REVISTA DE MATEMÁTICA: TEORÍA Y APLICACIONES 2014 21(2) : 283–294
CIMPA – UCR ISSN: 1409-2433
MORTALITY AMONG YOUNG NICARAGUAN
IMMIGRANTS TO COSTA RICA: AN APPLICATION
OF GEOGRAPHICALLY WEIGHTED STATISTICAL
REGRESSION
MORTALIDAD ENTRE LOS INMIGRANTES
NICARAGÜENSES EN COSTA RICA: UNA
APLICACIÓN DE LA REGRESIÓN GEOGRÁFICA
PONDERADA
ROGER E. BONILLA∗ JUAN B. CHAVARRÍA†
Received: 7/May/2013; Revised: 21/May/2014;
Accepted: 10/Jun/2014
∗Escuela de Estadística, Universidad de Costa Rica, 2060 San José, Costa Rica. Fax: (506)
2511-6483, (506) 2511-6500. E-mail: roger.bonilla@ucr.ac.cr
†Misma dirección que/same address as R. Bonilla. E-mail: jchavarr@fce.ucr.ac.cr
283
284 R.E. BONILLA – J.B. CHAVARRÍA
Abstract
This paper applies a geographically weighted statistical regression
(GWR) model to young Nicaraguan immigrant homicides in Costa Rica
during the period 1998-2008 and identifies possible covariates. The pa-
rameters for the GWR model are:
Yi(g) = β0(g) + β1(g)x1 + β2(g)x2 + . . .+ βk(g)xk + ε,
which may be obtained from the solution of
β(g) = (XTW (g)X)−1XTW (g)Y.
The GWR model is a more adequate model than the classic models, such
as the log-linear Poisson model. In the GWR model, poverty was the most
significant variable. The map of the estimators associated with the percent-
age of poor households suggests that the relationship between poverty and
mortality by homicide for young Nicaraguan immigrants is stronger in the
Caribbean region and neighboring zones. When the GWR model was run
for homicides among young Costa Ricans, this effect was not observed, as
it was among Nicaraguan immigrants.
Keywords: geographically weighted regression (GWR); spatial correlation; homi-
cides; Costa Rica; immigration.
Resumen
Este trabajo aplica un modelo de regresión estadística especial pon-
derada (GWR) a los homicidios de inmigrantes nicaragüenses jóvenes en
Costa Rica en el período 1998-2008 e identifica sus posibles covariables.
Los parámetros del modelo GWR
Yi(g) = β0(g) + β1(g)x1 + β2(g)x2 + . . .+ βk(g)xk + ε,
β(g) = (XTW (g)X)−1XTW (g)Y.
El modelo GWR es un modelo más adecuado con respecto a modelos clási-
cos como el log-lineal de Poisson. En el modelo GWR la variable pobreza
resultó la más significativa. El mapa de los estimadores asociados con el
porcentaje de hogares pobres sugiere que la relación entre la pobreza y la
mortalidad por homicidios de jóvenes nicaragüenses es más fuerte en el
Caribe y zonas aledañas. Cuando el modelo GWR se aplicó a los homi-
cidios entre jóvenes costarricenses, este efecto no se observó.
Palabras clave: regresión estadística espacial ponderada (GWR); correlación
espacial; homicidios; Costa Rica; inmigración.
Mathematics Subject Classification: 91B72, 62J99, 62J05.
Rev.Mate.Teor.Aplic. (ISSN 1409-2433) Vol. 21(2): 283–294, July 2014
MORTALITY AMONG YOUNG NICARAGUAN IMMIGRANTS 285
1 Introduction
The geographically weighted statistical regression model (GWR) allows differ-
ent relationships between the dependent variable and independent covariates ex-
isting in different geographic points (x, y) (municipalities) within a geographic
space (Brunsdon et al. 1996). The GWR model produces a set of β estimators for
a regression in each geographic unit: the value for the geographic unit plus the
values for the neighboring municipalities, weighted by distance. In each regres-
sion, the observations are weighted by a function related to the distance of the
geographic unit for which the regression is being carried out. The function used
depends on an estimate of the bandwidth obtained by a Monte Carlo simulation
procedure [7], [4].
The function for the weights Wj used in the GWR model has the following
form [5]:
Wj = exp
(
−dj
b2
)
Where di is the Euclidean distance from the centroid of the ith municipality
and each one of the centroids of the other municipalities where the regression is
being carried out so
arg minb =
n∑
i=1
(yi − y6=i(b))
2,
where y6=i(b) is the fitted value of the GWR model using a parameter b omiting
the observation i from the calibration process.
