Table 3 Models and related equations in estimating the prevalence of malaria and the effect of covariates

Space–time smoothing estimate

(a) Prevalence estimation with weight

\({p}_{it}=\frac{\sum_{k=1}^{{n}_{it}}{w}_{k}{y}_{k}}{\sum_{k=1}^{{n}_{it}}{w}_{k}}\)

(1)

(b) True prevalence

\(logit\left({p}_{it}\right)|{\lambda }_{it}\sim N\left({\lambda }_{it},{V}_{it}^{L}\right)\)

\({\lambda }_{it}=\beta +{\beta }_{it}\left(rural\right)+\left(\lambda +{\gamma }_{it}\right)\left(urban\right)+{\alpha }_{it}+{\epsilon }_{it}+\frac{1}{\sqrt{\tau }}\left(\sqrt{\left(1-\phi \right)}{S}_{i}+\sqrt{\phi }{e}_{i}\right)+{\delta }_{it}\)

\(\phi\) is the spatial fraction(2)

(c) PC Prior

The prior for \(\phi\) is not on a close form. The prior on \(\tau\):

\(\pi \left(\tau \right)=\frac{\lambda }{2}{\tau }^{-3/2}{e}^{-\lambda {\tau }^{-1/2}},\tau >0,\lambda >0\) (3)

\({y}_{j}\)(a) was defined to be the cluster’s outcome and\({w}_{k}\), if available, the weight associated with the cluster\(k\). \(i\in I\), \(I\) is the partition of the region, and \({n}_{it}\) is the number of clusters at time \(t\) in the area\(i\)

(b) the true prevalence was estimated using \(logit\left({p}_{it}\right)\).\(\beta\) is the intercept for a cluster identified as in rural area, \(\beta +\lambda\) intercept for urban area. \({\beta }_{it}\) and \({\gamma }_{it}\) are temporal main time-varying effects. \({\alpha }_{it}\) is the structured time trends, \({\epsilon }_{it}\) is the temporal term (\(iid\) XE “\({\text{iid}}\)”), \({S}_{i}\)is the structured spatial trends, \({e}_{i}\) the unstructured spatial terms and \({\delta }_{it}\) the interaction between space and time

(c) The priors were defined on the hyperparameters \(\phi\) and \(\tau\) for the for the structured and instructured effect [26]

For time effect \(\alpha\), the following was used: \({\alpha }_{it}\sim N\left({\alpha }_{t-1},{\sigma }^{2}\right)\) (random walk order 1) or \(N\left(2{\alpha }_{t-1}+{\alpha }_{t-2},{\sigma }^{2}\right)\) (random walk order 2). The auto-regressive model AR(1) is defined as \({\alpha }_{t}=\phi {\alpha }_{t-1}+{\omega }_{t} {\omega }_{t}\sim N\left(0,{k}^{\left(-1\right)}\right)\), \(t\ge 2\) and \({\alpha }_{1}\sim N\left(0,{\tau }_{1}\right)\), \({\tau }_{1}=k\left(1-{\phi }^{2}\right)\). The spatial structure of \({\delta }_{it}\) is modelled by the ICAR model:

\({\delta }_{it}|{\delta }_{jt},{\tau }_{\delta }\sim N\left(\frac{1}{{n}_{i}}\sum_{j\in {n}_{\partial i}}{\delta }_{jt},\frac{{t}_{\delta }}{{n}_{i}}\right)\),

\(\partial i\)= the set of neighbours for the area \(i\), \({n}_{\partial i}\) number of these neighbours

SPTOLS

\({y}_{it}\sim \mathcal{N}\left({\eta }_{it},{\sigma }_{e}^{2}\right)\)

\({\eta }_{it}=\sum_{j=1}^{p}{\beta }_{j}{x}_{ji}+{\omega }_{it}, {\omega }_{it}=a{\omega }_{i\left(t-1\right)}+{\xi }_{it}, \left|a\right|<1\)(4)

