Skip to main content

Estimating the seasonally varying effect of meteorological factors on the district-level incidence of acute watery diarrhea among under-five children of Iran, 2014–2018: a Bayesian hierarchical spatiotemporal model


Under-five 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 district-level 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), infrastructure-to-household 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 high-risk 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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Data availability

Meteorological dataset analyzed during the current study is available in the IRIMO repository, Elevation data can be obtained from https://www.calcmaps.commap-elevation/. The raw data of different stages of Iran National Population and Housing Census and the Households Income and Expenditure Survey (HIES) is available at 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:



World Health Organization


Institute for Health Metrics and Evaluation


Under-five acute watery diarrhea


Global Burden of Disease

SDG 3:

Sustainable Development Goal 3


Bayesian hierarchical modeling


Markov chain Monte Carlo


Integrated nested Laplace approximation


Ministry of Health and Medical Education


Infrastructure-to-household size ratio


Years of schooling


Iran Meteorological Organization


Households Income and Expenditure Survey


Statistical Center of Iran


Principal component analysis


Demographic and Health Surveys


Standardized incidence ratios


Local indicators of spatial association


Incidence rate


Independent and identically distributed


Intrinsic conditional autoregressive




First-order random walk


Deviance information criterion

\({P}_{DIC}\) :

Effective number of parameters for DIC


Watanabe–Akaike Information Criterion

\({P}_{WAIC}\) :

Effective number of parameters for WAIC


Logarithmic score


Credible interval


Mean credible interval width


Mean absolute error


Standard deviation


Confidence interval


Interquartile range




Incidence rate ratio


  • Abuzerr S, Nasseri S, Yunesian M, Hadi M, Zinszer K, Mahvi AH, Nabizadeh R, Abu Mustafa A, Mohammed SH (2020) Water, sanitation, and hygiene risk factors of acute diarrhea among children under five years in the Gaza Strip. J Water Sanit Hyg Dev 10(1):111–123

  • Adane M, Mengistie B, Medhin G, Kloos H, Mulat W (2017) Piped water supply interruptions and acute diarrhea among under-five children in Addis Ababa slums, Ethiopia: a matched case-control study. PLoS ONE 12(7):e0181516.

    CAS  Article  Google Scholar 

  • Agegnehu MD, Zeleke LB, Goshu YA, Ortibo YL, Mehretie Adinew Y (2019) Diarrhea prevention practice and associated factors among caregivers of under-five children in Enemay District, Northwest Ethiopia. J Environ Public Health 2019:5490716.

    Article  Google Scholar 

  • Agustina R, Sari TP, Satroamidjojo S, Bovee-Oudenhoven IM, Feskens EJ, Kok FJ (2013) Association of food-hygiene practices and diarrhea prevalence among Indonesian young children from low socioeconomic urban areas. BMC Public Health 13(1):1–12

  • Alebel A, Tesema C, Temesgen B, Gebrie A, Petrucka P, Kibret GD (2018) Prevalence and determinants of diarrhea among under-five children in Ethiopia: A systematic review and meta-analysis. PLoS ONE 13(6):e0199684.

    CAS  Article  Google Scholar 

  • Alemayehu B, Ayele BT, Kloos H, Ambelu A (2020a) Individual and community-level risk factors in under-five children diarrhea among agro-ecological zones in southwestern Ethiopia. Int J Hyg Environ Health 224:113447.

    Article  Google Scholar 

  • Alemayehu B, Ayele BT, Melak F, Ambelu A (2020) Exploring the association between childhood diarrhea and meteorological factors in Southwestern Ethiopia. Sci Total Environ 741:140189

  • Anselin L (1995) Local indicators of spatial association—LISA. Geogr Anal 27(2):93–115

  • Anwar MY, Warren JL, Pitzer VE (2019) Diarrhea patterns and climate: a spatiotemporal Bayesian hierarchical analysis of diarrheal disease in Afghanistan. Am J Trop Med Hyg 101(3):525–533

  • Ashkenazi S, Schwartz E (2020) Traveler’s diarrhea in children: new insights and existing gaps. Travel Med Infect Dis 34:101503.

    Article  Google Scholar 

  • Azage M, Kumie A, Worku A, C. Bagtzoglou A, Anagnostou E (2017) Effect of climatic variability on childhood diarrhea and its high risk periods in northwestern parts of Ethiopia. PLoS One 12(10):e0186933

  • Berde AS, Yalçın SS, Özcebe H, Üner S, Karadağ-Caman Ö (2018) Determinants of childhood diarrhea among under-five year old children in Nigeria: a population-based study using the 2013 demographic and health survey data. Turk J Pediatr 60(4):353–360.

