Malnutrition among children is a major public health problem in developing countries including Bangladesh [1–10]. Globally children with moderate and severe acute malnutrition are approximately 60 million and 13 million respectively [2]. Between 8 and 11 million under-five children also die each year in the world [2, 9]. More than 50% of these deaths are attributed to malnutrition, which are mostly preventable through economic development and public health measures [2]. Although Bangladesh has already achieved a remarkable progress in reducing child malnutrition from 68% in the late 1980s to 41% in 2007 [6, 11] and under-five mortality [12], still malnutrition is a common problem in this country [13, 14]. It is one of the countries with very high burden of malnutrition [5, 14]. The underlying cause for 60% of the under-five deaths is malnutrition in Bangladesh [5].

Malnutrition among children is a critical problem because its effects are long lasting and go beyond childhood. It has both short- and long-term consequences [6, 8, 10]. For instance, malnourished as compared to non-malnourished children are physically, emotionally and intellectually less productive and suffer more from chronic illnesses and disabilities [6, 15, 16]. Malnutrition among children depends on complex interactions of various factors reflecting socio-demographic, environmental, reproductive, institutional, cultural, political and regional factors [3, 4, 6, 13, 14, 17–19]. Already many studies have been conducted to find the predictors of malnutrition in Bangladesh and elsewhere [7, 8, 20–28].

Poverty is found to be strongly associated with child malnutrition [4]. Although the relationship between economic ability and malnutrition is complex, a number of studies have illustrated that children of poorer households tend to be more undernourished than children of wealthier ones [3, 13, 22, 24–26, 29]. Parental education is also identified as a strong precdictor of child malnutrition [14, 28]. However, the association of maternal education is relatively stronger than parental education [14, 20, 29]. Other determinants of child malnutrition may include social deprivation [23], rural-urban place of residence [18, 30], religions [31, 32], number of children [1, 18], source of drinking water [1] and toilet facility [3, 33].

Most of the abovementioned studies, that addressed the topics of child malnutrition, used categorical outcome variables and applied logistic regression (multinomial/binary) [7, 10, 14, 27–29, 34] models to find the predictors of child malnutrition. For instance, Mashal et al. [27] used multivariable logistic regression to identify the risk factors of malnutrition among children in Afghanistan. Mueller et al. [34] applied logistic regression to study the relationship of malnutrition with morbidity and mortality among West African children. Ojiako et al (2009) used Tobit model to find the determinants of malnutrition among preschool children in Nigeria [19]. Sometimes we see studies which addressed non-negative outcome variables like number of children in a household [35, 36] and number of accidents [37]. Recently different types of regression based on Poisson distribution namely standard Poisson regression model, negative binomial regression model and generalized Poisson regression (GPR) model have been used to model such kind of count variables [36–38]. However, applications of these models are based on certain assumptions. For instance, standard Poisson regression model assumes equal mean and variance of the dependent variable. In reality, often this equality assumption is not true because the variance could be higher than mean (over-dispersion property) or lower than mean (under-dispersion property). Ignorance of these properties may produce biased standard errors and inefficient estimates of regression parameters, although the estimates of the standard Poisson model are still consistent. The negative binomial regression model is more flexible than the standard Poisson model and is frequently used to analyse outcome variable with over-dispersion. The GPR model, on the other hand, can capture both over- and under-dispersion properties of the data, which make this model even more flexible [37]. However, a count variable like number of children in a household often shows under-dispersion property [35]. Our data also shows the under-dispersion property. Therefore, we applied the GPR model to our data. The first objective was to demonstrate the applicability of this model as an alternative to study the child malnutrition in Bangladesh. The second objective was to find some predictors of the count variable defined as the number of under-five malnourished children in a Bangladeshi family.

The estimated mean $\left(\overline{Y}=0.626\right)$ and variance (s _y ² = 0.369) of the outcome variable reveal the under-dispersion property of the data. In the total sample, 16.9 percent of the under-five children are malnourished. Table 1 provides descriptive statistics for all predictors. According to this table, illiteracy rate is lower among mothers of children (26 percent) as compared to their fathers (32 percent). In contrast, the rate of higher education is higher among fathers of children (13.3 percent) than their mothers (8.6 percent). Nearly 30 percent of the families have access to the mass media. About one-fourth of the families (26 percent) have no toilet facility. The mean number of children ever born to a woman is 2.67.