In each Monte Carlo simulation the values are distributed randomly through-
out the space and the GWR model is repeated. Once the bandwidth calibration
process is completed, the GWR model uses the best-fit bandwidth to adjust the
GWR model in each geographic unit.
In modeling count data, such as homicides among Nicaraguan immigrants,
on the basis of traditional models, such as the Poisson log-linear model, there can
be problems capturing all of the relevant sources of variation, because of spatial
autocorrelation. The Poisson log-linear model assumes that the coefficients do
not vary locally. The GWR model takes into account this spatial autocorrelation
[6], [23].
Homicides among young Nicaraguan immigrants to Costa Rica
A preliminary analysis of the core information for this study allowed us to iden-
tify that homicides are the predominant cause of death among young Nicaraguan
Rev.Mate.Teor.Aplic. (ISSN 1409-2433) Vol. 21(2): 283–294, July 2014
286 R.E. BONILLA – J.B. CHAVARRÍA
Table 1: Standardized mortality rates (× 100 000) and deaths by causes of deaths among
Nicaraguan immigrants and Costa Ricans. Costa Rica 1998–2008.
Costa Ricans Nicaraguan Immigrants Relative
Causes of deaths Rate Deaths Rate Deaths risk
Infectious diseases 1,95 251 2,65 35 1,4
Cancer 12,18 1564 9,62 125 0,8
Chronic respirat diseases 1,66 212 1,490 20 0,9
Cardiovascular diseases 5,71 726 6,63 86 1,2
Diabetes 0,62 77 0,45 6 0,7
Alcoholism 1,84 232 1,95 25 1,1
Motor vehicle accidents 20,02 2575 23,43 311 1,2
Other fatalities 8,82 1147 15,04 194 1,7
Suicides 9,84 1265 10,64 137 1,1
Homicides 13,36 1578 24,72 325 2,0
Other causes 15,65 2009 15,86 205 1,0
Source: Instituto Nacional de Estadística y Censos, (2009).
immigrants from 15 to 34 years of age (Table 1). This coincides with the litera-
ture consulted ([13], [12], [19], [20], [22], [18]), which emphasizes that the main
causes of death among immigrants are external causes, particularly homicides,
although the relationship between migration and external causes of deaths is not
clear.
In Costa Rica, up to this point, the only studies of mortality among Nicaraguan
immigrants have been [12] and more recently [13], but they do so in a general
context. The researchers find the same results as those found in other countries:
high mortality from external causes (accidents and homicides) versus a low mor-
tality from disease related deaths. Due to the age groups studied, homicides and
accidents are the main causes of death among immigrants and Costa Ricans. A
large proportion of the Nicaraguan immigration to Costa Rica consists of indi-
viduals in these age groups and their mortality continues without having been
studied [12].
2 Methods and data
2.1 Mathematical notation
Define the classical linear model with k covariates with a sample of size n as
Y = βTX + ε,
Rev.Mate.Teor.Aplic. (ISSN 1409-2433) Vol. 21(2): 283–294, July 2014
MORTALITY AMONG YOUNG NICARAGUAN IMMIGRANTS 287
where Y = (Y1, . . . , Tn)T ∈ Rn, β = (β0, . . . , βk)T ∈ Rk+1,

 1 X11 . . . X1k. . . . . . . . . . . .
1 Xn1 . . . Xnk


and ε ∈ Rn is some vector with E[ε|X ] = 0, can be extended as follows:
Yi(g) = β0(g) + β1(g)x1 + β2(g)x2 + . . .+ βk(g)xk + ε,
where (g) ∈ R2 is a vector which indicates that the parameter will be estimated
at a location whose geographic coordinates are given by the vector g [9], [8].
That model allows local variation among the estimators.
The parameters for a traditional linear regression model can be resolved by
solving:
β = (XTX)−1XTY.
So that to estimate the parameters of a GWR model we have to use the following
weighting scheme:
β(g) = (XTW (g)X)−1XTW (g)Y.
The weighting factors are selected in such a way that the observations that
are close to an estimation point have more influence on the results than those ob-
servations that are far from that point. The results from the parameter estimates
can be presented on maps to study regional variations. By using GWR models,
it is assumed that geographic autocorrelation exists.
2.2 Data and general procedures
The study population consists of young immigrants (from 15 to 34 years of age)
that died in Costa Rica due to homicide between 1998 and 2008, as reported in
the national vital statistics system available at the National Institute of Statistics
and Census [15]. The geographic unit was the Costa Rican municipality and the
geographic location of the municipality was the geometric centroid geographic
coordinate according to the Lambert North cartography system [14].