    Article  Google Scholar 

  • Besag J, York J, Mollié A (1991) Bayesian image restoration, with two applications in spatial statistics. Ann Inst Stat Math 43(1):1–20

  • Bivand R, Altman M, Anselin L, Assunção R, Berke O, Bernat A, Blanchet G (2015) Package ‘spdep’. The Comprehensive R Archive Network

  • Blangiardo M, Cameletti M (2015) Spatial and spatio-temporal Bayesian models with R-INLA. John Wiley & Sons, Hoboken

  • Blangiardo M, Boulieri A, Diggle P, Piel FB, Shaddick G, Elliott P (2020) Advances in spatiotemporal models for non-communicable disease surveillance. Int J Epidemiol 49:I26–I37.

    Article  Google Scholar 

  • Budhathoki SS, Bhattachan M, Yadav AK, Upadhyaya P, Pokharel PK (2016) Eco-social and behavioural determinants of diarrhoea in under-five children of Nepal: a framework analysis of the existing literature. Tropical Medicine and Health 44(1):1–7.

    Article  Google Scholar 

  • Carlton EJ, Eisenberg JNS, Goldstick J, Cevallos W, Trostle J, Levy K (2014) Heavy rainfall events and diarrhea incidence: the role of social and environmental factors. Am J Epidemiol 179(3):344–352.

    Article  Google Scholar 

  • Chou W-C, Wu J-L, Wang Y-C, Huang H, Sung F-C, Chuang C-Y (2010) Modeling the impact of climate variability on diarrhea-associated diseases in Taiwan (1996–2007). Sci Total Environ 409(1):43–51

  • De Albuquerque C, Roaf V (2012) On the right track: good practices in realising the rights to water and sanitation.

  • Delahoy MJ, Cárcamo C, Huerta A, Lavado W, Escajadillo Y, Ordoñez L, Vasquez V, Lopman B, Clasen T, Gonzales GF (2021) Meteorological factors and childhood diarrhea in Peru, 2005–2015: a time series analysis of historic associations, with implications for climate change. Environ Health 20(1):1–10

  • Diouf K, Tabatabai P, Rudolph J, Marx M (2014) Diarrhoea prevalence in children under five years of age in rural Burundi: an assessment of social and behavioural factors at the household level. Global Health Action 7(1):24895.

    Article  Google Scholar 

  • D’Souza RM, Hall G, Becker NG (2008) Climatic factors associated with hospitalizations for rotavirus diarrhoea in children under 5 years of age. Epidemiol Infect 136(1):56–64.

    CAS  Article  Google Scholar 

  • Dunn G, Johnson GD, Balk DL, Sembajwe G (2020) Spatially varying relationships between risk factors and child diarrhea in West Africa, 2008–2013. Math Popul Stud 27(1):8–33.

    Article  Google Scholar 

  • Eesteghamati A, Gouya M, Keshtkar A, Najafi L, Zali MR, Sanaei M, Yaghini F, El Mohamady H, Patel M, Klena JD, Teleb N (2009) Sentinel hospital-based surveillance of Rotavirus diarrhea in Iran. J Infect Dis 200(SUPPL. 1):S244–S247.

    Article  Google Scholar 

  • Elliott P, Savitz DA (2008) Design issues in small-area studies of environment and health. Environ Health Perspect 116(8):1098–1104.

    Article  Google Scholar 

  • Feleke H, Medhin G, Kloos H, Gangathulasi J, Asrat D (2018) Household-stored drinking water quality among households of under-five children with and without acute diarrhea in towns of Wegera District, in North Gondar, Northwest Ethiopia. Environ Monit Assess 190(11):1–12.

  • Gelman A, Hwang J, Vehtari A (2014) Understanding predictive information criteria for Bayesian models. Stat Comput 24(6):997–1016.

    Article  Google Scholar 

  • Gómez-Rubio V (2020) Bayesian inference with INLA. CRC Press

  • Hashizume M, Armstrong B, Hajat S, Wagatsuma Y, Faruque ASG, Hayashi T, Sack DA (2007) Association between climate variability and hospital visits for non-cholera diarrhoea in Bangladesh: effects and vulnerable groups. Int J Epidemiol 36(5):1030–1037.

