Modeling Determinants of Low Birth Weight for Under Five-Children in Ethiopia

The child weight was first dichotomized based on the cut-off points as described in literature review leading to the binary response (Table 1).

The variables that were considered in the research and expected to be the risk factors of LBW, were grouped in to maternal, socio-economic, demographic, and health and environmental factors (Table 2).

A range of techniques has been developed for analyzing data with categorical response variables. Marginal models are among the statistical models widely used to model clustered or repeated data. The primary objective of marginal model is to analyze the population-averaged effects of the given factors. It includes Generalized Estimating Equations (GEE) and Alternating Logistic Regression (ALR).

GEE approach is used to account for the correlation between responses of interest for subjects from the same cluster. It is non-likelihood method that captures the association within clusters in terms of marginal correlations [5]. Liang and Zeger [6] proposed GEE for clustered as well as repeated data, which require only the correct specification of correlation structure. The model for GEE based on generalized linear models and working correlation structure is given by:where \(g({\pi }_{j})\) is logit link function, \(X_{j}=(n_{j}\times p)\) dimensional vector of known covariates, \({\beta }=(1 x p)\) dimensional vector of unknown parameter, and \(E(Y_{j})={\pi }_{j}\) is expected value of the responses \(Y_{j}\) in cluster j which is binomially distributed as \(Y_{j}{\sim }bin(n_{j},{\pi }_{j})\).

GEE is not likelihood approach, rather it is quasi likelihood based. The parameter \({\beta }\) is estimated by solving estimating equations which consist of the working correlation matrix \(R_{j}\) and matrix with the marginal variances on the main diagonal and zeros elsewhere \(A_{j}\). The score equation used to estimate the marginal regression parameters while accounting for the correlation structure is given by:

ALR is an extension of GEE which measures pair wise association of two observations in the same cluster. ALR extends classical GEE in the sense that precision estimates for both the regression parameters \({\beta }\) and the association parameters \({\alpha }\). Moreover with ALR inferences can be made about pair wise associations between subjects [5]. Let \({\gamma }_{jkl}\) be the log odds ratio between outcomes \(Y_{jk}\) and \(Y_{jl}\), and Let \({\pi }_{jk}=P(Y_{jk}=1)\) and \({\upsilon }_{jkl}=P(Y_{jk}=1,Y_{jl}=1)\), then the association of the two responses is defined as [5]:Assume \({\gamma }_{jkl}={\alpha }\), then the pairwise log odds ratio \({\alpha }\) is the regression coefficient in logistic regression of \(Y_{jk}\) on \(Y_{jl}\).

Rather than maximum likelihood, ALR is also quasi likelihood based. Let \({\xi }_{j}\) be a vector with elements \({\xi }_{jkl}=E(Y_{jk}/(Y_{jl}=y_{jl}))\) and let \(R_{j}\) be the vector of residual with elements \(R_{jkl}=Y_{jk}-{\xi }_{jkl}\). Let \(S_{j}\) a vector of diagonal matrix with diagonal element \({\xi }_{jkl} (1-{\xi }_{jkl})\) and let \(W_{j}\) denote the matrix of \({\frac{{\partial }{\xi }_{j}}{{\partial }{\alpha }}}\). Finally, let \(A_{j}=Y_{j}-{\pi }_{j}\), \(B_{j}=cov(Y_{j} )\), \(C_{j}={\frac{{\partial }{\pi }_{j}}{{\partial }{\beta }}}\). Then the alternating logistic regression parameter \({\delta }=({\beta },{\alpha })\) is the simultaneous solution of the following unbiased estimating equations [6].The above estimating equation are solving for \({\beta }\) and \({\alpha }\) by using Gauss-Seidel procedure algorithm.

Generalized linear mixed models is one parts of cluster specific models. It extends ordinary regression by allowing non-normal responses and a random component with the link function of the mean. Assumed conditionally on q-dimensional random effects \(b_{j}\) to be drawn independently from N(0, D), The outcomes \(y_{ij}\) of \(Y_{j}\) are independent with the density of the formThen the generalized linear mixed model [5]; with logit link is defined aswhere \(j=1,2,\ldots ,m\) \(E(Y_{ij}/b_{j})={\pi }_{ij}\), is the mean response vector conditional on the random effects \(b_{j},\) for subjects in cluster j and \(X_{ij}\) and \(Z_{ij}\) are p-dimensional and q-dimensional vectors of known covariate values. \(\beta\) is a p-dimensional vector of unknown fixed regression coefficients, and \(\phi\) is a scale parameter. The random effects \(b_{j}\) are assumed follow a multivariate normal distribution with mean 0 and covariance matrix D.

