Background: In Ethiopia, child malnutrition is one of the most serious public health problem and the highest in the world. The overall prevalence of underweight among children under five years old in Ethiopia was 25% and 7% are severely and moderate underweight in 2014 respectively. Malnutrition in sub-Saharan Africa contributes to high rates of childhood morbidity and mortality. However, little information on the nutritional status of children is available from informal settlements. The primary aim of this study was then to determine the determinants of children malnutrition in Ethiopia using Bayesian generalized linear model. The overall prevalence of underweight among children in Ethiopia was 36.4.

Methods: Data was obtained from 2011 Ethiopian Demographic and Health Survey (EDHS). Bayesian Generalized regression model was used to identify the effects of selected socioeconomic, demographic, health and environmental covariates. Bayesian approach with Markov Chain Monte Carlo (MCMC) technique was used.

Results: The analysis result revealed that out of the 11, 654 number cases examined in this study 32.8% of male children were underweight and 33.6% of female children were underweight. It was found that the covariates succeeding birth interval, sex of child, child by choice not by chance, vaccination and cough were the most important determinants of children nutritional status in Ethiopia.

Conclusion: Inference is the fully Bayesian and classical generalized linear model based on recent Markov chain Monte Carlo techniques.Some of the socioeconomic, demographic and environmental determinants included in the study were found to be statistically significant. The result was suggested that for reducing childhood malnutrition, due emphasis should be given in improving the knowledge and practice of parents on appropriate young child feeding practice and frequent growth monitoring together with appropriate and timely interventions.

Keywords: nutrition, weight-for-age, malnutrition, anthrometric measurement, mcmc, bayesian generalized linear model

Source of data

This study is based on the 2011 Ethiopian Demographic and Health Survey (EDHS) data obtained from the Central Statistics Agency of Ethiopia. In this study, to assess the nutritional status of children, we have three anthropometric indices: weight-for-age, height-for-age and weight-for-height which are considered as the indicators of nutritional status.

Variables of the study

In this study socio-economic, demographic, health and environmental characteristics are to be proximate determinants of nutritional status. In our application on childhood nutritional status the variables are Weight-for-age (underweight) which is measured by Z-score and body mass index(BMI). Body mass index is a measure of weight adjusted for height. It is calculated as weight in kilograms divided by the square of height in meters.⁸ These variables are continuous. Since the nutritional status of children(weight-for age) and body mass index are affected by a number of factors, the predictor variables to be analyzed in this study as determinants of nutritional status of children are grouped as socio-economic, demographic, health and environmental factors. These factors include the child"s age, sex of child, birth order, preceding birth interval, duration of breast feeding, region, wealth index, mother education level, types of place of residence, number of household member, source of drinking water, ever had vaccination and children by choice not by chance.

UV-Vis analysis of products generated after Tubantin Brown GGL treatment

Decolorization percentage of TB GGL after treatment by binary mixture (H. desertis G11 and M. trichothecenolyticum TL13) was analyzed by UV-Vis spectrophotometer (Shimadzu UV-1800 PC model Kyoto, Japan) by scanning centrifuged cell-free supernatant samples in the range 200–800 nm. The supernatant used was obtained after centrifugation 10000 rpm for 15 min at 4°C.

$p\Theta y\frac{py\Theta p\Theta }{py}$

As _1,_2..._j $randomand$ p(y)= $\int py\Theta p\Theta d\Theta$ is fixed and constant.

Thus the posterior distribution can be expressed as:

$p\Theta y\propto py\Theta p\Theta$    (1)

Bayesian generalized linear regression model

When the effect of covariates on the response variables is linear we follow Bayesian Gaussian linear regression parametric approach. One of the most popular and useful tools in data analysis is the linear regression analysis model. Consider the normal linear regression model in which a response variableis related to one or more explanatory or predictor variables $X_1\mathrm{,...,}{X}_{p-1}$. For a random sample of n individuals, the model becomes:

$y_i{\beta }_0+{\beta }_1{x}_{i1}+{\beta }_2{x}_{i2}+\mathrm{...}+{\beta }_{p-1}{x}_{ip-1}+{\epsilon }_i$   (2)

$p\beta \mathrm{,}{\sigma }^{2}y\frac{py\beta \mathrm{,}{\sigma }^{2}p\beta \mathrm{,}{\sigma }^2}{{\displaystyle \int py\beta \mathrm{,}{\sigma }^{2}p\beta \mathrm{,}{\sigma }^{2}d\beta \mathrm{,}{\sigma }^2}}$   (3)

Prior distribution

In Bayesian inference, the unknown function $f_j$ , j =1,...,p , the fixed effects parameters $\gamma {\gamma }_1\mathrm{,...,}{{\gamma }_r}^{}$,aswell as the variance parameterar ${\tau }^2$ are considered as random variables and have to be supplemented by appropriate prior assumptions. In the absence of any prior knowledge we assume independent diffuse priors ${\gamma }_j\propto const$, j=1,...,r for the parameters of fixed effects. In this study, for the continuous covariates child age in month and Mother age at first birth those assumed to have a possibly nonlinear effect on the Z-score are modeled non-parametrically as cubic P-spline with second order random walk priors.

