We have already considered in some detail the analysis of a multi-centre trial of treatments for hypertension in Sections 1.3 and 2.5. Here, we discuss more fully the interpretation of the results from these analyses and consider estimates of treatment effects by centre.

Results from fixed and random effects analyses of DBP are summarised in Table 5.1. An initial fixed effects model including centre-treatment effects was also fitted. However, overall treatment effects were not estimable in this model, because all treatments were not received at every centre (see Table 1.1 in Section 1.3). This model gave a non-significant p-value for centre-treatment

Model |
Fixed effects |
Random effects |
Method |

1 |
Baseline, treatment, centre - |
OLS | |

2 |
Baseline, treatment |
Centre |
REML |

3 |
Baseline, treatment |
Centre, treatment-centre |
REML |

Treatment effects (SEs) | |||

Model |
Baseline |
A - B A - C |
B - C |

1 |
0.22 (0.11) |
1.20(1.24) 2.99 (1.23) |
1.79 (1.27) |

2 |
0.22 (0.11) |
1.03 (1.22) 2.98 (1.21) |
1.95 (1.24) |

3 |
0.28 (0.11) |
1.29 (1.43) 2.93 (1.41) |
1.64(1.45) |

Variance components | |||

Model |
Centre |
Treatment-centre |
Residual |

1 |
- |
— |
71.9 |

2 |
7.82 |
- |
70.9 |

3 |
6.46 |
4.10 |
68.4 |

effects (p = 0.19) and thus the usual 'fixed effects' approach would have been to remove centre-treatment effects to give Model 1.

The centre-treatment variance component is positive in Model 3 and this leads to an increase in the treatment standard errors over the fixed effects models (as indicated by the variance formulae given in Section 5.2). However, note that the baseline standard error is similar between the models. This is because baseline effects are estimated at the residual error level and not at the centre-treatment level. Results from Model 3 can be related to the potential population of centres. Since there are 29 centres, there are no problems arising from an inadequate number of DF for the variance components, and we can be confident in presenting these results if global inference is required.

The centre variance component is positive in Models 2 and 3 and therefore some information on treatments will be recovered from the centre error stratum. However, the standard errors in Model 2 are only slightly smaller than in Model 1, indicating that only a small amount of information has been recovered. Also, some of the improvement in the standard error may be due to the smaller residual variance which has resulted from the use of this model.

Plots of the centre and centre-treatment effects from Model 3 were used in Section 2.5 to assess the normality of the random effects and to check whether any centres were outlying. Additionally, we can now take differences between the centre-treatment effects to calculate treatment effect estimates for each centre. In Model 3 these will be shrunken towards the overall treatment mean. We illustrate this by calculating the treatment difference A-C for just the first eight centres in the study. Unshrunken fixed effects estimates are also calculated for comparison using results from the initial model (fitting treatment, centre and centre-treatment effects as fixed).

Centre |
Number of patients |
Fixed model |
Random model |

1 |
39 |
3.81(3.34) |
3.19(2.94) |

2 |
10 |
-5.67(6.80) |
1.56(3.40) |

3 |
8 |
25.66(7.63) |
6.05(3.40) |

4 |
12 |
-0.12(5.90) |
2.42(3.39) |

5 |
11 |
14.12(7.21) |
4.56(3.40) |

6 |
5 |
2.89(8.37) |
3.03(3.39) |

7 |
18 |
7.38(4.82) |
4.22(3.31) |

8 |
6 |
-4.68(8.33) |
2.15(3.39) |

It can be seen that, in general, shrinkage is towards the overall treatment difference of 2.92 (although this is not the case for all centres, because the models make different adjustments for baseline effects). The relative shrinkage (i.e. (fixed estimate - random estimate)/(fixed estimate)) is usually greatest for the smaller centres. The standard errors of the random effects estimates are smaller than those of the fixed effects estimates, because the random effects model utilises information on the treatment effects in the full sample as well as information from the centre of interest. By contrast, the fixed effects standard errors do not utilise the full sample information and are larger, because they are calculated using only information from the centre of interest. This also causes the fixed effects standard errors to vary greatly between the centres, because they are directly related to the centre sizes. It is difficult to determine whether any of the centres are outlying using the fixed effects estimates, because they need to be considered bearing in mind centre size. For example, at centre 3 a very large treatment difference is given by the fixed estimate, but the shrunken random estimate appears acceptable. We note that the standard errors of the random effects estimates have increased by around 20% from those reported in the first edition of this book, by use of the Kenward-Roger option.

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