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QUANTITATIVE METHODS
Spatial Risk Patterns-ANOVA: Multivariate Analysis Of
Suicide-Related Emergency Calls
P. Escobar-Hernandeza,b, A. L´opez-Qu´ıleza, F. Palm´ı-Peralesa, M. Marcob
aDepartment of Statistics and Operational Research, Universitat de Val`encia, Spain;
bDepartment of Social Psychology, Universitat de Val`encia, Spain
ARTICLE HISTORY
Compiled October 29, 2024
ABSTRACT
Multivariate spatial disease mapping has become a pivotal part of everyday practice
in social epidemiology. Despite the existence of several specifications for the relation
between different outcomes, there is still a need for a new strategy that focuses on
comparing the spatial risk patterns of different subgroups of the population. This
paper introduces a new approach for detecting differences in spatial risk patterns
between different populations at risk, using suicide-related emergency calls to study
suicide risks in the Valencian Community (Spain).
KEYWORDS
SRP-ANOVA; Suicide; Emergency Calls; INLA; Multivariate
1. Introduction
The analysis of social problems across spatial and temporal contexts has emerged
as an essential aspect of social epidemiology. These models, originally intended for
disease mapping (Lawson, Banerjee, Haining & Ugarte , 2016), have proven effective
in mapping various social issues, such as suicide, intimate partner violence, or
child maltreatment. Moreover, it is a common practice to include sociodemographic
covariates, known as ecological regression. (Kemp , 2005).
In practical applications, small geographic regions are often selected, creating
challenges due to sparse data with minimal observations per area. In these cases,
it is assumed that nearby areas are more similar than distant ones, which helps to
smooth the associated risk for each area. Bayesian hierarchical models are useful in
modeling the risk surface, leveraging strength from neighboring regions to achieve
local smoothness through spatial random effects. These models are formulated using
Gaussian Markov Random Fields (Rue & Held , 2005, GMRF), typically considering
neighbors areas that share a common boundary.
When multiple variables, outcomes or groups are recorded in each area, the
resulting dataset transforms into a multivariate dataset. For example, different
studies have been published analyzing the spatial overlap of intimate partner violence
P. Escobar-Hernandez. Email: pablo.escobar@uv.es
arXiv:2410.21227v1 [stat.AP] 28 Oct 2024
against woman (IPVAW) and child maltreatment (Gracia, L´opez-Qu´ılez, Marco &
Lila , 2018), crimes against women (Vicente, Goicoa & Ugarte , 2021), vehicle theft,
larceny, and burglary (Chung & Kim , 2019), disadvantage variables (Quick & Luan
, 2021), police calls reporting street-level violence and behind-closed-doors crime
(Marco, Gracia, L´opez-Qu´ılez & Lila , 2021) and substantiated and unsubstantiated
child maltreatment referrals (Marco, Maguire-Jack, Gracia & L´opez-Qu´ılez , 2020).
There are two main categories of multivariate models depending on the nature
of the relation between the different outcomes: models based on multivariate con-
ditional auto-regressive (MCAR) distributions (Carlin & Banerjee , 2002; Gelfand
& Vounatsou , 2003), a generalization of the conditional auto-regressive (CAR)
distribution, and spatial factor analysis models (Wang & Wall , 2003).
In general multivariate models, such as the MCAR, the possible relations between
the different groups of interest are included in the model using the precision matrix.
For instance, the MCAR distribution can be viewed as a conditionally specified proba-
bility model for interactions between space and a given characteristic of interest. Thus,
MCAR takes into account dependence across space and between groups/diseases.
However, practical difficulties arise from MCAR’s complex dependence structure:
most interaction’s effects would be weakly identified by the data, so the dependence
structure could not be properly identified. Strong prior distributions may improve
identifiability but most of the time there is no genuine prior information (Zhang,
Hodges & Banerjee , 2010).
On the contrary, in spatial factor analysis the different outcomes are considered
to have one or more underlying, unobserved common spatial factors, that are
estimated and combined using weights to determine the geographical pattern of a
given outcome. A special case of spatial factor models are shared component models,
where the spatial pattern for each different outcome is usually a combination of a
shared common spatial effect and some individual specific spatial effects (Held, Hohle
& Hofmann , 2005; Palm´ı-Perales, G´omez-Rubio, Bivand, Cameletti & Rue , 2023).
