The infant and child mortality rate is one indicator of health in a given community or country. 39 5.1 Results of fitting a joint vulnerability model (community effect and household effects) to the 2011 Uganda DHS data to identify factors influencing under-five survival in Uganda.
Background
Background of the study area
Okwero et al. (2010) described the Ugandan economy as one of the fastest growing economies in the world. Despite this growth, Uganda's per capita income of $320 in 2007, the country still ranks among the poorest countries of the world.
Literature review
He highlighted factors such as the gender of the child (males were at higher risk of death), place of residence (northern Uganda recorded the highest mortality rate), birth interval (less than 24 months was associated with high under-five mortality) . and mother's education (no primary education was associated with a higher risk of child death before age five). Rutstein et al (2000) showed that internally comparable data obtained from research programs such as the Demographic and Health Survey (DHS) identified some key factors that help explain trends in child and infant mortality.
Basic survival analysis concepts
- The survival and the hazard functions
- The survival function
- The Hazard function
- The relationship between the hazard and the survival functions
- Non-parametric methods
- The Empirical survival function
- The Kaplan-Meier estimator
Survival data are summarized using hazard and survival function estimates (Le and Le, 1997, Miller Jr, 2011). It allows a simple and quick estimation of the survival function in the presence of censoring.
Study objectives
- Exponential distribution
- Weibull distribution
- Log-normal distribution
- Log-logistic distribution
- The Exponential and Weibull regression models
- Semi-parametric models
- The Cox-proportional hazard model
The probability density function of the Weibull distribution for different values of . c) The shapes of the Weibull hazard function for some selected values of the shape parameter. Important statistical distributions for a time to event random variable 20. a) The probability density function of the logistic distribution for different values of.
Estimation of unknown parameters in both parametric and semi-parametric
The test statistic has a chi-square distribution with p degrees of freedom, where p is the dimension of the information matrix I(βββ). Given that ˆβββi is the regression parameter estimateβββi, then ˆhi = exp (βββi) is the hazard ratio estimate of a covariateXi.
Variables and best fitting model selection
Fitting a standard Cox-proportional hazard model on the 2011
However, it is not possible to include all Kaplan-Meier plots in this thesis for all covariates included in the study, but by fitting a univariate Cox-proportional hazard model, the hazard ratios have enough information to provide about the distribution of the hazard. on a given covariate. Based on the Scoenfeld residuals, these factors cannot be fitted into the final cox proportional hazard model to obtain the adjusted hazard ratios. Other variables such as the child's gender fulfill this assumption and can therefore be fitted into the final cox-proportional hazard model.
Table (4.4) shows that the child's gender and the number of births in the last year were strongly associated with a high under-five mortality rate. Using the Akaike Information Criterion (AIC) of Akaike (1973), the best-fitting standard Cox proportional hazard model consisted of five variables including father's education, child's gender, number of births in the last year, mother's age group and household head's gender, Table (4.5 ). From Table (4.5), there is sufficient evidence to conclude that the gender of the household head, the number of births in the last year and the gender of the child influence the survival of children under the age of five in Uganda.
The frailty term captures the total effects of all factors affecting the child's risk of death, which are not included in the standard Cox proportional hazard model presented in chapter 3.
Model development
These models help explain the relationship between individuals in a particular cluster or the difference between individuals in different clusters. The concept of vulnerability was introduced by Vaupel and Stallard (1979), which shows that some individuals are more vulnerable, susceptible or at greater risk than others, although they appear similar when we take into account observable or measurable characteristics such as gender, age and weight . The main assumption of a vulnerability model is that the information about the hidden internal or external factors is contained in the form and structure of the hazard function and in the form of the vulnerability distribution (Hanagal, 2011).
The univariate frailty model
The distributions of frailty
Equation (5.6) shows that the average hazard or the unconditional hazard depends on the frailty distribution. Most of the researchers assume a Gamma fragility, but fragility can be assumed to follow other distributions which include;. The gamma frailty model is the most frequently used frailty model for the following reasons;.
Since the frailty term is a positive random variable, it makes the gamma distribution the most suitable choice for the frailty term;. When the shape parameter is equal to 1 (α = 1), the distribution becomes exponential with parameter β, and for large values of α the distribution assumes a bell shape identical to that of a normal distribution. Note: The mean and variance of the gamma distribution can be obtained by using the first and second derivatives of the Laplace transform, respectively.
