Analyses

The four analysis models shown in Table 5.3 are considered. Models 1 and 2 were fitted using fixed effects models (GLMs). In these models the data are analysed in binomial form (i.e. using the trial-treatment frequencies). However, identical results would have been obtained had the data been analysed in Bernoulli form (i.e. using one observation per patient). Model 3 is a random effects model and the data are analysed in Bernoulli form using a GLMM. Using this form allows the 'shrunken' treatment effects at each trial to be estimated using PROC GLIMMIX. However, very similar results can be obtained by analysing the data as binomial frequencies, since there are no baseline values and no categories are uniform. We illustrate this by fitting Model 3 (a) which is identical to Model 3 except that the data are in binomial form and the dispersion parameter is fixed at one (allowing variance at the residual level to be modelled by the trial-treatment variance

Table 5.3 Variance component and treatment odds ratio estimates.

Model

Method (data form)

Fixed effects

Random effects

1

GLM (binomial)

Treatment, trial, trial-treatment

2

GLM (binomial)

Treatment, trial

-

3

P-L (Bernoulli)

Treatment

Trial, trial-treatment

3(a)

P-L (binomial)

Treatment

Trial, trial-treatment

4

Bayes (Bernoulli)

Treatment

Trial, trial-treatment

Variance components

Dispersion

Treatment odds

Model

Trial

Trial-treatment

parameter

ratio (95% CI)

p-value

1

-

1F

0.63 (0.47 — 0.84)

0.002

2

-

-

1F

0.66 (0.56 — 0.79)

<0.0001

3

1.42

0.16

0.98

0.60(0.34 — 1.05)

0.07

3(a)

1.43

0.16

1F

0.60(0.34 — 1.05)

0.07

4

1.32

0.21

1F

0.60(0.35 — 1.01)

0.06

Note: F = parameter is fixed.

Note: F = parameter is fixed.

component). Model 4 is the same random effects model as Model 3 but is fitted using the Bayesian approach. This was done using the Gibbs sampler in the BUGS package (see Section 2.3.5) in the same way as described for the multi-centre analysis performed in Section 2.5 (Model 5).

Odds ratios and confidence intervals are calculated by exponentiating the treatment difference estimates and their confidence intervals on the logit scales (see Section 3.3.9). In recent versions of SAS, this can be done using relevant options. The treatment effect is tested using asymptotic Wald chi-squared tests in Models 1 and 2, and an F test in Models 3 and 3(a) (see Section 3.3.8 for details on GLM and GLMM significance testing). In Model 4 twice the probability of the treatment difference being greater than zero is taken to provide a 'Bayesian' p-value (see Section 2.3.3).

In Models 3 and 3a, the Kenward-Roger method has been used to adjust for the fixed effects standard error bias. This has had the effect of both increasing the standard error and modifying the DF for the F test as compared with the results presented in the first edition of this book.

5.7.2 Results

The results are shown in Table 5.3. Results from Models 1 and 2 do not take account of any additional variation in the treatment effect between trials and therefore shouldbe formally related only to the trials included. The trial-treatment interaction was highly significant in Model 1 (p = 0.0006) and this would cast doubt on any formal extrapolation of the results from these models. In Model 1 the overall treatment effect is calculated as an unweighted average of the treatment effects at each trial (see Section 5.2). Such an estimate is clearly inappropriate, since the trial sizes differ widely. This problem does not arise in Model2, where centre-treatment effects are omitted. Note that in the above models the 95% confidence intervals are based on exponentiating the treatment estimate ±1.96 x SE, because of the asymptotic normality of the estimate.

Models 3, 3(a) and 4 take account of the extra variation in the treatment effect between trials by fitting trial and trial-treatment effects as random. In these models we are assuming that the random effects are normally distributed, and as the variance components are estimated with eight DF, the confidence intervals are more appropriately estimated as treatment estimate ±t8,0.975 x SE. Model 3(a) gives very similar results to Model 3, indicating that it makes little difference here whether the data are analysed in Bernoulli or binomial form. The trial variance component is fairly large in all of the models, indicating that the overall incidence of pre-eclampsia varies greatly between trials. Thus, it is likely that quite different inclusion criteria were used for the trials or that pre-eclampsia was defined differently by the different practitioners. The positive trial-treatment components indicate some variation in the treatment effect across trials. This is reflected in the size of the treatment confidence intervals, which are wider than those in Models 1 and 2. The results from these analyses can be generalised with some confidence to the full population of pre-eclampsia sufferers.

The Bayesian analysis (Model 4) gives almost identical treatment ORs and confidence intervals to the pseudo-likelihood analyses (Models 3 and 3(a)). The differences in variance component estimates between Models 3 and 4 are not unexpected, since we have taken them to be the medians of the marginal posterior distributions in Model 4, whereas in Model 3 the estimates are the values which maximise the pseudo-likelihood surface.

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