Optimal Crossover Designs and the Problem of Correlated Outcomes

1. Optimal Crossover Designsand the Problem of Correlated Outcomes

2. Goal of clinical trial Measure TREATMENT (DRUG) EFFECTS. We assess the accuracy of the measurements by the VARIANCE (of certain BLUE estimators). A small variance is good. An alternative (reciprocal), more intuitive, formulation of the goal: Collect treatment-effect INFORMATION. A large amount of treatment-effect information is good.

3. Group 1 (n1 subjects) Group 2 (n2 subjects) Treatment Sequence 1 Treatment Sequence 2 Period 1 A B Period 2 B A Alice Andy Arnie Arthur Amy Arlene Betty Bob (A) 10 10 10 9 9 10 (B) 9 10 (B) 9 8 9 8 8 9 (A) 10 8 The Classical Crossover Design Classical Crossover Design with data The General Classical Crossover Design

4. Crossover designs with two or more periods:The Big Advantage and the Problems . (1) (  The period effect. (2)The carryover effect. Correlated data. BIG ADVANTAGE Each subject becomes his own control. PROBLEMS The period effect (a small problem). The carryover effect (a CONTROVERSIAL issue). Correlated data (aBIG, BIG problem).

5. Response Models A RESPONSE MODEL gives the probability of a given subject’s response. This probability depends on five unknown things:  Period Effects, Subject Effects, Carryover Effects, Treatment Effects + the COVARIANCE MATRIX. The Covariance Matrix controls the correlation (or dependence) of a subject’s observations.  Two different models: The Random Subjects Effects Model The Fixed Subjects Effects Model Optimality and Response Models. Optimality and the Carryover Controversy.

6. n1 n2 1 7 1.14 2 6 .67 3 5 .53 4 4 .5 An easy optimality computationfor the classical crossover design. Find the n1 and n2, such that n1+n2=8, that minimizes the variance of the BLUE estimator of the difference of treatment effects.

7. What is an optimal design? An optimal design is the very best design! It has the most accurate treatment-effect estimators. It has the most treatment-effect information.

8. n1 n1 n2 n2 n3 n3 n4 n4 n5 n6 n7 n8 n9 A A A A B A B B B B C C C A A B B A C B A B C A B C Extended Crossover Designs: more than 2 treatment sequences Extended Crossover Designfor 2 treatments Extended Crossover Design for 3 treatments

9. n1 n2 n3 n4 n5 n6 n7 n8 A B A B A B A B A B A B B A B A A B B A A B B A For three or more periods, optimality depends on the Covariance Matrix Repeated Measurements Design – 2 treatments, 3 periods)

10. n1 n1 n2 n2 n3 n3 n4 n4 A B A B A B A B A B A B B A B A A B B A A B B A Dual Balanced Design, p=3,t=2 Total n = 2(n1+n2+n3+n4)

11. n1 n4 n1 n4 n1 n4 n4 n2 n4 n2 n4 n2 n2 n5 n5 n2 n5 n2 n5 n3 n5 n3 n5 n3 n3 n3 n3 A A B A B C B A C A C B B A A C B C B A C A C B B C C A B C B C A C A A A B B B B C C C A C B A C B A C A B B A B C C A C B A C B A C C A B C B A A B A A B B C C Symmetric Design, p=3,t=3 Total n = 3 n1+6(n2+n3+n4+n5)

12. Design Efficiency Design Efficiency = Efficiency measureshow good the design is – compared to the optimal design. If efficiency = 0, the design is the worst possible. If efficiency = 1, the design is optimal – the best possible.

13. Design n1 n2 n3 n4 WORST EFFICIENCY 3.4.1 (J&K) 0 0 3 3 0 3.4.2 (J&K) 0 3 0 3 .495 3.4.3 (J&K) 0 3 3 0 .125 3.6.1 (J&K) 0 2 2 2 .417 Dual Balanced Designs and their Worst (?) Efficiencies p=3,t=2,n=12

14. Worst (?) efficiencies of four symmetric designs p=3,t=3,n=36 Design n1 n2 n3 n4 n5 WORST EFFICIENCY PL30(EP) 0 0 0 6 0 0 BAL(EP) 4 0 0 4 0 0 KM 0 0 0 0 6 .444 MY DESIGN 0 0 0 1 5 .417 Symmetric designs and their worst (?) efficiencies p=3, t=3, n=36

15. Linear Optimality Equations Linear Equations for approximately optimal dual-balanced designs (p=3, t=3): n1=n2=n3=0 0=r4n4 +r5n5 n=n4+n5

16. Strategies for finding optimal designs for correlated outcomes (a) Assume covariance matrix is of a special form. (b) Competing designs is not the full set of designs. (c) The “efficiency-optimal design” approach. (d) The response-adaptive design approach.

17. 5 5 5 5 5 5 1 1 1 1 1 1 A B B A B C A B C A B C C A B B C A C A B B C A B C A C A B C A B B A A My Design UniversallyOptimal Repeated Measurements Design, p=3, t=3, n=36

18. A A B B C C B C A C A B C B C A B A The Kunert-Martin “optimal” design 3 treatments, 3 periods, NO REPEATED TREATMENTS

19. n/4 n/4 n/4 n/4 A B A B A B B A B A B A Matthews’“efficiency-optimal” design for autoregressive covariance matrices. Two treatments, three periods

20. Responsive Adaptive Designs In the usual clinical trial, an experimental subject is allocated, using a randomization procedure,to his/her treatment sequence. In a response-adaptive clinical trial, a new subject is allocated, using the responses of the previous subjects,to his/her treatment sequence.

21. Adaptive Allocation Rule for Symmetric Designs Assign all the new subjects to the i-th symmetric block for which the i-th quadratic, evaluated at theta, is maximum. The coefficients of the quadratic function corresponding to the i-th symmetric block depend on V, the covariance matrix. theta= the x for which the sum of ni*(the i-th quadratic(x)) is minimized. theta depends on V.

22. Efficiency vs. number of subjects in adaptive dual-balanced designs obtained by the adaptive allocation rule in 1000 simulated experiments. p=3, t=2, V is the bad covariance matrix.

23. Efficiency vs. # of subjects in dual balanced designs obtained by the adaptive allocation rule in 1000 simulated experimentsp=3,t=2, V autoregressive.

24. Efficiency vs. # of subjects in adaptive symmetric designs obtained by the adaptive allocation rule in 1000 simulated experiments.p=3,t=3, V autoregressive.

25. The General Adaptive Allocation Rule Given design d, assign all new subjects to a treatment sequence t that maximizes Tr=Trace of matrix the treatment effects information matrix of design d

26. A-,D- and E-efficiencies vs # of subjects in adaptive designs obtained by the general adaptive allocation rule in 1000 simulated experiments. p=3, t=3, V=I.

27. The sample (N=24 and 192) distribution function of -2Ln[LR] and ChiSquare(2) p=3, t=3, V autoregressive with rho=-.05