Biology Reference
In-Depth Information
For the Shewhart detector, optimal detection can be achieved by
r
(
s
) =
L
(
s
,
s
).
This detector is good at detecting large process shifts quickly, but if the shift
is small but sustained, accumulating deviations over time is necessary in
order to detect the change. The likelihood ratio based cumulative sum origi-
nally proposed by Page (1954) is one method to deal with accumulation and
is advantageous for detecting sustained shifts. It uses
rs
() max{
=
1
≤ ≤: , .
tsLst
( }
When the
y
t
are independent and identically distributed discrete random
variables, such count data CUSUM detectors are well investigated (see, for
example, Hawkins and Olwell (1998)). However, biosurveillance data often
exhibit seasonal variations and time trends that violate the assumption of an
identical distribution. As in Höhle et al. (2009), let
s
fy
fy
(
;
;
θ
θ
)
∑
rs
() max
=
log
i
i
1
0
,
(
)
1
≤≤
ts
it
=
(12.4)
where we have used the log-likelihood ratio (LLR) instead of the likelihood
ratio. Let θ
0
denote the in-control and θ
1
the out-of-control parameters. If θ
0
and θ
1
are known, Equation 12.4 can be written in recursive form as follows:
fy
fy
(
;
;
θ
θ
)
,
r
=
0
and
r
=
ax
0
, +
r
log
s
s
1
0
for
s
≥ .
1
0
s
s
−
1
(
)
(12.5)
One sees that for time points with LLR>0, that is, evidence against in-control,
the LLR contributions are added up. On the other hand, no credit in the direc-
tion of the in-control is given because
r
s
cannot get below zero.
In practical applications, the in-control and out-of-control parameters are,
however, hardly ever known beforehand. A typical procedure in this case
is to use historical phase 1 data for the estimation of θ
0
with the assump-
tion that these data originate from the in-control state. This estimate is then
used as plug-in value in the above LLR. Simultaneously, the out-of-control
parameter θ
1
is specified as a known function of θ
0
, for example, as a known
multiplicative increase in the mean. Developing appropriate count data time
series models together with statistical inference for the estimation of θ
0
and
θ
1
in a statistical process control framework is thus an important aspect of
performing biosurveillance.
As we suspect the number of persons in the eight age groups to shift
toward older age during the years, we want to take the population size of
the eight age strata into account in our monitoring. We do so by using data
