Abstract
Underfive years old acute watery diarrhea (U5AWD) accounts for most diarrheal diseases’ burden, but little is known about the adjusted effect of meteorological and socioeconomic determinants. A dataset containing the seasonal numbers of U5AWD cases at the district level of Iran is collected through MOHME. Accordingly, the districtlevel standardized incidence ratio and Moran’s I values are calculated to detect the significant clusters of U5AWD over sixteen seasons from 2014 to 2018. Additionally, the author tested twelve Bayesian hierarchical models in order to determine which one was the most accurate at forecasting seasonal number of incidents. Iran features a number of U5AWD hotspots, particularly in the southeast. An extended spatiotemporal model with seasonally varying coefficients and space–time interaction outperformed other models, and so became the paper’s proposal in modeling U5AWD. Temperature demonstrated a global positive connection with seasonal U5AWD in districts (IRR: 1.0497; 95% CrI: 1.0254–1.0748), owing to its varying effects during the winter ((IRR: 1.0877; 95% CrI: 1.0408–1.1375) and fall (IRR: 1.0866; 95% CrI: 1.0405–1.1357) seasons. Also, elevation (IRR: 0.9997; 95% CrI: 0.9996–0.9998), piped drinking water (IRR: 0.9948; 95% CrI: 0.9933–0.9964), public sewerage network (IRR: 0.9965; 95% CrI: 0.9938–0.9992), years of schooling (IRR: 0.9649; 95% CrI: 0.944–0.9862), infrastructuretohousehold size ratio (IRR: 0.9903; 95% CrI: 0.986–0.9946), wealth index (IRR: 0.9502; 95% CrI: 0.9231–0.9781), and urbanization (IRR: 0.9919; 95% CrI: 0.9893–0.9944) of districts were negatively associated with seasonal U5AWD incidence. Strategically, developing geoinformation alarm systems based on meteorological data might help predict U5AWD highrisk areas. The study also anticipates increased rates of U5AWD in districts with poor sanitation and socioeconomic level. Therefore, governments should take appropriate preventative actions in these sectors.
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Data availability
Meteorological dataset analyzed during the current study is available in the IRIMO repository, https://data.irimo.ir/login/login.aspx. Elevation data can be obtained from https://www.calcmaps.commapelevation/. The raw data of different stages of Iran National Population and Housing Census and the Households Income and Expenditure Survey (HIES) is available at https://www.amar.org.ir/english. The other datasets generated and/or analyzed during the current study are not publicly available due to MOHME policies but are available from the corresponding author on reasonable request.
Code availability
All the software codes regarding model specifications are available here: https://www.rinla.org/home.
Abbreviations
 WHO:

World Health Organization
 IHME:

Institute for Health Metrics and Evaluation
 U5AWD:

Underfive acute watery diarrhea
 GBD:

Global Burden of Disease
 SDG 3:

Sustainable Development Goal 3
 BHM:

Bayesian hierarchical modeling
 MCMC:

Markov chain Monte Carlo
 INLA:

Integrated nested Laplace approximation
 MOHME:

Ministry of Health and Medical Education
 IHSR:

Infrastructuretohousehold size ratio
 YOS:

Years of schooling
 IRIMO:

Iran Meteorological Organization
 HIES:

Households Income and Expenditure Survey
 SCI:

Statistical Center of Iran
 PCA:

Principal component analysis
 DHS:

Demographic and Health Surveys
 SIRs:

Standardized incidence ratios
 LISAs:

Local indicators of spatial association
 IR:

Incidence rate
 iid:

Independent and identically distributed
 ICAR:

Intrinsic conditional autoregressive
 BYM:

BesagYorkMollié
 RW1:

Firstorder random walk
 DIC:

Deviance information criterion
 \({P}_{DIC}\) :

Effective number of parameters for DIC
 WAIC:

Watanabe–Akaike Information Criterion
 \({P}_{WAIC}\) :

Effective number of parameters for WAIC
 LS:

Logarithmic score
 CrI:

Credible interval
 MCrIW:

Mean credible interval width
 MAE:

Mean absolute error
 SD:

Standard deviation
 CI:

Confidence interval
 IQR:

Interquartile range
 HH:

HighHigh
 IRR:

Incidence rate ratio
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Acknowledgements
The author gratefully acknowledges Dr. Håvard Rue, Dr. Finn Lindgren, and Dr. Elias T. Krainski for their valuable suggestions and technical supports in Rinla discussion group (https://groups.google.com/g/rinladiscussiongroup). The author also appreciates Dr. Babak Eshrati for extracting the disease dataset from MOHME’s communicable diseases surveillance system.
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Appendix
Appendix
This appendix provides an overview of the 12 models used throughout the article. It is worth noting that all models use the same notations to condense explanations and facilitate comparisons across models:
Model 1: The ordinary multivariate regression model (nonspatial and nontemporal) as:
where \({\beta }_{0}\) is the intercept; \({X}_{k,it}\) is the value of \(k\) th covariate in district \(i\), season \(t\); and \({\beta }_{k}\) is the estimated coefficient of \(k\) th covariate, showing its linear fixed effect through space and time.
Model 2: The spatial ecological regression model as:
where \({u}_{i}\) and \({v}_{i}\) are the spatial unstructured and structured random effect in district \(i\), respectively; \({\mathcal{N}}_{i}\) is the number of districts sharing boundaries with \(i\) th district; and \({\sigma }_{i}^{2}\) is the variance between \(i\) th district’s effect and neighbors. In every model, we selected an independent and identically distributed (iid) Gaussian distribution as diffuse prior for \({u}_{i}\) and the intrinsic conditional autoregressive (ICAR) structure as the prior distribution of \({v}_{i}\) (Eqs. 9–10). \({u}_{i}+{v}_{i}\) togetherly construct a prominent convolution model known as Besag York Mollié (BYM) (Besag et al. 1991).
Model 3: The time series regression model as:
where \({\gamma }_{t}\) and \({\phi }_{t}\) are the temporal unstructured and structured random effect in year \(t\), respectively. In every model, we selected an iid Gaussian distribution as noninformative prior for \({\gamma }_{t}\) (Eq. 12). We utilized the seasonal latent model available in RINLA as the prior of \({\phi }_{t}\), assuming \(\sum_{t=1}^{4}{\phi }_{t}=0\). We formulated the density for seasonal prior in Eq. 13.
Model 4: A seasonally varying coefficient model containing nonstationary random effects for meteorological variables,
which can be stated more straightforwardly as follows:
where \({X}_{1,it}\), \({X}_{2,it}\), and \({X}_{3,it}\) represent rainfall, temperature, and wind speed of district \(i\) in season \(t\); \(\{{X}_{4,it},\dots ,{X}_{11,it}\}\) are the remaining covariates that are just linear predictors having fixed effects; \(\{{\xi }_{1,t},{\xi }_{2,t},{\xi }_{3,t}\}\) are the variablespecific seasonally varying structured random effects for which we specified cyclic firstorder random walk (RW1) priors (Eqs. 16–17).
Model 5: A spatiotemporal model (combining model 2 and model 3),
Model 6: The spatial ecological regression with seasonally varying coefficients for meteorological variables (combining model 2 and model 4),
Model 7: The time series regression model with seasonally varying coefficients for meteorological variables (combining model 3 and model 4),
Model 8: The spatiotemporal model with seasonally varying coefficients for meteorological variables (combining models 2–4),
Model 9: Model 8 + space–time interaction between spatially unstructured (\(u\)) and temporally unstructured (\(\gamma\)) random effects (referred to as interaction type I (KnorrHeld 2000)),
where \({\delta }_{it}\) is the mentioned interaction term in district \(i\), season \(t\); \({I}_{u}\) and \({I}_{\gamma }\) are 429 × 429 and 16 × 16 identity matrices, respectively and \(\otimes\) is the Kronecker product operator used to build a block matrix out of \({I}_{u}\) and \({I}_{\gamma }\).
Models 9–12 have the same formulation for \({\mu }_{it}\) and only the distribution of \({\delta }_{it}\) differs.
Model 10: Model 8 + space–time interaction between spatially unstructured (\(u\)) and temporally structured (\(\phi\)) random effects (interaction type II) as:
where \({R}_{\phi }\) is a 16 × 16 matrix with RW1 structure.
Model 11: Model 8 + space–time interaction between spatially structured (\(v\)) and temporally unstructured (\(\gamma\)) random effects (interaction type III),
where \({R}_{v}\) is a 429 × 429 matrix with ICAR proximity structure which we specified the prior in Eq. 10.
