Recall the crossover design set up as a Latin rectangle: Sequence=Subject A B C A B C 3 C A B B C A

Transcription

1 D. More on Crossover Designs: # periods = # trts Recall the crossover design set up as a Latin rectangle: Period Sequence=Subject A B C A B C 2 B C A C A B 3 C A B B C A With one subject per sequence, each takes the treatments in a unique order. 636

2 Usually, though, multiple subjects are assigned to each sequence: Period Sequence A B C A B C 2 B C A C A B 3 C A B B C A so we have nesting: subjects(sequence). 637

3 Now this looks like a pseudo-split plot design. How? whole plot unit: whole plot factor: split plot unit: split plot factor: Why a pseudo split plot? 638

4 What is labeled treatment in our split-plot analysis is actually a mixture of treatment effects carry-over effects As for other Latin square designs, usually any interactions are ignored, e.g. we assume the treatment effects are the same for all sequences and for all periods. 639

5 Crossover designs are built to avoid the confounding of which two effects? We have previously ignored the possibility of carryover effects (assumed then to be 0), but now we will estimate them. When is a design balanced for first-order carryover effects? 640

6 These are called first-order carryover effects. Is the three period design shown earlier balanced for second-order carryover effects? 641

7 Model: Y ijk = µ + α i + b ij + γ k + τ d(i,k) + λ c(i,k 1) + e ijk i = 1,..., n orders or patterns or sequences j = 1,..., r subjects per pattern k = 1,..., p periods d = 1,..., t treatments c = 1,..., t treatment carry-over 642

8 α i = b ij iid N(0, σb 2) γ k = τ d(i,k) = λ c(i,k 1) = e ijk iid N(0, σe 2) We define λ c (i,0) = 0 i (k = 1). Effects can be fixed or random as needed, and we assume random effects are independent of each other. 643

9 Let s consider some expected values to help make sense of these terms. First Pattern = A B C i = 1 E[Y 1j1 ] = µ + α 1 + γ 1 + τ 1 k = 1 E[Y 1j2 ] = k = 2 E[Y 1j3 ] = k = 3 644

10 Second Pattern = B C A i = 2 E[Y 2j1 ] = k = 1 E[Y 2j2 ] = k = 2 E[Y 2j3 ] = k = 3 and so on. 645

11 V ar[y ijk ] = Cov[Y ijk, Y ijk ] = 646

12 If we are willing to assume there are no carryover effects, then we can still use the Latin square or rectangle models and ANOVA tables. If we are not willing, then we must use this model and its ANOVA tables. 647

13 Source df Whole plot = subject (Between subjects) Sequence n 1 Subject(sequence) n(r 1) Split plot = period (Within subjects) Period p 1 Treatment t 1 Carryover t 1 Error (nr 1)(p 1) 2(t 1) Total nrp 1 648

14 What term is missing compared to a split plot analysis? We are taking that term s SS, removing a treatment SS and a carryover SS, and putting the rest into Error SS. The treatment and carryover effects are not orthogonal to each other; they do not occur in all possible combinations. What is an example? 649

15 Thus, treatment effect estimates must be adjusted for carryover effects. Likewise, carryover effects must be adjusted for treatment effects. Tests: Tests are constructed using the General Linear F-test approach. There are no easy sums of squares formulas to write down in an ANOVA table. It is generally recommended that a non-significant carryover term not be dropped. 650

16 Tests for period differences and pattern differences are not usually carried out, since they are not of interest. Some authors believe that the appropriate tests for these two effects come from Type I sums of squares, rather than the General Linear F-test - Type III approach. If the Type III approach is preferred for period, then a p 1 df contrast must be used to create the test. Period is confounded partly with carryover, so a Type III test which fully adjusts for carryover will not use a sum of squares for period that is big enough. 651

17 Estimation: Least square means must be used. Diagnostics: As before 652

18 When should crossover designs be used? 653

19 How do we choose a crossover design? Not all Latin squares or rectangles are balanced for first order carryover effects, even though they are balanced for treatment effects. If t = # treatments is an even number, then one Latin square is sufficient. If t = # treatments is an odd number, then two carefully chosen Latin squares are needed. 654

