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Multilevel analysis of infant mortality and its risk factors in South Africa

interpreted the results. V Adjiwanou supervised the study design and analysis, revised the manuscript and interpreted the
results.
Conflict of Interest
This paper is part of an MPhil degree thesis at the university of Cape Town in 2014. The authors declare that there is no
conflict of interest.

Ethics
Consent was obtained from all participating persons in the study.

Appendix
Methods of Parameter Estimation

There are two commonly used estimation methods for multilevel logistic regression models: quasi-likelihood (QL)
approach and Bayesian approach with Markov Chain Monte Carlo (MCMC) methods (Goldstein, 2011). In QL approach,
the non-linear logistic regression equation is estimated first using a Taylor series expansion which approximates a
nonlinear function by an infinite series of terms (Breslow et al., 1993). If the Taylor series is expanded about the fixed
and the random parameters, then the estimation is known as penalised quasi-likelihood (PQL) (Breslow et al.). Once
the quasi-likelihood has been formed, unbiased estimates of the random parameters can be found by applying either
iterative generalised least squares (IGLS) or restricted generalised least square (RGLS) which are estimation procedures
in the case of continuous response variables (Goldstein, 2011). On the other hand, the Bayesian approach using MCMC
estimation methods can be used by first specifying starting values prior distributions for each of the model parameters
and then sequentially sampling subsets of parameters from their conditional posterior distributions using Markov chain. A
discussion and technical details of MCMC estimation methods for multilevel models can be found from in Browne (2003)
and Goldstein (2011)The MCMC procedure followed by MLwiN–software dedicated for multilevel modelling and used
by this research–by default assigns flat prior distributions to the parameters of the model. That is, for fixed terms
and for random terms, where is a very small number. After assigning initial values, usually
estimates from QL methods, the MCMC procedure in MLwiN then performs the simulations in two phases. In the initial
burn-in period it runs until the chain converges to its stationary distribution; and in the next stage (monitoring period)
th
th
it runs so that the means and standard errors of the parameters are estimated. The 2.5 and 97.5 quantiles of the chains
provide Bayesian 95% credible intervals in order to make inferences concerning the estimated parameters, serving the
same purpose as 95% confidence intervals. For fitting the aforementioned model, the number of iterations run is 1000 in
the burn-in period and 10 000 for the monitoring period.
After running the model, residuals at municipal and province level (estimates of and ) are calculated so that the
underlying assumptions, such as normality and constant variance of residuals, be investigated with the help diagnostic
plots. Furthermore, as part of model diagnostics, the trace of the chains, autocorrelations (AC) and partial autocorrelations
(PAC) functions at iteration t and t-k having accounted for iterations , and Monte Carlo standard errors
(MCSE) are investigated for each of the posterior distributions of the parameter in the model. For the model to be good
it is expected that the traces be not skewed, the AC and PAC functions be less correlated and the MCSE be close to zero.
Increasing the number of iterations produces better results in all these dimensions. A comprehensive detail of parameter
estimation and model diagnostics using MCMC simulations methods can be found from MLwiN manual (Rasbash,
Charlton, Browne et al., 2012).

Model Diagnostics
A three-level logistic regression model is fitted on the survival status of children born twelve months before the census.
The parameters of the model are estimated using the Bayesian MCMC procedure by running the simulation for 1000
burn-in and 10000 monitoring period. After fitting the model, the reasonableness of the parameter estimates are assessed
by looking at some diagnostics plots including the autocorrelation plots of successive iterations of the chains and Monte
Carlo standard error plots for checking convergence of the posterior distributions. These are done for each of the fixed and
random terms in the model. Some of these plots are given in the annex from Figure (1) to Figure (f). The assumptions of
normality of the residual terms at municipality and province level are approximately maintained.

52 International Journal of Population Studies | 2017, Volume 3, Issue 2