Multilevel time series modelling of antenatal care coverage in Bangladesh at disaggregated administrative levels
Section 5. Selected models and model prediction

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5.1   MTS model for ANC0

No transformation for the input series of the direct estimates or the FH estimates is considered. The following fixed effect components are included in the selected models for MTS-I, MTS-II, and MTS-III:

$$1+\text{Division}+yr\mathrm{.}c+\text{Division}*yr\mathrm{.}c\mathrm{,}(5.1)$$

where $yr\mathrm{.}c$ denotes the standardized quantitative year variable, which defines a fixed effect linear trend. Similarly $\text{Division}*yr\mathrm{.}c$ defines a fixed effect linear trend for each separate division. The random effects part of the three models is shown in Table 5.1. If multiple varying effects are modeled, then there is a choice between scalar, diagonal or full covariance matrix $V$ in (4.3). For variation over time, second order random walks $\text{RW2\_Division}$ and $\text{RW2\_District}$ were finally selected. White noise components are considered but not included in the final model since it did not further improve the model fit. 

The linear predictor of the selected model can be written, element-wise for district $i$ and year $t,$ as

$${\eta }_{it}={\beta }^{\prime }{x}_{it}+{\nu }_i+z_t{\nu }_i^{\mathrm{<mo\; stretchy="false">(</mo>}yr\mathrm{<mo\; stretchy="false">)</mo>}}+u_{it}+u_{j\mathrm{<mo\; stretchy="false">[</mo>}i\mathrm{<mo\; stretchy="false">]</mo>}t}^{\mathrm{<mo\; stretchy="false">(</mo>}div\mathrm{<mo\; stretchy="false">)</mo>}}+s_i\mathrm{,}(5.2)$$

where $\beta$ is the vector of fixed effects corresponding to the covariates $x_{it}$ as specified in (5.1), ${\nu }_i$ are random intercepts varying by district, $z_t$ denotes the $yr\mathrm{.}c$ variable for year $t,$ and ${\nu }_i^{\mathrm{<mo\; stretchy="false">(</mo>}yr\mathrm{<mo\; stretchy="false">)</mo>}}$ are the corresponding random slopes varying by district. These random intercepts and slopes are jointly distributed as 

$$\left(\begin{array}{c}{\nu }_i\\ {\nu }_i^{\mathrm{<mo\; stretchy="false">(</mo>}yr\mathrm{<mo\; stretchy="false">)</mo>}}\end{array}\right)\stackrel{\text{iid}}{{\displaystyle ~}}N\left(\left(\begin{array}{c}0\\ 0\end{array}\right)\mathrm{,}\left(\begin{array}{cc}{\sigma }_I^2& \rho {\sigma }_I{\sigma }_S\\ \rho {\sigma }_I{\sigma }_S& {\sigma }_S^2\end{array}\right)\right).(5.3)$$

The second-order random walk effects at district and division level are distributed as 

$$\begin{array}{ll}{u}_{it}-2u_{i\mathrm{<mo\; stretchy="false">(</mo>}t-\mathrm{1<mo\; stretchy="false">)</mo>}}+u_{i\mathrm{<mo\; stretchy="false">(</mo>}t-\mathrm{2<mo\; stretchy="false">)</mo>}}\hfill & \stackrel{\text{iid}}{{\displaystyle ~}}N\mathrm{<mo\; stretchy="false">(</mo>0,}{\sigma }_{R2}^2\mathrm{<mo\; stretchy="false">)</mo>}\hfill \\ u_{j\mathrm{<mo\; stretchy="false">[</mo>}i\mathrm{<mo\; stretchy="false">]</mo>}t}^{\mathrm{<mo\; stretchy="false">(</mo>}div\mathrm{<mo\; stretchy="false">)</mo>}}-2u_{j\mathrm{<mo\; stretchy="false">[</mo>}i\mathrm{<mo\; stretchy="false">]</mo>}\mathrm{<mo\; stretchy="false">(</mo>}t-\mathrm{1<mo\; stretchy="false">)</mo>}}^{\mathrm{<mo\; stretchy="false">(</mo>}div\mathrm{<mo\; stretchy="false">)</mo>}}+u_{j\mathrm{<mo\; stretchy="false">[</mo>}i\mathrm{<mo\; stretchy="false">]</mo>}\mathrm{<mo\; stretchy="false">(</mo>}t-\mathrm{2<mo\; stretchy="false">)</mo>}}^{\mathrm{<mo\; stretchy="false">(</mo>}div\mathrm{<mo\; stretchy="false">)</mo>}}\hfill & \stackrel{\text{iid}}{{\displaystyle ~}}N\left(\mathrm{0,}{\mathrm{<mo\; stretchy="false">(</mo>}{\sigma }_{R2}^{\mathrm{<mo\; stretchy="false">(</mo>}div\mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">)</mo>}}^2\right)\mathrm{,}\hfill \end{array}(5.4)$$

where $j\mathrm{<mo\; stretchy="false">[</mo>}i\mathrm{<mo\; stretchy="false">]</mo>}$ should be read as division $j$ to which district $i$ belongs. Finally, the spatial effects $s_i$ are distributed as 

