Statistical Methods in Medical Research 4ed

drop-out. In this it is supposed that complete data are observed on an individual
up to and including measurement occasion D 1, where 2 D k ‡ 1, with all
subsequent outcomes being missing: D ˆ k ‡ 1 indicates that the individual did
not drop out. This reduces the modelling problem, as the missing value process
has been greatly restricted. While this undoubtedly limits the practical utility of
the method, it is an important case and also allows progress in a methodologically
challenging area.
Diggle and Kenward (1994) and Diggle (1998) suggest that the missing value
process, which can now also be called a drop-out process, can be modelled by
log
Pr…D ˆ d jD d, y†
1 Pr…D ˆ d jD d, y† ˆ ad ‡ Ps
12.6 Longitudinal data 447
jˆ0
b jyd j, …12:57†
where s is a prespecified integer and for notational convenience we take yj ˆ 0
for j 0. This model is not the only one that could be considered, but it is
plausible; it assumes that the probability of drop-out at any given stage can
depend on the outcomes up to and including that stage, but not on the future of
the process. If all the bs are 0 then the process is MCAR, and if b 0 ˆ 0 the process
is MAR. If neither of these simplifications obtains then the log-likelihood for the
observed data has an extra term over those in (12.56), which accommodates the
way the parameters b and u jointly determine aspects of the data. This will
involve contributions from probabilities in (12.57) but this depends on the
unobserved value yd, so it is necessary to compute an expression for
P(D ˆ d jD d, y o) by integrating the probability in (12.57) over the conditional
distribution of the unobserved value, f …yd jy o†. By this means valid inferences
can be drawn about u even when data are missing not at random.
The MCAR, MAR classification of Little and Rubin (1987) is naturally
expressed by factorizing f …y, r† as f …rjy†f …y† and then specifying particular
forms for the first factor; these are known as selection models. Another approach
would be to use the alternative factorization, f …yjr†f …r†, which gives rise to
pattern mixture models, so called because they view the joint distribution as a
mixture of different distributions on the outcomes, one distribution for each
pattern of missing data. This view, which is extensively discussed in Little (1993),
Hogan and Laird (1997), Diggle (1998) and Kenward and Molenberghs (1999),
gives an alternative and in many ways complementary approach to the problem.
The reliance of the MAR, MCAR classification on the form of f …rjy† means that
the extent to which this classification can be carried over into pattern mixture
models is not transparent. Molenberghs et al. (1998) have shown that for dropout,
but not for more general patterns of missing data, MAR models can be seen
to correspond to certain classes of pattern mixture model.
In order to make valid inferences about f …y, r† from incomplete data …y o, r†,
various models have to be constructed or assumptions made. A fundamental