3.2 Space-time models with two-level spatial random effects
3.4.1 Data generation
Counts are generated using a Poisson distribution with mean eitrit where the ex-
pected number of cases is 1, 3, 5 or 10 giving rise to four different subscenarios. A common true spatio-temporal log-risk surface has been generated as the sum of a FLA random term (ξi), a SLA geographical effect (ψj(i)) -provinces in Scenario 1
and health areas in Scenario 2-, a global temporal effect (γt) and a SLA space-time
interaction effect (δj(i)t) i.e.,
log rit= ξi+ ψj(i)+ γt+ δj(i)t for j = 1, . . . , m; t = 1, . . . , T. (3.6)
Scenario1
To create a scenario where the geographical distribution of risks varies among the provinces, m = 4 independent random effects have been generated from a LCAR distribution with different amount of spatial variability, i.e, the spatial random effect ξ = (ξ1, ξ2, ξ3, ξ4)0 has been generated as
ξj ∼ N (0, [τξj(λξjRξj+ (1 − λξj)Inj)]
−1
), for j = 1, . . . , 4
where Rξj is the spatial neighborhood matrix of municipalities within each province,
Inj is the identity matrix of size nj (number of municipalities in province j) and λξj
takes the values {0.1, 0.4, 0.6, 0.9}. The precision components τξj have been chosen
so that the random effects ξj take values in the range [−0.415, 0.405], which is equivalent to exp(ξj) ∈ [0.66, 1.5]. Note that 1.5 means that the risk of that area is 50% higher than in the overall study area, and 0.66 (1/1.5) means the same 50% but lower than the overall study area.
As only four SLA are considered in this scenario, the province level random effect ψj(i) has been fixed in order to keep the simulated log-risks into a controlled range.
62 Two-level spatially structured models
Municipality level spatial pattern ξi
0.66 0.73 0.81 0.90 1 1.10 1.22 1.35 1.5 5 10 15 20 0.9 1.0 1.1 1.2 e xp ( γt ) Temporal pattern
Figure 3.5: Map of the municipality level spatial patterns of mortality risk (left) and global temporal pattern (right) in Scenario 1.
Specifically, it has been set up as
ψ = (0.2 × 10n1, −0.2 × 10n2, 0.1 × 10n3, −0.1 × 10n4)0,
where 1nj are vector of ones of length nj, i.e., the random effect is defined so that
municipalities within the same province take the same value.
The temporal random effect γt has been randomly generated from a first order
random walk
γ ∼ N (0, [τγRγ]−),
where Rγ is the T × T structure matrix of a RW1. The precision component τγ
has been chosen so that γt takes values in the range [−0.415, 0.405], similar to
the spatial case. To imitate a temporal pattern corresponding to a possible real scenario, a smooth version of this random effect has been considered. Both spatial and temporal patterns are shown in Figure 3.5. Finally, province level space-time interactions δj(i)t have been generated from different parametric trend shapes for
each province (see Figure 3.6).
Scenario2
In this scenario, the number of SLAs has been increased to m = 13, corresponding to the number of health areas of Navarre and the Basque Country. Similarly to Scenario 1, m independent LCAR distribution random effects have been generated
3.4 Simulation study 63 5 10 15 20 −0.2 −0.1 0.0 0.1 0.2 Álava 5 10 15 20 −0.2 −0.1 0.0 0.1 0.2 Gipuzkoa 5 10 15 20 −0.2 −0.1 0.0 0.1 0.2 Navarre 5 10 15 20 −0.2 −0.1 0.0 0.1 0.2 Biscay
Figure 3.6: Province level space-time interactions δj(i)t for the generated log-risks
surface in Scenario 1.
for the municipality level spatial random effect ξ = (ξ1, ξ2, . . . , ξm)0, where λξj takes
the same values as in Scenario 1, {0.1, 0.2, 0.6, 0.9}, in such a way that health areas within the same province have a common value. However, unlike Scenario 1, a LCAR distribution with a spatial smoothing parameter equal to λψ = 0.75 has been
considered for the SLA random effect. Once again, the precision components τξj
and τψ have been chosen so that random effects ξj and ψ take values in the range
[-0.415,0.405]. The sum of both spatial patterns is plotted on the left inFigure 3.7. For the global temporal pattern, the same random effect γt defined for Scenario
1 has been used. Finally, specific temporal trends have been generated for each health area to determine the space-time interaction term δj(i)t (see right hand side
plot in Figure 3.7). These second-order polynomial trends change gradually from a U-shaped curve for the north-westernmost region (Ezkerraldea-Enkarterri) to an inverse-U shaped curve for the south-easternmost health area (Tudela).
The final true spatio-temporal risk surfaces defined in Equation (3.6) for both Scenarios 1 and 2 are shown inFigure 3.8. The amount of variability explained by the interaction term is about 20% in both cases. The simulated counts are generated from a Poisson distribution with mean µit = eitrit giving rise to eight different
subscenarios (four different expected values for each scenario). For each subscenario thirty data sets have been generated and five different models have been fitted: a spatio-temporal LCAR model (see Equation (1.4)) and four two-level models; in particular, a two-level model A (with FLA space-time interactions), two-level models B and D (with SLA interactions), and finally, a two-level structure model F with m independent spatial random effects, ξj ∼ N (0, [τξj(λξjRξj + (1 − λξj)Inj)]
−1) (with
SLA interactions). Models C and E are not considered because after fitting models A and B, model B (with SLA interactions) is always better (seeTable 3.5). In all these models, a LCAR and a RW1 prior have been used for the municipality-level spatial random effect ξi and the structured temporal effect γt, respectively. Type II and
64 Two-level spatially structured models
Spatial pattern: ζi=exp(ξi+ ψj(i))
0.62 0.70 0.79 0.89 1 1.13 1.27 1.43 1.60 5 10 15 20 −0.2 −0.1 0.0 0.1 0.2 Spatio−temporal pattern δj(i)t
Figure 3.7: Map of the sum of municipality level ξi and basic health area level ψj(i)
spatial patterns of mortality risk (left) and SLA space-time interactions δj(i)t (right)
for the generated log-risks surface in Scenario 2.
Type IV space-time interactions have been fitted for both municipality and province level interactions. In Type II interactions, temporal trends are different from region to region but do not have any structure in space while in Type IV interactions the temporal trends are likely to be similar for adjacent regions. Finally, for SLA spatial random effects ψj(i), an exchangeable prior N (0, τψ−1Im) has been considered for the
province level random effect in Scenario 1, while a LCAR prior has been used for the health area level random effect in Scenario 2.