3.4.1 Introduction and models fitted

The multi-centre trial of treatments for lowering blood pressure introduced in Section 1.3 is used again here. An adverse event, 'cold feet', is analysed as a binary variable and observations at the final or last attended visit are used. Cold feet was, in fact, recorded on a scale of 1-5:1 = none, 2 = occasionally, 3 = on most days, 4 = most of the time, 5 = all of the time. A binary 'cold feet' variable was created by taking categories 1 and 2 as negative and categories 3-5 as positive. The frequencies of cold feet by treatment and centre are shown in Table 3.1. In this trial 'cold feet' was recorded at baseline so, in order to include a baseline covariate in the model (and so reduce between-patient variation), we will analyse the data in Bernoulli form.

Table 3.1 indicates that there are several zero frequencies of cold feet and these will lead to uniform centre and centre-treatment categories. This in turn may cause variance component bias (see Section 3.3.5) and it is not clear whether a random effects model will be satisfactory. Here, we will fit a variety of models (see Table 3.2) and discuss their strengths and weaknesses. In practice, only Model 1 is likely to be considered as a fixed effects model since the large number of uniform categories will cause problems in estimating satisfactory treatment effects in Models 2 and 3 (as discussed in Section 3.4.2). In Model 4, centre effects are fitted as random and in Model 5 both centre and centre-treatment effects are fitted as random. Model 5 takes into account the random variation in the treatment effect between centres and results can be related with more confidence to the 'population' of potential centres. Models 4 and 5 are both fitted using pseudo-likelihood.

Models 6 and 7 are the same as Models 4 and 5 except that they are fitted using a Bayesian model with non-informative priors to obtain a joint (posterior) distribution of the model parameters. The Bayesian models are set up in a similar way to the normal example described in Section 2.5 except that Bernoulli distributions are now assumed for the observations. Again, normal distributions with zero means and very large variances (of 1000) were used as non-informative

Table 3.1 Frequencies of cold feet by treatment and centre.

Treatment

Treatment

Table 3.1 Frequencies of cold feet by treatment and centre.

Centre |
A |
B |
C |
Total |

1 |
3/13 |
5/14 |
1/12 |
9/39 |

2 |
2/3 |
0/4 |
0/3 |
2/10 |

3 |
0/3 |
0/3 |
0/2 |
0/8 |

4 |
1/4 |
1/4 |
0/4 |
2/12 |

5 |
1/4 |
3/5 |
0/2 |
4/11 |

6 |
0/2 |
1/1 |
1/2 |
2/5 |

7 |
0/6 |
1/6 |
0/6 |
1/18 |

8 |
1/2 |
0/1 |
1/2 |
2/5 |

9 |
- |
- |
0/1 |
0/1 |

11 |
0/4 |
1/4 |
0/4 |
1/12 |

12 |
0/3 |
1/3 |
0/4 |
1/10 |

13 |
1/1 |
0/1 |
0/2 |
1/4 |

14 |
0/8 |
2/8 |
1/8 |
3/24 |

15 |
1/4 |
0/4 |
0/3 |
1/11 |

18 |
0/2 |
0/2 |
0/2 |
0/6 |

23 |
1/1 |
- |
0/2 |
1/3 |

24 |
- |
- |
0/1 |
0/1 |

25 |
0/3 |
0/2 |
0/2 |
0/7 |

26 |
0/3 |
1/4 |
0/3 |
1/10 |

27 |
- |
1/1 |
0/1 |
1/2 |

29 |
1/1 |
- |
0/1 |
1/2 |

30 |
0/1 |
0/2 |
0/2 |
0/5 |

31 |
0/12 |
0/12 |
0/12 |
0/36 |

32 |
1/2 |
0/1 |
0/1 |
1/4 |

35 |
0/2 |
0/1 |
- |
0/3 |

36 |
0/9 |
5/6 |
0/8 |
5/23 |

37 |
0/2 |
0/1 |
1/2 |
1/5 |

40 |
0/1 |
1/1 |
- |
1/2 |

41 |
0/2 |
0/1 |
0/1 |
0/4 |

Total |
13/98 |
23/92 |
5/93 |
41/283 |

priors for the fixed effects (baseline and treatment), and inverse gamma distributions with very small parameters (of 0.0001) were used as non-informative prior distributions for the centre and centre-treatment variance components. Note that this prior specification for the variance components ensures that negative variance component samples cannot be obtained. Ten thousand samples were taken using the Gibbs sampler using the package BUGS (see Section 2.3.5) to repeatedly sample conditionally from the posterior distribution of the model parameters. The values sampled were then used directly to obtain parameters estimates, standard deviations, probability intervals and 'Bayesian' p-values in exactly the same way as described for the example given in Section 2.5.

Model |
Fixed effects |
Random effects |
Method |

1 |
Baseline, treatment |
— |
GLM |

2 |
Baseline, treatment, centre - |
GLM | |

3 |
Baseline, treatment, centre-treatment - |
GLM | |

4 |
Baseline, treatment |
Centre |
P-L1 |

5 |
Baseline, treatment |
Centre, centre-treatment |
P-L1 |

6 |
Baseline, treatment |
Centre |
Bayes |

7 |
Baseline, treatment |
Centre, centre-treatment |
Bayes |

1P-L = |
pseudo-likelihood. | ||

3.4.2 |
Results | ||

Estimates of the variance components and fixed effects for each model |
are shown | ||

in Table 3.3. | |||

Table 3.3 Estimates of variance components and fixed effects (on the logit scale). | |||

Variance components | |||

Model |
Centre Treatment-centre Dispersion parameter |
-21og(L) | |

1 |
- |
l.OO^l.Ol) |
178.17 |

2 |
- |
- 1.001 (0.79) |
147.78 |

3 |
- |
- 1.001 (0.53) |
100.05 |

4 |
0.09 |
- 0.94 |
- |

5 |
0.00 |
1.88 0.54 |
- |

6 |
0.022 |
- 1.001 |
- |

7 |
0.032 |
0.142 1.001 |
- |

Treatment effects (SEs) | |||

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

1 |
2.97(0.49) |
-0.77(0.44) 0.94(0.60) |
1.70 (0.57) |

2 |
2.67(0.55) |
-0.98(0.49) 1.05 (0.65) |
2.03 (0.63) |

3 |
3.09 (0.80) |
-- |
- |

4 |
2.91 (0.48) |
-0.77(0.43) 0.93 (0.58) |
1.70 (0.56) |

5 |
3.06 (0.46) |
-0.59 (0.57) 1.18 (0.66) |
1.78 (0.66) |

6 |
3.05 (0.51) |
-0.81 (0.45) 1.00 (0.61) |
1.80(0.59) |

7 |
3.17(0.56) |
-0.68 (0.53) 1.12(0.69) |
1.80(0.65) |

1 Dispersion parameter is fixed at one (value in brackets is its estimate).

2 Estimates are median values from the marginal posterior distributions.

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