Expanding on the idea of shared spatial effects and increasing in complexity
there is a series of different formulations (Gelman , 2002; Hodges, Cui, Sargent &
Carlin , 2007; Mar´ı-Dell’Olmo, Mart´ınez-Beneito, Gotsens & Pal`encia , 2014; Nobile
& Green , 2000; Zhang, Hodges & Banerjee , 2010), under the name of Smoothed
Analysis-Of-Variance (SANOVA). SANOVA selects for effects that are large, removing
most of those that are small (Zhang, Hodges & Banerjee , 2010). In contrast with
MCAR models, that rely on estimating weakly identifiable parameters, SANOVA
focuses on smoothing interactions, thus providing more stable and reliable results.
Among this formulation, Zhang, Hodges & Banerjee (2010) considers series of
orthogonal contrasts between diseases, i. e., a set of linear combinations of the out-
comes we wish to model. Subsequently, Mar´ı-Dell’Olmo, Mart´ınez-Beneito, Gotsens
& Pal`encia (2014) proposed a reformulation that relies on using spatially structured
random effects for each of the contrasts instead of employing such effects for the
independent modeling of every one of the groups. Since the linear combinations are
defined by the user manually, the spatial dependence assumed is restricted to the
combinations chosen. This is a reasonable assumption in most cases, since often the
spatial pattern is not specific to a given group but rather due to the distribution of
2
some risk factors which may influence more than one group. In consequence, we can
attribute spatial dependence to certain combinations of the risk for some groups.
Moreover, this formulation allows for extensions such as multivariate ecological
regression or spatio-temporal models.
Lastly, Botella-Rocamora, Mart´ınez-Beneito & Banerjee (2015) and Mart´ınez-
Beneito, Botella-Rocamora & Banerjee (2017) extended the idea of the SANOVA
model using a multidimensional approach, known as M-Model. The proposal is
based on using linear combinations of proper CAR spatial effects, adjusting different
loadings for each group. The M-Model takes its name from the Mmatrix that de-
fines the aforementioned loadings of the different CAR spatial effects for each outcome.
The differences between multidimensional and SANOVA have been discussed
extensively. Mart´ınez-Beneito, Botella-Rocamora & Banerjee (2017) addressed
the main issues with SANOVA in comparison to multidimensional approaches,
such as the M-Model. In particular, multidimensional modeling does not need a
specific selection of contrasts for making geographical comparisons, eliminating the
contrast-dependency found in SANOVA results. In addition, SANOVA would have to
include several interaction terms between contrasts to achieve the same flexibility as
multidimensional models, thus losing the advantage of being less parameterized.
However, in some particular datasets, where data may not be sufficiently informa-
tive, the numerical approximations can fail to properly estimate the rich covariance
structure of multidimensional approaches. In addition, the interpretation of the M
matrix that defines the loadings for each different CAR spatial effect for each outcome
is not straightforward, hindering the conclusions that can be extracted regarding
spatial-risk patterns of the different groups of interest. Finally, even if the model is
perfectly specified and data is informative enough, the computational cost of using
this type of models increases exponentially the number of spatial units. This implies
that, although theoretically the number of outcomes that can be jointly analyzed
is unlimited, in practice the computational cost of extending the model to a large
geographical area increases significantly.
After reviewing the current state of the art, we considered that there was a place
for an ANOVA-like specification, somewhere between the original Smoothed-ANOVA
(Mar´ı-Dell’Olmo, Mart´ınez-Beneito, Gotsens & Pal`encia , 2014) and his final gener-
alization of the M-Model (Mart´ınez-Beneito, Botella-Rocamora & Banerjee , 2017).
This new modelization, designated as Spatial Risk Patterns-ANOVA (SRP-ANOVA),
aims to determine whether different stratum of a given population presents different
spatial risk patterns for a given outcome. Specifically, we will focus on a 2-way
ANOVA, where each group has 2 different categories, adding up to 4 possible different
groups.
The remainder of the paper is structured as follows. In Section 2 we present a
detailed description of the new methodology proposed, named Spatial Risk Patterns-
ANOVA (SRP-ANOVA). Section 3 contains the specifications of the implementation
from a bayesian approach. Section 4 includes the analysis of spatial risk patterns in
suicide, using suicide related-emergency calls as outcome, including 3 different groups
(COVID-19 period, gender and type of caller) combined in pairs. Finally, Section 5
contains the discussion and conclusions extracted from this paper.