The general form of the priors

Suppose thatis the vector of corresponding function evaluations at observed values of x. Then, the general form of the prior for $f$ is

$f{\tau }^2\propto exp-\frac{1}{2{\tau }^2}{f}^{}Kf$   (4)

${\tau }^2~IGa\mathrm{,}b$

$p{\tau }^{2}a\mathrm{,}b{{\tau }^2}^{-a-1}exp\frac{b}{{\tau }^2}$

Common choices for $a$ and $b$ are $a=1$ and $b=0.005$ (or $b=0.0005$ ). Alternatively, one may take $a=b=0.001$.

Priors for fFixed effects

As indicated above for the parameter vector $\gamma$ of fixed effects we choose a diffuse prior ${\gamma }_j\propto const$, j=1,...,r Another choice would be to work with a multivariate Gaussian distribution $\gamma ~N{\gamma }_0\mathrm{,}{\sigma }_{{\gamma }_0}$. In this study, diffuse priors will be used for the fixed effects.

Priors for continuous effects under linear setup

First and second order random walk priors

Bayesian P-splines

Any smoother depends heavily on the choice of smoothing parameter, and for P spline in a mixed (fixed and continuous) framework. A closely related approach for continuous covariates is based on the P-splines approach introduced by Eilers and Marx (1996). The basic assumption of this approach is that the unknown functionca $f_j$ be approximated by a spline of degree l with equally spaced knots $x_{min}={\xi }_0<{\xi }_1<...{\xi }_{r-1}<{\xi }_r=x_{max}$ within the domain of $x_j$.

Posterior inference

In this case, the specific form of the posterior depends only on the parameterization of the regression terms in the model. Then, we use Markov Chain Monte Carlo (MCMC) simulations to draw samples from the posterior and statistical inference is done by means of Markov Chain Monte Carlo techniques in a full Bayesian setting. Now we restrict the presentation to models with predictor. Full Bayesian inference is based on the entire posterior distribution. Letbe the vector of all unknown parameters, then the posterior is given by.

$p\mathrm{<mo\; stretchy="false">(</mo>}\alpha \mathrm{<mo>/</mo>}y\mathrm{<mo\; stretchy="false">)</mo>}\propto L\mathrm{<mo\; stretchy="false">(</mo>}y\mathrm{,}{\beta }_1\mathrm{,}{\tau }_1\mathrm{,}{\beta }_2\mathrm{,}{\tau }_2\mathrm{,...,}{\beta }_p\mathrm{,}{\tau }_p\mathrm{,}\gamma \mathrm{<mo\; stretchy="false">)</mo>}{\displaystyle \prod _{j\mathrm{<mo>=</mo>1}}^p}p\mathrm{<mo\; stretchy="false">(</mo>}{\beta }_{j\mathrm{<mo>/</mo>}{\tau }_j^2}p\mathrm{<mo\; stretchy="false">(</mo>}{\tau }_j^2\mathrm{<mo\; stretchy="false">)</mo>}$

$\propto L\mathrm{<mo\; stretchy="false">(</mo>}y\mathrm{,}{\beta }_1\mathrm{,}{\tau }_1\mathrm{,}{\beta }_2\mathrm{,}{\tau }_2\mathrm{,...,}{\beta }_p\mathrm{,}{\tau }_p\mathrm{,}\gamma \mathrm{<mo\; stretchy="false">)</mo>}$

${\displaystyle \prod _{j\mathrm{<mo>=</mo>1}}^p}\frac{1}{{\mathrm{<mo\; stretchy="false">(</mo>}{\tau }_j^2\mathrm{<mo\; stretchy="false">)</mo>}}^{rk\mathrm{<mo\; stretchy="false">(</mo>}\frac{k_j}{2}\mathrm{<mo\; stretchy="false">)</mo>}}}exp\mathrm{<mo\; stretchy="false">(</mo>}-\frac{1}{2{\tau }_j^2}{\beta }_j^{\mathrm{<mo>\text{'}</mo>}}{K}_j{\beta }_j\mathrm{<mo\; stretchy="false">)</mo>}{\displaystyle \prod _{j\mathrm{<mo>=</mo>1}}^p}{\mathrm{<mo\; stretchy="false">(</mo>}{\tau }_j^2\mathrm{<mo\; stretchy="false">)</mo>}}^{-a_j-1}exp\mathrm{<mo\; stretchy="false">(</mo>}-\frac{b_j}{{\tau }_j^2}\mathrm{<mo\; stretchy="false">)</mo>}$

Model diagnostic

Once a model has been developed, we now would like to know how effective the model is in describing the outcome. This is referred to as goodness of fit. The most common ways of checking goodness of fit are: diagnosis for convergence and mixing and posterior-predictive check. For convergence tests, the researcher used time series plot.