Model 12: Model 8 + space–time interaction between spatially structured (\(v\)) and temporally structured (\(\phi\)) random effects (interaction type IV),
Finally, it should be noted that all betweencovariate interaction effects have been entered into the models above using a forward approach. However, they did not show any subtle effects, and followingly we decided not to include them in main analyses.
We briefly introduce eight widely used model fit metrics that were utilized to compare the models in the study:
Deviance information criterion
Deviance information criterion (DIC) (Spiegelhalter et al. 2002) is a predictive accuracy measure consisting two components given by,
where \(y\) and \(\theta\) are the data and likelihood parameters vector, respectively; \(D\) is the expected deviance of the model; \(\overline{D }\) is the posterior mean of deviance and \({P}_{D}\) is the effective numbers of parameters. \({P}_{D}\) is defined via:
where \(D(\overline{\theta })\) is the evaluated deviance for the parameters posterior mean.
DIC is the most used measure in evaluating Bayesian models. In summary, lower values of \(\mathrm{DIC}\) indicate better performance as the first component (\(\overline{D }\)) indicates goodness of model fit and, simultaneously, the effective number of parameters (\({P}_{D}\)) is the penalty component for model complexity.
Watanabe–Akaike information criterion
Watanabe–Akaike information criterion (WAIC, also known as Widely Applicable Information Criterion) is developed by Watanabe (Watanabe 2010) to compare Bayesian models’ predictive accuracy given by,
where the first component implies log pointwise posterior predictive density and \({P}_{W}\) is the effective numbers of parameters. \({P}_{W}\) is defined via:
The overall structure and interpretation of WAIC is similar to DIC. Based on recent BHM studies, WAIC offers a number of benefits over DIC, since it employs a more genuine Bayesian approach by making the use of the whole posterior distribution (instead of posterior mean) in calculating the outofsample expectation. Also, it is preferable in case of singular models with nonidentifiable parameterization. On the other side, WAIC is less often utilized in practice due to the extra computational work required and potential challenges in dealing with structured data (Gelman et al. 2014).
Crossvalidated logarithmic score
A Bayesian leaveoneoutcrossvalidation measure is conditional predictive ordinate (CPO) (Pettit 1990) which is calculated for every observation as follows:
where \({y}_{it}\) is the observation in district \(i\), season \(t\), and \({y}_{it}\) is the vector of observations after removing \({y}_{it}\). Crossvalidated logarithmic score (LS) is defined as logscore of CPO values via:
where \(I\) and \(T\) are the number of districts and seasons, respectively. Lower values of \(\mathrm{LS}\) show higher predictive ability.
95% credible interval coverage
A summary statistic used for model selection is 95% credible interval coverage which is given by,
where \({\widehat{y}}_{97.5,i}\) and \({\widehat{y}}_{2.5,i}\) are the 97.5% and 2.5% quantiles of the posterior samples estimated in district \(i\), season \(t\), respectively and \(I(.)\) represents the indicator function. Closer values of coverage to 100 imply better fit.
Mean credible interval width (MCrIW)
Mean credible interval width (MCrIW) is another evaluation statistic defined as:
Higher values of \(\mathrm{MCrIW}\) demonstrate bigger uncertainty in the fitted values of the model.
Mean absolute error (MAE)
One of the most common metrics used to evaluate model fitness in the literature is mean absolute error (MAE). It is defined as:
Lower values of \(\mathrm{MAE}\) indicate better model fit. It should be noted that observing extremely large \(\mathrm{MAE}\) values is not necessarily a good clue and may declare overfitting issues.
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Masinaei, M. Estimating the seasonally varying effect of meteorological factors on the districtlevel incidence of acute watery diarrhea among underfive children of Iran, 2014–2018: a Bayesian hierarchical spatiotemporal model. Int J Biometeorol 66, 1125–1144 (2022). https://doi.org/10.1007/s00484022022639
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Issue Date:
DOI: https://doi.org/10.1007/s00484022022639
Keywords
 Climate change
 Acute diarrhea
 Spatial epidemiology
 Child health
 Bayesian hierarchical modeling
 INLA