3
2. Spatial Risk Patterns-ANOVA (SRP-ANOVA)
Let Ei
gbe the number of persons at risk in area i(i= 1, ..., I ) and group g
(g= 1, ..., G), with Yi
gcorresponding to the number of cases observed in area iin
group g.θi
grepresents the underlying true-area-group specific relative risk, as long as
we can assume that the observations are conditionally independent random sample
from a given probability distribution from the exponential family, typically a Poisson
distribution:
Yi
g∼Poisson Ei
gθi
g.(1)
When estimating the expected values, we need to address the difference between
multivariate and multidimensional analysis. In multivariate analysis, we are interested
in studying several outcomes jointly. For example, child maltreatment and suicide
rates, or two independent diseases like lung cancer and liver cancer. Therefore, Ei
g
is calculated individually for each outcome, i.e., the expected number of cases for a
given outcome in a given area depends on the population of the area and the total
number of cases for that given outcome.
In contrast, when we are using multidimensional analysis with an ANOVA-like
design, we focus on the incidence of a given disease in different subgroups of the
population. This is the case, for example, when we are studying the incidence of a
given disease in male vs female population, or using different age groups. In this
case, Ei
gis calculated combining groups because the outcome is the same for each
group and, therefore, we expect all groups to have the same risk if there is no
difference among them. In consequence, for a given group in a given area, the number
of expected cases depends on the sum of all the cases in all the groups and the
population of the area divided by the number of groups.
However, it is important to note that whenever we use groups that are not equally
represented in the general population, we should use the actual population at risk
for each group. For example, if we are interested in a particular outcome and the
difference between the local and foreign populations, the expected values must take
into account the differences in population between the two groups.
The underlying risk θi
gis modeled in a log-scale, taking advantage of the capability
of INLA for running models with several likelihoods (G´omez-Rubio , 2020). In
particular, the log-relative risk for each group is treated as an independent model,
but we are able to estimate a given spatial effect for one of the groups and then
add it as a fixed copy to the rest of the groups. This allows us to sequentially add
different shared spatial effects, that are estimated using a given group as reference and
then built-in the rest of the groups. All the spatial effects included follow a Besag’s
specification with flat uniform prior distributions as priors assigned to the standard
deviation parameters (Palm´ı-Perales, G´omez-Rubio, Bivand, Cameletti & Rue , 2023).
Specifically, we propose a sequential modelization strategy, where we increase the
number of spatial effects on each step and determine if the fitting of the model has
improved enough to justify the increase in complexity using DIC (Spiegelhalter,
4
Best, Carlin & Van der Linde , 2002). Since DIC penalizes excessive complexity, the
inclusion of a spatial effect that does not significantly modify the previous shared
spatial effects will be punished, giving a worse DIC score. This allows us to use DIC
values to detect which spatial effects are different enough from zero and, consequently,
which factors are responsible for some of the spatial differences observed within the
given outcome of interest. Another possible option would be using WAIC (Watanabe
, 2010) as a comparison metric, but since WAIC has being reported to have issues
with correlated data (Gelman, Hwang & Vehtari , 2013), such as spatial or temporal
data, we have opted for using DIC. Nevertheless, we present the WAIC as well in
Section 3.
