We consider the four-period, four-treatment, cross-over trial described by Jones and Kenward (1989). Three drugs, A, C and D, and a placebo B were compared to assess their effect on cardiac output, measured by the left ventricular ejection time (LVET). Each treatment was given for one week, with a one-week washout period between treatments. Observations were made at the end of each treatment period. Fourteen patients were used in the trial, yielding 56 observations. To demonstrate the use of the mixed models approach, we have arbitrarily set 13 of

Table 7.4 Estimates of variance components and treatment effects. Standard errors of estimates appear in brackets.

Fixed patients Random patients

Patients |
- |
2721 |

Residual |
1667 |
1657 |

Treatment differences | ||

A - B |
77.4(20.1) |
72.5(19.9) |

A - C |
36.8(17.3) |
32.8(17.1) |

A - D |
77.3(19.5) |
74.6(19.4) |

B-C |
-40.6(19.9) |
-39.7(19.5) |

B-D |
-0.1(21.2) |
2.1(21.0) |

C-D |
40.4(18.7) |
41.8(18.5) |

Including carry-over | ||

Variance components | ||

Patients |
- |
2750 |

Residual |
1840 |
1831 |

Treatment differences | ||

A-B |
76.0(22.6) |
69.1(22.2) |

A-C |
40.5(20.4) |
32.1(20.0) |

A-D |
84.9(22.5) |
79.0(22.3) |

B-C |
-35.4(23.0) |
-37.0(22.4) |

B-D |
8.9(25.5) |
9.9(25.0) |

C-D |
44.3(23.0) |
46.9(22.5) |

the 5 6 observations to be missing. The results of four analyses are presented in Table 7.4, from the combinations of inclusion or exclusion of carry-over effects, and handling the patient effects as fixed or random.

All pairwise comparisons of the carry-over effects were non-significant and will not be considered further, or the details presented. We see that between-patient variation is moderate, being around 50% higher than the residual variance component. The random effects analysis without carry-over effects produces an average 1% reduction in the standard errors of the paired treatment comparisons compared with the fixed effects model, a modest but worthwhile gain. Comparing these estimates with the analyses in which carry-over was also fitted, two points are clear. First, the standard errors of the mean treatment differences are larger when carry-over terms are present, irrespective of whether a fixed effects model is fitted, or whether the patient term is regarded as random. Secondly, the reduction in the standard error of the mean treatment difference by fitting patient effects as random is larger when carry-over terms are also fitted. This latter result is general, and we will see more dramatic differences in later examples.

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