Application of ordinal logistic regression analysis in determining risk factors of child malnutrition in Bangladesh
 Sumonkanti Das^{1}Email author and
 Rajwanur M Rahman^{2}
DOI: 10.1186/1475289110124
© Das and Rahman; licensee BioMed Central Ltd. 2011
Received: 23 January 2011
Accepted: 14 November 2011
Published: 14 November 2011
Abstract
Background
The study attempts to develop an ordinal logistic regression (OLR) model to identify the determinants of child malnutrition instead of developing traditional binary logistic regression (BLR) model using the data of Bangladesh Demographic and Health Survey 2004.
Methods
Based on weightforage anthropometric index (Zscore) child nutrition status is categorized into three groupsseverely undernourished (< 3.0), moderately undernourished (3.0 to 2.01) and nourished (≥2.0). Since nutrition status is ordinal, an OLR modelproportional odds model (POM) can be developed instead of two separate BLR models to find predictors of both malnutrition and severe malnutrition if the proportional odds assumption satisfies. The assumption is satisfied with low pvalue (0.144) due to violation of the assumption for one covariate. So partial proportional odds model (PPOM) and two BLR models have also been developed to check the applicability of the OLR model. Graphical test has also been adopted for checking the proportional odds assumption.
Results
All the models determine that age of child, birth interval, mothers' education, maternal nutrition, household wealth status, child feeding index, and incidence of fever, ARI & diarrhoea were the significant predictors of child malnutrition; however, results of PPOM were more precise than those of other models.
Conclusion
These findings clearly justify that OLR models (POM and PPOM) are appropriate to find predictors of malnutrition instead of BLR models.
Keywords
Ordinal logistic regression model Proportional odds model Partial proportional odds model Binary logistic regression model Anthropometric index Child malnutritionBackground
Malnutrition is one of the most important causes for improper physical and mental development of children. Child malnutrition still remains a public health problem in developing countries like Bangladesh [1, 2]. It is an underlying cause of child morbidity and mortality. Twothirds of childhood deaths occurred due to malnutrition in Bangladesh [3]. From Bangladesh Demographic and Health Survey (BDHS) 2007, it is investigated that 43% children are stunted, and 41% are underweight in Bangladesh [4]. According to WHO, these levels of stunting and underweight are above the threshold of "very high" prevalence [5]. The level of wasting (17%) also shows that children in Bangladesh were in "serious severity" [4, 5]. Using BDHS 2004 data, a study observed that nearly three fifths children were malnourishedeither stunted, wasted or underweight [6]. The identification of factors for child malnutrition is still the interest of many researchers. Various methods are applied to uncover the factors of child malnutrition. Among them logistic regression analysis has got most preference in previous studies [7–10]. In most of the studies, the response variable was considered as binary (nourished and undernourished); consequently the binary logistic regression model was applied in all the cases. However, the nutrition status of a child is usually classified as nourished, moderately malnourished and severely malnourished. When the researchers are interested to find the determinants of malnutrition and severe malnutrition, two separate binary logistic regression (BLR) models are required to develop by grouping the response variable into two categories [7]. This task is tedious and cumbersome due to estimation and interpretation of more parameters. However, the researcher may consider the response variable as ordinal and may apply ordinal logistic regression model for the same purpose. A few studies have been done using ordinal logistic regression model (OLR) to identify the predictors of child undernutrition [11]. In many epidemiological and medical studies, OLR model is frequently used when the response variable is ordinal in nature [12–17]. The study has made an effort to identify the predictors of child malnutrition as well as severe malnutrition for under five Bangladeshi children by developing an ordinal logistic regression model.
Ordinal Logistic Regression Model
There are several occasions when the outcome variable is polychotomous. Such outcome variable can be classified into two categoriesmultinomial and ordinal. While the dependent variable is classified according to their order of magnitude, one cannot use the multinomial logistic regression model. A number of logistic regression models have been developed for analyzing ordinal response variables [12, 18–24]. Moreover, when there is a need to take several factors into consideration, special multivariate analysis for ordinal data is the natural alternative. There are various approaches, such as the use of mixed models or another class of models, probit for example, but the ordinal logistic regression models have been widely used in most of the previous research works [18, 19, 25–33]. There are several ordinal logistic regression models such as proportional odds model (POM), two versions of the partial proportional odds modelwithout restrictions (PPOMUR) and with restrictions (PPOMR), continuous ratio model (CRM), and stereotype model (SM). The most frequently used ordinal logistic regression model in practice is the constrained cumulative logit model called the proportional odds model [18, 33–35].
The POM is the most widely used in epidemiological and biomedical applications but POM leads to strong assumptions that may lead to incorrect interpretations if the assumptions are violated [28]. If the data fail to satisfy the proportional odds assumption, a valid solution is fitting a partial proportional odds model [36]. Another simple and valid approach to analyze the data is to dichotomize the ordinal response variable by means of several cutoff points and use separate binary logistic regression models for each dichotomous response variable [37]. However, Gameroff suggested that the second procedure should be avoided if possible because of the loss in statistical power and the reduced generality of the analytical solution [17].
Methods
Data and Variables
The study has utilized the nationwide data of BDHS 2004 where completed and plausible anthropometric data were available for 6005 (weighted) children [38]. Weightforage anthropometric index is an excellent overall indicator of a population's nutritional health status. Moreover, weightforage is a composite index of weightforheight and heightforage [4]. So the study considered only weightforage anthropometric index instead of weightforheight and heightforage to measure the children nutrition status. Child nutrition status was categorized into three groupsseverely undernourished (< 3.0 Zscore), moderately undernourished (3.0 to 2.01 Zscore) and nourished (≥2.0 ZScore). Thus nutrition status is an ordinal response variable grouped from a continuous variable.