Random-effects models were fitted by maximization of the marginal likelihood, obtained by integrating out the random effects. The Laplace approximation method [5] has been designed to approximate integrals of the form \(I={\int }{e}^{Q(b)}db\) where,Q(b) is known, unimodal, and bounded function of a q-dimensional variable b.

Under the GEE, model building strategy is started by fitting a model containing all possible covariates in the data. This was done by considering two different working correlation assumptions (exchangeable and independence). In order to select the important factors related to LBW, the backward elimination procedure was used. The full model for the probability of getting LBW of ith child from jth cluster, (\(\pi _{ij}\)) was fitted as:The subscripts in each covariate is defined as, M = Male, Mi = middle, Ri = Rich, U = Urban, 1 = 20–39, 2 = 40–49, Y = Yes, 1+ = 1–4, 5+ = five and above, W = Widowed, D = divorced, Mo = Moderate, Se = severe, Pr = Primary, Sec = Secondary, 10+ = ten and above, 11+ = eleven and above.

After fitting the model, covariates with the largest p value are removed and the model was refitted with the rest of the covariates sequentially. Then, residence, ever had terminated pregnancy, birth order and preceding birth interval are the covariates excluded from the model. The QIC values of full model and reduced models are 4011.6165 (which is found in appendix) and 3986.4033 respectively. Then it turned out that the model with sex, wealth status, age of mother, number of antenatal care, marital status, vaccination, anemia level and mothers’ education level was the most parsimonious model.

Then, from Table 4, exchangeable working correlation assumption was found to be plausible since the two standard errors were closer to each other with correlation parameter (\(\alpha =0.0857\)). Therefore, the final proposed generalized estimating equation model for low birth weight is given as:Parameter estimates and their corresponding empirically corrected standard errors alongside the p values from the final GEE model are presented in Table 5.

Model building for ALR is follows the same procedure in GEE model building strategy. First ALR model is fitted using all proposed covariates. Then the covariate with the large p value is removed. Residence, ever had terminated pregnancy, birth order and preceding birth interval are removed covariates with (p value \(> 0.05\)). The QIC values of both saturated and reduced models are 4011.8139 and 3986.1527 respectively.Using the selected covariates and the association parameter \(\alpha\), alternating logistic regression (ALR) model that provides information about pair wise association of observations between two different individuals within the same cluster was fitted.Therefore, the final proposed ALR model included the association parameter for low birth weight is given as:

Table 6 presents parameter estimates and their corresponding empirically corrected standard errors alongside the p values from ALR model. Each parameter \(\beta _{j}\) reflects the effect of factor \(X_{j}\) on the log odds of the probability of being born with LBW, statistically controlling all the other covariates in the model. Then, the odds ratio of variables is calculated as the exponent of \(\beta _{j}\) i.e. odds ratio = \(e^{\beta _{j}}\) The ALR analysis from Table 6 suggests that, sex of child is significantly related to birth weight of child. After controlling all other variables in the model the odds that a male child born with LBW is \(\exp (\beta _{1})=\exp (-0.3461)=0.7074\) (95% CI 0.6074, 0.8239) times lower than the female child. This means the probability that male child born with LBW is 29% lower than that of female. As it has been seen from the result of the ALR model, mothers wealth status is statistically significant on birth weight of child. The estimated odds that child born to a mother who are from highest wealth status is \(\exp (-0.3522) =0.7031\) (95% CI 0.5766, 0.8572) times less likely to have low birth weight compared to the reference group. This implies that the probability of LBW is reduced by 29% for children whose their mother are from highest wealth status when compared with children whose their mothers are from lowest wealth status. In this study, middle wealth status has no significant effect on LBW of children. There is also a strong association between age of mother and birth weight of child. This implies that, after adjusting all other predictor variables in the model, the estimated odds that child born to a mother who are from age group 20–39 is exp(− 1.0008) = 0.3675 (95% CI 0.2242, 0.6024) times lower to have low birth weight compared to reference age group (15–19). This means percentage of low birth weight is decreased by 63% for children whose their mothers are in age group 20–39 when compared to children whose their mothers are in early age group. The estimated odds that child born to a mother who are from age group 40–49 is exp(0.8581) = 0.4239 (95% CI 0.2512, 0.7154) times lower to have low birth weight when compared to reference age group. This means percentage of low birth weight is decreased by 57% for children whose their mothers are in age group 40–49 when compared to children whose their mothers are in early age group. The results also indicate a negative association between LBW and the number of antenatal care visits. The results suggest that the higher the number of antenatal visits, the lower the odds of LBW. The odds that a child born to mother who follow antenatal care for more than five times is exp(−0.2832) = 0.7533 (95%CI 0.5573, 0.8886) times lower to have low birth weight compared to one whose mother do not follow antenatal care. This implies that low birth weight is reduced by 25% for children whose their mothers follow antenatal care for more than five times. As we can see from the analysis, following antenatal care for less than five times has no significant effect on LBW of child. Another significant ingredient of LBW is marital status of mother. Mothers who are divorced are more likely to deliver child with LBW than mothers who are married. The odds of LBW for divorced mother is \(\exp (0.3402)=1.4052\) (95% CI 1.0153, 1.9450) times higher as compared to reference group. This implies LBW of baby increased by 40% for divorced mothers when compared to married mothers. Statistically significant association has been seen between vaccination and LBW of child. The odds that a child born to vaccinated mother is \(\exp (-0.2582) =0.7724\) (95% CI 0.6325, 0.9431) times lower to have low birth weight compared to one whose mother is not vaccinated. This implies LBW is decreased by 22% for children whose their mothers are vaccinated. Statistically significant association has been seen between LBW and anemia level. The odds that a child born to mother who moderately suffered from anemia is \(\exp (0.2293)=1.2577\) (95% CI 1.0641, 1.4866) times higher to have low birth weight. And the odds that a child born to mother who severely suffered from anemia is \(\exp (0.6874)=1.9885\) (95% CI 1.3738, 2.8785) times higher to have low birth weight compared to one whose mother is not suffered from anemia. This implies that the percentage of delivering child with LBW is increased by 26% and 99% respectively for moderately anemic and severely anemic mothers compared to not anemic mothers. The analysis from Table 6 suggests that, education is significantly related to LBW of children. After controlling all other variables in the model, the odds that mother whose her education level is primary deliver a child with LBW is \(\exp (-0.1962)=0.8218\) (95% CI 0.6682, 0.8952) times lower when compared to the reference group. This shows LBW is reduced by 18% for children whose their mothers education level is primary compared to children whose their mothers are not educated. The ALR model also presents the estimated constant log odds ratio (alpha) which, provide information about the association between individual observations within the same cluster. The estimated pair wise odds ratio relating two responses from the same cluster is exp(0.4107) = 1.5078 (95% CI 1.2693, 1.7912). Thus, the value of alpha which is greater than one indicates that, the associations is found to be significant (p value < 0.0001) and this means that there is a strong positive association between individual children regarding LBW in the same cluster.

