1. Plenary Data Analysis Session PSI Conference Bristol 2006 John Matthews School of Mathematics and Statistics University of Newcastle upon Tyne

2. Five Period Crossover volunteer study  Active treatments, A to F – six doses of a new compound  Two control treatments, c 1, c 2, namely a positive control S - a standard treatment already on the market a negative control P - a placebo with zero dose of the active compound  Alternating study, two cohorts, each of 10 volunteers

3.  A to F are increasing doses d j in g  Exact doses not taken into account in analysis but doses must escalate  Contrasts d j - c i of equal and primary interest: i = 1,2 and j = 1,…,6 ABCDEF 103060150250400

4.  Within each period, data (FEV 1 ) collected at baseline and at six times post administration  Main interest is in response at 12 hours  One volunteer withdrew after three periods (missed P and F), otherwise data complete  Substantial washout – no carryover, ?period effect?

5. Quick and thoughtless analysis  Fixed subjects effects  Period effect  No period to period baseline

6. ANOVA table Df Sum Sq Mean Sq F value Pr(>F) factor(subject) 19 50.672 2.667 115.4127 <2.e-16 factor(period) 4 0.120 0.030 1.2969 0.28017 factor(Rx) 7 0.467 0.067 2.8880 0.01067 * Residuals 67 1.548 0.023  No strong evidence of period effect  Some treatment effect – largely because of difference between +ve and –ve controls

7. Mean effects Dose Mean Difference (l) from negative control (baseline as covariate) SEP 10-0.087-0.0770.0690.0620.210.22 30-0.052-0.0450.0700.0630.460.48 60-0.039-0.0090.0670.0610.570.88 150-0.046-0.0340.0680.0610.500.58 250-0.029 0.0700.0630.680.65 400-0.043-0.0580.0740.0670.560.39 S0.1250.1580.0490.0440.0130.0008

8. Design V’rs Cohort 1 V’rs Cohort 2 1,2 PASCE 11,12 PBSDF 3,4 SACEP 13,14 SBDFP 5,6 ACPES 15, 16 BDPFS 7,8 APCSE 17, 18 BPDSF 9.10 ASCPE 19,20 BSDPF

9. Can a better design be found? 1. Need to establish criteria for how good a design is 2. Not all aspects are numerical 3. Practical constraints 4. Statistical criteria

10. Practical constraints  Doses must escalate  Don’t start too high  Dose increments should not be too large  Two cohorts - cohort 2 investigated while cohort 1 rests. Study to finish in 3/12

11. Statistical criteria  Model can be written as  Hence where

12. Variance of contrasts  Want to consider variance of estimate of   This might be thought to be C -1 but C is singular  Therefore use g-inverse C -  C - is not unique but  If A is a c  t matrix of contrasts of interest then dispersion of contrasts, AC - A, is well defined (need rank(C)=7=t-1 if all contrasts to be estimable)

13. Contrast matrix A SPABCDEF 1000000 100 0000  1000000 01 00000 010 0000  0100000

14. Statistical Criterion  If all else is equal, we prefer a design with lower mean variance for a contrast of interest, i.e. minimise trace( AC - A)  Might want to minimise max{( AC - A) ii } but this is not pursued here

15. Practical improvements to design  Split doses into {A,C,E} and {B,D,F} to permit alternating design  Also ensures that dose increments are not large  Given doses must escalate there is little room for use of different designs  Flexibility about when controls given  Once these are chosen, sequences are defined

16. Design controls Cohort 1Cohort 2 PSPS SPSP PSPS PSPS SPSP

17. Control disposition  Same pattern in two cohorts  P before S in 12 volunteers  trace( AC - A)=2.429  There are 5 C 2 = 10 different unordered pairs of places in a sequence  Allowing for order there are twenty possible sequences for each cohort

18. Possible control sequences PS PS PS PS PS PS PS PS PS PS  Type ‘PS’ sequences  Further 10 sequences with S preceding P, type ‘SP’ sequences  Fill in gaps with either {A,C,E} or {B,D,F}  Allocate {A,C,E} to 10 sequences and {B,D,F} to remainder – allows alternation and close to balance on volunteers

19. Allocation method 1 PS PS PS PS PS PS PS PS PS PS  40 possible sequences  10 with {A,C,E} in sequences shown  10 with {A,C,E} and Type ‘SP’ sequences  20 as above but with {B,D,F} not {A,C,E}  Choose random 20 from these 40. Perhaps search for a ‘good’ set

