Variable of the study: In this study, the potential determinant factors expected to be correlated with pregnancy-related death are included as variables.
The predictor variables: Many explanatory variables were used as predictors of maternal mortality. The explanatory variables that included in this study were:
Method of data analysis
The statistical model that used for this data to analysis was the Bayesian multilevel logistic model. The Markov chain Monte-Carlo (MCMC) method is a general simulation method for sampling from posterior distributions and computing posterior quantities of interest [
18]. Sampling process of MCMC approaches is pretty heavy but has no bias and, so, these methods are preferred when accurate results are expected, without regards to the time it takes [
24]. The association between water pipe dependence and chronic obstructive pulmonary disease, by comparing frequentist and Bayesian methods’ results show as Bayesian approach have advantages over the frequentist one, particularly in case of a low power of the frequents analysis [
38].
The data collection procedure is the hierarchical level or structures that means the levels are nested one another; Kinds of the literature indicated that the Bayesian models are given preference because the technique is more robust and precise than the traditional (classical) statistics since it is usually criticized based on the priors and information from the likelihood. Thus, this collective information has been strengthening the better determination of the parameter [
34,
35]. Bayesian multilevel logistic regression model has been helped to facilitate the representation of complex multilevel data structures, the variation among the women and the region the specification of objective priors and also gives us more informative and quantitative answers than any of the standard frequents approaches [
39], thus why the reason for selecting this model. MLwiN 2.02 version software was adopted for the analysis of this study.
Multilevel Logistic Regression Model: Multilevel hierarchical modeling explicitly accounts for the clustering of the units of analysis, individuals nested within groups. The study helps for examination of the effects of group level and individual level variation- of observations. We further simplify the presentation by assuming there is a women-level predictor and regional level factor of maternal mortality. Multilevel models are statistical models which allow not only independent variable at any level of a hierarchical structure but also at least one random effect above level one group. A multilevel logistic regression model can account for lack of independence across levels of nested data (i.e., individuals nested within regions). For simplicity of presentation two-level models for this study, i.e., models accounting for women-level and regional -level effects. In this data structure, level-1 is the women level and level-2 is the regional level. Within each level-2 unit, there is nj in the jth region.
The standard assumption is that Y
ij has a Bernoulli distribution [
10]. Then, the two-level models are given by:
$$\log it\left( {\mathop \pi \nolimits_{ij} } \right) = \log \left[ {\frac{{\mathop \pi \nolimits_{ij} }}{{\mathop {1 - \pi }\nolimits_{ij} }}} \right] = \mathop \beta \nolimits_{0j} + \sum\nolimits_{h = 1}^{k} {\mathop \beta \nolimits_{hj} \mathop x\nolimits_{ijk} } \;$$
(2)
$$i = 1,2,3,\mathop n\nolimits_{j} ,h = 1,2,...k,j = 1,2,...,11\;$$
(2.1)
$$\mathop \beta \nolimits_{0j} = \mathop \beta \nolimits_{0} + \mathop u\nolimits_{0j}$$
(2.2)
$$\mathop \beta \nolimits_{1j} = \mathop \beta \nolimits_{1} + \mathop u\nolimits_{1j} ,...,\mathop \beta \nolimits_{kj} = \mathop \beta \nolimits_{k} + \mathop u\nolimits_{kj}$$
(2.3)
$$\log it\left( {\mathop \pi \nolimits_{ij} } \right) = \log \left[ {\frac{{\mathop \pi \nolimits_{ij} }}{{\mathop {1 - \pi }\nolimits_{ij} }}} \right] = \mathop \beta \nolimits_{0} + \sum\nolimits_{h = 1}^{k} {\mathop \beta \nolimits_{hj} } \mathop x\nolimits_{ijk}$$
(3)
\(\mathop x\nolimits_{i} = \left( {\mathop x\nolimits_{1ij} ,\mathop x\nolimits_{2ij} ,...,\mathop x\nolimits_{kij} } \right)\), represent the first and the second level covariates, for variable k.