\({x}_{mi}\) is the mth variable values at cluster \(i\), \({\sigma }_{e}^{2}\) the variance for the measurement error modelled through a Gaussian process spatially and serially uncorrelated. Here \({\omega }_{it}\) is an AR(1). Coefficient \(a\) is spatially correlated. \({\xi }_{it}\) is a GF (Gaussian Field) independent in time, with mean \(0\) such that for \({t}_{1}={t}_{2}\), \(Cov\left({\xi }_{i{t}_{1}},{\xi }_{i{t}_{2}}\right)\) is non-null and is the Matèrn covariance function defined as above. The PC prior was used for the autocorrelation parameter \(a\) such that the probability of the standard deviation to be greater than 0 is 0.9. For the remaining parameters, loggamma distributions were used with mean 1, and precision of 0.00001 or 0.001 as also used by Musenge et al. [25]

GAM

\({y}_{it}\sim \mathcal{N}\left({\eta }_{it},{\sigma }_{e}^{2}\right)\)

\({\eta }_{it}=\sum_{j=1}^{p}{f}_{j}\left({x}_{ji}\right)+{v}_{it}, {v}_{it}=\rho {v}_{i\left(t-1\right)}+{u}_{it}\)

• Space time correlated GAM: \(\rho\) is estimated

Replicated model: \(\rho\) = 0 (5)

The prior specification for the functions \({U}_{j}\) is \({f}_{j}\) for all \(j\) is a random walk process of order 2 (RW(2)). \(f\left(t\right)=2f\left(t-1\right)-f\left(t-2\right)+u\left(t\right)\), \(f\left(t\right)=f\left({x}_{\left(t\right)}\right)\), \({x}_{\left(t\right)}\) is an ordered statistic, with \(u\left(t\right)\sim N\left(0,{\tau }^{2}\right)\) and diffuse prior set on \(f\left(1\right)\) and \(f\left(2\right)\)

Spatial econometrics

\(y=X\beta +u\); with \(u=\rho Mu+e\)

with \(e\sim MVN\left(0,{\sigma }^{2}{I}_{n}\right)\)

(6)

SEM, lag on the error:

The average total impact (direct) is the posterior marginal of the coefficient \({\beta }_{r}\)

\(y=\rho Wy+X\beta +e\);\(e\sim MVN\left(0,{\sigma }^{2}{I}_{n}\right)\)

(7)

The SLM,

\(y=\rho Wy+X\beta +WX\gamma +e\); \(e\sim MVN\left(0,{\sigma }^{2}{I}_{n}\right)\)(8)

The SDM:

The average impact: \({n}^{-1}tr\left({\left({I}_{n}+\rho W\right)}^{-1}\right){\beta }_{r}+{n}^{-1}tr\left({\left({I}_{n}+\rho W\right)}^{-1}W\right){\gamma }_{r})\),

\(y=X\beta +WX\gamma +u\); \(u=\rho Wu+e\), \(e\sim MVN\left(0,{\sigma }^{2}{I}_{n}\right)\)(9)

SDEM:

The average total impact: \({\beta }_{r}+{\gamma }_{r}\)

These are implemented in INLA as spatial latent model by conditioning on \(\rho\) to allow the model to take the suitable form (53). In this estimation, a Gaussian prior was used for \(log\left(\frac{\rho }{1-\rho }\right)\), and a normal prior for \(\beta\) with mean \(0\) and very large variance

In the implementation, the variance of the likelihood was set to \({e}^{-15}\) [32]. Therefore, it is not possible to compare models fitted within INLA, but the obtained DIC can perhaps be compared with the one obtained with the other models such as the GAM

SPDE

\({\left({k}^{2}-\Delta \right)}^{\alpha /2}\left(\tau f\left(x\right)\right)=W\left(x\right)\)

with

\(cov\left(f\left(0\right),f\left(x\right)\right)=\frac{{\sigma }^{2}}{{2}^{v-1}\Gamma \left(v\right)}{\left(k\parallel x\parallel \right)}^{v}{K}_{v}\left(k\parallel x\| \right)\)

(10)

f is a weak solution

W(x) is a Gaussian white nose process, \(\Delta\) is the Laplacian, \(k\) is a s

cale parameter related to the range \(r\), \(\tau\) affects the variance of \(f\), \(\alpha\) is related to the smoothness of \(f\). This solution is approximated by a GMRF through triangulation

The range is the distance after which the spatial correlation is small, i.e. almost null or less than 0.1. The default value for \(\alpha\) was \(2\). Specification on the range is provided in Supplementary material

\(r=\frac{\sqrt{8}}{k}\)