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

Abstract

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.

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 (non-spatial and non-temporal) as:

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

(7)

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:

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

(8)

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

(9)

$${v}_{i}|{v}_{j\ne i}=Normal(\frac{1}{{\mathcal{N}}_{i}}\sum_{j=1}^{{\mathcal{N}}_{i}}{v}_{j},\frac{{\sigma }_{i}^{2}}{{\mathcal{N}}_{i}})$$

(10)

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:

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

(11)

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

(12)

$$\pi ({\phi }_{t}|{\sigma }_{\phi }^{2})\propto \mathrm{exp}(-\frac{1}{2{\sigma }_{\phi }^{2}}\sum_{t=1}^{13}{({\phi }_{t}+{\phi }_{t+1}+{\phi }_{t+2}+{\phi }_{t+3})}^{2})$$

(13)

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,

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

(14)

which can be stated more straightforwardly as follows:

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

(15)

$${\xi }_{t}|{\xi }_{t-1}\sim Normal({\xi }_{t-1},{\sigma }_{\xi }^{2})$$

(16)

$$\sum_{t=1}^{4}{\xi }_{k,t}=0, k=\mathrm{1,2},3$$

(17)

$$\pi ({\xi }_{t}|{\sigma }_{\xi }^{2})\propto \mathrm{exp}(-\frac{1}{2{\sigma }_{\xi }^{2}}\sum_{t=2}^{T}{({\xi }_{t}-{\xi }_{t-1})}^{2})$$

(18)

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. 16–17).

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

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

(19)

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

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

(20)

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

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

(21)

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

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

(22)

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)),

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

(23)

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

(24)

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:

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

(25)

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),

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

(26)

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),

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

(27)

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,

$$\mathrm{DIC}=\overline{D }+{P}_{D}$$

(28)

$$\overline{D }={E}_{\theta }(D|y)$$

(29)

$$D=-2\mathrm{log}(p\left(y|\theta \right))$$

(30)

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:

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

(31)

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,

$$\mathrm{WAIC}=-2\sum_{i=1}^{n}\mathrm{log}(p({y}_{i}|y))+2{P}_{W}$$

(32)

where the first component implies log pointwise posterior predictive density and ${P}_{W}$ is the effective numbers of parameters. ${P}_{W}$ is defined via:

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

(33)

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:

$${\mathrm{CPO}}_{it}=p({y}_{it}|{y}_{-it})$$

(34)

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:

$$\mathrm{LS}=-\frac{1}{IT}\sum_{i,t}log({\mathrm{CPO}}_{it})$$

(35)

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,

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

(36)

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:

$$\mathrm{MCrIW}=\frac{1}{IT}\sum_{i,t}({\widehat{y}}_{97.5,it}-{\widehat{y}}_{2.5,it})$$

(37)

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:

$$\mathrm{MAE}=\frac{1}{IT}\sum_{i,t}|{\widehat{y}}_{it}-{y}_{it}|$$

(38)

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