Table 2 shows the summary results of bivariate analyses between outcome and predictor variables. All the predictors except religion show significant associations with outcome variable.

The estimated value of the dispersion parameter (α) and its standard deviation from equation (i) are -0.30266 and 0.000439, respectively. Our Null hypothesis formulated as H ₀ : α = 0, (Z=-14.4405 and p < 0.05) is rejected at 5% level of significance.

According to the results of GPR analysis (Table 3), variables namely mother’s education, father’s education, wealth index, toilet/sanitation, sources of drinking water, as well as total number of children are statistically associated with the child malnutrition in a family. The incidence rate of children suffering from malnutrition is higher for mothers having no education (IRR=1.39; 95% CI=1.09-1.78) and 1-5 years education (IRR=1.37; 95% CI= 1.08-1.74) as compared to mothers with higher education. Similar results are also found for father’s education. The incidence rate of malnutrition among children is estimated to be nearly 1.64 times higher in the lowest quintile than the highest quintile. Children of middle and fourth quintiles also show higher incidence rate of malnutrition as compared to the children of highest quintile. The children who drink piped water and tubewell water are nearly 21% and 29% less likely to experience malnutrition than the children drinking other sources of water (such as dug well water, unprotected well, surface water, unprotected spring, river or dam or lake or ponds or canal, rain water, etc). Similarly, toilet facility is strongly associated with malnutrition status of children. A child from a family having no sanitary toilet facility has 1.43 times higher incidence rate of experiencing malnutrition than a child with toilet facility. The total number of children ever-born in a family and the malnourished children are also positively associated.

Our study demonstrates that the GPR model is an ideal alternative to study the malnutritional status of children defined as the number of under-five malnourished children in a family. This model is a good alternative because most of the results of this study are found to be consistent with the findings of many other studies [1, 3, 7, 10, 14, 19, 26, 28, 31, 33, 44, 45].

According to the GPR model, the malnutritional status of children is insignificant between rural and urban areas. This finding is both consistent [14] and contradictory [18]. A similar study in Vietnam [18] reports higher level of malnutrition in rural areas than urban areas. Some possible causes according to this study are lack of economic, socio-cultural, healthcare and intuitional facilities in rural areas [18]. Like in Vietnam, rural areas of Bangladesh also suffer from limited infrastructure and facilities in terms of modern healthcare services, sanitation, education, electricity and economic facilities. Particularly health services are concentrated in urban areas than rural areas [12]. Although urban-rural disparity in terms of child malnutrition is negligible in Bangladesh, this is not the case for many other indicators. For instance, one recent study reports remarkable urban-urban disparities for antenatal care service, age at marriage, fertility and child mortality in Bangladesh [46]. Another study in Vietnam also reports higher age at marriage, smaller family size and lower mortality rate of children in urban areas as compared to rural areas. What are the reasons for this inconsistency? One of the possible reasons might be related to the adjustment of the GPR model by other potential predictors, namely household wealth and maternal education [30]. Inclusion of other predictors along with place of residence into the same model may reduce the strength of urban-rural nutritional disparity. However, further elaborative research is warranted in this regard.

Many studies suggest that mother education is linked with child health outcomes. The relationship of maternal education with child malnutrition is more demonstrable than paternal education, health service availability, and socioeconomic status [14, 20, 29, 47]. However, some studies also show parental educational effects on child nutrition [21, 48]. Our study finds a significant positive relationship between mother’s education and child nutrition. This result is consistent with many other studies [1, 3, 7, 8, 10, 14, 18, 19, 26, 28, 31, 44, 45, 47, 49]. Such a relationship could exist because maternal schooling is strongly associated with good child care and good health. Women with higher as compared to lower educational level are more likely to raise their family income, which helps the families to provide more quality diets and better healthcare to their children. Additionally, educated mothers can efficiently use limited household resources and available healthcare facilities, limit their family size, maintain better health promoting behaviours and provide healthcare to their children [18, 47]. All these factors positively contribute to the child nutrition.