20. Method 1  For original design trace( AC - A)=2.429  Method 1 ensures no particular degree of balance  Optimal row-column designs are uniform on periods and subjects, i.e. each treatment appears equally often on each volunteer and in each period  Cannot achieve this but can we get ‘close’?  If we achieve a certain balance on volunteers, ‘closeness’ can be measured by treatment by period incidence matrix

21. Example of a Treatment Period Incidence Matrix PSABCDEF 44aa0000 44bbcc00 44ddeeff 4400ccbb 440000aa

22. Treatment  Period Incidence Matrix for original design PSABCDEF 44660000 44442200 44006600 44002244 44000066

23. Allocation method 2 PS PS PS PS PS PS PS PS PS PS  Allocate {A,C,E} to 5 randomly chosen ‘PS’ sequences

24. Allocation method 2 PSACE PASCE PS PACES APSCE PS PS PS ACPES PS  Allocate {A,C,E} to 5 randomly chosen ‘PS’ sequences

25. Allocation method 2 PSACE PASCE PS PACES APSCE PS PS PS ACPES PS  Allocate {A,C,E} to 5 randomly chosen ‘PS’ sequences  Allocate {B,D,F} other 5 ‘PS’ sequences

26. Allocation method 2 PSACE PASCE PBDSF PACES APSCE BPDSF BPDFS BDPSF ACPES BDFPS  Allocate {A,C,E} to 5 randomly chosen ‘PS’ sequences  Allocate {B,D,F} other 5 ‘PS’ sequences

27. Allocation method 2 PSACE PASCE PBDSF PACES APSCE BPDSF BPDFS BDPSF ACPES BDFPS  Allocate {A,C,E} to 5 randomly chosen ‘PS’ sequences  Allocate {B,D,F} other 5 ‘PS’ sequences  This gives full replication of ‘PS’ sequences  Allocate {A,C,E} to the 5 ‘SP’ sequences analogous to the ‘PS’ sequences just allocated to {B,D,F}  Allocate {B,D,F} to remaining ‘SP’ sequences  Gives balance over periods of two sets of doses

28. Treatment  Period Incidence Matrix for all method 2 designs PSABCDEF 44660000 44333300 44114411 44003333 44000066

29. Method 2 results  trace =2.3168  Variances of contrasts versus P given right (same as versus S) Original design Method 2Ratio A0.20680.19771.05 B0.20680.19771.05 C0.19380.18371.05 D0.19380.18371.05 E0.20680.19771.05 F0.20680.19771.05

30. Further method  Method 2 imposes balance but does not allow duplication of sequences  May be merit in allowing this

31. Allocation method 3 APCES   Choose a ‘PS’ sequence at random and allocate {A,C,E}

32. Allocation method 3 APCES  BSDFP   Choose a ‘PS’ sequence at random and allocate {A,C,E}  Allocate {B,D,F} to corresponding ‘SP’ sequence

33. Allocation method 3 APCES  BSDFP  SACPE   Choose a ‘PS’ sequence at random and allocate {A,C,E}  Allocate {B,D,F} to corresponding ‘SP’ sequence  Allocate {A,C,E} to ‘reverse’ ‘SP’ sequence

34. Allocation method 3 APCES  BSDFP  SACPE  PBDSF   Choose a ‘PS’ sequence at random and allocate {A,C,E}  Allocate {B,D,F} to corresponding ‘SP’ sequence  Allocate {A,C,E} ‘reverse’ ‘SP’ sequence  and {B,D,F} to the analogous ‘PS’ sequence

35. Allocation method 3 APCES  BSDFP  SACPE  PBDSF   Choose a ‘PS’ sequence at random and allocate {A,C,E}  Allocate {B,D,F} to corresponding ‘SP’ sequence  Allocate {A,C,E} ‘reverse’ ‘SP’ sequence  and {B,D,F} to the analogous ‘PS’ sequence  This allocates 4 volunteers – repeat a further 4 times, sampling with replacement at first step

36. Allocation method 3: chosen design ACEPS 33 PACSE 11 ACPES 11 plus other sequences as in method 3  trace =2.262

37. Treatment  Period Incidence Matrix for method 3 design PSABCDEF 55550000 44224400 22332233 44004422 55000055

38. Method 3 results  trace =2.262  Variances of contrasts versus P given right (same as versus S) Original design Method 3Ratio A0.20680.18851.10 B0.20680.18851.10 C0.19380.18831.03 D0.19380.18831.03 E0.20680.18851.10 F0.20680.18851.10

39. Conclusions  Little room for manoeuvre in design of dose-escalating studies  Positioning of controls is about limit  Nevertheless worth doing – proposed change equivalent to 10% reduction in variance at no cost  Work to be done to extend existing work on comparison with multiple controls to allow for other constraints