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TIME 1 2 3 4 x Observed data

missing given observed under MAR Line of mean of conditional distribution of

Predicted mean values under MAR

Predicted mean values under MNAR x

δ

2δ

Figure 6.5: Schematic illustration of increasing the rate of decline by δ after withdrawal

6.5 Pattern mixture approach with longitudinal data via MI

The above approach has the potential to be extended to longitudinal data in several ways. Perhaps the easiest is to start with the conditional distribution of the missing given the observed data under MAR, and then modify this after a patient has dropped out.

A natural approach is to suppose that patients who withdrawal have a different, usually poorer response than predicted by MAR. For example in an asthma trial, we might suppose that FEV1

improves more slowly (or declines more quickly) after withdrawal. If the change in rate of decline is denoted δ , then the conditional mean for the first response after withdrawal is reduced by δ , the second by 2δ and so on. This is schematically illustrated in Figure6.5.

As discussed above, if possible we can elicit from experts the mean and variance of δl in the

treatment group l, which is assumed to be normally distributed, and Cor (δl, δl0), for all treat- ment groups l and l0. In practice, the theory above shows the widest confidence intervals occur when the correlation is zero (assuming it is not negative), and useful information can be obtained by assuming the distribution of δ is the same in both arms.

We can use this approach with multiple imputation as follows. First, we create the K imputations under MAR. Suppose we have two treatment groups. Then, for each imputation, k, we sample

µ d1k d2k ¶ ∼ N µµ δ1 δ2 ¶ , µ σ₁2 σ12 σ12 σ22 ¶¶ .

For each patient, in each treatment arm, l = 1, 2, for each imputation, we then decrease/increase the first MAR imputed observation by dlk, the second by 2dlk and so on. We then analyse the

resulting datasets and combine the estimates using Rubin’s rules. If the time between observa- tions is not constant, we may want to change the multipliers of d from 1, 2, 3, . . . , to maintain a linear change. We can handle interim missing observations by decreasing them by dl, or

simply leaving them with their MAR imputed values. The latter is consistent with a different mechanism driving interim missing data and patient withdrawal.

Coefficient Parameter estimates when δ =

0 (MAR) −0.01 −0.02 −0.03 −0.04

Treatment −0.086 (0.021) −0.088 (0.026) −0.098 (0.026) −0.095 (0.028) −0.101 (0.029)

Baseline 0.89 (0.021) 0.89 (0.021) 0.89 (0.021) 0.89 (0.021) 0.89 (0.022) Intercept 0.14 (0.042) 0.12 (0.047) 0.11 (0.048) 0.09 (0.051) 0.08 (0.051)

Table 6.5: Estimates of treatment effect at the final time point when patients who withdraw have each subsequent MAR imputed FEV1 value reduced by δ , 2δ and so on. The reference group

is those on active treatment. All values are in litres

As the theory above suggests, the confidence intervals are going to be narrower the greater the correlation between the d’s for different treatment arms. In practice, information on this corre- lation is unlikely to be available, so as the correlation is unlikely to be negative, a conservative approach is to set it to zero, i.e. set σ12= 0. Likewise often, in the absence of prior information,

we can set δ1= δ2.

EXAMPLE6.3 Isolde data: sensitivity analysis through MI

We illustrate the above approach which is implemented in a SAS macro by Roger(2006) (see Appendix C). Recall there is an active and placebo arm in this trial. We will use the above approach, sampling for imputation k = 1, . . . , 50, (d1k, d2k) from

µ d1k d2k ¶ ∼ N µµ δ δ ¶ , µ σ2 0 0 σ2 ¶¶ .

We compare the results of multiple imputation under MAR with values −10, −20, −30, −40 ml for δ and σ2= 0.0052. Note that the Isolde trial has a large number of patients (just under half) with some interim missing data. Imputed interim missing data is left unchanged by the macro (in other words it is assumed the MAR model is correct). Only when a patient has withdrawn do we start to change the MAR imputations.

Table 6.5shows the results, for the treatment estimate at the final time point. We used K = 50 imputations. We see that the treatment estimate increases with δ , but the standard error also increases, so that the significance of treatment slightly declines (z = −4.1 for δ = 0 and −3.5 for δ = 0.04). The intercept also decreases as δ increases. This makes sense as the MNAR model reduces the FEV1, but more patients withdraw, and withdraw earlier, in the placebo arm. Thus

the effect of increasing the rate of decline when patients withdraw results in a greater reduction in final FEV1 values in the placebo arm than the active arm, and hence a larger estimated

treatment effect. The increase in standard error, resulting from the increased variability of the final FEV1’s, warns that this sensitivity analysis cannot be interpreted to say that the MAR

analysis underestimates the significance of the treatment effect. ¤ 6.5.1 Further points

This approach can be readily generalised to discrete data. In that case, though, we have to work with the mean of the underlying distribution (Binomial, Poisson etc.) rather than working

6.6 Pattern-mixture models and intention to treat analyses 137