The following variables based on an empirical model were included in the
database:
• Response Variable: Number of young immigrant homicides in the munic-
ipality (y) (years 1998 to 2008).
Rev.Mate.Teor.Aplic. (ISSN 1409-2433) Vol. 21(2): 283–294, July 2014
288 R.E. BONILLA – J.B. CHAVARRÍA
• Exposure Variable: Population of young Nicaraguan immigrants (census
2000). Factors (covariates predictors)
Demographic.
• Percentage of Nicaraguan immigrants in the municipality (x1) (census
2000).
• Percentage of persons in the municipality between the ages of 15 and 34
(youth) (x2) (census 2000).
• Sex ratio of the municipality (x3) (census 2000).
Geographic.
• Percentage of households in the municipality classified as in the urban
zone (x4) (census 2000).
Socio-economic characteristics.
• Percentage of households in the municipality with at least one unmet basic
need (UBN) (x5) (census 2000).
• Percentage of the economically active population in the municipality work-
ing in each of the three economically productive sectors (primary, sec-
ondary, tertiary) (x6, x7, x8) (census 2000).
The data were processed with the STATA statistical package [21], GWR
module. The GeoDA package [2]1 was also be used for the geographic statistical
regression. Finally, the MapInfo package [16] was used for the map presentation.
3 Results
Figure 1 presents the mortality rate from homicide among young immigrants on
a percentile scale. According to the map, the highest homicide rates (90th per-
centile and higher) are found in the municipalities of the southern zone: Dota,
Tarrazú, Parrita, Aguirre and Pérez Zeledón, as well as in the municipality of
Limón (on the Caribbean coast), in Carrillo (Guanacaste), in Palmares, and San
Pablo de Heredia, in the Central Valley. Moran’s I was used as the statistics
to evaluate global spatial autocorrelation [1]. Geographic autocorrelation of the
homicide rates for young immigrants is significant (Moran’s I = 0.0976). The
1http://geodacenter.asu.edu/software/
Rev.Mate.Teor.Aplic. (ISSN 1409-2433) Vol. 21(2): 283–294, July 2014
MORTALITY AMONG YOUNG NICARAGUAN IMMIGRANTS 289
GWR model takes into account the spatial autocorrelation, since the model co-
efficients vary locally.
Figure 1: Mortality rate from homicide for young Nicaraguan immigrants (percentiles).
Costa Rica 1998–2008. Legend: Red, more than 45.9; Orange, from 24.5 to
45.9 Yellow, from 9.9 to 24.5 Blue, from 0.0 to 9.9.
Table 2 presents the GWR model regression results, as well as those from
the classic model. The test for bandwidth suggests that the GWR model offers a
better description of the homicide rates among immigrants than a classic Poisson
log-linear regression model with the same covariates (p < 0.000). The best-fit
bandwidth is 142.9 kilometers. This means that the GWR model will use the
municipalities located within a distance of 142.9 kilometers to estimate the re-
gression. In practical terms, this means that the GWR regression is better than
the classical model of a log-linear Poisson regression, because in the other case,
the bandwidth would be greater, covering the country as a whole [5]. Any band-
width that is less than one-half of the maximum distance between a point of the
periphery (borders) of Costa Rica and another, indicates that the GWR models
Rev.Mate.Teor.Aplic. (ISSN 1409-2433) Vol. 21(2): 283–294, July 2014
290 R.E. BONILLA – J.B. CHAVARRÍA
are more adequate than the classical Poisson log-linear regression models. Tra-
ditional models, such as the Poisson log-linear model, ignore the bandwidth. To
have an idea of the dimension of the bandwidth in this case, it is approximately
the distance between the capital, San José and the city of Limón (approximately
141 kilometers).
Table 2: Incidence rate ratio from the classic Poisson log-linear regression model and
test for non-stationarity of the coefficients from the GWR model. The response
variable is the number of deaths by homicide for Nicaraguan immigrant or
Costa Rican youth by municipality in Costa Rica. Costa Rica 1998–2008.a
Classic GWR Modelb
Sociodemographic covariates model Nicaraguan Costa Ricans
immigrants
Percentage of immigrants (x1) 0.96* 0.01 0.00
Sex Ratio (x3) 1.01* 0.01 0.00
Percentage Urban Population (x4) 1.01* 0.01 0.00
Percentage Poor Households (x5) 1.04** 0.01* 0.01
Percentage Adults (35+ years) (x2) 1.05 0.02 0.01
% EAPc in Secondary Sector (x7) 1.05 0.01 0.01
% EAP in Tertiary Sector (x8) 1.05* 0.01 0.02
Number of cells (N ) 81 81 81
Goodness of fit
χ2 92.10
Prob > χ2 0.08
Bandwidth 142986 142986
p value 0.00 0.00
aThe exposure variable is the population of Nicaraguan immigrant or Costa Rican
youths, as may be the case.
bNon-stationarity test for the coefficient. Value si. The test for significance of
non-stationarity of the estimators is a test that indicates whether the variable
is significant within the GWR model.
cEAP = Economically Active Population.