Unlike in the marginal models, (GEE and ALR) where parameters are treated as population averages, in the GLMM analysis, parameter interpretation is based on specific subjects or cluster. The parameter interpretation is conditional on the random effects, which is common for all individual children in the same cluster. Given the same random intercept \(b_{j}\), the estimated odds of LBW of child is \(\exp (-0.3815) =0.6828\) (95% CI 0.5815, 0.8017) times lower for male child when compared to female child in the same jth cluster keeping constant the other fixed effect variable in the model. This implies the probability of low birth weight is 32% less likely for male child than female child in the same cluster at the given random effect. In the same way, the estimated odds that a child born to a mother who are from highest wealth status is \(\exp (-0.3304)=0.7186\) (95% CI 0.5778, 0.8939) times lower to have low birth weight compared to the reference group in the same cluster. This shows that the probability of LBW is reduced by 28% for children whose their mother are from highest wealth status when compared with children whose their mothers are from lowest wealth status. The estimated odds that child born to a mother who are from age group 20–39 is \(\exp (-1.1031)=0.3318\) (95% CI 0.2031, 0.5419) times lower to have low birth weight compared to reference age group (15–19). This means percentage of low birth weight is decreased by 67% for children whose their mothers are in age group 20–39 when compared to children whose their mothers are in early age group in the same cluster. The estimated odds that child born to a mother who are from age group 40–49 is \(\exp (-0.9378)=0.3914\) (95%CI 0.2303, 0.6653 times lower to have low birth weight when compared to reference age group. This means percentage of low birth weight is decreased by 61% for children whose their mothers are in age group 40–49 when compared to children whose their mothers are in early age group in the same cluster. At the given constant random effect, The odds that a child born to mother who moderately suffered from anemia is exp(0.2459) = 1.2787 (95% CI 1.0751, 1.5210) times higher to have low birth weight. And the odds that a child born to mother who severely suffered from anemia is exp(0.7822) = 2.1862 (95% CI 1.5303, 3.1236) times higher to have low birth weight compared to one whose mother is not suffered from anemia. This shows that the probability that mothers deliver child with LBW for mothers who are moderately anemic is 28% more likely than mothers who are not anemic and the probability that mothers deliver child with LBW for severely anemic mothers is two folds more likely than mothers who are not anemic. The interpretation of other predictor variables can be done in a similar manner.

Figure 1 below revealed that the residuals versus observation with CLID number suggested that residuals are symmetric around zero (i.e. positive and negative residuals are almost equal). Q–Q plots for normality of random effects at regional and cluster levels illustrates that the random effects are normally distributed with mean zero and variance covariance matrix D. Thus, the fitted GLMM model is fine for LBW data.