\(\mathop \beta \nolimits_{{}} \left( { = \mathop \beta \nolimits_{0} ,\mathop \beta \nolimits_{1} ,...,\mathop \beta \nolimits_{k} } \right)\), are the regression parameter coefficient.
\(\left( {\mathop u\nolimits_{0j} ,\mathop u\nolimits_{1j} ,...,\mathop u\nolimits_{kj} } \right)\), is the random effect of the model parameter at level two. With the assumption, \(\mathop u\nolimits_{hj}\) follows a normal distribution with mean zero and variance \(\mathop \sigma \nolimits_{u}^{2}\) without \({\mathrm{U}}_{\mathrm{hj}}\) \(\mathop u\nolimits_{hj}\) the above equation can be the single-level logistic regression. That means the 1st equation is the single level logistic model and the 2nd equation is two levels model. Therefore conditional on \(\mathop u\nolimits_{0j} ,\mathop u\nolimits_{1j} ,...,\mathop u\nolimits_{kj}\) the yij can be assumed to be independently distributed as Bernoulli random variables.
\(\mathop x\nolimits_{i} = \left( {\mathop x\nolimits_{1ij} ,\mathop x\nolimits_{2ij} ,...,\mathop x\nolimits_{kij} } \right)\), represent the first and the second level covariates, for variable k.
\(\left( {\beta = \mathop \beta \nolimits_{0} ,\mathop \beta \nolimits_{1} ,...,\mathop \beta \nolimits_{k} } \right)\), are the regression parameter coefficient. \(\mathop u\nolimits_{0j} ,\mathop u\nolimits_{1j} ,...,\mathop u\nolimits_{kj}\), is the random effect of the model parameter at level two. With the assumption, \(\mathop u\nolimits_{hj}\) follows a normal distribution with mean zero and variance \(\mathop \sigma \nolimits_{u}^{2}\).
Bayesian multilevel analysis of empty model (null model)
It is the probability distribution for group-dependent probabilities without taking further explanatory variables into account.
The null model is defined as:
$${\text{logit }}\left( {\pi_{ij} } \right) = \beta_{0} + U_{0j}$$
(4)
The index i indicates individual for level one, j indicates region for level two, \({U}_{0j}\) is level two errors, \({\beta }_{0}\) is the overall average of maternal mortality.
Bayesian multilevel analysis of random intercept model
In the random intercept model the intercept is the only random effect meaning that the groups (region) differ with respect to the average value of the response variable.
The random intercept model expresses the logit of
\({\pi }_{ij}\) is a sum of the linear function of explanatory variables and given as:
$$logit(\pi_{ij} ) = {\text{log}}\left[ {\frac{{\pi_{ij} }}{{1 - \pi_{ij} }}} \right] = \beta_{0j} + \beta_{1} X_{1ij} + \cdots + \beta_{k} X_{kij} = \beta_{0j} + \mathop \sum \limits_{h = 1}^{k} \beta_{hij}$$
(5)
where the intercept term
\({\beta }_{0j}\) is assumed to vary randomly and is given by the sum of an average intercept
\({\beta }_{0}\) and group-dependent deviations
\({U}_{0j}\) that is
\({\beta }_{0j}={\beta }_{0}\)+
\({U}_{0j}\) as a result.
$${\text{logit}}(\pi_{ij} ) = \beta_{0} + \mathop \sum \limits_{h = 1}^{k} \beta_{hj} x_{hij} + U_{0j}$$
(6)
where
\({\beta }_{0}\)+
\(\sum_{h=1}^{k}{\beta }_{hj}{x}_{hij}\) is the fixed part of the model and
\({U}_{0j}\) is the random or stochastic part of the model.
Bayesian multilevel analysis of random coefficients model
The multilevel random effect coefficients logistic regression model is based on linear models for the log odds that include random effects for groups or other higher levels.
