Bayesian inference and prior specification.

Bayesian inference was performed using Markov chain Monte Carlo (MCMC) methods implemented in JAGS (Just Another Gibbs Sampler). The following prior distributions were specified:

• Regression coefficients:for the continuous variables andfor the categorical variables.
• Observation variance:
• State variance:
• Initial state:

These priors are weakly informative, chosen for their ability to identify the model without influencing posterior estimates. The scale of the priors was informed by prior neonatal mortality studies and time-series modeling literature. Four independent MCMC chains were run, each with 1,000 burn-in iterations followed by 5,000 sampling iterations. Convergence was assessed through trace plots, posterior density plots, and the Gelman–Rubin diagnostic, with indicating satisfactory convergence. Posterior means and 95% credible intervals (CrIs) were reported for all the model parameters.

Nature of and rationale for the model.

The study opted to formulate a dynamic linear model to monitor the impact of the SDG interventions on neonatal mortality, as dynamic linear models constitute a flexible class of models that can effectively address typical features in policy monitoring and large datasets.

For instance, this study aimed to construct a dynamic linear model to assess the situation after the onset of the SDGs, having previously evaluated the situation. To achieve this objective, this study used a general Gaussian dynamic linear model, as presented in Eq 1:

(1)

where represents the observation for the state at time t.

denotes a function of the x covariates for the state at time t specified as

(2)

is a vector of time-carrying parameters

is a transition matrix depicting how the model parameters evolve over time.

and are the observation and evolution errors, respectively.

On the basis of (1), the model is formulated as follows (3):

(3)

Where:

represents the district neonatal mortality rate at time t.

are the m covariates, including the maternal health care policies and interventions, in which case a code 0 is used for Before and 1 for After the introduction of a given policy/intervention during the observation time for a given policy or intervention.

is the interaction covariate between the covariate, including the maternal health care policies and interventions, with as their respective coefficients,

denotes the time-varying intercept.

are the time-varying coefficients associated with each of the m covariates, including maternal health care policies and interventions.

and are the observation and evolution error terms, respectively, where and .

Model parameter estimation.

A Bayesian approach using the Kalman filtering technique was used for estimating the study model parameters because the Bayesian approach focuses on how a particular part of the data depends on the other data parts since this study had to compare the neonatal mortality situation in the country before and after the introduction of the SDGs, implying that past values had a considerable influence on future values in this study.

Considering that Bayesian inference has its roots in Bayes rule, as noted in the studies by [13] and [14], to obtain the final parameter estimates, this study made use of the theorem through the posterior density function as expressed in (4);

(4)

The final parameter estimates are obtained using a quadratic mean loss function (squared error loss) method, which involves obtaining a full posterior distribution of parameters and then calculating their average as the final parameter estimate.

Rewriting the formulated model by the study presented in (3), which is expressed in the form of two equations, observation and state equations, we obtain (5);

(5)

where

is the observed data is a p-dimensional design vector of covariates and the interaction terms.

is a p-dimensional time-varying parameter vector (for both covariates and the interaction terms).

and are the observation and evolution error terms, respectively, where and .

Since the formulated model is a multivariate Gaussian state space model, based on the standard results of a multivariate Gaussian distribution pertaining to its marginal and conditional distributions, the random vector of parameters also has a Gaussian distribution, and the same applies to the other respective marginal and conditional distributions.

Therefore, since all the relevant distributions are Gaussian, and bearing in mind that the process of Bayesian inference involves passing from a prior distribution, , to a posterior distribution, , we estimated the model parameters through the Kalman filtering technique to obtain the means and variances to finally derive the parameter estimates (posterior means) through the following procedure:

1. Obtaining the prior distribution (based on the evolution equation reflected in the second line of (3)).

Considering the fact that a dynamic linear model is a Gaussian time space model, we used a normal prior distribution for a p-dimensional state vector by first obtaining the initial values where , and we then attained one step ahead of the prediction of the parameter vector distribution by first obtaining the mean as given in (6)

(6)

Then, we obtained the predicted covariance in (7) denoted as

(7)

Therefore, which is the prior distribution.