The sequential steps are the following:
•M0: the outcome of interest does not exhibits a spatial risk pattern. This initial
model includes an individual intercept for each given group that accounts for
the 4 different baseline risk levels α1−4, as well as individual unstructured effects
ωi
1−4representing the heterogeneity effects for each area in each group:
log θi
1=α1+ωi
1
log θi
2=α2+ωi
2
log θi
3=α3+ωi
3
log θi
4=α4+ωi
4
(2)
•M1: this models contemplates the possibility that each group presents spatial
patterns different enough to justify using one for each group. Essentially, since
there is no shared effect, this model is an independent univariate specification for
each group. Spatial structured effects ϕi
1−4represent the individual spatial effects
for each group, spatial unstructured effects ωi
1−4represent the heterogeneity
effects for each group and α1−4are the intercepts that define the baseline risk
level for each group:
log θi
1=α1+ωi
1+ϕi
1
log θi
2=α2+ωi
2+ϕi
2
log θi
3=α3+ωi
3+ϕi
3
log θi
4=α4+ωi
4+ϕi
4
(3)
•M2: this model is the first one to include shared spatial effects. The notation
hereafter for the shared spatial effects is the following: ϕi
jk represents the spa-
tial effect estimated using level jof the first factor considered, and level kof
the second factor considered. The groups are included in the following order: g1
corresponds to θi
11,g2corresponds to θi
12,g3corresponds to θi
21 and g4corre-
sponds to θi
22. In this particular model, the outcome of interest presents a spatial
risk pattern ϕi
11, estimated using the first level of the first factor and the first
level of the second factor. Moreover, each group has a different baseline level,
represented by α1−4, and an individual unstructured term ωi
1−4:
5
log θi
1=α1+ωi
1+ϕi
11
log θi
2=α2+ωi
2+ϕi
11
log θi
3=α3+ωi
3+ϕi
11
log θi
4=α4+ωi
4+ϕi
11
(4)
•M3: the outcome of interest exhibits a spatial risk pattern ϕi
11 estimated using the
first level of the first factor and the first level of the second factor, and this pattern
is shared for the rest of the groups. Moreover, the first factor exhibits certain
particularities that modify the general spatial risk and is, therefore, shared by
those two groups, in this case represented by ϕi
21. In this case, ϕi
21 is estimated
using the second level of the first factor, and the first level of the second factor.
As usual, the 4 different groups present a baseline level represented by α1−4and
the unstructured term ωi
1−4:
log θi
1=α1+ωi
1+ϕi
11
log θi
2=α2+ωi
2+ϕi
11
log θi
3=α3+ωi
3+ϕi
11 +ϕi
21
log θi
4=α4+ωi
4+ϕi
11 +ϕi
21
(5)
•M4: the outcome of interest presents a spatial risk pattern ϕi
11 estimated using
the first level of the first factor and the first level of the second factor, that is
shared for the rest of the groups. Moreover, the second factor exhibits certain
particularities that modify the general spatial risk and is, therefore, shared by
those two groups, in this case represented by ϕi
12. In this case, ϕi
12 is estimated
using the first level of the first factor, and the second level of the second factor.
As usual, the 4 different groups present a baseline level represented by α1−4and
the unstructured term ωi
1−4:
log θi
1=α1+ωi
1+ϕi
11
log θi
2=α2+ωi
2+ϕi
11 +ϕi
12
log θi
3=α3+ωi
3+ϕi
11
log θi
4=α4+ωi
4+ϕi
11 +ϕi
12
(6)
•M5: the outcome of interest presents a spatial risk pattern ϕi
11 estimated using
the first level of the first factor and the first level of the second factor. Moreover,
the first factor exhibits certain particularities that modify the general spatial
risk and is, therefore, shared by those two groups, in this case represented by
ϕi
21. In addition, the second factor presents different spatial risk factors that
are not reflected by neither the general share spatial effect nor the first factor
spatial effect but do not depend on the level of the first factor. In this case, this
is represented again with ϕi
12. Finally, the 4 different groups present a baseline
level represented by α1−4and the unstructured term ωi
1−4:
6
log θi
1=α1+ωi
1+ϕi
11
log θi
2=α2+ωi
2+ϕi
11 +ϕi
12
log θi
3=α3+ωi
3+ϕi
11 +ϕi
21
log θi
4=α4+ωi
4+ϕi
11 +ϕi
21 +ϕi
12
(7)
•M6: this final model implies that there is an overall share spatial risk pattern
for all the groups, the first factor presents an specific underlying risk factor that
modifies the share common factor and there is an interaction between the first
and the second factor such that, for each level of the first factor exists a different
underlying spatial pattern for the second factor. In terms of formulation, this
translates to:
log θi
1=α1+ωi
1+ϕi
11
log θi
2=α2+ωi
2+ϕi
11 +ϕi
21
log θi
3=α3+ωi
3+ϕi
11 +ϕi
12
log θi
4=α4+ωi
4+ϕi
11 +ϕi
12 +ϕi
22
(8)
where:
◦α1−4are the 4 different intercepts that represent the baseline risk level for
each group.
◦ωi
1−4are the 4 different unstructured heterogeneity spatial effects.
◦ϕi
11 captures an overall-shared spatial effect, estimated using the first level
of the first factor and the first level of the second factor.
◦ϕi
21 captures the possible spatial variation explained by the second level of
the first factor.
◦ϕi
12 captures the possible difference in the spatial risk pattern caused by
the interaction of the first level of the first factor, and the second level of
the second factor.