$$s_i|s_{i^{\prime }\ne i}\stackrel{\text{ind}}{{\displaystyle ~}}N\left(\frac{1}{a_i}{\displaystyle \sum _{i^{\prime }\in nb\mathrm{<mo\; stretchy="false">(</mo>}i\mathrm{<mo\; stretchy="false">)</mo>}}}{s}_{i^{\prime }}\mathrm{,}\frac{1}{a_i}{\sigma }_{Sp}^2\right)\mathrm{,}(5.5)$$

where $a_i$ is the size of the set $nb\mathrm{<mo\; stretchy="false">(</mo>}i\mathrm{<mo\; stretchy="false">)</mo>}$ of neighbouring districts of district $i.$ Priors for the covariance matrix in (5.3) and the other variance parameters are chosen as described in Section 4.1. For identifiability of the model components, the following constraints are imposed: 

$$\begin{array}{llll}{\displaystyle \sum _{t\mathrm{<mo>=</mo>1}}^T}{u}_{it}\mathrm{<mo>=</mo>}0\hfill & \text{and}\hfill & {\displaystyle \sum _{t\mathrm{<mo>=</mo>1}}^T}t{u}_{it}\mathrm{<mo>=</mo>}0\hfill & \text{for}\text{all}\text{districts}i\mathrm{,}\hfill \\ {\displaystyle \sum _{t\mathrm{<mo>=</mo>1}}^T}{u}_{j\mathrm{<mo\; stretchy="false">[</mo>}i\mathrm{<mo\; stretchy="false">]</mo>}t}^{\mathrm{<mo\; stretchy="false">(</mo>}div\mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo>=</mo>}0\hfill & \text{and}\hfill & {\displaystyle \sum _{t\mathrm{<mo>=</mo>1}}^T}t{u}_{j\mathrm{<mo\; stretchy="false">[</mo>}i\mathrm{<mo\; stretchy="false">]</mo>}t}^{\mathrm{<mo\; stretchy="false">(</mo>}div\mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo>=</mo>}0\hfill & \text{for}\text{all}\text{divisions}j\mathrm{,}\hfill \\ {\displaystyle \sum _{i\mathrm{<mo>=</mo>1}}^{M_d}}{s}_i\mathrm{<mo>=</mo>}0.\hfill & \hfill & \hfill & \hfill \end{array}(5.6)$$

Note that RW2 trends are specified at division and district levels, both with a scalar variance structure. A division level trend is shared by all underlying districts. Deviations of each district from this division-level trend is modeled with RW2 trends at district level. This is a parsimonious alternative to borrow strength over time and space, compared to modelling RW2 trends at the district level only with a full covariance matrix (Boonstra and van den Brakel, 2019).

5.2   MTS model for ANC4

The square-root transformation is applied to the input series of the direct and FH estimates of ANC4 for models MTS-I, MTS-II, and MTS-III. For MTS-I the GVF (3.3) is applied to the transformed standard errors to obtain the variance matrix $\Sigma ,$ as explained at the end of Subsection 3.5. For the fixed effect component a factor variable called “Region” has been created based on the degree of urbanization following Rahman, Mohiuddin, Kafy, Sheel and Di (2019). The variable has four levels; 1 for three big cities Dhaka, Chittagong and Gazipur, 2 for other nine regional big cities (Barisal, Bogra, Comilla, Khulna, Mymensing, Narayanganj, Rajshahi, Rangpur, Sylhet), 3 for three hilly districts (Bandarban, Khagrachhari and Rangamati) and 4 for the remaining districts. This variable mainly helped to adjust the estimates for the three hilly districts which have very few (even no) information in the considered seven surveys. The final model has the following fixed effects components: 

$$1+\text{Division}+yr\mathrm{.}c+\text{Region}\text{.}(5.7)$$

The interaction between “Division” and “yr.c” (like in the ANC0 model) was found to be insignificant in the ANC4 model. The random effect components for ANC4 model shown in Table 5.2 are very similar to those used for the model of ANC0 (shown in Table 5.1). A local level trend instead of smooth trend at division level (RW1_Division in Table 5.2) has been considered since the smooth trend component (RW2_Division, as in Table 5.1) resulted in some bias in the national and divisional trends. Also, the model with RW1_Division component gives better scores for the information criteria compared to the model with RW2_Division component. White noise components are considered but not included in the final model since it did not further improve the model fit.

Alternatively, the model can be expressed as in (5.2), where now $\beta$ and $x_{it}$ correspond to the fixed effects specification (5.7). The only other difference is that the division-level trends are now modelled as a first-order random walk:

$$u_{j\mathrm{<mo\; stretchy="false">[</mo>}i\mathrm{<mo\; stretchy="false">]</mo>}t}-u_{j\mathrm{<mo\; stretchy="false">[</mo>}i\mathrm{<mo\; stretchy="false">]</mo>}\mathrm{<mo\; stretchy="false">(</mo>}t-\mathrm{1<mo\; stretchy="false">)</mo>}}\stackrel{\text{iid}}{{\displaystyle ~}}N\left(\mathrm{0,}{\mathrm{<mo\; stretchy="false">(</mo>}{\sigma }_{R1}^{\mathrm{<mo\; stretchy="false">(</mo>}div\mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mtext>\hspace{0.17em}</mtext><mo\; stretchy="false">)</mo>}}^2\right)\mathrm{,}(5.8)$$

where for identifiability reasons the constraint ${\displaystyle {\sum }_{t\mathrm{<mo>=</mo>1}}^T}{u}_{jt}^{\mathrm{<mo\; stretchy="false">(</mo>}div\mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo>=</mo>}0$ is imposed for all division $j.$ As in the case of ANC0, RW1 trends are specified at division and RW2 trends at the district levels, both with a scalar variance structure as a parsimonious way to borrow strength over time and space.

