## Example Balaams design

This is a well-known example initially described by Hunter et al. (1970). The aim was to determine the effect of amantadine (treatment A) on subjects suffering from

Group |
Subject |
Baseline |
Period 1 |
Period 2 |

1AA |
1 |
14 |
12.50 |
14.00 |

2 |
27 |
24.25 |
22.50 | |

3 |
19 |
17.25 |
16.25 | |

4 |
30 |
28.25 |
29.75 | |

Mean |
22.50 |
20.56 |
20.63 | |

2BB |
1 |
21 |
20.00 |
19.51 |

2 |
11 |
10.50 |
10.00 | |

3 |
20 |
19.50 |
20.75 | |

4 |
25 |
22.50 |
23.50 | |

Mean |
19.25 |
18.13 |
18.44 | |

3AB |
1 |
9 |
8.75 |
8.75 |

2 |
12 |
10.50 |
9.75 | |

3 |
17 |
15.00 |
18.50 | |

4 |
21 |
21.00 |
21.50 | |

Mean |
14.75 |
13.81 |
14.63 | |

4BA |
1 |
23 |
22.00 |
18.00 |

2 |
15 |
15.00 |
13.00 | |

3 |
13 |
14.00 |
13.75 | |

4 |
24 |
22.75 |
21.50 | |

5 |
18 |
17.75 |
16.75 | |

Mean |
18.60 |
18.30 |
16.60 |

Parkinsonism. The trial was placebo controlled (treatment B). After a run-in period of one week during which baseline information was recorded, there were two four-weekly treatment periods, without a washout period. Weekly scores (0-4) were recorded for each of 11 physical signs and the data presented in Table 7.7 give the weekly average total scores in each treatment period. Seventeen patients were randomised, and the data have no missing values.

Table 7.8 presents the results of analyses with and without inclusion of a carry-over term, and with patient effects fitted as fixed or random.

An immediate point to note from the two mixed models is the very high patient variance component compared with the residual variance component (Table 7.8). This immediately suggests that little gain in efficiency will accrue from between-patient information. This is confirmed by comparison of the treatment standard errors, where even in the model where carry-over is fitted the reduction is only 4%. In most situations, where the between-patient variation is less extreme, the existence of the AA and BB treatment groups would lead us to expect greater benefits from the mixed models approach.

In this study there is no evidence of any carry-over effect, and most statisticians would choose to report the model which excludes carry-over. However, having

Table 7.8 Estimates of variance components and treatment effects. Standard errors of estimates appear in brackets.

Fixed patients

Random patients

Ignoring carry-over Variance components Patients

Residual (within patients) Treatment difference

Including carry-over Variance components Patients

Residual (within patients) Treatment difference Carry-over difference

1.05

1.12

chosen a design for its optimal properties in estimating both treatment and carry-over effects, there is a strong case for reporting the fuller model.

The presentation here has been restricted to the analysis of the results in the two treatment periods, and the fact that baseline observations were also recorded has been ignored. Jones and Kenward (1989) present additional analyses utilising this baseline data, and in particular use interactions with the baseline to test for an effect of the baseline level on the treatment effect, period effect and carry-over effect. Although none of these terms was statistically significant at the 10% level of significance, they found indications that the treatment differences were higher with greater baseline levels. These authors also handled carry-over in a more involved way than we have employed in our analyses. They allowed for the possibility that carry-over would be different in those on the AA or BB sequence from those on the AB or BA sequence, but this term in the analysis of variance was clearly non-significant.

The conclusions from the trial will, in this instance, be qualitatively similar whichever of the previously described analytical methods is used, as long as carryover is ignored in estimating treatment effects. Amantadine produces a reduction in the physical signs of Parkinson's disease which is statistically significant at the 5% level. Note, however, that inclusion of carry-over terms in the model produces a substantial increase in the standard error of the treatment effect, leading to non-significance of the treatment effect.

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