Since our prior has been derived on the basis of a normal distribution, we can write its probability density function as given in (8):

(8)

1. ii. Obtaining the likelihood distribution function (based on the observation equation reflected in the first line of (3)).

We start this step by obtaining the predicted (one step) distribution function of (the observed value):

Let

Nevertheless, we first obtained the mean,

(9)

We then obtained the variance,

(10)

Therefore, , which is the observed value likelihood distribution.

Since the observed value variable follows a normal distribution, we can write its likelihood function probability density function as follows;

(11)

1. iii. Obtaining the posterior distribution

From (4), an expression derived from the Bayes theorem, we can obtain the posterior probability density function by multiplying (11) by (8) and hence obtaining the probability density function of the posterior distribution function as given in (12);

(12)

Since both the prior and likelihood distributions are Gaussian, the resulting posterior is also Gaussian. If we let the posterior , from the first property of a Gaussian distribution pertaining to the probability density function, we can obtain the values of by simplifying the right-hand side of (12);

(13)

From (13), since the posterior , it implies that;

Therefore,

From this expression, we used a quadratic mean loss function (squared error loss), which involves computing the full posterior distribution of the parameters and averaging them to obtain the final parameter estimate.

Model validation.

To validate the model, we conducted residual tests on autocorrelation, heteroscedasticity, and normality using the Durbin–Watson test, the Breusch Pagan test, and the Shapiro–Wilk test, respectively. The following properties were checked for:

1. Autocorrelation: The residuals should not be autocorrelated.
2. Homoscedasciticity: The residuals should have a constant variance.
3. Normality: The residuals are expected to be normally distributed.

To verify the above model properties, tests were carried out using the following hypotheses:

1. : Residuals are not autocorrelated vs. : Residuals are autocorrelated.
2. : Residuals have constant variance Vs : Residuals do not have constant variance.
3. : Residuals are normally distributed vs. : Residuals are not normally distributed.

Test statistics, critical values, and the decision rule.

While testing for autocorrelation of residuals, the Durbin–Watson test was used. To obtain its test statistic, the following formula was used:

(14)

where

1. = the residuals from the regression model.
2. T = the number of observations.

With respect to the critical value and the decision rule, we assume that the DW (value) ranges between 0 and 4; since a DW value approaching 2 indicates no autocorrelation, a value approaching 0 indicates positive autocorrelation, and a value approaching 4 indicates negative autocorrelation, taking into consideration the probability value of the test statistic in comparison to the level of significance.

The Breusch Pagan test was used to assess the homoscedasticity of residuals, whereby to obtain its test statistic, the following formula was used:

(15)

where

1. n = number of observations

= R-squared value obtained by regressing the squared residuals from the original regression on the independent variables, their squares, and their cross products. This regression is known as auxiliary regression.

With respect to the critical value and the decision rule, because the test statistic of Breusch Pagan follows a chi-square distribution with degrees of freedom equal to the number of independent variables in the auxiliary regression, the test statistic value was compared with the critical (tabulated) value to ascertain whether to reject the null hypothesis basically considering the significance level and the reported probability value of the test statistic.

The Shapiro–Wilk test was used to determine if the residuals were normally distributed, whereby the following formula was used to obtain its test statistic, denoted as W:

(16)

where

= ordered residuals from smallest to largest.

= mean of the residuals.

= constants derived from the expected values of the order statistics of a standard normal distribution computed using statistical software.

n = number of observations.

The critical value and the decision rule were determined on the basis that W ranges between 0 and 1; values of W close to 1 suggest that the residuals are approximately normally distributed, whereas lower values indicate deviations from normality.

Furthermore, The performance of the Bayesian dynamic linear model formulated in this study has also been evaluated in our separate, independently published study, which compares it with mixed-effects interrupted time-series models using real-world neonatal mortality data from Uganda (2015–2023). Whereas the present work emphasizes model formulation and Monte Carlo simulation-based validation, the other study focuses on empirical comparison [16].