◦ϕi
22 captures the possible difference in the spatial risk pattern caused by
the interaction of the second level of the first factor, and the second level
of the second factor.
It is straightforward to notice that, in several of the models, the order of the groups
could have an impact on the estimation of the spatial effects. In particular, M0and
M1can not be affected since essentially the specification is equivalent to an univariate
scenario. In M2the reference group could produce different spatial patterns, which
would affect the performance of the model. Therefore, it is reasonable to test the 4
different models where the reference group is exchanged.
In M3and M4we need to test if depending on the reference group for the given
factor studied the overall fitting of the model is affected. This implies that we need
to consider two different possibilities for each one of them.
M5can be viewed as a combination of models M3and M4. Therefore, we need to
consider the two possibilities of each specification depending on the reference level
7
for each factor. Since we are working with 2-level factors, this adds up to 4 different
models.
Finally, M6contemplates the interaction between the two factors. In this case,
in addition to the 4 possibilities considered in M5, we must also take into account
that, depending on the order of inclusion of the factors, the estimation of the spatial
effects will vary. This is cause by the nature of the interaction: the first factor is used
for the spatial effects ϕ11 and ϕ12, which are shared by different groups, whereas
the individual spatial effects that arise from the interaction among the two factors
are not shared by other groups. In consequence, if the factor that is considered
initially varies, the decomposition of the spatial variance may be equivalent, but
the results are not. Consequently, we need to consider both possibilities. All of
this adds-up to 8 different possibilities that need to be tested. A general overview
of the spatial effects included and the number of combinations can be found in Table 1.
Table 1. Total number of combinations tested for each model.
Model Spatial Structured Effects Combinations of groups
M0: Intercepts - 1
M1: Individual spatial effects ϕ1,ϕ2,ϕ3,ϕ41
M2: Shared Effect ϕ11 4
M3: F1L2 Effect ϕ11,ϕ21 2
M4: F2L2 Effect ϕ11,ϕ12 2
M5: F1L2 Effect + F2L2 Effect ϕ11,ϕ12 ,ϕ21 2x2
M6: F1L2 Effect * F2L2 Effect ϕ11,ϕ12 ,ϕ21,ϕ22 2x2x2
Total Number 22
The principle behind this strategy is simple, with a similar approach to an ordinary
ANOVA design, giving the researcher a tool to effectively compare spatial risk patterns
for different groups and select the model that better fits the actual existing differences
between the subgroups of interest. Furthermore, there is no initial supposition about
the relationship between the different groups and, since the underlying structure is
relatively simple, we are capable of testing all the options within a sensible amount
of time.
3. Bayesian Inference with INLA
This modelization is implemented within a Bayesian approach through the Inte-
grated Nested Laplace Approximation (INLA) (Rue, Martino & Chopin , 2009). INLA
aims to approximate the posterior distribution, in contrast with asymptotically exacts
methods methods like the Markov Chain Monte Carlo (MCMC) methods, using a
combination of Laplace approximations and numerical integration. It provides a fast
and accurate way to estimate the posterior mean, variance, and other parameters
without the need for extensive simulation. This makes INLA particularly attractive
for complex hierarchical models, such as those encountered in disease mapping,
spatial statistics, and other fields where efficient computation is essential.
In particular, we designed a custom function that estimates all the aforementioned
models simultaneously. To ensure that the different spatial effects are identifiable and
8
comparable, each random spatial effect is estimated with a sum-to-zero constraint
and scaled to have an average variance of 1.
Different specifications for the random spatial effects could be implemented, which
may affect the results obtained. Specifically, we have opted for using ICAR spec-
ifications, choosing flat uniform distributions for the standard deviation as prior’s
distributions (Palm´ı-Perales, G´omez-Rubio, Bivand, Cameletti & Rue , 2023).
4. Analysis of Suicide-Related Emergency-Calls
Suicide poses a significant social and public health challenge, resulting in over
700,000 deaths globally each year (WHO , 2021). Official data is showing an in-
creasing trend in the last decades and there is a rising concern about the impact
of COVID-19, (Garc´ıa-Fern´andez et al. , 2003) and the subsequent restrictive
policy measures adopted on mental health (WHO , 2022). In the pursuit of a
more profound comprehension of suicide and the risk factors associated with it,
epidemiological research has become an increasingly popular field of study (Zalsman
et al. , 2017). Most studies have traditionally analyzed spatial and spatio-temporal
patterns aggregated for the entire population but in recent years several studies
have analyzed the spatial patterns of male and female suicide outcomes (Cayuela,
Cerame del Campo, L´opez-S´anchez, Rodr´ıguez Dom´ınguez & Cayuela Dom´ınguez ,
2020; Chang et al. , 2011; Lin, Hsu, Gunnell, Chen & Chang , 2019; Marco et al. , 2024).
