On the Analysis of Crossover Designs Dallas E. Johnson Professor Emeritus Kansas State University

1. On the Analysis of Crossover Designs Dallas E. Johnson Professor Emeritus Kansas State University

2. dejohnsn@ksu.edu 785-532-0510 (Office) 785-539-0137 (Home) Dallas E. Johnson 1812 Denholm Dr. Manhattan, KS 66503-2210

15. Note that and

18. To answer these kinds of questions, Shanga simulated two period/two treatment crossover experiments satisfying four different conditions: (1) no carryover and equal variances (C0V0), (2) no carryover and unequal variances(C0V1), (3) carryover and equal variances (C1V0), and (4) carryover and unequal variances (C1V1).

19. Each of 1000 sets of data under each of these conditions was analyzed four different ways assuming: (1) no carryover and equal variances (C0V0), (2) no carryover and unequal variances(C0V1), (3) carryover and equal variances (C1V0), and (4) carryover and unequal variances (C1V1).

20. PROCMIXED; TITLE2'EQUAL VARIANCES'; CLASSES SEQ PERIOD TRT PERSON; MODEL PEF=SEQ TRT PERIOD/DDFM=SATTERTH; REPEATED TRT/SUBJECT=PERSON(SEQ) TYPE=CS; LSMEANS TRT /PDIFF; RUN; PROCMIXED; TITLE2'UNEQUAL VARIANCES'; CLASSES SEQ PERIOD TRT PERSON; MODEL PEF=SEQ TRT PERIOD/DDFM=SATTERTH; REPEATED TRT/SUBJECT=PERSON(SEQ) TYPE=CSH; LSMEANS TRT /PDIFF; RUN;

21. Tests for equal treatment effects.

22. Tests for equal treatment effects.

23. NOTE: Failing to assume carryover when carryover exists invalidates the tests for equal treatment effects and the invalidation generally gets worse as the

32. Goad and Johnson (2000) showed: (1) If  satisfies the H-F conditions, then the traditional tests for treatment and period effects are valid for all crossover experiments both with and without carryover.

33. (2) There are cases where the ANOVA tests are valid even when  does not satisfy the H-F conditions. (a) In the no carryover case, tests for equal treatment effects are valid for the six sequence three period/three treatment crossover design when there are an equal number of subjects assigned to each sequence. (b) In the no carryover case, tests for equal period effects are valid only when the H-F conditions be satisfied

34. (b) The traditional tests for equal treatment effects and equal period effects are valid for a crossover design generated by t-1 mutually orthogonal tt Latin squares when there are equal numbers of subjects assigned to each sequence. (c) The traditional tests for equal treatment effects, equal period effects, and equal carryover effects are likely to be invalid in the four period/four treatment design regardless of whether carryover exists or not.

35. Cases where the validity of ANOVA tests are still in doubt. (4) When carryover exists, the tests for equal carryover effects are not valid unless  satisfies the H-F conditions. (5) When there are unequal numbers of subjects assigned to each sequence, the ANOVA tests are unlikely to be valid unless  satisfies the H-F conditions.

36. Goad and Johnson (2000) provide some alternative analyses for crossover experiments. Consider again, the three period/three treatment crossover design in six sequences.

37. Question: Suppose the variance of a response depends on the treatment, but that the correlation is the same between all pairs of sequence cells. That is, for Sequence 1, the covariance matrix is:

38. Shanga simulated three period/three treatment crossover experiments satisfying four different conditions: (1) no carryover and equal variances (C0V0), (2) no carryover and unequal variances(C0V1), (3) carryover and equal variances (C1V0), and (4) carryover and unequal variances (C1V1).

39. Each of 1000 sets of data under each of these conditions was analyzed four different ways assuming: (1) no carryover and equal variances (C0V0), (2) no carryover and unequal variances(C0V1), (3) carryover and equal variances (C1V0), and (4) carryover and unequal variances (C1V1).

40. TITLE1'CRSOVR EXAMPLE - A THREE PERIOD/THREE TRT DESIGN'; TITLE2'ASSUMES CARRYOVER AND UNEQUAL VARIANCES'; PROCMIXED; CLASSES SEQ PER TRT PRIORTRT SUBJ; MODEL Y = SEQ TRT PER PRIORTRT/DDFM=KR; LSMEANS TRT PER PRIORTRT/PDIFF; REPEATED TRT/SUBJECT=SUBJ TYPE=CSH; RUN;

41. Tests for equal treatment effects.

42. Tests for equal treatment effects.

43. Tests for equal treatment effects.

44. Tests for Carryover

45. Tests for Carryover

46. In the three treatment/three period/six sequence crossover design, Shanga also considered testing Shanga claimed that his tests were LRTs, but Jung (2008) has shown that they are not LRTs. Nevertheless, Shanga's tests had good power for detecting unequal variances.

47. That’s All For Now!