Study materials

This study uses children anthropometric data collected in the nationwide 2011 Bangladesh Demographic and Health Survey (BDHS) [16]. The country was stratified into 20 strata according to rural and urban enumeration areas of 7 divisions. A nationally representative sample is drawn following a two-stage stratified sampling design: 600 clusters (393 from rural and 207 from urban areas) are drawn at the first stage and then 30 HHs were systematically selected from the selected enumeration areas which are called clusters in the BDHS survey. A total of 17,141 HHs from where 17,842 ever married woman were selected to collect socio-economic, demographic, environmental, and health care related information. Anthropometric measures age, height, and weight are collected for the children aged under 5 years. In total about 8281 children under age five at the interview date were selected for measuring height and weight, however measurements were completely collected from 7826 children (few were absent or refused to provide height and weight), of which 7647 children had plausible anthropometric information for calculating anthropometric measures. As the 2011 BDHS data, the same children data has been utilized in this study. The characteristics of the study population are detailed in the 2011 BDHS report [16].

Child nutritional status

In 2011 BDHS data, three anthropometric indices HAZ, WAZ, and WHZ are calculated based on WHO 2006 Child Growth Standards [17]. These anthropometric measures are routinely analyzed to provide assessment of child nutritional status [18]. The HAZ represents the chronic nutrition measure of under five children. A child is defined as stunted when his/her HAZ is less than −2.00 SD. Let y _i denotes the HAZ of a child and z _i indicates the nutrition status of the child: nourish (z _i  = 0 when y _i  ≥  − 2.0) or stunted (z _i  = 1 when y _i  <  − 2.0). Also, let the vector x _i denotes the values of the explanatory variables which are assumed linearly related to the interval-scale response variable Yand to a logit link function of the probability of being malnourished. Under this consideration, the development of the linear model (LM) and interval-scale MCA (IS-MCA) model for Y, and binary logistic model (BLogM) and nominal-scale MCA (NS-MCA) model for Z are briefly discussed in the following sub-sections.

Linear regression analysis

A linear regression model (LM) is fitted using either least squares or maximum likelihood (ML) method considering the underlying assumptions including linearity of explanatory variables with response variable. Goodness of a fitted linear model is assessed mainly by F-test for overall model and R-squared measures. The performance of the fitted model can also be measured by comparing the observed nutrition status (based on observed HAZ) with the predicted nutrition status (based on predicted HAZ). The proportion of children classified correctly in such way is termed as correct classification rate in this paper.

Logistic regression analysis

Logistic regression model predicts the probability of a child being malnourished instead of predicting his/her nutrition measure given the values of explanatory variables. A binary logistic model (BLogM) can be written as \( {\mathrm{log}}_e\left[{\pi}_i/\left(1-{\pi}_i\right)\right]={\mathbf{x}}_i^T\boldsymbol{\upbeta} \) where \( {\pi}_i={p}_r\left({z}_i=1|{\mathbf{x}}_i^T\right) \) \( ={e}^{{\mathbf{x}}_i^T\boldsymbol{\upbeta}}/\left(1+{e}^{{\mathbf{x}}_i^T\boldsymbol{\upbeta}}\right) \) is the conditional probability of z _i  = 1 given \( {\mathbf{x}}_i^T \). This BLogM is fitted using ML method with a suitable iterative process such as Newton’s method [19]. The goodness of a fitted BLogM is assessed mainly by an R-squared statistic and a goodness-of-fit statistic. The mostly used R-squared statistics available in common statistical software are McFadden [20] and Cox and Snell R-squared statistics [21] (Nagelkerke’s pseudo R-squared statistics [22] in SPSS). The Hosmer-Lemeshow (H-L) test, the most commonly used goodness-of-fit test for BLogM, assures linearity between the log-odds and the explanatory variables [23]. An alternative measure of goodness-of-fit is to compare the observed nutrition status to the predicted nutrition status based on the fitted BLogM, which helps to find out false negative and false positive classification rates [23]. In SPSS, an overall classification rate is reported based on a cut-off point of \( \widehat{p}=0.50 \). Overall performance of the BLogM model can be assessed by comparing classification rates obtained from full and null models. This assessment is relevant to the area under receiver operating characteristic curve of a fitted model [24]. Significance of a predictor can be assessed by likelihood ratio, Wald, and score tests [19], which are asymptotically equivalent [25].

Multiple classification analysis

The linear relationship between response and explanatory variables is in question when the explanatory variables are nominal in nature. The MCA is a multivariate technique which relaxes this linearity assumption [15] and assess the interrelationship through an additive model. The MCA determines the effect of each predictor on the response before and after adjustments for its inter-correlations with other predictors. Each category of a nominal explanatory variable (factor) is considered as an independent predictor (dummy variable), and uncorrelated to other explanatory variables. The advantage of MCA over multiple linear regression analysis is that it can handle any form of interrelationships between the explanatory and the response variables. Also, the similar additive model can be developed for either interval or nominal-scale response variable with the same explanatory variables.