In order to determine whether distinct risk groups exhibit disparate spatial risk
patterns, we will employ the SRP-ANOVA strategy, as outlined in the preceding
section. Our response variable will be suicide-related emergency calls in the Valencian
Community (Spain), spanning the period from 2018 to 2023, distributed in 542
municipalities. In particular, data will be divided per period (preCOVID-19 vs
postCOVID-19), gender of the person in crisis (male vs female) and type of person
making the call (person in crisis vs bystander). The aforementioned factors will be
combined in pairs, with the objective of studying the spatial overlap for each of the
three combinations of categories (gender vs period, caller vs period and gender vs
caller). All the modelization has been performed using an ASUS TUF Gaming F15
laptop, with a 11th Gen Intel Core i7 2.30GHz processor and 16GB of RAM.
Specifically, the distribution of calls for each combination of groups is the following:
•Gender vs Period: our dataset presents 8788 calls from males preCOVID-19,
10908 calls from females preCOVID-19, 17189 calls from males postCOVID-19
and 22686 calls from females postCOVID-19. From the number of calls is
straightforward to detect that some groups present significantly higher number
of calls, which will translate in relative risk maps where most municipalities
present higher values.
•Caller vs Period: our dataset presents 4062 calls from person in crisis preCOVID-
19, 15921 calls from bystander preCOVID-19, 8177 calls from person in crisis
postCOVID-19 and 31969 calls from bystander postCOVID-19. In this case the
groups are more unbalanced, which suggests that the relative risks maps should
present more acute differences.
9
•Gender vs Caller: for this set of groups our dataset contains 5554 calls from
male person in crisis, 6322 calls from female person in crisis, 20152 calls from
male bystanders and 27126 calls from female bystanders. In this case person in
crisis and bystanders present the greatest differences, so we expect male and
female groups to present similar relative risk maps for both person in crisis and
bystanders.
4.1. Gender VS Period
Firstly, we have analyzed if the spatial risk pattern of male and female persons in
crisis changed with COVID-19 and the measures adopted. In order to balance the
groups we have used the same time span before and after COVID-19. Furthermore,
it has been assumed that the underlying population at risk is identical for all four
groups, given that the discrepancy between the male and female populations prior to
and following the onset of the pandemic is minimal.
The best 5 models in terms of DIC are presented in Table 2. For this combination
of factors, 4 of them correspond to M5. In particular, the best model is a M5
specification using preCOVID-19 and Male as basis levels. The order of the factors in
this case is not relevant.
Spatial effects for the best model are represented in Figure 1. The overall shared
effect oscillates between -0.31 and 0.32 and presents a low risk cluster of municipalities
in the north of the region, and some more in the inland south. The middle of the region
and the coastline contain most of the areas with greater risks.
Figure 1. Spatial Effects of the b est model for Gender VS Period in suicide-related emergency calls.
The postCOVID-19 effect, oscillating between -0.09 and 0.13, implies that COVID-
19 produced an increase in the risk for areas in the north and a decrease for areas in
the south, with two clear clusters. Nevertheless, the size of this effect is considerably
smaller.
10
Finally, the Female effect ranges between -0.32 and 0.32, suggesting that in this
case the effect of the gender is greater than the period effect. In particular, the spatial
effect demonstrates a distinct concentration of areas with negative values along the
central and northern coastline, as well as a significant aggregation of municipalities
with elevated risk in the central and southern inland regions.
Table 2. Top 5 Gender VS Period models in terms of DIC.
Model Combination DICs WAICs CPU (sec)
M5preCOVID-19 + Male 9131.4 9166.8 8.17
M5postCOVID-19 + Male 9131.6 9171.1 7.91
M5preCOVID-19 + Female 9132.5 9167.1 8.55
M6Female * preCOVID-19 9132.6 9166.7 10.54
M5postCOVID-19 + Female 9133.1 9171.6 8.37
Figure 2 illustrates the adjusted relative risks. In this case, the group with the
highest risks is the female population postCOVID-19, with several high-risk munici-
palities located in the inland middle of the region. Furthermore, the distribution of
high-risk areas is similar for males postCOVID-19. Finally, both the preCOVID-19
groups present mostly low-risk areas, with the exception of individual municipalities.