*Significant at 5%.
**Significant at 1%.
The significance test for non-stationarity for the estimators is a test indicating
whether the variable is significant within the GWR model [5]. The test provided
only one significant variable, which is the percentage of poor households. The
Rev.Mate.Teor.Aplic. (ISSN 1409-2433) Vol. 21(2): 283–294, July 2014
MORTALITY AMONG YOUNG NICARAGUAN IMMIGRANTS 291
relationship between the number of young Nicaraguan immigrant homicides and
the percentage of poor households varies among the municipalities in Costa Rica;
it can be interpreted as follows: in certain regions of Costa Rica, there are social
or economic phenomena that provoke a modification in the relationship between
poverty and homicides. Further social research in these regions of Costa Rica
is needed. The value for Moran’s I was calculated with the residuals from the
GWR model (0.0543), which was not significant at 5%. The Null Hypothesis
of non-existence of geographic autocorrelation in the residuals from the GWR
model cannot be rejected.
Figure 2: GWR model. Estimators associated with % of poor households (percentiles).
Dependent variable = Mortality rate by homicide for young Nicaraguan im-
migrants. Costa Rica 1998–2008. Legend: Red, more than 0.058; Orange,
from 0.051 to 0.058; Yellow, from 0.045 to 0.051; Blue sky, from 0.023 to
0.045; Deep blue, from 0.016 to 0.023.
Rev.Mate.Teor.Aplic. (ISSN 1409-2433) Vol. 21(2): 283–294, July 2014
292 R.E. BONILLA – J.B. CHAVARRÍA
The map of the estimators associated with the percentage of poor households
(Figure 2) suggests that the relationship between poverty and mortality by homi-
cide for young Nicaraguan immigrants is stronger in the Caribbean region and
neighboring zones, where the main center is the municipality of Matina. In the
municipalities in the Central Valley, Sarapiquí and the Southern Zone, there is an
intermediate relationship between poverty and mortality by homicide for young
immigrants. In the rest of the country, this relationship is low, where it’s lowest
point is in the municipalities of Golfito and Osa, close to the Panamanian border.
When the GWR model was run for homicides among young Costa Ricans,
this effect was not observed, as it was among Nicaraguan immigrants (Table 2).
In particular, the covariates, percentage poor households, was not significant at
5%. This indicates that the relationship between poverty and homicides interacts
differently among Nicaraguan immigrants and Costa Ricans, since it affects the
former more and this association is significant.
4 Final discussion
The results from this study demonstrate that the GWR model is a more adequate
model for describing homicides among young Nicaraguan immigrants than clas-
sical models, such as Poisson log-linear regression since the GWR model incor-
porates spatial autocorrelation. GWR models allows calculate local estimators
versus classical models that do not incorporate spatial autocorrelation. Up to
what point are these results valid and reliable? One element that could lessen
the validity of the results obtained has to do with aggregate statistics. The fact
that the data are aggregated at the municipality level presents two serious prob-
lems: (1) The Numerator Effect and (2) the Ecological Fallacy. In regards to
the Numerator Effect, in this paper, we have used the municipalities as our unit
of analysis, since it is a geographic unit that guarantees a sufficient number of
cases for the numerator. Ideally, we would have used smaller administrative
units, such as the districts in Costa Rica. However, this was not possible, pri-
mordially because of the fact that there would have been numerous zeros in the
numerators [11], [3]. For that reason, the municipality was chosen as the ge-
ographic unit for analysis. Secondly, in the analysis in this paper we run into
the Ecological Fallacy, which is present in research using aggregated statistics.
The fallacy consists of affirming that the individual has the characteristics of the
statistical aggregate to which the individual belongs ([17], [10]). Another effect
of the ecological fallacy is that some of the relationships, for example the inverse
relationship between poverty and homicides, are difficult to explain upon aggre-
gating the data. These relationships can only be observed at the individual level.