In MCA model, a coefficient is assigned to each category of each explanatory variable in such way that the response value for an individual is the sum of the coefficients assigned to all categories that represent the individual characteristics, grand mean of the response and a random error term. Thus MCA models for Y and nominal-scale nutrition status Z can be expressed by the same model as ν _j … n  = μ + a _j  + b _k  +  .  …  + e _j … n , where ν _j … n is the response value for a child who falls into j ^th category of predictor A, k ^th category of predictor B and so on; μ is the grand mean, a _j is the added effect of j ^th category of predictor A (difference between μ and mean response value of j ^th category of A), b _k is the added effect of k ^th category of predictor B (difference between μ and mean response value of k ^th category of B); e _j … n is the error. The coefficients are estimated via a technique like iterative least squares method. The diagnostic of the fitted model can be done by checking whether all the predictors can explain a significant proportion of variation. For details please see Andrews et al. [15] and Nagpaul [26].

For assessing the importance of a factor (the degree of relationship), two correlation ratios called eta (η _i ) and beta (β _i ) statistics are calculated from the model before and after the adjustment of other predictors respectively [27]. Eta and beta values indicate the proportion of variation in the response variable accounted by each predictor. The beta value indicates the importance of a predictor on the response variable based on which the predictors can be ranked [28]. The comparison between eta and beta values helps one to examine whether the importance of a predictor in a bivariate situation remains in a multivariate design. The specific forms of eta and beta for a predictor are illustrated by Nagpaul [26]. The gain of MCA over regression analysis is estimating the effect of each predictor on the response variable with or without considering the effects of all other predictors without any constraints [26].

Explanatory variables

A number of explanatory variables at children level (age, birth weight, and birth interval), mother’s level (education and nutrition status), household level (wealth status and family size), community level (rural and urban areas), and regional level (division) are considered in the study for developing all the LM, BLogM, IS-MCA, and NS-MCA models. The household and community information were collected from the household head, while children and mother’s information were collected from the mothers. The considered explanatory variables are identified as significant predictors of child malnutrition in many child malnutrition researches [8, 29–35] by developing BLogM models considering Z as response variable. In this study, linear and IS-MCA models are developed to see how the considered explanatory variables influence the interval-scale HAZ score. The considered LM, BLogM, IS-MCA and NS-MCA models are developed using the LM, LOGIT and ANOVA with MCA functions of SPSS (22.0 version) respectively.

Significance and association of the explanatory variables with child malnutrition

The fitted LM shows that the mean HAZs for the older-groups children are about 0.90 Z-score lower than the mean HAZ of infants. The IS-MCA model also shows that the adjusted predicted mean (APM) of HAZ are about −0.94 Z-score for infants and more than −1.80 Z-score for older children (Table 1). The fitted BLogM shows that children aged 12–23 months have the higher risk of being malnourished compared to infants and then the risk decreases gradually with the age categories (Table 2). The NS-MCA model confirms the results of BLogM: the lowest adjusted predicted proportion (APP) of stunted children in the infant group (20.8%) and highest proportion in the 12–23 group (48.9%). In case of child birth weight, the fitted models show that the small birth weight children have about 0.46 SD lower HAZ, about 16% higher APP and more than double risk of being stunted (APM: -1.94, APP: 49.5 & OR: 2.1) compared to the children with large birth weight (APM: -1.49, APP: 33.6 & OR: 1). For birth interval, the coefficients in LM and the APMs in IS-MCA model decrease significantly with the decrease of birth interval category. The MCA models also show that the children with less than 2-years birth interval have higher stunting (APP: 47%) and lower mean HAZ (APM: -1.89 SD). It is noted that here the reference group represents children with first birth order or 48+ month previous birth interval.

The fitted LM and IS-MCA show increasing regression coefficients and APMs respectively (Table 1), while BLogM and NS-MCA show decreasing odds ratios and APPs respectively with the improvement of mother’s education (Table 2). The models show that the illiterate mothers’ children have about 0.50 lower HAZ and 1.9 times higher risk of being stunted compared to those of higher educated mothers (APMs: −1.77 and −1.29 SD; APPs: 44.0% and 32% for illiterate and higher). In case of mothers’ body mass index (BMI), Table 2 shows that mothers with lower BMI have higher risk of having stunted children (OR: 1.31 and APP: 46.0%) compared to mothers with higher BMI (OR: 0.79 and APP: 33%).

The estimated regression coefficients and mean HAZs are found to increase with the order of household economic status (Table 1). The results of BLogM and NS-MCA models (Table 2) indicate that the children living in poorest households are more likely to be stunted (OR = 3.0 and APP: 51%) than those of richest households (OR = 1.0 and APP: 28%). For family size, BLogM and NS-MAC models show slightly higher odds ratios (OR: 1.19 and 1.19) and APPs (43% and 43%) for the small and large families.

The fitted models indicate that the children living in rural areas have slightly lower mean HAZ and risk of being stunted (APM: -1.64, APP: 39.71) than those living in urban area (APM: -1.72, APP: 42.84). However, the unadjusted predicted mean HAZ (Urban: −1.46 SD and Rural: −1.76 SD) and unadjusted predicted proportion of stunted children (Urban: 35.23% and Rural: 43.07%) are found lower in urban areas in the IS-MCA and NS-MCA models respectively. The analysis by regional settings “Division” indicates that the children of Rajshahi division had significantly lower risks of being malnourished (OR: 0.67 & APM: -1.44) compared to the other divisions, while children of Sylhet division are more vulnerable (OR: 1.17, APM: -1.80).

Ranking of the risk factors

Goodness of the fitted models