Figure 2. Relative Risks adjusted for the best model of Gender VS Period in suicide-related emergency calls.
4.2. Caller VS Period
Secondly, we will test if the spatial risk pattern differ taking into account the type
of caller (person in crisis or bystander), pre and postCOVID-19. Just like in the
previous section, the groups present the same time span before and after COVID-19.
In addition, since a given person could be both a bystander or a person in crisis, the
population at risk remains the same for all the groups.
The best 5 models in terms of DIC are presented in Table 3. For this combination of
factors, 4 of them correspond to M6. Specifically, the best model is a M6specification
using person in crisis and preCOVID-19 as basis levels. In this case, the order of the
11
factors is relevant, and for the best model factor 1 is caller and factor 2 is COVID-19.
Table 3. Top 5 Caller VS Period models in terms of DIC.
Model Combination DICs WAICs CPU (sec)
M6bystander * postCOVID-19 8571.7 8593.1 7.89
M6preCOVID-19 * bystander 8575 8587.6 7.81
M6postCOVID-19 * bystander 8575.1 8590.9 8.06
M6bystander * preCOVID-19 8594.8 8628 6.34
M5preCOVID-19 + bystander 8360.2 8648.7 7.09
Spatial effects for the best model are represented in Figure 3. The overall shared
effect ranges between -1.29 and 1.36, presenting a clear cluster of higher risk areas in
the center of the region and specially closer to the coastline on the east. Moreover,
there are two clusters north and south inland with negative values for the spatial
effects.
Figure 3. Spatial Effects of the b est model for Caller VS Period in suicide-related emergency calls.
On the contrary, the spatial effects for person in crisis show positive values in
the north and south inland, predominantly rural areas. In this case, the size of the
effect is respectable, oscillating between -0.82 and 2.15, suggesting the presence of
municipalities where the risk for person in crisis is considerably higher.
Bystander preCOVID-19 interaction effect presents an extremely clear north-south
distribution, with smaller values that slightly modify the previous spatial patterns,
ranging between -0.09 and -0.12. Finally, the person in crisis preCOVID-19 interaction
effect exacerbates the patterns observed in the general effect, presenting the most
extreme values ranging between -2.01 and 3.05.
Adjusted relative risks are represented in Figure 4. As expected by the number of
counts, the bystander postCOVID-19 group presents high risk values in most of the
region, with the exception of a cluster of areas in the inland north and the inland south,
which coincides with the spatial effects observed. Bystander preCOVID-19 presents
several moderately high risk areas in the coastline, but the rest of the region has low
12
risk. Finally, both person in crisis pre and postCOVID-19 have low risk in all the
region, with the exception of several individual municipalities.
Figure 4. Relative Risks adjusted for the best mo del of Caller VS Period in suicide-related emergency calls.
4.3. Gender VS Caller
Lastly, we will analyze whether the spatial risk pattern of male and females in
crisis varies depending on whether the person making the call is either a bystander
or a person in crisis. In this case, it is assumed that the underlying risk population is
equal for all groups, given that an individual may be both a bystander and a person
in crisis, and that the female and male populations are balanced.
The best 5 models in terms of DIC are presented in Table 4. For this combination
of factors, all of them correspond to M6. In particular, the best model uses male and
bystander as basis levels, with gender being the first factor and caller the second factor.
Table 4. Top 5 Gender VS Caller models in terms of DIC.
Model Combination DICs WAICs CPU (sec)
M6Male * bystander 8658.4 8684.4 9.71
M6Female * bystander 8662.7 8678.3 7.65
M6bystander * Female 8686.9 8711.1 7.84
M6bystander * Male 8720.3 8752 8.11
M6person in crisis * Female 8733.3 8727.2 9.37
Figure 5 represents the spatial effects adjusted for the best model. For the overall
shared effect, there is a low risk zone in the inland north, and a high risk zone in the
south of the region. Moreover, the municipalities in the middle of the region exhibit
greater values as well. In this case, this effect ranges between -1.34 and 1.39.
