Available online at https://journal.
com/index.
php/ijqrm/index International Journal of Quantitative Research and Modeling e-ISSN 2721-477X p-ISSN 2722-5046 Vol.
No.
2, pp.
229-235, 2026
Negative Binomial Regression with Climatic and Sociodemographic Covariates for Modeling Overdispersed Dengue Hemorrhagic Fever Counts:
Evidence from Bandung City.
Indonesia Rifki Saefullah1*.
Natasya Tamarysma Putri2.
Mugi Lestari3 Master's Program of Mathematics.
Faculty of Mathematics and Natural Sciences.
Universitas Padjadjaran.
Jatinangor.
West Java.
Indonesia Management Study Program.
Faculty of Economics and Business.
Universitas Informatika Dan Bisnis Indonesia.
Bandung.
Indonesia *Corresponding author email: rifki23008@mail.
Abstract Dengue Hemorrhagic Fever (DHF) remains one of the most prevalent vector-borne diseases in Indonesia, with Bandung City consistently reporting high annual incidence.
Count regression models have been widely applied in disease epidemiology.
however, many studies default to Poisson regression without testing for overdispersion, which violates a fundamental modeling assumption when variance exceeds the mean.
This study proposes a Negative Binomial Regression (NBR) framework that jointly incorporates climatic variables .
onthly rainfall, mean temperature, relative humidit.
and sociodemographic covariates .
opulation density, drainage quality index, vegetation cove.
to model weekly DHF case counts across 30 sub-districts of Bandung City from 2019 to 2023.
Overdispersion was formally assessed using the Cameron-Trivedi test.
Incidence Rate Ratios (IRR.
and 95% confidence intervals were estimated for all predictors.
Model selection was performed via AIC.
BIC, and likelihood ratio tests against a Poisson baseline.
Results demonstrate significant overdispersion .
ispersion parameter I confirming the appropriateness of NBR over Poisson regression.
Monthly rainfall ( ), lagged one-week ), and population density emerged as significant positive predictors, while drainage quality index was protective.
The NBR model achieved substantially lower AIC .
0 vs 3.
and BIC .
0 vs 3.
compared to Poisson.
These findings provide quantitative evidence for spatiotemporal DHF surveillance and can guide targeted vector-control resource allocation in urban West Java.
Keywords: Negative Binomial Regression.
Dengue Hemorrhagic Fever.
Overdispersion.
Incidence Rate Ratio.
Bandung City.
Climatic Variable Introduction Dengue Hemorrhagic Fever (DHF) is an arboviral disease transmitted primarily by Aedes aegypti mosquitoes and represents a significant public health burden in tropical and subtropical nations (Sirisena & Noordeen, 2014.
WHO,
Indonesia is classified among the hyperendemic countries, with the national Ministry of Health recording a consistently high incidence rate nationwide.
Bandung City in West Java Province, with its dense urban population of 5 million residents, has been identified as one of the highest-burden municipalities in the country.
Understanding the quantitative relationship between DHF transmission dynamics, climatic drivers, and sociodemographic characteristics is essential for designing evidence-based intervention programs (Sundari et al.
In the epidemiological literature, count regression methods are the standard analytical framework when the outcome variable represents discrete non-negative events.
Poisson regression assumes equidispersion .
ean equals however.
DHF case counts in urban settings frequently exhibit overdispersion where the observed variance exceeds the mean due to heterogeneous exposure risk, spatial clustering of breeding sites, and unobserved confounders (Jaya et al.
, 2.
Ignoring overdispersion in Poisson models produces underestimated standard errors, inflated test statistics, and misleading inference about covariate effects.
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Negative Binomial Regression (NBR) extends the Poisson model by introducing an additional dispersion parameter, making it more suitable for overdispersed count outcomes (Sutriyawan et al.
, 2.
While NBR has been applied in dengue modeling in Thailand.
Brazil, and Malaysia, its systematic application with combined climatic and sociodemographic predictors across urban Indonesian sub-districts remains underexplored in the quantitative research Previous studies have examined DHF gender differences (Maharani et al.
, 2.
and applied Poisson-based frameworks for infectious disease counts.
however, none have formally tested overdispersion and compared Negative Binomial against Poisson within a multivariate climatic-sociodemographic covariate structure.
This paper addresses that gap with a rigorous model selection approach and spatial IRR profiling across Bandung sub-districts.
Novelty: .
First application of Cameron-Trivedi overdispersion testing to DHF count data in Bandung City.
Integration of a composite Drainage Quality Index (DQI) as a novel sociodemographic covariate.
Spatial IRR profiling across 30 sub-districts enabling district-level risk stratification.
Temporal lag analysis of self-exciting DHF dynamics within the NBR framework.
Literature Review Dengue Count Regression Models Count data models form the cornerstone of infectious disease epidemiology.
The Poisson regression model assumes that the conditional variance equals the conditional mean, an assumption frequently violated in practice.
When variance exceeds the mean .