The relationship between economic inequality and children nutritional status is investigated by many studies [3, 26, 29, 45, 50]. Generally the greater degree of economic disparity is associated with higher mortality [51]. The relationship between economic disparity and malnutrition at the national level is not straightforward, because better economy at the national level does not necessarily mean better health care for all. Social and economic disparity in a country may differently influence the accessibility to food and healthcare services including the burden of disease. A number of studies have illustrated that children from poorer households are more likely to be malnourished than children from wealthier households [3, 22, 24, 25]. Social deprivation is also linked with a child's nutritional status [23]. In Bangladesh the nutritional status of children differs in different economic classes [26], which can be attributed to the fact that rich families have more ability to allocate necessary resources for their children than poor families. Understandably allocation of more resources to their children improves their health conditions by reducing multiple health risks.

Our finding reveals a strong positive association between number of children ever born to a woman and the number of under-five malnourished children in a family. These results are consistent with the findings of other study [1, 18]. Generally families with more children experience more economic strain for food consumption and hence they are more likely to suffer from poor nutritional status. In other words, inadequate allocation of household resources among many children may lead to the low nutritional status. Particularly poor families cannot fulfil the nutritional requirements of the children. Families with more children generally devote less time to take care of their children [18]. Because of negative impacts of higher fertility on nutritional status of children, increasing birth interval should be an important strategy to improve the nutritional status among the under-five children.

In Bangladesh men are generally the main earner of a family, although employment opportunities are increasing for women due to the flourishing garment sectors. Income of the family is strongly associated with the type of father’s occupation. Normally fathers with more prestigious job have higher income than fathers with low level jobs and therefore children from the higher income families should have better nutritional status. However, this is not the case in our study. The insignificant association of father’s occupation with the nutritional status of children can be explained by the lack of proper nutritional knowledge and confounding effect of education of parents. Like father’s occupation, religion does not play any significant role in explaining the nutritional status of the under-five children in Bangladesh.

In our study factors namely source of drinking water and type of toilet also show significant association with child malnutrition. Similar results are reported by Pongou et al [3]. These are plausible because access to safe drinking water and hygienic toilet are the pre-conditions for maintaining good hygiene and nutrition among children. The incidence of various water-borne illnesses can be reduced with the improved supply of drinking water [15]. Therefore increasing access to the safe drinking water and hygienic sanitation are important to improve the nutritional status among under-five children.

This study has several strengths. The use of nationally representative data is one of the important strengths. Our findings could be reliable because of the large sample. Application of the GPR model as an alternative of other methods is another strength. Inclusion of right predictors into the model based on previous studies also increases the strength of the study. However, this study is not free from limitation. Firstly, all inherent limitations associated with the cross-sectional data are also true here. Another limitation of the study is that the model does not include regional and cultural variables, which are also reported as significant predictors [13]. Sex of the children is also not included in the model. Finally some socioeconomic variables are strongly correlated with each other (e.g. wealth index and education), which may produce biased estimates because of multicollinearity [13]. Exclusion of important predictors due to non-availability may also alter our findings.

Our study demonstrates that the GPR model is an appropriate model to identify predictors affecting the nutritional status of children in Bangladesh. Father’s and mother’s education, wealth index, source of drinking water of the household, toilet facility, and total number of children ever born to a woman are significantly associated with child malnutrition in Bangladesh. Various strategies are reported by many studies [4–6, 11, 13, 44]. Increasing educational facilities for mothers and fathers can improve the child nutrition. Facilitating access to safe drinking water and sanitation for poor families is also necessary to improve the child nutrition. Since higher fertility (i.e. number of children ever born to a woman) has a negative impact on child nutrition, government should implement policies to limit family size by increasing birth space [44]. Comprehensive and concerted nutritional interventions such as exclusive breastfeeding, complementary feeding, supplementation of micronutrients to children and mother, hygiene interventions, and management of severe malnutrition are also needed to improve child nutrition [4, 8, 11]. Other strategies such as public transportation to carry food and relief programmes for the disadvantaged groups are important to reduce child malnutrition [13]. Addressing inequity and general deprivation and implementation of other health programmes are also necessary to reduce malnutrition among children [4]. We should keep in mind that adequate nutrition of children is a prerequisite to build a healthy and productive nation [11]. In addition, to achieve the millennium development goal of halving child undernutrition by 2015, Bangladesh needs to scale up target-oriented programmes such as poverty-reduction income generating interventions and improvement of public food transports for the poor population and disadvantaged regions [13].