    Article  Google Scholar 

  • IHME (2020) Global Burden of Disease Collaborative Network. Global Burden of Disease Study 2019 (GBD 2019) Results. Institute for Health Metrics and Evaluation Seattle. Accessed 14 July 2021

  • Knorr-Held L (2000) Bayesian modelling of inseparable space-time variation in disease risk. Stat Med 19(17–18):2555–2567.;2-%23

    CAS  Article  Google Scholar 

  • Kumi-Kyereme A, Amo-Adjei J (2016) Household wealth, residential status and the incidence of diarrhoea among children under-five years in Ghana. J Epidemiol Glob Health 6(3):131–140.

    Article  Google Scholar 

  • Li R, Lai Y, Feng C, Dev R, Wang Y, Hao Y (2020) Diarrhea in under five year-old children in nepal: a spatiotemporal analysis based on demographic and health survey data. Int J Environ Res Public Health 17(6):2140.

    Article  Google Scholar 

  • Mallick R, Mandal S, Chouhan P (2020) Impact of sanitation and clean drinking water on the prevalence of diarrhea among the under-five children in India. Child Youth Serv Rev 118:105748.

    Article  Google Scholar 

  • Martínez-Bello DA, López-Quílez A, Torres-Prieto A (2017) Bayesian dynamic modeling of time series of dengue disease case counts. PLoS Negl Trop Dis 11(7):e0005696.

    Article  Google Scholar 

  • Masinaei M, Eshrati B, Yaseri M (2020) Spatial and spatiotemporal patterns of typhoid fever and investigation of their relationship with potential risk factors in Iran, 2012–2017. Int J Hyg Environ Health 224:113432.

    Article  Google Scholar 

  • McMichael AJ, Campbell-Lendrum DH, Corvalán CF, Ebi KL, Githeko A, Scheraga JD, Woodward A (2003) Climate change and human health: risks and responses. World Health Organization

    Google Scholar 

  • Mertens A, Balakrishnan K, Ramaswamy P, Rajkumar P, Ramaprabha P, Durairaj N, Hubbard AE, Khush R, Colford JM Jr, Arnold BF (2019) Associations between high temperature, heavy rainfall, and diarrhea among young children in rural Tamil Nadu, India: A Prospective Cohort Study. Environ Health Perspect 127(4):47004

  • Miettinen OS (1974) Proportion of disease caused or prevented by a given exposure, trait or intervention. Am J Epidemiol 99(5):325–332.

    CAS  Article  Google Scholar 

  • Misriyanto E, Sitorus RJ (2020) Analysis of environmental factors with chronic diarrhea in toddlers in Jambi City in 2019. Int J Sci Soc 2(4):300–310

  • Moors E, Singh T, Siderius C, Balakrishnan S, Mishra A (2013) Climate change and waterborne diarrhoea in northern India: impacts and adaptation strategies. Sci Total Environ 468–469:S139–S151.

    CAS  Article  Google Scholar 

  • Moraga P (2017) SpatialEpiApp: A Shiny web application for the analysis of spatial and spatio-temporal disease data. Spat Spatio-Temp Epidemiol 23:47–57.

    Article  Google Scholar 

  • Mukabutera A, Thomson D, Murray M, Basinga P, Nyirazinyoye L, Atwood S, Savage KP, Ngirimana A, Hedt-Gauthier BL (2016) Rainfall variation and child health: effect of rainfall on diarrhea among under 5 children in Rwanda, 2010. BMC Public Health 16(1):1–9.

  • Mulatya DM, Ochieng C (2020) Disease burden and risk factors of diarrhoea in children under five years: evidence from Kenya’s demographic health survey 2014. Int J Infect Dis 93:359–366.

    Article  Google Scholar 

  • Musengimana G, Mukinda FK, Machekano R, Mahomed H (2016) Temperature variability and occurrence of diarrhoea in children under five-years-old in Cape Town metropolitan sub-districts. Int J Environ Res Public Health 13(9):859.

    Article  Google Scholar 

  • Negesse Y, Taddese AA, Negesse A, Ayele TA (2021) Trends and determinants of diarrhea among under-five children in Ethiopia: cross-sectional study: multivariate decomposition and multilevel analysis based on Bayesian approach evidenced by EDHS 2000–2016 data. BMC Public Health 21(1):1–16.