Negative Binomial Regression provides a more flexible alternative by adding a gamma-distributed random effect to the Poisson mean, effectively modeling the extra variation (Hilbe.
Let YA denote the DHF case count for sub-district in week .
Under the Negative Binomial distribution, the probability mass function is:
where is the conditional mean, is the dispersion parameter, and the variance is the model collapses to Poisson.
Climatic Drivers of DHF Transmission A substantial body of literature documents the association between climatic variables and dengue incidence.
Rainfall creates standing water that serves as mosquito breeding habitat.
temperature influences larval development rates and adult vector competence.
relative humidity affects adult mosquito survival.
These relationships are nonlinear and often exhibit time lags of one to three weeks between environmental exposure and observed case counts.
Sociodemographic Determinants Beyond climate, sociodemographic factors shape individual and community-level vulnerability.
Population density increases vector-host contact rates.
Inadequate drainage infrastructure promotes permanent breeding sites.
Studies in Indonesian urban settings highlight drainage quality and household density as key modifiable risk factors.
This study operationalizes drainage quality through a composite index constructed from official municipal infrastructure data.
Materials and Methods Data Sources and Study Area The study area comprises 30 sub-districts .
of Bandung City.
West Java Province.
Indonesia.
Weekly DHF case data from January 2019 through December 2023 .
weekly observations per sub-district, subdistrict-week observation.
were obtained from the Bandung City Health Department (Dinas Kesehata.
Climatic data .
aily rainfall, mean temperature, relative humidit.
were sourced from the Bandung Meteorological Station of BMKG (Badan Meteorologi.
Klimatologi, dan Geofisik.
and aggregated to weekly resolution.
Population density and drainage infrastructure data were obtained from BPS-Statistics Bandung City .
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Variable Operationalization The dependent variable is the weekly count of laboratory-confirmed or clinically diagnosed DHF cases per subdistrict.
Independent variables include:
A Rainfall .
m/wee.
: total weekly precipitation A Temperature (AC): weekly mean temperature A Population Density .
ersons/kmA): census-derived sub-district density A Drainage Quality Index (DQI): composite score .
Ae.
derived from proportion of sub-district with adequate open drainage channels, closed drainage coverage, and routine maintenance schedule A Vegetation Cover (%): NDVI-derived estimate of green area per sub-district A Lag-1 and Lag-2 Cases: autoregressive terms capturing transmission dynamics.
Overdispersion Testing Overdispersion was assessed using the Cameron-Trivedi auxiliary regression test (Cameron & Trivedi, 1.
Under the null hypothesis of equidispersion (Poisso.
, the auxiliary regression is:
should yield A significant positive confirms overdispersion and motivates use of NBR.
Negative Binomial Regression Specification The Negative Binomial log-linear model specifies:
is an offset term .
ogarithm of the at-risk populatio.
to account for differing sub-district population Parameters were estimated by Maximum Likelihood using the BFGS optimization algorithm (Zeileis et al.
Standard errors are robust .
andwich-typ.
to account for residual heteroskedasticity.
Model Selection and Validation Model selection employed Akaike Information Criterion (AIC).
Bayesian Information Criterion (BIC), and likelihood ratio test (LRT) comparing NBR against the Poisson baseline.
Goodness-of-fit was assessed via Pearson chi-square statistic and residual diagnostics (Q-Q plots.
Pearson residual plot.
Spatial heterogeneity in IRRs across sub-districts was examined by plotting sub-district-specific predicted incidence rates.
Results and Discussion Temporal Distribution of DHF Cases and Rainfall Figure 1 presents the monthly DHF case distribution across 2019-2023 and the corresponding rainfall-DHF scatter plot with regression trend line.
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Figure 1: (To.
Monthly DHF case distribution by year, 2019-2023.
Bandung City.
(Botto.
Scatter plot and linear trend of monthly rainfall versus DHF case counts ( DHF incidence exhibits a clear seasonal pattern, with peak transmission from November through March coinciding with the wet season when monthly rainfall exceeds 250 mm.
The correlation between rainfall and DHF cases across all sub-district-month observations is , confirming that precipitation is a primary driver of transmission dynamics.
The time-series also reveals inter-annual variation, with 2020 and 2021 displaying higher baseline case counts, potentially reflecting pandemic-era surveillance shifts and vector control disruptions.
Overdispersion Assessment and Model Comparison Figure 2 illustrates the distributional fit of Poisson versus Negative Binomial models to the empirical case count data and compares model selection criteria.
Figure 2: (Lef.
Histogram of observed weekly DHF counts with Poisson and Negative Binomial fitted distributions.
(Righ.
Model selection criteria comparison: AIC.
BIC, log-likelihood, and deviance.
The Cameron-Trivedi test yields C , strongly rejecting equidispersion and confirming that a Negative Binomial model is statistically warranted.
The estimated dispersion parameter is C As shown in Figure 2 .
eft pane.
, the Negative Binomial distribution closely tracks the Saefullah et al.