Consider a model with group-specific regression of logit of the success probability logit(
\({\pi }_{ij}\)) on a single level-one explanatory variable X.
$$logit(\pi_{ij} ) = {\text{log}}\left[ {\frac{{\pi_{ij} }}{{1 - \pi_{ij} }}} \right] = \beta_{0j} + \mathop \sum \limits_{h = 1}^{k} \beta_{hij} + U_{0j} + \mathop \sum \limits_{h = 1}^{k} U_{{{\varvec{hj}}}} X_{hij}$$
(7)
The term
\(\sum_{h=1}^{k}{U}_{{\varvec{h}}{\varvec{j}}}{X}_{hij}\) can be regarded as a random interaction between group and the explanatory variables. The random intercept variance, Var(
\({U}_{0j}\)) =
\({\sigma }_{0}^{2}\), the random slope variance, Var(
\({U}_{1j}\)) =
\({\sigma }_{1}^{2}\) and the covariance between the random effects, Cov(
\({U}_{0j}\);
\({U}_{1j}\)) =
\({\sigma }_{01}^{2}\) are called variance components [
41].
Likelihood function
The key ingredients to a Bayesian analysis are the likelihood function, which reflects information about the parameters contained in the data, and the prior distribution, which quantifies what, is known about the parameters before observing data
$${\raise0.7ex\hbox{${Y_{ij} }$} \!\mathord{\left/ {\vphantom {{Y_{ij} } {\pi_{ij} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${\pi_{ij} }$}} \propto Bernoulli\left( {\pi_{ij} } \right)$$
$$\begin{gathered} {\text{logit}}\left( {\pi_{ij} } \right) = {\text{log}} \hfill \\ \left[ {\frac{{\pi_{ij} }}{{1 - \pi_{ij} }}} \right] = \beta_{0j} + \mathop \sum \limits_{h = 1}^{k} \beta_{{hj{ }}} X_{hij} + U_{0j} + \mathop \sum \limits_{h = 1}^{k} U_{hj} X_{hij} \hfill \\ \end{gathered}$$
(8)
Prior distribution
The prior distribution is a probability distribution that represents the prior information associated with the parameters of interest. It is a key aspect of a Bayesian analysis. Even if there is prior knowledge about what we are examining, in some cases we might prefer not to use this and let the data speak for themselves. In this case, we wish to express our prior ignorance into the Bayesian system. This leads to non-informative priors.
The posterior distribution
All Bayesian inferential conclusions are based on the posterior distribution of the model generated. Using the prior and likelihood function above the full conditional distribution of posterior parameter
\({\beta }_{0}\),
\({\beta }_{1}\), …,
\({\beta }_{k}\) is given by:
$$p\left( {\beta _{h} \left| {\Omega _{u} ,U_{{oj}} ,y_{{ij}} } \right.} \right) \propto \prod\limits_{{ij}} {\pi _{{ij}}^{{y_{{ij}} }} } \left( {1 - \pi _{{ij}} } \right)^{{1 - y_{{ij}} }}$$
(9)
Estimation techniques
Markov Chain Monte Carlo (MCMC) Methods: The use of Markov chain Monte Carlo (MCMC) methods to evaluate integral quantities has exploded over the last 15 years. To solve a variety of "unsolvable" problems in Bayesian inference we used the Markov Chain Monte Carlo approach [
31].Whenever the frequentist results were clear cut (due to a large sample size or a strong association), performing the MCMC method helped to increase the accuracy of the results by narrowing the credible interval, but did not change the direction of hypothesis acceptance. The initial definition required is that of a more primitive concept that underlies for the second MC which is called Markov chains [
9]. Although MCMC methods can be used for both frequentist and Bayesian inference, it is more common and easier to use them for Bayesian modeling and MCMC estimation has been restricted to a subset of the potential models that can be fitted in
MLwiN [
9].
Metropolis–Hastings algorithm: Metropolis–Hastings algorithm is a Markov chain Monte Carlo (MCMC) method for obtaining a sequence of random samples from a probability distribution. The Metropolis–Hastings algorithm works by generating a sequence of sample values. In this thesis the posterior doesn’t look like any distribution we know (no Conjugate) and some (or all) of the full conditionals do not look like any distributions we know (no Gibbs sampling for those whose full conditionals we don’t know. This is why we were interested to use Metropolis–Hastings algorithm.