The female effect oscillates between -0.34 and 0.44, slightly modifying the overall
shared effect obtained with male bystander as reference group. This effect presents
a greater risk in the north, especially in the coastline, and a lower risk in the south,
13
especially inland.
Figure 5. Spatial Effects of the b est model for Gender VS Caller in suicide-related emergency calls.
The spatial effect for the male person in crisis interaction reflects a big clus-
ter of high risk municipalities in the mid inland of the region, ranging between
-1.29 and 4.01. Finally, the female person in crisis effect ranges between -0.99
and 2.09, showing some high risk groups of municipalities in the inland north and
south, with a cluster of lower risk municipalities in the central inland part of the region.
Lastly, Figure 6 represents the adjusted relative risks for the best model. In this case,
male and female bystanders present similar patterns. Both groups exhibit clusters of
high risk areas in the inland central part of the region, as well as the whole coastline.
Female and male person in crisis present most of the municipalities as low risk, which
is to be expected considering the number of counts for each group.
Figure 6. Relative Risks adjusted for the best model of Gender VS Caller in suicide-related emergency calls.
14
5. Discussion
Taking advantage of our formulation, we have been able to detect clearly defined
spatial risks patterns for the different subgroups studied, with an extremely reason-
able computation cost. The results presented analyzing suicide-related emergency
calls provide a complementing view for the existing literature of this major social
problem. Most of the studies published so far have relied on either mortality data or
suicide-related health records (Bridge et al. , 2018; Congdon , 2011; Helbich, Plener,
Hartung & Bl¨uml , 2017), with some studies using suicide-related emergency calls
(Lersch , 2020; Marco, Gracia, L´opez-Qu´ılez & Lila , 2018; Marco et al. , 2024). By
the time of this publication, this is the first article where several emergency calls
from different subgroups of the population have been modeled and compared using
multivariate disease mapping. The potential causal factors underlying the observed
patterns remain to be identified. However, it is evident that different population
strata are exposed to different risk factors, which are themselves spatially distributed.
Further research should consider the addition of sociodemographic covariates, with
the potential for investigation into whether the regression coefficients remain consistent
across all of the considered subgroups. Nevertheless, this would create new identifica-
tion problems, with spatial confounding occurring between the regression coefficients
and the different shared spatial effects included (Urdangarin, Goicoa & Ugarte , 2022).
Furthermore, the number of groups and categories that can be analyzed using the
proposed formulation is a notable limitation. Future research could be conducted
with the aim of increasing the number of groups and categories per group, or both.
Nevertheless, it is evident that any of these options would significantly increase the
number of models that must be considered. Specifically, the addition of groups would
introduce higher-order interactions, which could be challenging to identify if the
dataset is not highly informative. The increase in categories per group would necessi-
tate testing whether all categories are distinct or not, both within and between groups.
Another possible expansion of this formulation would be including a temporal di-
mension, leading to the development of multivariate spatio-temporal formulations. In
this case, it is necessary to consider different specifications, given that both space and
time could interact with the categories of the different groups included. Furthermore,
spatio-temporal models typically include an interaction term between space and
time. Despite the existence of four distinct types, as outlined by (Knorr-Held , 2000),
the prevalent approach involves utilizing the Type IV interaction. This interaction
enables each spatio-temporal unit to influence the overall spatial and temporal effects,
resulting in a highly intricate precision matrix, which our formulation is trying to
avoid in the first place.
In conclusion, this publication presents a new strategy for modelling multidimen-
sional data, whereby several subgroups of the population may exhibit disparate spatial
patterns. The principal advantages are a straightforward design, which facilitates the
communication of results to a broader audience of social psychology researchers, and
the low computational cost. The code of the function, as well as the code for the
figures, can be found in https://github.com/VdaK1NG/SRP-ANOVA.
15
Acknowledgments
We wish to thank the 112 Valencian Community Emergency Service (Valencia Agency
of Security and Response to Emergencies) for providing the dataset.
Disclosure statement
The authors of this article declare no conflict of interest.
Funding
This study was funded by the Social Observatory La Caixa Foundation
(LCF/PR/SR21/52560010). ALQ and FPP thank support by the grant PID2022-
136455NB-I00, funded by Ministerio de Ciencia, Innovaci´on y Universidades of Spain
(MCIN/AEI/10.13039/501100011033/FEDER, UE) and the European Regional De-
velopment Fund.
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