Biomedical Engineering Reference
In-Depth Information
13.3.6
Multiple Imputation Approaches
For these approaches we rewrite the scores ci
i
as a weighted average of scores
for each possible interval:
X
c
i
=
w
ij
C
j
;
(13.9)
j=1
where
S
0
(t
j
) S
0
(t
j+1
)
S
0
(t
j
) S
0
(t
j+1
)
!
C
j
=
is the rank-like score associated with the interval (t
j
;t
j+1
], and
S
0
(t
j
) S
0
(t
j+1
)
S
0
(`
i
) S
0
(r
i
)
!
w
ij
= If`
i
t
j
< ri}
i
g
:
Fay and Shih (2012) showed that for CIA data, the distribution of the imputed
scores, say C
i
, does not asymptotically depend on the assessment times A
i
,
even when the data have ATD.
For the multiple imputation method, we create b data sets by imputation.
For the first data set at the i-th observation, we sample from C
1
;:::;C
m
with probabilities wi1,
i1
;:::;w
im
and treat the resulting imputed score C
i
as
an exactly observed response. Then using those imputed values, we calculate
the standard logrank or Wilcoxon{Mann{Whitney ecient score U and its
associated variance V , estimated in the usual way by Martingale methods
(Huang et al., 2008). We repeat this process b times, letting the values of U
and V derived from the j-th imputed data set be U
j
and V
j
, respectively.
Then we make inferences by assuming that
U
p
V
P
j=1
U
j
and
is normally distributed, where U =
1
b
P
j=1
V
j
b
!
P
j=1
U
j
U
U
j
U
0
b 1
!
V =
:
We could also do a similar calculation using the permutational variance or
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