Page 209 - Modern Spatiotemporal Geostatistics

190 Modern Spatiotemporal Geostatistics — Chapter 9

with both the health effect and the risk factor; etc.). This decision seems to
be justified by the fact that—as it will be shown in Example 9.15 below—the
incorporation of PMi 0 distribution as a possible confounder has little effect on
the quantitative assessment of the temperature exposure-mortality association.
As a result, condition (Hi.) was assumed valid at this stage of the analysis; the
consideration of confounders is examined in Example 9.15. Regarding the main
spatiotemporal condition of Postulate 9.1, its application requires death-rate
predictions based on: (a) soft death-rate data only; and (b) on the combination
of soft death-rate and hard temperature-exposure data. Both predictions (a)
and (b) are obtained with the help of BME techniques. Death-rate predictions
were first obtained using soft death-rate data which were available in the form
of intervals; i.e. D(s, t) € (I, u), where / is a lower rate value and u is a
upper value. Examples of such interval soft data are shown in Figure 9.8. Let
fo (Dk) denote the posterior pdf of the death rates Dk = D(sk, t k) which are
predicted at space/time points (sk, tk) using death-rate data at neighboring
counties as well as at the same county but during different days. For illustra-
tion, the pdf obtained for County 7 during the day tk = t? = 415 is shown
in Figure 9.9. On the basis of the fjj (D?) of Figure 9.9, the most probable
death-rate prediction (mode) DT\D — 2.78 (deaths per 100,000 people per
day) was found, as well as other desirable estimates. Next, using both the
interval death-rate data and the hard temperature-exposure data, the vector
BME approach provided the posterior pdf fox(Di) for County 7 during the
same day tk = ti = 415 (shown in Fig. 9.9).
The fDx(D-r) is clearly an improvement over fo(D^. It leads, e.g.,
to the mode death-rate prediction of DT\DX — 3.35 (deaths per 100,000
people per day) that is closer to the actual death rate D-j = 3.77 than the
£>7|£) previously calculated (£)y was assumed unknown during the prediction
process). According to the PEP criterion, these results support an associa-
tion between colder temperatures and death rates at the population level. To
proceed further with the study of the temperature exposure-death-rate associ-
ation, death-rate predictions D k\D and D k\DX were produced at all counties
in the state for all days of the 1996 winter season 5. As before, while the
Dk\D predictions were obtained at each space/time point using only death-
rate data, the D k\ox predictions were calculated using death-rate data as
well as temperature data. On the basis of these predictions, the death-rate
prediction errors ek\D = \Dk- Dk\D\ and ek\DX = \Dk - Dk\ox\ were
calculated at each space/time point (sk, tk). The association between expo-
sure to colder temperature and death rate was then investigated by computing
the PEP parameter f3ox of Equation 9.48 above (in %), where EDX and ED,
respectively, are the arithmetic averages of the ek\ox ar|d ek\D values over
a spatiotemporal domain consisting of all counties and a time window of 30
days. According to the PEP criterion, negative /?£>A-values would support the
cold temperature exposure-death-rate association (whereas zero values would
not support such an association).