Acabados decorativos

We could develop the above model to show the inclusion of more than one covariate predictive of withdrawal. Rather than do this, however, we show this is equivalent to extending the model to allow for a post-randomisation variable that is predictive of withdrawal.

The key observation is that our models are all built around assuming the joint distribution of all variables is multivariate normal. We parameterise this distribution in such a way that the effect of treatment adjusted for baseline is readily estimable. However, the multivariate normal distribution does not depend on time in any way. Therefore there is no difference between including baseline variables predictive of withdrawal and post-randomisation values predictive of withdrawal. We simply include them as another response in the model, estimating a separate mean for each treatment group. As in the example above, if the data are complete, our model gives the same estimated treatment effect as an ANCOVA fitted to the response and baseline data. If the data are not complete, our model gives a sensible estimate of the treatment effect under MAR.

Variable Estimate Std. Error t-value Pr(> |t|)

Intercept 0.642 0.233 2.753 0.006

treatment: placebo -0.047 0.039 -1.205 0.229

baseline FEV1 -0.034 0.045 -0.750 0.453

BMI 0.014 0.004 3.138 1.78×10−3

mean exacerbation rate -0.069 0.015 -4.750 2.53×10−6

sex: male -0.072 0.047 -1.528 0.127

age -0.004 0.003 -1.365 0.173

Table 3.14: Log odds ratios from a logistic regression of patient withdrawal (0=withdrawal) on baseline variables and exacerbation rate

EXAMPLE3.7 Isolde data: adjusting for post-randomisation exacerbation rate

For each patient, we calculate their mean exacerbation rate as their total number of exacer- bations before withdrawal divided by their time on trial. We then included this in a logistic regression of withdrawal before the end of the study, together with treatment, BMI, sex, age and baseline FEV1. As Table3.14shows, patients with high exacerbation rates are much more

likely to withdraw; this is therefore an important variable to adjust for in a MAR analysis. Figure3.2shows that exacerbation rate is quite non-normal. As our model is multivariate nor- mal, and we are not concerned with interpreting changes in exacerbation rates directly, we use √

exacerbation rate, which is more normally distributed (right panel, Figure3.2).

To fit this model, we follow the previous arrangement of data (Table3.12), giving Table3.15. Note the extra response (4), for mean exacerbation rate, which has to have a different mean for the two treatment groups (newtreat= 6, 7). Fitting this model, using in turn mean ex- acerbations and their square root, gives the results in Table 3.16. Comparing the second row with Table3.13, we see the estimated treatment effect is closer to that obtained before adjusting for BMI, but we now have fractionally more information. The difference between the results in Table 3.16is very small; the assumption of normality for exacerbation rate does not appear important in this example. Nevertheless, it is preferable, where possible, to transform variables to approximate normality. Finally, the results underline the importance of adjusting for all the key predictors of withdrawal, including post-randomisation ones. ¤ So far we have obtained estimates of treatment conditional only on baseline. If we wish to condition on variables that we have included as responses, as previously observed, we simply fit a single mean for that variable across treatment groups, following the logic of (3.3). Of course, fully observed baseline values can be included as covariates in the model in the usual way. We will generally need an interaction of such a covariate with newtreat. The precise form this takes must be carefully chosen to ensure the desired conditioning on (adjustment for) each of the response variables.

The difference in handling baseline variables predictive of withdrawal, and post-randomisation variables such as exacerbations, is that with baseline variables we can either condition on (i.e.

3.5 Missing baseline/follow-up: handling additional covariates predictive of missing data 67

Mean exacerbation rate

Frequency 0 2 4 6 8 10 12 0 50 100 150 200 250 300

mean exacerbation rate

Frequency 0.0 1.0 2.0 3.0 0 50 100 150 200

Figure 3.2: Histograms of mean exacerbation rate, and its square-root

adjust for) them, or marginalise over them. Conditioning on them may sometimes provide more precise estimates. Further, usually there are relatively few baseline values missing and the reason for them being missing does not depend on treatment. Thus conditioning on them by including them as a covariate in the analysis (and thus implicitly excluding from the analysis patients with them unobserved) is unlikely to bias the estimate of treatment effect. However, we do not usually wish to condition on post-randomisation responses. E.g. in Isolde, we do not want an estimate of the effect of treatment on FEV1 conditional on exacerbations. Thus,

we have to marginalise over post-randomisation responses, by including them in the model as shown above.

3.5.3 Summary

1. Baseline and post-randomisation variables predictive of withdrawal can be handled in the same way.

2. To obtain treatment estimates unadjusted for them, we include them as additional re- sponses with separate means for each treatment group.

3. Usually, baseline variables predictive of withdrawal have relatively few missing values. Therefore, to adjust treatment estimates for them, it is best to include them as covariates. 4. If they have many missing values, include them as an additional response, but with the

Variables

patient treatment response response newtreat identifier group indicator

1 2 1 22.3 1 1 2 2 0.98 3 1 2 3 1.30 4 1 2 4 1.3 6 2 1 1 25.3 2 2 1 2 1.46 3 2 1 3 missing 5 2 1 4 0.3 7 3 2 1 23.4 1 3 2 2 missing 3 3 2 3 2.97 4 3 2 4 0.5 6 ... ... ... ... ...

Table 3.15: Data arrangement for estimating treatment effect, extending Table 3.12 to include mean exacerbation rate