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empirical frequency distribution, particularly in capturing the right tail of high-count weeks that the Poisson model severely underestimates.
Table 1: Model Comparison Statistics Criterion Poisson Neg.
Binomial i (Improvemen.
Decision AIC NBR preferred BIC NBR preferred Log-Likelihood NBR preferred LRT Reject Poisson Deviance/df NBR adequate Negative Binomial Regression Results Table 2 presents the full NBR coefficient estimates.
Incidence Rate Ratios (IRR), confidence intervals, and pvalues.
Table 2: Negative Binomial Regression Estimates (Dependent Variable: Weekly DHF Coun.
Predictor Intercept Rainfall .
m/wee.
Temperature (AC) Population Density Drainage Quality Index Vegetation Cover (%) Lag-1 Cases Lag-2 Cases Dispersion () Note: *** sub-district-week observations.
SE = Robust standard error.
Each additional millimeter of weekly rainfall is associated with a increase in expected DHF counts ( ), while each degree Celsius rise in temperature corresponds to a The Lag-1 Cases coefficient yields , indicating strong self-exciting dynamics: a sub-district with 10 additional cases this week is expected to have more cases next week, all else equal.
The Drainage Quality Index demonstrates a statistically significant protective effect , implying that a one-unit improvement in the composite drainage score reduces expected weekly case counts by approximately Spatial IRR Profiling and Forest Plot Figure 3 presents the Incidence Rate Ratios for the top 15 highest-risk sub-districts and the forest plot of regression coefficients with 95% confidence intervals.
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Figure 3: (Lef.
Horizontal bar chart of IRRs by sub-district, color-coded by risk tier.
(Righ.
Forest plot of regression coefficients with 95% confidence intervals.
Spatial analysis reveals substantial heterogeneity in sub-district-level risk.
Sub-districts with high population density, low drainage quality, and proximity to riverways show the highest predicted IRRs, exceeding relative to the city average.
This stratification can directly inform the Bandung City vector control unit in prioritizing fogging operations, larval source reduction, and community health education campaigns.
Residual Diagnostics Figure 4 presents model diagnostic plots confirming adequacy of the Negative Binomial specification.
Figure 4: (Lef.
Pearson residuals vs fitted values.
(Righ.
Normal Q-Q plot of Pearson residuals.
The Pearson residuals show no systematic pattern against fitted values, with the vast majority falling within (Figure 4, left pane.
The Q-Q plot indicates approximate normality of residuals (Figure 4, right pane.
, supporting the validity of inferential statistics.
The model deviance-to-degrees-of-freedom ratio of 1.
86 is consistent with adequate fit for a Negative Binomial model.
Discussion This study demonstrates that Negative Binomial Regression substantially outperforms the Poisson model for weekly DHF count data in Bandung City, as evidenced by lower AIC/BIC values, a significantly higher loglikelihood, and adequately scaled Pearson deviance.
The formal Cameron-Trivedi overdispersion test provides the statistical basis for model selection rather than relying on ad hoc visual inspection of variance-mean relationships, which represents a methodological advance over prior urban DHF studies in Indonesia.
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The significant positive effect of lagged cases .
oth Lag-1 and Lag-.
confirms the self-exciting nature of dengue transmission, consistent with compartmental models that incorporate vector population dynamics.
This finding implies that early detection and rapid response within a two-week window is critical to interrupt amplification Health authorities should incorporate near-real-time case surveillance data as inputs to predictive risk The novel Drainage Quality Index demonstrates a statistically significant protective association.
While drainage quality is not a conventional variable in dengue regression models, its significant negative IRR suggests that infrastructure-level interventions may have measurable epidemiological benefits beyond the direct effects captured by temperature and rainfall alone.
Future studies should validate the DQI construction methodology and explore spatial spillover effects across adjacent sub-districts.
Vegetation cover did not reach statistical significance in the full model ( ), which may reflect collinearity with temperature or the coarse spatial resolution of NDVI estimates derived from satellite imagery at the sub-district Future work could employ higher-resolution vegetation data at the neighborhood scale.
Conclusion This paper presented a Negative Binomial Regression framework for modeling overdispersed DHF case counts across 30 sub-districts of Bandung City over a five-year period.
Key contributions are: .
formal statistical demonstration that Poisson regression is inappropriate for this dataset due to significant overdispersion.
identification of rainfall, temperature, population density, drainage quality, and temporal lags as significant determinants of DHF incidence.
generation of sub-district-level Incidence Rate Ratio profiles for spatial risk and .
introduction of a composite Drainage Quality Index as a novel covariate with quantifiable public health implications.
The findings provide actionable evidence for resource allocation in the Bandung City public health system.
Subdistricts with high predicted IRRs and low drainage quality scores should be prioritized for pre-season vector control interventions, particularly in October-November before peak transmission.
Future research directions include zeroinflated negative binomial extensions.
Bayesian hierarchical formulations with spatial random effects, and integration with climate forecast models for prospective DHF outbreak risk prediction.
References