Model selection and comparison

Model selection is to select the best model among several choices based on evaluation of the performance of the models. One immediate question is how to measure goodness of fit? There are many choices. The simplest one is residual sum of squares.⁹

$\sum (y_i-{\hat{y}}_i)$

Where $y_i$ is the observed responses and ${\hat{y}}_i$ is the fitted value. This measure is also called mean squared error, and commonly used as the default measure. For GLM, deviance is a quality of fit for a model that depends on the likelihood. The researcher skip the details and the underlying theory of the definition, but pointing out that the statistical software can compute the quantity automatically. The next question is why we should consider model selection? There are several reasons. First, people tend to believe or can understand simpler models with fewer predictors and less complicated structure. Second, one can certainly add more and more features into the model without screening and get better and better fit, till perfect fit, but the problem is over fitting. Remember that the researcher want to find the best-predicting model not the best fitting model. In this study, AIC(Akaike Information Criterion), BIC(Bayesian Information Criterion), CV(Cross-Validation) and DIC (The deviance information criterion) are used to compare the linear frequent and the linear Bayesian approach with their smallest values to be better model. In this study, authors used R software(version 3.4.0) to analyze the determinant of factors under five years children nutritional status in Ethiopia.

Descriptive analysis of the data

As one can see from Table 1, among the cases examined in this study of male children were underweight and of female children were underweight. The highest prevalence of child malnutrition was observed among children whose preceding birth interval was less than 24 months ( 35%) unlike to the lowest prevalence of child malnutrition which was recorded from children whose preceding birth interval is 48 and above (31.9% ). The proportion of underweight children as we can seen from Table 1, it is also differs by type of place of residence: urban and rural. Accordingly, higher numbers of underweight children (33.6% )) reside in rural areas, and relatively less numbers of underweight children (31.2% ) reside in urban centers. The evidence given in Table 1 shows that the percentage nutritional statuses of children who had diarrhea recently seem to be high probability of underweight than those children which had no diarrhea recently. Likewise, one can observe that the percentages nutritional status of children who had fever recently seemed high probability of underweight than those children who have no fever recent (33.9% ,33.1% , respectively). The evidence given in Table 1 shows that the percentages nutritional status of children in households who had used other water sources seemed to be more underweight than those children in households which used water from public source. The family education level has also it"s effect on the nutritional status of children that a mother with higher educational level(father education level) had a child with better nutritional status. Since the percentages of underweight is decreasing from no education to secondary and above education level. Children who had disease are affected by malnutrition e.g child who had cough is more affected than child who had not cough as we observe from Table 1.

From Table 2 the highest malnourished region are SNNP, Oromia, Gambela, Affar and Tigray. From all of these region the smallest malnourished region is Harari. The place of residence has also its impact as result the highest malnourished lives in rural comparing with urban.

Bayesian generalized linear regression model

Results of generalized linear regression analysis weight-for-age revealed that, the intercept is estimated to be -692.832. It can be interpreted as the value you would predict for weight for age if all predictor variables are set to minimum. We would expect an average Z-sore value of approximately -692.832 for weight for age (under weight). Results of generalized linear model analysis weight for age shows that sex of child is significant(estim -145.487, p-value 0.0315)which indicate that, Weight-for-age(underweight) of a female is lowered by 145.487 as compared to a male child. Children by choice not by chance is significant( estim -214.050, p-value 0.0432) that is Underweight of children by choice not by chance is lowered by 214.0.050 as compared to children by chance. Mother education level at secondary and above is significant level. And it is positively related to weight-for-age of children (estim 838.0805, p-value 0.0873). Weight-for-age declines with increasing mother education level. Ever had vaccination is significant(estim -194.487, p-value 0.0270). Weight-for-age of children who had ever vaccination is lowered by 194.487 as compared to no had never vaccination. Had cough is significant. Weight-for-age of children who had cough is increased by 282.807 as compared to who had no cough. Duration of breast feeding who had ever breasted, not currently breastfeeding is positively related to weight-for-age (etim 314.196, p-value=0.0655), significant at level. Weight-for-age declines with increase in duration of breast feeding of a child. Results of generalized linear model for frequentist approach for body mass index shows that the intercept value is approximately -553.208 , the y-intercept, can be interpreted as the value you would predict for body mass index if all predictor variables are zero. We would expect an average value of approximately -553.208 for body mass index (malnourished). However, this is only a meaningful interpretation if it is reasonable that all the predictors equal to zero, and if the data set actually included values for all predictors that were near 0. If neither of these conditions are true, then really has no meaningful interpretation. It just shift the regression line in the right place. The current study revealed that, Child age in month is significant at level. Body mass index of child is negatively related to child age(estm 6.340, p-value 0.0812). Which implies as child age increases the body mass index is deceased. Succeeding birth interval is positively related to body mass index (etim 6.558, p-value=0.0722), significant at level. Body mass index increase with increasing in succeeding birth interval of a child. Sex of child (gender) is significant. Body mass index of a female is lowered by 140. as compared to a male child. Children by choice not chance is significant. Body mass index of child is negatively related to children by choice not chance. Child ever had vaccination is significant. Body mass index of child is negatively related to ever had vaccination. Child who had cough is significant at . Body mass index of child is positively related to child who had cough. Duration of beast feeding is significant. Body mass index of child is positively to duration of breast feeding.