20 EX: t = 5 Period Pattern A B C D E A B C D E 2 B C D E A C D E A B 3 D E A B C B C D E A 4 E A B C D E A B C D 5 C D E A B D E A B C 655

21 Crossover designs which are balanced for all orders of carryover effects can be done by using t 1 orthogonal Latin squares. Recall the design for t = 3: Period Pattern A B C A B C 2 B C A C A B 3 C A B B C A Now superimpose the two squares on top of each A B C other: B C A C A B 656

22 A in square 1 occurs once each with,, and in square 2. B in square 1 occurs once each with,, and in square 2. C in square 1 occurs once each with,, and in square

23 A complete orthogonal set of Latin squares requires t(t 1) subjects at least. t(t 1) allows for one subject per pattern. Unless second-order or higher order carryover effects are thought to be a serious problem, these designs are not used. 658

24 Randomization: Crossover designs are not randomized in the way Latin square designs are randomized, even when the design is based on a Latin square or rectangle. How is a Latin square or rectangle randomized? How should we randomize for a crossover design? 659

25 E. More on Crossover Designs: p t Extra Period Designs: p = t + 1 We saw then when p = t, the treatment and carryover effects were not orthogonal. A consequence of this is that the carryover effects can t be estimated with as much precision as the treatment effects. 660

26 When p = t, we can never observe the carryover effect of treatment A on treatment A. We can only observe the carryover effect of treatment A on treatments B, C, etc. By adding one extra period to the design, we can estimate the carryover effect of each treatment on itself. 661

27 Period Pattern A B C A B C 2 B C A C A B 3 C A B B C A 4 Each carryover effect appears an equal number of times in each pattern. Each carryover effect appears an equal number of times with each treatment. 662

28 Now carryover and treatment effects are orthogonal. However, the design is no longer balanced for treatments within pattern. Such designs should be used only if strong first-order carry-over effects are expected. Another disadvantage is that blinding of subjects or researchers to treatments may be broken during the last period if they know that the last period has the same treatment as the preceeding. 663

29 Fewer period designs: p < t Designs where each subject only receives a subset of the treatments are particularly attractive when dropout is expected or when the periods are of long duration or when lengthy washout periods are needed. Subjects are now like incomplete blocks, not complete blocks. Such designs can be derived by deleting > 1 periods from a complete set of orthogonal Latin squares or by a Youden square. 664

30 Such designs are not attractive when it is thought that there are interactions between treatments and periods, or between treatments and patterns. In such a case, you want your design to contain all combinations of treatments following other treatments. 665

31 F. Crossover Designs when t = 2 Two-period two-treatment crossover designs are very common, but have one drawback: carryover effects are confounded with other study effects. They cannot be tested from our earlier model: Y ijk = µ + α i + b ij + γ k + τ d + λ c + e ijk Period Pattern A B 2 B A 666

32 Consider expected values: E[Y 1j1 ] = µ + α 1 + γ 1 + τ 1 E[Y 1j2 ] = E[Y 2j1 ] = E[Y 2j2 ] = Now we can impose the usual fixed effects constraints: 2 i=1 = 0 2 k=1 γ k = 0 2 d=1 τ d = 0 2 c=1 λ c = 0 667

33 What do we get? E[Y 1j1 ] = E[Y 1j2 ] = E[Y 2j1 ] = E[Y 2j2 ] = How many parameters are there in total? How many Ŷ i k are there? 668

34 So we have a problem. Some effect must be assumed zero and dropped in order to fit a model: period effect? pattern effect? carryover effect? 669

35 Switchback Design: Period Pattern A B 2 B A 3 A B which enables estimation of all effects. 670

36 Extra Period Design: Period Pattern A B 2 B A 3 B A which also enables estimation of all effects. 671

37 The extra period design is strongly preferred over the switchback design. The extra period design results in orthogonal treatment and carryover effects. The switchback design results in a high degree of correlation between the estimates of τ d and λ c. The extra period design results in more efficient treatment and carryover effect estimates. 672

38 More details on all crossover designs can be found in Jones and Kenward (1989) Design and Analysis of Crossover Trials. 673