## Example ABBA crossover design

We illustrate the AB/BA cross-over design with results from an unpublished study comparing two diuretics in the treatment of mild to moderate heart failure. After initial screening for suitability, there was a period of not less than one day, and not more than seven days, where diuretic treatment was withheld. Immediately prior to randomisation to either the AB sequence of treatment, or the BA sequence, baseline observations were taken. Each treatment period lasted for five days, with an immediate transfer to the second treatment after the first treatment period was completed. As a washout was not employed between treatments, observations made in the first two days of each treatment period were not utilised in the analysis of the trial. The primary outcome measures were the frequency of micturition and the subjective assessment of urgency. As neither of these are suitable for illustrating the analysis of normally distributed data, we will use instead a secondary effectiveness variable, namely oedema status, together with diastolic blood pressure (DBP). Oedema status is formed by the sum of the left and right ankle diameters. The DBP was calculated from the mean of three readings. Both of these variables are measured prior to randomisation, and at the end of each treatment period.

In total, 101 patients were recruited for the study, but seven withdrew prior to randomisation. Of the remaining 94 patients, only two failed to complete both treatment periods. Therefore, in order to illustrate the alternative methods of analysis, we have systematically removed approximately one in five of the observations from the second period. The structure of the data as analysed is shown in Table 7.1.

For each of our outcome variables, four analyses have been performed. In all of them, a treatment effect and a period effect were included as fixed. In two of the

Table 7.1 Data structure for a cross-over trial comparing two diuretics in patients with heart failure.

Baseline values Post-treatment values

Table 7.1 Data structure for a cross-over trial comparing two diuretics in patients with heart failure.

Baseline values Post-treatment values

Patient |
Treatment |
Period |
Oedema |
DBP |
Oedema |
DBP |

1 |
B |
1 |
45 |
60 |
45 |
55 |

1 |
A |
2 |
45 |
60 |
45 |
60 |

2 |
A |
1 |
51 |
50 |
48 |
60 |

2 |
B |
2 |
51 |
50 |
48 |
65 |

3 |
A |
1 |
53 |
70 |
50 |
70 |

3 |
B |
2 |
53 |
70 |
52 |
80 |

4 |
B |
1 |
49 |
68 |
47 |
60 |

4 |
A |
2 |
49 |
68 |
47 |
60 |

5 |
A |
1 |
46 |
65 |
45 |
60 |

6 |
A |
1 |
61 |
95 |
60 |
95 |

6 |
B |
2 |
61 |
95 |
59 |
97 |

Model |
Fixed effects Random effects | |

1 |
Treatment, period, patient |
- |

2 |
Treatment, period |
Patient |

3 |
Treatment, period, patient, baseline |
- |

4 |
Treatment, period, baseline |
Patient |

Treatment effect: A-B (SE) | ||

Model |
Oedema |
DBP |

1 |
0.304(0.120) |
0.812 (0.775) |

2 |
0.301 (0.120) |
0.926 (0.765) |

3 |
0.304(0.120) |
0.812 (0.775) |

4 |
0.309 (0.118) |
1.013 (0.748) |

Variance components (SE) | ||

Model |
Patient Residual Patient |
Residual |

1 |
- 0.530 - |
22.19 |

2 |
66.825 0.530 76.77 |
22.25 |

3 |
- 0.530 - |
22.19 |

4 |
3.763 0.526 25.60 |
21.91 |

models the baseline level was also included in the model as a covariate. Whether or not the baseline is included in the model, separate models are considered with the patient effect being fitted either as random or as fixed. The results of the models are summarised in Table 7.2.

Examination of the variance component terms shows that for all models the patient term is larger than the residual term. This indicates that there may have been substantial benefits from employing a cross-over design rather than a parallel group design. We note that this is particularly striking for oedema status. Note also the effect of including the baseline as a covariate in the analysis. This has the effect of reducing the size of the patient variance component term in Model 4. The implications of this are that the benefits of the cross-over are somewhat reduced when a (highly correlated) baseline covariate is available and, conversely, that the use of a mixed model is likely to be most helpful in these circumstances if there are missing values.

We see this in the estimates of the treatment standard errors. Comparison of Models 1 and 2 for the oedema status shows that the standard errors are identical (to the number of digits reported), indicating that the between-subject variation is so large that recovery of between-subject information is ineffective. With inclusion of the baseline level as a covariate we see that Model 3 gives the same result as Model 1. This result is well known, showing that a single baseline has no effect on a fixed effects analysis. It does, however, produce a small reduction in the treatment standard error when a mixed model is fitted, showing that some between-subject information has been utilised.

The results for DBP show the recovery of between-subject information more clearly because of the relatively smaller between-subject variation. Here, we see a detectable reduction in the treatment standard error, even when baselines are not used, and with the inclusion of baselines, a reduction of around 4% in the standard error is seen with the mixed models approach. This gain is modest but worthwhile.

The greatest advantage of the mixed models approach will unfortunately be gained in situations where a cross-over trial shows little benefit over a parallel group study, i.e. where the between-subject variance component is small relative to the residual variance component.

Such a situation occurs in a trial reported by Jones and Kenward (1989). In this two-period, cross-over trial an oral mouthwash was compared with a placebo mouthwash. There were two six-week treatment periods with a three-week washout period separating them. The outcome variable reported was the average plaque score per tooth, with each tooth being assessed on an integer scale from zero to three. Results were presented for the 34 patients with data from both treatment periods. Interestingly, these data arose from a trial in which 41 patients were randomised, and 38 completed the trial. For the purposes of this illustration we have deleted the second observation from five randomly selected patients from the 34 with complete data.

Two models were fitted to the data using PROC MIXED. In both, a treatment effect and a period effect were included as fixed. In one, the patient effect was fitted as random and in the other it was fitted as fixed. The results are shown in Table 7.3.

Examination of the variance component terms shows that the patient term is appreciably smaller than the residual term. This indicates that the benefit of employing a cross-over design rather than a parallel group design may be small. The estimate of the period effect (not shown) is small, but in accord with our recommendation in the previous section we retain it in the model. The main interest, of course, lies in the estimates of the treatment difference and the associated standard errors. Both analyses demonstrate a clear advantage to using the active mouthwash. For our purposes in comparing the results of the two analytical strategies it is the standard errors which interest us, because it is purely a matter of chance which method gives the larger point estimate of the treatment

Fixed patients |
Random patients | |

Variance components | ||

Patients |
- |
0.029 (0.018) |

Residual |
0.069 |
0.066 (0.017) |

Treatment difference (SE) |
0.25 (0.069) |
0.24 (0.065) |

effect. We see that the use of the random effects model has reduced the standard error of the estimate of the treatment difference by around 6%.

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