Risk models derived from environmental data have been widely shown to be effective in delineating geographical areas of risk because they are intuitively easy to understand. We present a new method based on distances, which allows the modelling of continuous and non-continuous random variables through distance-based spatial generalised linear mixed models. The parameters are estimated using Markov chain Monte Carlo maximum likelihood, which is a feasible and a useful technique. The proposed method depends on a detrending step built from continuous or categorical explanatory variables, or a mixture among them, by using an appropriate Euclidean distance. The method is illustrated through the analysis of the variation in the prevalence of Loa loa among a sample of village residents in Cameroon, where the explanatory variables included elevation, together with maximum normalised-difference vegetation index and the standard deviation of normalised-difference vegetation index calculated from repeated satellite scans over time.
BoussinesqMGardonJKamgnoJ. Relationships between the prevalence and intensity of Loa loa infection in the Central province of Cameroon. Ann Trop Med Parasitol2001; 95: 495–507.
2.
DigglePJThomsonMCChristensenOF. Spatial modelling and the prediction of Loa loa risk: decision making under uncertainty. Ann Trop Med Parasitol2007; 101: 499–509.
3.
KamgnoJBouchiteBBaldetT. Study of the distribution of human filariasis in West province of Cameroon. Bull Soc Pathol Exot1997; 90: 327–330.
4.
ThomsonMConnorSD’AlessandroU. Predicting malaria infection in Gambian children from satellite data and bednet use surveys: the importance of spatial correlation in the interpretation of results. Am J Trop Med Hyg1999; 61: 2–8.
5.
NelderJAWedderburnRM. Generalized linear models. J R Stat Soc, Ser A1972; 135: 370–384.
6.
LiangKYZegerSL. Longitudinal data analysis using generalized linear models. Biometrika1986; 73: 13–22.
7.
BreslowNEClaytonDG. Approximate inference in generalized linear mixed models. J Am Stat Assoc1993; 88: 9–25.
8.
DigglePJRibeiroPJ. Model-based geostatistics, New York: Springer Series in Statistics, 2007.
9.
LairdJHWareNM. Random-effects models for longitudinal data. Biometrics1982; 38: 963–974.
10.
StiratelliRLairdNWareJH. Random-effects models for serial observations with binary response. Biometrics1984; 40: 961–971.
11.
SchallR. Estimation in generalized linear models with random effects. Biometrika1991; 78: 719–727.
12.
DigglePTawnAMoyeedR. Model based geostatistics. Appl Stat1998; 49: 299–350.
13.
ZhangH. On estimation and prediction for spatial generalized linear mixed models. Biometrics2002; 58: 129–136.
14.
ArenasCCuadrasC. Recent statistical methods based on distances. Contrib Sci2002; 2: 183–191.
15.
CuadrasCArenasC. A distance based regression model for prediction with mixed data. Commun Stat A –Theory Methods1990; 19: 2261–2279.
16.
CuadrasCArenasCFortianaJ. Some computational aspects of a distance-based model for prediction. Commun Stat –Simul Comput1996; 25: 593–609.
17.
ChristensenOF. Monte Carlo maximum likelihood in model-based geostatistics. J Comput Graph Stat2004; 13: 702–718.
18.
ChristensenOFDigglePJRibeiroPJAnalysing positive-valued spatial data: the transformed Gaussian model. In: MonestiezPAllardDFroidevauxR (eds). geoENV III – geostatistics for environmental applications, Boston: Kluwer Academic Publishers, 2001, pp. 287–298.
19.
GeyerCJThompsonEA. Constrained Monte Carlo maximum likelihood for dependent data (with discussion). J R Stat Soc, Ser B1992; 54: 657–699.
20.
GeyerCJ. On the convergence of Monte Carlo maximum likelihood calculations. J R Stat Soc, Ser B1994; 56: 261–274.
21.
SteinslandI. Parallel exact sampling and evaluation of Gaussian Markov random fields. Comput Stat Data Anal2007; 51: 2969–2981.
22.
ThomsonMCObsomerVKamgnoJ. Mapping the distribution of Loa loa in Cameroon in support of the African Programme for Onchocerciasis Control. Filaria J2004; 3: 7–7.
23.
CressieN. Statistics for spatial data, Revised ed. New York: John Wiley & Sons Inc, 1993.
24.
McCullaghPNelderJ. Generalized linear models, London: Chapman Hall, 1989.
25.
CuadrasCMDistance analysis in discrimination and classification using both continuous and categorical variables. In: DodgeY (ed). Recent developments in statistical data analysis and inference, North-Holland, Amsterdam: Elsevier Sciencie Publisher, 1989, pp. 459–474.
26.
GowerJ. Adding a point to vector diagrams in multivariate analysis. Biometrika1968; 55: 582–585.
RoyleJAWikleCK. Efficient statistical mapping of avian count data. Environ Ecol Stat2005; 12: 225–243.
29.
HφjbjerreM. Profile likelihood in directed graphical models from BUGS output. Stat Comput2003; 13: 57–66.
30.
McCullochCESearleSRNeuhausJM. Generalized, linear, and mixed models, 2nd ed. New Jersey: John Wiley & Sons, 2008.
31.
GowerJ. A general coefficient of similarity and some of its properties. Biometrics1971; 27: 857–874.
32.
TakougangIMeremikwuMWanjiS. Rapid assessment method for prevalence and intensity of Loa loa infection. Bull World Health Organ2002; 80: 852–858.
33.
WanjiSTendongforNEsumM. Heterogeneity in the prevalence and intensity of loiasis in five contrasting bioecological zones in Cameroon. Trans R Soc Trop Med Hyg2003; 97: 182–187.
34.
ChristensenOWaagepetersenR. Bayesian prediction of spatial count data using generalized linear mixed models. Biometrics2002; 58: 280–286.
35.
BojEDelicadoPFortianaJ. Distance-based local linear regression for functional predictors. Comput Stat Data Anal2010; 54: 429–437.
36.
ShumwayRHStofferDS. Time series analysis and its applications, New York: Springer-Verlag, 2000.
