We now illustrate how the framework we have set up extends naturally to include more follow- up visits. Although the primary analysis (were no data missing) is often ANCOVA at the last follow-up visit, when patients withdraw before the end of the study there is clearly valuable information in the previous observed values. Specifically, the previous observed values can provide valuable information about the reason for withdrawal, for example those who are per- forming poorly are more likely to withdrawal.
We can simply include the additional responses in the model, exactly the same way as mean exacerbation rate was included above, extending our newtreat variable accordingly. Assuming data are MAR, such an analysis is attractive because
1. if there were no missing observations, the estimated treatment effect, standard error, etc. would agree exactly with estimates obtained from an ANCOVA using only data from the final follow-up visit;
2. when end of follow-up data are missing, we make the best use of other measurements, under the missing at random assumption, and
3.6 Extension to longitudinal follow-up 69 Estimated treatment effect Std. error d.f. t-statistic p-value
0.08794 0.02306 310 3.81 0.0002
0.08856 0.02301 312 3.85 0.0001
Table 3.16: Estimated 3 year treatment effect, including mean exacerbation in the model. Row 1: results using exacerbation rate; row 2: results using square-root exacerbation rate
3. interim missing observations, and observations missing after patient withdrawal, are both handled without any additional work.
EXAMPLE3.8 Isolde study: analysis including all follow-up data
We illustrate this approach with Isolde. Our estimate of interest is the treatment effect at 3 years, adjusted for baseline. We have already seen that withdrawal depends on baseline BMI and exacerbations. Even so, including 6 month FEV1 in the model shown in Table3.14shows
the odds of not withdrawing are 3% higher for each 100 ml increase in 6 month FEV1(p=0.03).
We therefore extend the analyses above to include data from all follow-up measurements, to- gether with the number of exacerbations reported at each visit. For patients who withdrawal, this variable is sometimes reported after their last per-protocol FEV1 measure. As a patient’s
withdrawal is often triggered by an exacerbation leading to a visit to the GP who advises with- drawal, this provides potentially valuable information in support of MAR withdrawal. Note, though, that exacerbations are not that close to normally distributed, even with a square-root transformation. In the absence of joint non-linear and linear modelling (which itself raises further questions about appropriate covariance structures), as before we use the square-root of exacerbation data.
As only one baseline FEV1 value is, in fact, missing, we use baseline as a covariate in this
analysis. This means we do not need a variable that estimates a joint mean for baseline and different means for different treatment groups for other variables. The data are arranged as shown in Table 3.17. Here, newtreat shows the order of the response variable: 1 = BMI; 2–7 = number of exacerbations since last follow-up, recorded half yearly from 6 months to 3 years, and 8–13 = FEV1 recorded at half yearly follow-up visits from 6 months to 3 years. In the SAS
PROC MIXED analysis, we include newtreat as a class variable, together with ‘treatment’. We include a newtreat ‘treatment’ interaction to obtain the estimate of treatment at each follow- up visit. We further include a newtreat ‘baseline’ interaction to get a different adjustment for baseline at each time point. Likewise, to adjust for on age and sex, again with a different adjustment at each time point, we fit a newtreat ‘sex’ and newtreat ‘age’ interaction.
Table3.18 shows the results of fitting three models. Each has all the post-randomisation FEV1
measures as responses (6 scheduled for each patient). In addition, as BMI is predictive of with- drawal, but we do not wish to condition on it, all the models have BMI as an additional response. Models 1 and 2 are simplified. They replace the 6 follow-up exacerbation readings with a single
Variables
patient sex age baseline FEV1 treatment newtreat response
identifier (1=male) (years) (litres) (2=placebo)
1 1 63.98 0.98 2 1 22.3 1 1 63.98 0.98 2 2 0 1 1 63.98 0.98 2 3 1 1 1 63.98 0.98 2 4 1 1 1 63.98 0.98 2 5 2 1 1 63.98 0.98 2 6 0 1 1 63.98 0.98 2 7 0 1 1 63.98 0.98 2 8 1.30 1 1 63.98 0.98 2 9 1.15 1 1 63.98 0.98 2 10 1.03 1 1 63.98 0.98 2 11 0.98 1 1 63.98 0.98 2 12 0.96 1 1 63.98 0.98 2 13 1.10 2 2 64.33 0.89 1 1 25.3 2 2 64.33 0.89 1 2 0 2 2 64.33 0.89 1 3 0 2 2 64.33 0.89 1 4 0 ... ... ... ... ... ... ...
Table 3.17: Data arrangement for estimating treatment effect, including longitudinal follow-up data on exacerbations and FEV1
value,√mean exacerbation rate, derived as described in Example3.7. Thus models 1 and 2 fit an 8-dimensional normal distribution. Model 1 has the same covariance matrix for both treat- ment groups (36 parameters); but, as there is some evidence of more variability in the placebo group, model 2 fits a separate covariance matrix for each treatment group (72 parameters). Model 3 attempts to use all possible information about withdrawal in the exacerbation rate data. Instead of using mean exacerbation rate, we have as 6 additional responses the exacerbation rates observed at each of the clinic visits. Combined with the 6 FEV1 measures and BMI this
makes 13 responses per patient. Again, we fit an unstructured covariance matrix, this time with 91 parameters.
Reassuringly, all three models give similar results, suggesting that, after including mean exacer- bation rate in the model, there is little gained by including the number of exacerbations at each visit. As√mean exacerbation rate is roughly normal, but the number of exacerbations is very far from normal, models 1 and 2 are slightly preferable.
A further question of interest to the investigators was whether the treatment effect could be summarised by a straight line, for each treatment group, and whether the slope of these lines was different. All these patients have declining FEV1, but a slower rate of decline in the active
3.7 Inverse probability weighting methods 71