Generalized Poisson Spatial Autoregressive Modeling of Pneumonia Cases in Children Under Five in Tuban

Joshua Capri Gunawan Sihombing, Sutikno Sutikno, Achmad Choiruddin

Abstract


The number of pneumonia cases in children under five in Tuban Regency presents two significant data challenges, namely, overdispersion and spatial dependency. This study aims to develop and apply the Generalized Poisson Spatial Autoregressive (GPSAR) model to address both issues simultaneously. The model was estimated using a MLE-BHHH procedure and validated using 10-fold cross-validation (CV). The results confirm the model's validity and superiority. The GPSAR model outperformed the non-spatial GPR model in terms of goodness-of-fit (AIC: 1301.09 vs. 1312.67) and predictive accuracy (Out-of-Sample CV-RMSE: 8.451 vs. 8.716). Statistically, the structural parameters for spatial lag and overdispersion were highly significant. Two predictor variables, exclusive breastfeeding and complete basic immunization, were also found to be statistically significant factors. This research provides a robust regression framework for spatial count data exhibiting overdispersion and offers new insights into pneumonia case determinants in the region.

Keywords


Count Data; Overdispersion; Pneumonia; SAR Model; and Spatial Lag

Full Text:

PDF

References


[1] R. G. Bender et al., “Global, regional, and national incidence and mortality burden of non-COVID-19 lower respiratory infections and aetiologies, 1990–2021: A systematic analysis from the Global Burden of Disease Study 2021,” The Lancet Infectious Diseases, vol. 24, no. 9, pp. 974–1002, 2024. doi: 10.1016/S1473-3099(24)00176-2.

[2] D. Manik, W. P. J. Kaunang, and E. M. Mantjoro, “Distribusi kasus dan kematian akibat pneumonia pada balita di Indonesia tahun 2019–2023,” Profetif: Jurnal Kesehatan Masyarakat, vol. 9, no. 2, pp. 2972–2986, 2025. doi: 10.31004/prepotif.v9i2.44589.

[3] R. Aprilia, F. Faisal, I. Irwandi, S. Suharni, and E. Efriza, “Tinjauan literatur: Faktor risiko dan epidemiologi pneumonia pada balita,” Scientific Journal, vol. 3, no. 3, pp. 166–173, 2024. doi: 10.56260/sciena.v3i3.144.

[4] V. Alvionita, S. Sulfatimah, A. Astuti, and N. Nurfitri, “Hubungan status gizi dan status imunisasi dengan kejadian pneumonia pada bayi,” Ahmar Metastasis Health Journal, vol. 1, no. 4, pp. 137–143, 2022. doi: 10.53770/amhj.v1i4.92.

[5] J. Liu, “Sufficient dimension reduction for Poisson regression,” Econometrics and Statistics, vol. 34, pp. 109–119, 2025. doi: 10.1016/j.ecosta.2022.09.001.

[6] S. Subedi and U. J. Dang, “Multivariate Poisson lognormal distribution for modeling counts from modern biological data: An overview,” Computational and Structural Biotechnology Journal, vol. 24, pp. 1–13, 2025. doi: 10.1016/j.csbj.2025.03.017.

[7] S. Mardalena, P. Purhadi, J. D. T. Purnomo, and D. D. Prastyo, “Parameter estimation and hypothesis testing of multivariate Poisson inverse Gaussian regression,” Symmetry, vol. 12, no. 10, p. 1738, 2020. doi: 10.3390/sym12101738.

[8] S. Q. Aina, M. Fauziyah, and W. P. Nurmayanti, “Comparison of generalized Poisson regression and negative binomial regression models based on Akaike information criterion values,” Statistika, vol. 25, no. 1, pp. 86–101, 2025. doi: 10.29313/statistika.v25i1.5402.

[9] P. Puig, J. Valero, and A. Fernández-Fontelo, “Some mechanisms leading to underdispersion: Old and new proposals,” Scandinavian Journal of Statistics, vol. 51, no. 1, pp. 245–267, 2024. doi: 10.1111/sjos.12677.

[10] L. P. Fávero, R. d. F. Souza, P. Belfiore, H. L. Corrêa, and M. F. Haddad, “Count data regression analysis: Concepts, overdispersion detection, zero-inflation identification, and applications with R,” Practical Assessment, Research & Evaluation, vol. 26, p. 13, 2021. doi: 10.7275/44nn-cj68.

[11] S. M. Berliana, P. Purhadi, S. Sutikno, and S. P. Rahayu, “Parameter estimation and hypothesis testing of geographically weighted multivariate generalized Poisson regression,” Mathematics, vol. 8, no. 9, p. 1523, 2020. doi: 10.3390/math8091523.

[12] S. W. Tyas, L. A. Puspitasari et al., “Geographically weighted generalized Poisson regression model with the best kernel function in the case of the number of postpartum maternal mortality in East Java,” MethodsX, vol. 10, p. 102002, 2023. doi: 10.1016/j.mex.2023.102002.

[13] S. Manda, N. Haushona, and R. Bergquist, “A scoping review of spatial analysis approaches using health survey data in sub-Saharan Africa,” International Journal of Environmental Research and Public Health, vol. 17, no. 9, p. 3070, 2020. doi: 10.3390/ijerph17093070.

[14] P. Phang, S. Aslam, J. Labadin, and V. J. Jayaraj, “Spatial autocorrelation analysis of infectious disease incidence rates at state and district level using supra-adjacency weights matrix,” Health, vol. 13, no. 2, pp. 456–470, 2025. doi: 10.13189/ujph.2025.130217.