  • Omona S, Malinga GM, Opoke R, Openy G, Opiro R (2020) Prevalence of diarrhoea and associated risk factors among children under five years old in Pader District, northern Uganda. BMC Infect Dis 20(1):1–9.

  • Online K (2014) Report from the Department of Economics about the Results of Implementing the First Phase of Targeting: The People Have Become Poor, the Class Gaps Have Increased. Khabar Online. Accessed 14 July 2021 2014

  • Paredes-Paredes M, Okhuysen PC, Flores J, Mohamed JA, Padda RS, Gonzalez-Estrada A, Haley CA, Carlin LG, Nair P, DuPont HL (2011) Seasonality of diarrheagenic Escherichia coli pathotypes in the US students acquiring diarrhea in Mexico. J Travel Med 18(2):121–125

  • Pettit LI (1990) The conditional predictive ordinate for the normal distribution. J Roy Stat Soc: Ser B (methodol) 52(1):175–184

  • Piel FB, Fecht D, Hodgson S, Blangiardo M, Toledano M, Hansell AL, Elliott P (2021) Small-area methods for investigation of environment and health. Int J Epidemiol 49(2):686–699.

    Article  Google Scholar 

  • Prasetyo D, Ermaya Y, Martiza I, Yati S (2015) Correlation between climate variations and rotavirus diarrhea in under-five children in Bandung. Asian Pacif J Trop Dis 5(11):908–911.

    Article  Google Scholar 

  • Robert CP, Elvira V, Tawn N, Wu C (2018) Accelerating MCMC algorithms. Wiley Interdisc Rev: Comput Stat 10(5):e1435.

  • Rue H, Martino S, Chopin N (2009) Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations. J R Stat Soc Ser B Stat Methodol 71(2):319–392.

    Article  Google Scholar 

  • Singh RBK, Hales S, De Wet N, Raj R, Hearnden M, Weinstein P (2001) The influence of climate variation and change on diarrheal disease in the Pacific Islands. Environ Health Perspect 109(2):155–159.

    CAS  Article  Google Scholar 

  • Spiegelhalter DJ, Best NG, Carlin BP, Van Der Linde A (2002) Bayesian measures of model complexity and fit. J R Stat Soc Ser B Stat Methodol 64(4):583–616.

    Article  Google Scholar 

  • Tate JE, Burton AH, Boschi-Pinto C, Parashar UD, Agocs M, Serhan F, De Oliveira L, Mwenda JM, Mihigo R, Ranjan Wijesinghe P, Abeysinghe N, Fox K, Paladin F (2016) Global, regional, and national estimates of rotavirus mortality in children <5 years of age, 2000–2013. Clin Infect Dis 62:S96–S105.

    Article  Google Scholar 

  • Thiam S, Cissé G, Stensgaard AS, Niang-Diène A, Utzinger J, Vounatsou P (2019) Bayesian conditional autoregressive models to assess spatial patterns of diarrhoea risk among children under the age of 5 years in mbour, senegal. Geospat Health 14(2):321–328.

    Article  Google Scholar 

  • UNICEF (2016) Diarrhoea remains a leading killer of young children, despite the availability of a simple treatment solution. UNICEF data: monitoring the situation of children and women

  • Vaghefi SA, Keykhai M, Jahanbakhshi F, Sheikholeslami J, Ahmadi A, Yang H, Abbaspour KC (2019) The future of extreme climate in Iran. Sci Rep 9(1):1–11

  • Wang Y, Li J, Gu J, Zhou Z, Wang Z (2015) Artificial neural networks for infectious diarrhea prediction using meteorological factors in Shanghai (China). Appl Soft Comput 35:280–290

  • Wang X, Yue Y, Faraway JJ (2018) Bayesian regression modeling with INLA. Chapman and Hall/CRC, London

  • Wang H, Di B, Zhang T, Lu Y, Chen C, Wang D, Li T, Zhang Z, Yang Z (2019) Association of meteorological factors with infectious diarrhea incidence in Guangzhou, southern China: a time-series study (2006–2017). Sci Total Environ 672:7–15

  • Wasihun AG, Dejene TA, Teferi M, Marugán J, Negash L, Yemane D, McGuigan KG (2018) Risk factors for diarrhoea and malnutrition among children under the age of 5 years in the Tigray Region of Northern Ethiopia. PLoS ONE 13(11):e0207743.