5.3   Trend estimation

Trend estimates are computed based on the MCMC simulation results. In a first step, for each MCMC replicate, an $M$ -dimensional vector containing predictions at the most detailed level of all year-district combinations is computed as 

$${\eta }^{\mathrm{<mo\; stretchy="false">(</mo>}r\mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo>=</mo>}X{\beta }^{\mathrm{<mo\; stretchy="false">(</mo>}r\mathrm{<mo\; stretchy="false">)</mo>}}+{\displaystyle \sum _{\alpha }}{Z}^{\mathrm{<mo\; stretchy="false">(</mo>}\alpha \mathrm{<mo\; stretchy="false">)</mo>}}{v}^{\mathrm{<mo\; stretchy="false">(</mo>}\alpha \mathrm{,}r\mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{,}(5.9)$$

where superscript $\mathrm{<mo\; stretchy="false">(</mo>}r\mathrm{<mo\; stretchy="false">)</mo>}$ indexes the retained MCMC draws. Note that ${\eta }^{\mathrm{<mo\; stretchy="false">(</mo>}r\mathrm{<mo\; stretchy="false">)</mo>}}$ also includes predictions for the years without survey observations. Since a square root transformation was applied to the ANC4 series, initially the following back-transformation for the vectors ${\eta }^{\mathrm{<mo\; stretchy="false">(</mo>}r\mathrm{<mo\; stretchy="false">)</mo>}}$ was considered following Boonstra et al. (2021):

$${\theta }^{\mathrm{<mo\; stretchy="false">(</mo>}r\mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo>=</mo>}{\mathrm{<mo\; stretchy="false">(</mo>}{\eta }^{\mathrm{<mo\; stretchy="false">(</mo>}r\mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">)</mo>}}^2+{\left(\text{se}\mathrm{<mo\; stretchy="false">(</mo>}{\widehat{Y}}_{it}\mathrm{<mo\; stretchy="false">)</mo>}\right)}^2\mathrm{.}(5.10)$$

The second term on the right hand side is a (relatively small) bias correction using the transformed and smoothed standard errors. The bias correction stems from the fact that the design expectation of the direct estimates can be written as 

$$E\mathrm{<mo\; stretchy="false">(</mo>}\widehat{Y}\mathrm{<mo\; stretchy="false">)</mo>}\mathrm{<mo>=</mo>}E\left({\mathrm{<mo\; stretchy="false">(</mo>}\widehat{Y}\mathrm{<mo\; stretchy="false">)</mo>}}^2\right)\mathrm{<mo>=</mo>}E\left({\mathrm{<mo\; stretchy="false">(</mo>}\eta +E\mathrm{<mo\; stretchy="false">)</mo>}}^2\right)\mathrm{<mo>=</mo>}{\eta }^2+\text{var}\mathrm{<mo\; stretchy="false">(</mo>}E\mathrm{<mo\; stretchy="false">)</mo>,}$$

where $E$ is the vector of sampling errors after transformation, assumed to be normally distributed with standard errors $\text{se}\mathrm{<mo\; stretchy="false">(</mo>}{\widehat{Y}}_{it}\mathrm{<mo\; stretchy="false">)</mo>.}$ A difficulty with the data at hand is that the bias correction can only be applied to the survey years, since standard errors are only available for those years. Applying the bias correction only for the survey years distorts the trend estimates, as illustrated in Das, van den Brakel, Boonstra and Haslett (2021). In case of MTS-I model, the impact of this bias correction is most clear for those domains with zero direct estimates particularly for Chittagong hilly districts. The impact of the bias correction is less in case of MTS-II and MTS-III models since the estimated standard errors of the FH estimates are already smoothed enough and consistent. However, at national and division levels this bias correction causes some overestimation in some survey years for all the trends based on the MTS models. Therefore, the bias correction for the square root transformation is not applied in the trend estimates but only used in the calculation of cross-sectional FH estimates.

Trend estimates with their standard errors at the most detailed level of districts for all years are obtained by taking the mean and the standard deviation over the MCMC replications ${\eta }^{\mathrm{<mo\; stretchy="false">(</mo>}r\mathrm{<mo\; stretchy="false">)</mo>}},$ respectively. Trends at the divisional and national levels are obtained by aggregating each MCMC replication from the most detailed regional level of districts, using the number of ever-married women as a weighting variable. Subsequently, trend estimates and their standard errors are obtained by taking the mean and the standard deviation over these aggregated MCMC replications.

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