[15] T. Saffary et al., “Analysis of COVID-19 cases’ spatial dependence in US counties reveals health inequalities,” Frontiers in Public Health, vol. 8, p. 579190, 2020. doi: 10.3389/fpubh.2020.579190.

[16] K. Vilinová and L. Petrikovičová, “Spatial autocorrelation of COVID-19 in Slovakia,” Tropical Medicine and Infectious Disease, vol. 8, no. 6, p. 298, 2023. doi: 10.3390/tropicalmed8060298.

[17] A. Getis, “Reflections on spatial autocorrelation,” Regional Science and Urban Economics, vol. 37, no. 4, pp. 491–496, 2007. doi: 10.1016/j.regsciurbeco.2007.04.005.

[18] F. Famoye, J. T. Wulu, K. P. Singh et al., “On the generalized Poisson regression model with an application to accident data,” Journal of Data Science, vol. 2, no. 3, pp. 287–295, 2004. doi: 10.6339/JDS.2004.02(3).167.

[19] J. Cheng, Y. Cui, X. Wang, Y. Wang, and R. Feng, “Spatial characteristics of health outcomes and geographical detection of its influencing factors in Beijing,” Frontiers in Public Health, vol. 12, p. 1424801, 2024. doi: 10.3389/fpubh.2024.1424801.

[20] Y. Wang and K. M. Kockelman, “A Poisson-lognormal conditional-autoregressive model for multivariate spatial analysis of pedestrian crash counts across neighborhoods,” Accident Analysis & Prevention, vol. 60, pp. 71–84, 2013. doi: 10.1016/j.aap.2013.07.030.

[21] A. Rosilala and K. M. Kockelman, “Comparison of negative binomial SAR, SEM, and SARMA methods in modeling the number of malnutrition cases among toddlers in Central Java,” Jurnal Gaussian, vol. 14, no. 1, pp. 42–53, 2025. doi: 10.14710/j.gauss.14.1.42-53.

[22] N. A. Cruz, D. A. Romero, and O. O. Melo, “SAR models with specific spatial coefficients and heteroskedastic innovations,” arXiv preprint arXiv:2502.15580, 2025. doi: 10.48550/arXiv.2502.15580.

[23] J. Zhao and Y. Pu, “Characterization and estimation of heterogeneous spatial autocorrelation in spatial autoregressive models,” PLoS ONE, vol. 20, no. 7, e0327316, 2025. doi: 10.1371/journal.pone.0327316.

[24] P. Pfeiffer, J. Varga et al., “Quantifying spillovers of coordinated investment stimulus in the EU,” Macroeconomic Dynamics, vol. 27, no. 7, pp. 1843–1865, 2023. doi: 10.1017/S1365100522000487.

[25] M. Jankiewicz, “Analysis of spatial dependencies and spatial effects in the relationship between economic growth and unemployment in Europe,” Hungarian Geographical Bulletin, vol. 74, no. 2, pp. 163–176, 2025. doi: 10.15201/hungeobull.74.2.3.

[26] K. Wang et al., “Quantifying the spatial spillover effects of non-pharmaceutical interventions on pandemic risk,” International Journal of Health Geographics, vol. 22, no. 1, p. 13, 2023. doi: 10.1186/s12942-023-00335-6.

[27] S. Sutikno, P. Purhadi, F. Fachrunisah, and F. D. Cahyoko, “Estimation of parameters and hypothesis testing of multivariate spatial autoregressive model,” MethodsX, vol. 14, p. 103294, 2025. doi: 10.1016/j.mex.2025.103294.

[28] L. P. S. Pratiwi, N. P. N. Hendayanti, and I. K. P. Suniantara, “Perbandingan pembobotan seemingly unrelated regression – spatial durbin model untuk faktor kemiskinan dan pengangguran,” Jurnal Varian, vol. 3, no. 2, pp. 51–64, 2020. doi: 10.30812/varian.v3i2.596.

[29] D. M. Lambert, J. P. Brown, and R. J. G. M. Florax, “A two-step estimator for a spatial lag model of counts: Theory, small sample performance and an application,” Regional Science and Urban Economics, vol. 40, no. 4, pp. 241–252, 2010. doi: 10.1016/j.regsciurbeco.2010.04.001.

[30] StataCorp, *Stata Spatial Autoregressive Models Reference Manual*, College Station, TX: Stata Press, 2017.

[31] A. Choiruddin, F. F. Hannanu, J. Mateu, and V. Fitriyanah, “COVID-19 transmission risk in Surabaya and Sidoarjo: An inhomogeneous marked Poisson point process approach,” Stochastic Environmental Research and Risk Assessment, vol. 37, no. 6, pp. 2271–2282, 2023. doi: 10.1007/s00477-023-02393-5.

[32] A. Choiruddin, E. R. F. Sakti, T. D. A. Widhianingsih, J. Mateu, and K. Fithriasari, “Probabilistic deep learning for highly multivariate spatio-temporal log-Gaussian Cox processes,” IEEE Access, vol. 13, pp. 94761–94776, 2025. doi: 10.1109/ACCESS.2025.3570476.




DOI: https://doi.org/10.18860/cauchy.v10i2.37288

Refbacks

  • There are currently no refbacks.


Copyright (c) 2025 Joshua Capri Gunawan Sihombing, Sutikno Sutikno, Achmad Choiruddin

Creative Commons License
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Editorial Office
Mathematics Department,
Universitas Islam Negeri Maulana Malik Ibrahim Malang
Gajayana Street 50 Malang, East Java, Indonesia 65144
Faximile (+62) 341 558933
e-mail: cauchy@uin-malang.ac.id

Creative Commons License
CAUCHY: Jurnal Matematika Murni dan Aplikasi is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.