    CAS  Article  Google Scholar 

  • Watanabe S (2010) Asymptotic equivalence of Bayes cross validation and widely applicable information criterion in singular learning theory. J Mach Learn Res 11:3571–3594

    Google Scholar 

  • WHO (2017) WHO Fact Sheet. World Health Organization. Accessed 14 July 2021

  • Xu Z, Huang C, Turner LR, Su H, Qiao Z, Tong S (2013) Is diurnal temperature range a risk factor for childhood diarrhea? PLoS One 8(5):e64713

  • Xu Z, Liu Y, Ma Z, Toloo GS, Hu W, Tong S (2014) Assessment of the temperature effect on childhood diarrhea using satellite imagery. Sci Rep 4:5389

  • Zhang Y, Bambrick H, Mengersen K, Tong S, Hu W (2021) Using internet-based query and climate data to predict climate-sensitive infectious disease risks: a systematic review of epidemiological evidence. Int J Biometeorol 65(12):2203–2214.

    Article  Google Scholar 

Download references


The author gratefully acknowledges Dr. Håvard Rue, Dr. Finn Lindgren, and Dr. Elias T. Krainski for their valuable suggestions and technical supports in R-inla discussion group ( The author also appreciates Dr. Babak Eshrati for extracting the disease dataset from MOHME’s communicable diseases surveillance system.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Masoud Masinaei.

Ethics declarations

Ethics approval

The Iranian Ministry of Health and Medical Education (MOHME) waives the need for approval in analyses where only aggregated surveillance data are provided.

Data protection, confidentiality and privacy

Not applicable.

Consent to participate

Not applicable.

Consent for publication

Not applicable.

Competing interests

The author declares no competing interests.



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 (non-spatial and non-temporal) as:

$${\mu }_{it}={\beta }_{0}+\sum_{k=1}^{11}{\beta }_{k}{X}_{k,it}$$

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:

$${\mu }_{it}={\beta }_{0}+\sum_{k=1}^{11}{\beta }_{k}{X}_{k,it}+{u}_{i}+{v}_{i}$$
$$u\sim Normal(0,{\sigma }_{u}^{2}{I}_{u}^{-1})$$
$${v}_{i}|{v}_{j\ne i}=Normal(\frac{1}{{\mathcal{N}}_{i}}\sum_{j=1}^{{\mathcal{N}}_{i}}{v}_{j},\frac{{\sigma }_{i}^{2}}{{\mathcal{N}}_{i}})$$

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. 910). \({u}_{i}+{v}_{i}\) togetherly construct a prominent convolution model known as Besag York Mollié (BYM) (Besag et al. 1991).

Model 3: The time series regression model as:

$${\mu }_{it}={\beta }_{0}+\sum_{k=1}^{11}{\beta }_{k}{X}_{k,it}+{\gamma }_{t}+{\phi }_{t}$$
$$\gamma \sim Normal(0,{\sigma }_{\gamma }^{2}{I}_{\gamma }^{-1})$$
$$\pi ({\phi }_{t}|{\sigma }_{\phi }^{2})\propto \mathrm{exp}(-\frac{1}{2{\sigma }_{\phi }^{2}}\sum_{t=1}^{13}{({\phi }_{t}+{\phi }_{t+1}+{\phi }_{t+2}+{\phi }_{t+3})}^{2})$$

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 non-informative prior for \({\gamma }_{t}\) (Eq. 12). We utilized the seasonal latent model available in R-INLA 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,

$${\mu }_{it}={\beta }_{0}+\sum_{k=1}^{11}{\beta }_{k}{X}_{k,it}+\sum_{k=1}^{3}f({\xi }_{k,t}{X}_{k,it})$$

which can be stated more straightforwardly as follows:

$${\mu }_{it}={\beta }_{0}+\sum_{k=1}^{3}({\beta }_{k}+{\xi }_{k,t}){X}_{k,it}+\sum_{k=4}^{11}{\beta }_{k}{X}_{k,it}$$
$${\xi }_{t}|{\xi }_{t-1}\sim Normal({\xi }_{t-1},{\sigma }_{\xi }^{2})$$
$$\sum_{t=1}^{4}{\xi }_{k,t}=0, k=\mathrm{1,2},3$$
$$\pi ({\xi }_{t}|{\sigma }_{\xi }^{2})\propto \mathrm{exp}(-\frac{1}{2{\sigma }_{\xi }^{2}}\sum_{t=2}^{T}{({\xi }_{t}-{\xi }_{t-1})}^{2})$$

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 variable-specific seasonally varying structured random effects for which we specified cyclic first-order random walk (RW1) priors (Eqs. 1617).