Table 3 gives results for the fixed effects (categorical covariates) on the nutritional status of children under age five in Ethiopia. The output gives posterior means along with their standard deviations and credible intervals. Since the confidence interval do not include zero, sex of child, child by choice not by chance, ever had vaccination and had cough were found statistically significant at significance level. But, regions, mother education level, source of drinking water and duration of breast feeding were found statistically insignificant. One can observe that child who had vaccinated contributes to better nourishment for children under five age. In this study, the relative chance to be underweight for the children was found to decrease with the increase of vaccination for children. The children who had no vaccination were more likely to be malnourished as compared to children ever had vaccination. And also the analysis shows that female children are better nourished than male children. On the other hand, one can observe that underweight is higher for children who had cough than child who had no cough. So children who had cough should be treated at early stage. We can see the output matches up with frequentest(traditional) approach. As the sample size increase the result should converge to the same values. One of the benefits of the Bayesian perspective (for any analysis) is that it allows us to make credible interval statements. Credible intervals are similar to confidence intervals, but in the Bayesian framework, the interval REALLY IS believed to contain the true population parameter. For instance: a credible interval for a parameter being estimated is interpreted as, there is a probability that the actual parameter is included in that interval. This is because the interval is based on information from the posterior distribution, of for instance, one of the predictor"s coefficient posterior distribution. For example, we can say there is a probability that the true population value of the Child"s sex (Female) coefficient is between -271.255 and -11.280.

From Table 4 we also observe that number of significant variables are the same for weight for age and as the result those variables are affect body mass index. Body mass index of child sex(female) is lowered by -138.5870 comparing to male child which indicate the decrement of body mass index is highly observed in female children that can make female more susceptible to illness, impair the growth and development, cause infertility and the raise for different disease including death. Likewise from Table 4 child who had cough faces high body mass index comparing to child who had not cough and child ever vaccinated faces low body mass index. Which indicate only vaccination is not enough as result children should get sufficient macro nutrients. Generally, the Bayesian generalized linear model satisfy the normal assumption of the residual. the models in Bayesian approach MCMC post estimation diagnosis, it is also possible to extract sampling paths of parameters with function samples(), or to plot the samples directly. For instance, coefficient sampling paths for term sx(SBI) are displayed which is important to check the model convergence in the time series plot.

Model comparison

We analyzed the DHS data set using two different approaches. It is reasonable to ask which type of model is better. The two models are listed in the following:

Model 1 = Generalized linear regression model in frequentest

Model 2 = Generalized linear regression model in Bayesian Model selection is an important part of statistical analysis, and indeed is a central to the pursuit of science in general. Many authors have examined this question, rom both frequentist and Bayesian perspective, and many tools o selecting the "better model". The models proposed in this study are quite general and the model building process can be quite challenging. Currently, an automated procedure for Bayesian model selection is not available. However, a few recommendations are possible. Users should try to incorporate everything that is theoretically possible and different Bayesian models could be compared via the DIC.¹⁰ From Table 5, when we compare Model 1 and Model 2 via AIC, the most interesting fact here is that the better models in terms of AIC. Model 2 (Generalized linear regression model in Bayesian approach) since they have less value (28753.8) and (28710.6) and they are chosen by the data analyst. The point is that models 2 are a better models than models 1 (Gaussian linear regression model in frequentest approach).

Conclusion

1. Should give due attention to the high risk factors identified as contributing high risk of malnutrition in children under five years of age due emphasis.
2. Should be given better source of water for community in more facilitated manner.
3. And also male and female children should be equally treated and have similar care and support since female children were more likely to be underweight than male children in Ethiopia.
4. Finally, further findings should be incorporating for additional risk factors which are associated with childhood malnutrition in Ethiopia regional wise.

The authors gratefully acknowledge Central Statistics Agency for providing the data. This work was financially supported by the College of Natural Science, Hawassa University.

Author declares that there is no conflicts of interest.

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