Page 72 - Special Topic Session (STS) - Volume 4

STS563 Patrick Graham et al.

components of can be simulated sequentially using the following
decomposition

( , | , ∅, ) = ( | , ∅, )( | , , ∅, )( | , , , ∅, ).
An alternative approach to inference for the target population that is
simpler computationally uses the idea that at each setting of the covariates the
target population count, NT (x) can be approximated as

1 − ()
() = ()

1 − ()
where () = ()/( () + ()), which is the list over-
11
01
01
coverage, and ( ) is the list count at X = x. Justification for this

approximation is given in Graham and Lin (2019). A straightforward strategy
for implementing this approach involves weighting each list record by the ratio
(1− ( ) )
= . Estimated total population and sub-population counts can

(1− ( ))
then be obtained by summing the weights for individuals in the populations
of interest. The weights are a function of coverage model parameters, and
posterior inference therefore follows directly from the posterior distribution
for the coverage model parameters. In practice, we compute a set of weights
for each draw from the posterior for the coverage model parameters. Posterior
distributions for population counts can be obtained simply by computing the
relevant weighted counts for each set of weights. The administrative list
supplemented with a set of replicate weights also provides a convenient unit-
record representation of the target population. The variation across the
replicate sets of weights represents uncertainty due to estimating and
adjusting for under and over-coverage. In several simulated examples we have
found close agreement in estimates obtained from the weighting approach
and from directly estimating finite population counts. Consequently, in what
follows we concentrate on this weighting approach.
Since the weights that adjust for under and over-coverage depend only on
the coverage model parameters, our inference task is simplified because we
need only obtain the posterior distribution for these model parameters. As a
further simplification we use the conditional likelihood for ∅, which can be
computed directly from the observed data, in place of the full likelihood
which is difficult to compute because it involves integrating over the
missing data. The conditional likelihood for ∅ is

(∅) = ( ̃ ̃ = (0,0), , ∅)
|(
) ) ( + (1 − ( )∅ 11 ( )
)
( )∅ 11 ( ( )∅ 10 ( ∅ 01 (3)
= ∏ ∏ ∏
1 − (1 − ( ))∅ 10 ( ) 1 − (1 − ( ))∅ 10 ( ) 1 − (1 − ( ))∅ 10 ( )
: ̃ =(1.1) : ̃ =(1.0) : ̃ =(0,1)

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