Rev.Mate.Teor.Aplic. (ISSN 1409-2433) Vol. 21(2): 283–294, July 2014
MORTALITY AMONG YOUNG NICARAGUAN IMMIGRANTS 293
Notwithstanding, the results obtained are consistent with the literature that we
found, complementing the studies by [12] and [13]. The findings from this study
contribute to present useful evidence for an appropriate design for preventive
public policies and social programs that would benefit the young immigrants.
References
[1] Anselin, L. (1995) “Local indicators of spatial association – LISA”, Geo-
graphical Analysis 27: 93–115.
[2] Anselin, L.; Syabri, I.; Youngihn, K. (2005) “GeoDa: an introduction to
spatial data analysis”, Geographical Analysis 38(1): 5–22.
[3] Bailey, L.; Vardulaki, B.L.; Langham, J.; Chandramohan, D. (2005) Intro-
duction to Epidemiology. Open University Press, London.
[4] Bowman, A.W. (1984) “An alternative method of cross-validation for the
smoothing of density estimates”, Biometrika 71(2): 353–360.
[5] Brunsdon, C.; Fotheringham, A.S.; Charlton, M.E. (1996) “Geographically
weighted regression: a method for exploring spatial nonstationarity”. Geo-
graphical Analysis 28(4): 281–298.
[6] Clayton, D.G.; Bernardinelli, L.; Montomoli, G. (1993) “Spatial correla-
tion in ecological analysis”, Int. J. Epidemiol. 22(6): 1193–1202.
[7] Cleveland, W.S. (1979) “Robust locally weighted regression and smooth-
ing scatterplots”, Journal of the American Statistical Association 74(368):
829–836.
[8] Fotheringham, A.S.; Brunsdon, C.; Charlton, M.E. (2000) Quantitative Ge-
ography. Sage, London.
[9] Fotheringham, A.S.; Brunsdon, C.; Charlton, M.E. (2002) Geographically
Weighted Regression: The Analysis of Spatially Varying Relationships. Wi-
ley, Chichester.
[10] Greenland, S.; Robins, J. (1994) “Invited commentary: ecologic studies-
biases, misconceptions, and counterexamples”, American Journal of Epi-
demiology 139(8): 747–760.
[11] Hennekens, C.H.; Buring, J.E.; Mayren, S.L. (1987) Epidemiology in
Medicine. Little Brown & Company, Boston.
Rev.Mate.Teor.Aplic. (ISSN 1409-2433) Vol. 21(2): 283–294, July 2014
294 R.E. BONILLA – J.B. CHAVARRÍA
[12] Herring, A; Bonilla, R.; Borland, R.; Hill, K. (2008) “Patrones diferen-
ciales de mortalidad entre inmigrantes nicaragüenses y residentes nativos
de Costa Rica”, Población y Salud en Mesoamérica 6(1): article 2, 20 pp.
[13] Herring, A.; Bonilla, R. (2009) “Inmigrantes nicaragüenses en Costa
Rica: estado y utilización de servicios de salud”, Población y Salud en
Mesoamérica 7(1): article 4, 20 pp.
[14] Instituto Geográfico Nacional de Costa Rica (IGN) (1984) Proyección
Costa Rica Lambert Norte. San José, Costa Rica.
[15] Instituto Nacional de Estadística y Censos (2009) “Estadísticas vitales”.
En: http://censos.ccp.ucr.ac.cr, consultado 15/05/2009.
[16] MapInfo Corporation (1985) MapInfo Professional. MapInfo Corporation,
New York.
[17] Robinson, W.S. (1950) “Ecological correlations and the behavior of indi-
viduals”, International Journal of Epidemiology 38(2): 337–341.
[18] Sharma, R.D.; Michalowski, M.; Verma, R.B. (1990) “Mortality differ-
entials among immigrant populations in Canada”, International Migration
28(4): 443–450.
[19] Singh, G.K.; Miller, B.A. (2004) “Health, life expectancy and mortality
patterns among immigrant populations in the United States” Can. J. Public
Health 95(3): I14–I21.
[20] Sorenson, S.; Shen, H. (1999) “Mortality among young immigrants to Cal-
ifornia: injury compared to disease deaths”. Journal of Immigrant Health
1(1): 41–47.
[21] StataCorp (2005) STATA Package. Stata Corporation, College Station TX.
[22] Trovato, F. (1992) “Violent and accidental mortality among four immigrant
groups in Canada, 1970-1972”, Biodemography and Social Biology 39(1-
2): 82–101.
[23] Wolpert, R.L.; Ickstadt, K. (1998) “Poisson/gamma random field models
for spatial statistics”, Biometrika 85(2): 251–267.
Rev.Mate.Teor.Aplic. (ISSN 1409-2433) Vol. 21(2): 283–294, July 2014