner rules 6.4Applications 6.5 Block response-adaptive randomization

2. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 62 6.1 Unequal treatment allocation Suppose that treatment allocation is planned to be unequal, in an R:1 ratio to E:C then n E = Rn C so that n = n C + n E = n C (R + 1) Hence For unequal sized samples, the statistic Z of Lecture 2.5 is generalised to:

3. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 63 Hence so that For equal sized samples, we had:

4. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 64 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

5. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 65 Example: Normally distributed data  = 0.025 (one-sided)  z 1  = 1.96 1  = 0.9  z 1  = 1.282  = 1.8  R = 0.5 For (1:1) allocation Recruit 273 patients per treatment group

6. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 66 For other allocation ratios: R(R+1) 2 /4Rn 1 1544.87 21.125612.98 31.333721.95 41.563858.17 51.800980.76 The total number of patients increases as the allocation ratio increases The optimum value of R is 1

7. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 67 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

8. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 68 In group sequential trials, the test statistics B and V are chosen so that  B ~ N(  V, V)  increments (B i – B i  1 ) between interims are independent Essentially, we condition on the ancillary statistics V 1, V 2,... 0 V 1 V 2 V 3 V 4... then B 1, B 2,... will form a “Brownian motion” with drift  6.2 Varying the allocation ratio

9. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 69 Varying the allocation ratio n E :n C will effect the values of the V i but not the Brownian motion property, nor its drift, as (B i  B i  1 ) ~ N(  (V i  V i  1 ), V i  V i  1 ) 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 B i Robbins and Siegmund (1974) – normal case Robbins (1974) – binary case

10. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 610 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

11. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 611 Robbins (1952), Zelen (1969) Context: Comparison of E with C Patients give binary responses, parameters p E, p C 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

12. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 612 Example: PatientTreatmentOutcome 1ES 2EF 3CF 4ES

13. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 613 Randomised play-the-winner Wei and Durham (1978) Context: Comparison of E with C Patients give binary responses, parameters p E, p C Patients treated taken one at a time Responses are immediate Use an urn: u red balls; u blue balls;

14. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 614 Choose a ball at random: Red – treat next patient with E Blue – 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...

15. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 615  >  ≥ 0, so that success is rewarded and failure penalised Wei and Durham show that, if p E ≥ p C For example, if  = 0,

16. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 616 The RPW can be used with a fixed sample or a sequential design in order to reduce the sample size on the inferior treatment Wei and Durham suggest continuing until either S E + F C = r  select E or S C + F E = r  select C and explore the probability of correct selection The RPW can also be used when responses are delayed: the drawing of balls occurs when treatment assignment is to be made, the adding of balls when results are received

17. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 617 (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 p C  0.2 Design was RPW with u = 1,  = 0,  = 1 6.4 Applications

18. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 618 Babytreatmentoutcome#red#blue 011 1Esurvived21 2Cdied31 3Esurvived41 4E 51 5E 61 6E 71 7E 81 8E 91 9E 101 Esurvived111 Esurvived121 Esurvived131 Results:

19. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 619 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 Trial using 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)

20. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 620 (b)A trial in depressive disorder  Tamura et al. (1994) Treatments:fluoxetine (E) vs placebo (C) Response: SUCCESS = 50% reduction in HAMD 17 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

21. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 621 Month  65 mins > 65 mins #red#bluemEmE mCmC #red#bluemEmE mCmC 111331134 244132221 344103315 464268421 5844210564 6 66511642 716601171132 8211062191622 Total23222321 Urn composition at the middle of each month and monthly numbers of recruits:

22. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 622 Month  65 mins > 65 mins #red#bluemEmE mCmC #red#bluemEmE mCmC 111331134 244132221 344103315 464268421 5844210564 6 66511642 716601171132 8211062191622 #red > #blue as E is doing better, but the biased allocation failed to be reflected in the actual treatment choices

23. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 623 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

24. MPS/MSc in StatisticsAdaptive & Bayesian - Lect 624 Discussion Applications of response adaptive designs have so far been disappointing 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 next group of R + 1 patients must have R on E and 1 on C See Hu and Rosenberger (2006)

25. 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 MPS/MSc in StatisticsAdaptive & Bayesian - Lect 625 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

26. 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 MPS/MSc in StatisticsAdaptive & Bayesian - Lect 626

27. Decreasing probability of failure MPS/MSc in StatisticsAdaptive & Bayesian - Lect 627 Example: p C = 0.6 p E = 0.9 α = 0.025 1  β = 0.95 Sample sizeE(failures) Equal10025.0 RPW10720.8 Block RPW10020.6