The univariate semi-parametric Gamma frailty model
The multivariate semi-parametric frailty model
The shared frailty model
Frailty Modeling 47 A univariate frailty model on the other hand assumes a frailty term for each individual and this frailty term represents the individual's unmeasured or hidden covariates after considering the measured covariates. Let N denote the number of individuals in a given group with each individual in the group assigned to a group. Let the total number of groups be denoted by G such that, given the group i consisting of individuals, then;.
Uncensored or event occurred implies that the survival time of the individual in the data set was observed and therefore recorded, while censored or event did not occur implies that the event time was not observed and in this case the event time is assumed to be more than the recorded time (censored right). XXXij is a vector of covariates for individual j in the ith group, ui the unobserved covariates andh0(t) the baseline hazard function.
Estimating parameters in a semi-parametric frailty model
- The Expectation-Maximisation Algorithm (EM-Algorithm)
- The E-step
- The M-step
- Application of the EM algorithm on a shared Gamma frailty
- The penalised partial likelihood approach
- Implementation of the penalised partial likelihood on a Normal
- The markov chain monte carlo methods
Hanagal (2011) mentioned that the EM algorithm differs from the maximum likelihood estimation method (MLE) because it maximizes the full likelihood instead of the likelihood in equation (5.7). The EM algorithm first consists of the E-step, which is the expectation of the full data probability with respect to the missing information ZZZ given the observed data XXX and the initial parameter estimates (Small and Wang, 2003). 1977) proposed a modified form of the EM algorithm, the Generalized Expectation Maximization Algorithm (GEM), in cases where the solution to the M-step does not exist in closed form (Small and Wang (2003)).
At the M step, we plug the expected values of the frailty terms into the modified partial likelihood to obtain the estimates of βββ and h0. The penalized partial likelihood approach uses the random effectsui instead of the frailty terms zi = exp (ui). The maximization of the penalized partial probability consists of an outer loop and an inner loop.
Detailed information on the penalized partial likelihood and further understanding of the penalized partial likelihood approach we recommend a book by Duchateau and Janssen (2008).
The results from fitting the shared gamma frailty model on 2011 Uganda
The Weibull model
The Weibull distribution is the most commonly used distribution for modeling temporal event data due to its flexibility in determining the hazard function. A Bayesian approach 65 where δi is an indicator variable that has a value of 1 if it is error time and 0 if it is right censored.
The Weibull frailty model
The Bayesian approach 66 where ηηηij =βββ0+XXXTijβββ+zzzi,βββ ap×1 is vector of regression coefficients and βββ0 is denoted as intercept andXXXij is ap×1 covariate vector.
The INLA approach
- Data analysis using INLA for Bayesian inference
The main goal is to approximate the posterior bounds of the latent field, π(XXXi|YYY) and the posterior bounds of the hyperparameters, π(θθθ|YYY) and π(θj|YYY). Bayesian Approach 68 The posterior margin π(θθθ|YYY) of the hyperparameters θ is approximated using the Laplace approximation. Approximations of the posterior edges of the latent field are obtained using a finite summation.
We also assume a gamma prior with parameters 1 and 0.001 for the shape parameter α of the Weibull distributionα. The mean values presented in table (6.1) are the mean values of the estimated coefficients ˆβ for the covariates included in the model. From the results presented in Table (6.1), the fixed effects that have confidence intervals that include zero imply that the factors are not significant.
Factors that, according to the results presented in table (6.1), are strongly related to the survival of children under five years of age; the gender of the child, the number of births in the last year and the gender of the head of the household.
Bayesian frailty modelling
- Data analysis for the Weibull frailty model
- Results
The variance of the fragility term is 0.016, which is very close to zero and thus the results confirm that the community fragility is not significant. This implies that all the differences between the mortality rates of the children under the age of five are explained by the observed fixed covariates stated in the model. Children under the age of five in different households were exposed to different risks of death, whereby some of the children were more exposed to the risk than others.
The results indicate that gender of the child, gender of the household head and number of births in the last one year are strongly associated with the survival of children under the age of five. The short birth intervals depicted by the results may also be the result of the low use of modern family planning methods. This will reduce the death of children due to negligence by parents based on the gender of the child.
The biggest problem with it is that the quality of the report depends on the completeness and correctness of the birth and death anamnesis. To obtain the variance of the survival function estimate Sd(t), we use the delta method to estimate the survival function Sb(t). Appendix A. The delta method 86 using the delta method on the right-hand side of Eq.,.
