Biomedical Engineering Reference
In-Depth Information
visits; hospital admissions) longitudinally.
Two Manitoba Health les were utilized: hospital admission/discharge
records and a birth information le. From the birth le, data were obtained
on variables which were suspected of being associated with an increased risk
of childhood asthma, such as low birth weight, prematurity and neonatal
respiratory conditions (e.g., respiratory distress syndrome (RDS), transient
tachypnea of the newborn (TTN)). The birth and hospital les were linked
using the PHIN, assigned to each child at birth. Children in the (scal) 1984
birth cohort (i.e., born between April 1, 1984 and March 31, 1985) were
followed retrospectively until March 31, 1989 for hospitalizations resulting
from asthma (ICD-9: rubric 493). All newborns had at least 4 years of
observation, each being censored some time between ages 4 and 5. Further
details pertaining to data collection and record linkage are available in
Johansen et. al.
11
. We now discuss the statistical model of interest.
3. Model and Methods
We begin by dening the requisite notation. Let N
i
(t) be the total number
of events for subject i (i = 1; : : : ; n) as of time t. The censoring time for
subject i is given by C
i
and we dene = maxfC
1
; : : : ; C
n
g. The covariate
vector, which may contain time-dependent elements, is denoted by Z
i
(t).
The observed number of events is denoted N
i
(t) = N
i
(t^C
i
), where
a^b = minfa; bg. Event times for subject i are denoted T
i1
; : : : ; T
iN
i
,
where N
i
= N
i
(C
i
). Expressed in terms of stochastic integrals, N
i
(t) =
R
t
0
dN
i
(s), where dN
i
(s) = N
i
(s)N
i
(s) and sis the time instant
immediately preceding s. We assume that N
i
(t) is a counting process
(e.g., Chiang
3
), such that N
i
(t
2
)N
i
(t
1
) for t
2
> t
1
, dN
i
(t) =0 or 1, and
dN
i
(t)dN
j
(t) = 0 for i 6= j. The dN
i
(s) quantities are referred to as the
counting process increments.
The proportional means model
13;16
is given by:
i
(t)E[N
i
(t)jZ
i
] =
0
(t) expf
T
0
Z
i
g;
(1)
where
0
(t) is an unspecied baseline mean function and
0
is the parame-
ter of interest. In the case of time-dependent covariates, the proportional
rates model is given by:
d
i
(t)E[dN
i
(t)jZ
i
(t)] = d
0
(t) expf
0
Z
i
(t)g;
(2)
where d
0
(t) is the baseline rate function (the rate being interpreted as
the derivative of the mean). Model (1) is more restrictive in that it applies
to covariates that do not vary over time, Z
i
(t) = Z
i
for all t. Since it
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