Biology Reference
In-Depth Information
In Figure 12.5, we observe no specific patterns of the alarms across age strata,
except for single week alarms around the turn of the year 2007/2008 in age
groups <1, 1-4 and 75-84. Note that the 2007/2008 season in Denmark did
not exhibit any heavy influenza activity. Furthermore, the current surveil-
lance does correct for any population demographics using the linear trend
in Equation 12.2. It might, however, be worth investigating an additional
adjustment for population size in the eight age strata as these are expected
to change over the years.
As a further remark, in the notation of Section 12.3 the Farrington algo-
rithm does not utilize all available information at decision time s = t
0
, that
is,
r
= ′
with
′
y
s s
. The effects of seasonality are handled
robustly by using only “similar” weeks as reference values and hence no
explicit seasonal model is needed. Such an approach is, however, suboptimal,
if it is possible to adequately model the seasonal behavior as, for example, it
is done in Section 12.5.
Even though more than a single
y
s
is used to compute
r
(
y
s
) in the
Farrington algorithm, the decision in Equation 12.3 occurs by only com-
paring the current observation with the upper limit of the predictive
distribution. Hence, no accumulation of evidence against the in-control
situation occurs. In the next section, we reconsider this task from a statisti-
cal process control viewpoint and describe an approach taking accumula-
tion into account.
()
y
r
()
y
Farr
s
Farr
s
12.5 Negative Binomial CUSUM
Reconsidering Equation 12.1 more from the viewpoint of statistical process
control, the simplest class of detectors is the Shewhart detector, which for
r
(
y
s
) only utilizes information about the last time point, for example, by com-
paring the single
y
s
value to a fixed threshold value. In a parametric detec-
tion setup one assumes a known probability mass function (PMF)
f
(·;θ) for
y
s
,
which is parametrized by a parameter vector θ. If the parameter vector θ is
assumed to be known in the in-control and out-of-control state, an optimal
change-point detection can be achieved based on the partial likelihood ratio
(Frisén, 2003). Let
L(s,t)
with
s
≥
t
be the partial likelihood ratio between the
out-of-control and in-control models at time
s,
given that τ
= t
. Assuming
independence between the elements of
ys
when conditioning on the param-
eter θ, one obtains
∏
t
−
1
∏
s
fy
(
;
θ
)
fy
(
;
θ
)
s
f
(
y
y
| =
| >
=
τ
τ
t
)
fy
fy
(
;
θ
)
∏
i
0
i
1
Lst
()
,=
s
s
i
=
1
i
=
t
=
i
i
1
.
∏
s
f
(
s
)
(
;;
θ
0
)
fy
(
;
θ
)
it
=
i
0
i
=
1