Model 5: A spatiotemporal model (combining model 2 and model 3),

$${\mu }_{it}={\beta }_{0}+\sum_{k=1}^{11}{\beta }_{k}{X}_{k,it}+{u}_{i}+{v}_{i}+{\gamma }_{t}+{\phi }_{t}$$

Model 6: The spatial ecological regression with seasonally varying coefficients for meteorological variables (combining model 2 and model 4),

$${\mu }_{it}={\beta }_{0}+\sum_{k=1}^{3}({\beta }_{k}+{\xi }_{k,t}){X}_{k,it}+\sum_{k=4}^{11}{\beta }_{k}{X}_{k,it}+{u}_{i}+{v}_{i}$$

Model 7: The time series regression model with seasonally varying coefficients for meteorological variables (combining model 3 and model 4),

$${\mu }_{it}={\beta }_{0}+\sum_{k=1}^{3}({\beta }_{k}+{\xi }_{k,t}){X}_{k,it}+\sum_{k=4}^{11}{\beta }_{k}{X}_{k,it}+{\gamma }_{t}+{\phi }_{t}$$

Model 8: The spatiotemporal model with seasonally varying coefficients for meteorological variables (combining models 2–4),

$${\mu }_{it}={\beta }_{0}+\sum_{k=1}^{3}({\beta }_{k}+{\xi }_{k,t}){X}_{k,it}+\sum_{k=4}^{11}{\beta }_{k}{X}_{k,it}+{u}_{i}+{v}_{i}+{\gamma }_{t}+{\phi }_{t}$$

Model 9: Model 8 + space–time interaction between spatially unstructured (\(u\)) and temporally unstructured (\(\gamma\)) random effects (referred to as interaction type I (Knorr-Held 2000)),

$${\mu }_{it}={\beta }_{0}+\sum_{k=1}^{3}({\beta }_{k}+{\xi }_{k,t}){X}_{k,it}+\sum_{k=4}^{11}{\beta }_{k}{X}_{k,it}+{u}_{i}+{v}_{i}+{\gamma }_{t}+{\phi }_{t}+{\delta }_{it}$$
$$\delta \sim Normal(0,{\sigma }_{\delta }^{2}{({I}_{u}\otimes {I}_{\gamma })}^{-1})$$

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:

$$\delta \sim Normal(0,{\sigma }_{\delta }^{2}{({I}_{u}\otimes {R}_{\phi })}^{-1})$$

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),

$$\delta \sim Normal(0,{\sigma }_{\delta }^{2}{({R}_{v}\otimes {I}_{\gamma })}^{-1})$$

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),

$$\delta \sim Normal(0,{\sigma }_{\delta }^{2}{({R}_{v}\otimes {R}_{\phi })}^{-1})$$

Finally, it should be noted that all between-covariate 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,

$$\mathrm{DIC}=\overline{D }+{P}_{D}$$
$$\overline{D }={E}_{\theta }(D|y)$$
$$D=-2\mathrm{log}(p\left(y|\theta \right))$$

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:

$${P}_{D}=\overline{D }-D(\overline{\theta })$$

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:

$${P}_{W}=\sum_{i=1}^{n}\mathrm{var}(\mathrm{log}(p({y}_{i}|\theta )))$$

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 out-of-sample expectation. Also, it is preferable in case of singular models with non-identifiable 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).

Cross-validated logarithmic score

A Bayesian leave-one-out-cross-validation 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 \({y}_{it}\). Cross-validated logarithmic score (LS) is defined as log-score 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,

$$95\mathrm{\% coverage}=\frac{1}{IT}\sum_{i,t}I({y}_{it}\in \left[{\widehat{y}}_{2.5,it},{\widehat{y}}_{97.5,it}\right])\times 100$$

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.

Table 5

Table 5 Standardized incidence ratios (with 95% confidence interval) of U5AWD in districts with different characteristics based on five quantiles of every variable

Table 6

Table 6 High-High (HH) clusters of U5AWD at different levels of significance

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Masinaei, M. Estimating the seasonally varying effect of meteorological factors on the district-level incidence of acute watery diarrhea among under-five children of Iran, 2014–2018: a Bayesian hierarchical spatiotemporal model. Int J Biometeorol 66, 1125–1144 (2022).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Climate change
  • Acute diarrhea
  • Spatial epidemiology
  • Child health
  • Bayesian hierarchical modeling
  • INLA