Functional form of BLR, POM and restricted and unrestricted PPOM
Model  Functional form  Indication for use 

Binary Logistic Model (BLM)  ${\lambda}_{}\left(\underrightarrow{x}\right)=ln\left\{\frac{Pr\left(Y=1\underrightarrow{x}\right)}{Pr\left(Y=0\underrightarrow{x}\right)}\right\}={\alpha}_{}+\left({\beta}_{1}{x}_{1}+{\beta}_{2}{x}_{2}+...+{\beta}_{p}{x}_{p}\right)$  Response variable with two categories (Y = 0,1) 
Proportional Odds Model (POM)  $\begin{array}{l}{\lambda}_{j}\left(\underrightarrow{x}\right)=ln\left\{\frac{Pr\left(Y=1\underrightarrow{x}\right)+...+Pr\left(Y=j\underrightarrow{x}\right)}{Pr\left(Y=j+1\underrightarrow{x}\right)+...+Pr\left(Y=k\underrightarrow{x}\right)}\right\}=ln\left\{\frac{\sum _{1}^{j}Pr\left(Y=j\underrightarrow{x}\right)}{\sum _{j+1}^{k}Pr\left(Y=j\underrightarrow{x}\right)}\right\}\\ {\lambda}_{j}\left(\underrightarrow{x}\right)={\alpha}_{j}+\left({\beta}_{1}{x}_{1}+{\beta}_{2}{x}_{2}+...+{\beta}_{p}{x}_{p}\right),\mathsf{\text{j}}=\mathsf{\text{1,2,}}...\mathsf{\text{,k1}}\end{array}$  Originally continuous response variable, subsequently grouped, and valid proportional odds assumption 
Unrestricted Partial Proportional Odds Model (PPOMUR)*  $\begin{array}{l}{\lambda}_{j}\left(\underrightarrow{x}\right)=ln\left\{\frac{Pr\left(Y=1\underrightarrow{x}\right)+...+Pr\left(Y=j\underrightarrow{x}\right)}{Pr\left(Y=j+1\underrightarrow{x}\right)+...+Pr\left(Y=k\underrightarrow{x}\right)}\right\}=ln\left\{\frac{\sum _{1}^{j}Pr\left(Y=j\underrightarrow{x}\right)}{\sum _{j+1}^{k}Pr\left(Y=j\underrightarrow{x}\right)}\right\}\\ {\lambda}_{j}\left(\underrightarrow{x}\right)={\alpha}_{j}+\left[\left({\beta}_{1}+{\gamma}_{j1}\right){x}_{1}+...+\left({\beta}_{q}+{\gamma}_{jq}\right){x}_{q}+\left({\beta}_{q+1}\right){x}_{q+1}...+{\beta}_{p}{x}_{p}\right],\mathsf{\text{j}}=\mathsf{\text{1,2,}}...\mathsf{\text{,k1}}\end{array}$  Proportional odds assumption not valid 
Restricted Partial Proportional Odds Model (PPOMR)  $\begin{array}{l}{\lambda}_{j}\left(\underrightarrow{x}\right)=ln\left\{\frac{Pr\left(Y=1\underrightarrow{x}\right)+...+Pr\left(Y=j\underrightarrow{x}\right)}{Pr\left(Y=j+1\underrightarrow{x}\right)+...+Pr\left(Y=k\underrightarrow{x}\right)}\right\}=ln\left\{\frac{\sum _{1}^{j}Pr\left(Y=j\underrightarrow{x}\right)}{\sum _{j+1}^{k}Pr\left(Y=j\underrightarrow{x}\right)}\right\}\\ {\lambda}_{j}\left(\underrightarrow{x}\right)={\alpha}_{j}+\left[{\tau}_{j}\left\{\left({\beta}_{1}+{\gamma}_{j1}\right){x}_{1}+...+\left({\beta}_{q}+{\gamma}_{jq}\right){x}_{q}\right\}+\left({\beta}_{q+1}\right){x}_{q+1}...+{\beta}_{p}{x}_{p}\right],\mathsf{\text{j}}=\mathsf{\text{1,2,}}...\mathsf{\text{,k1}}\end{array}$  Proportional odds assumption not valid, and linear relationship for odds ratio (OR) between a covariable and the response variable 
The authority of DHS maintains all kinds of ethical standards and procedures for the survey and also takes informed consent from the survey respondents before the data collection. In addition, we have obtained approval from the DHS to use the data through the website of DHS. So no ethical approval is needed for the study from any other institutions.
Model Fitting
Since the response variable "nutrition status" is ordinal in nature (grouped from continuous variableweightforage anthropometric index), at first POM was formed without a careful assessment of the model adequacy. The chisquared score test for the proportional odds assumption [18, 36] was employed to see whether the main model assumption was violated or not. As the score test is often anticonservative (i.e., the resulting Pvalues are far too small) [13, 24, 36], we use other techniques to investigate the proportional odds assumption. We calculated single score tests for each covariate for checking whether proportional odds assumption is violated [24]. Graphical method has also been employed for checking the parallel slope assumptions for all covariables. In addition, separate binary logistic regression analyses have been conducted as a basis for more careful analysis [26]. We dichotomized the response variable taking account of the ordering by using cumulative probabilities. The response variable is dichotomized as "at least moderate undernutrition" with two categories '0' = no undernutrition & '1' = at least moderate undernutrition and "at least severe undernutrition" with two categories '0' = no undernutrition or moderate undernutrition and '1' = at least severe undernutrition. The overall goodnessoffit of the separate BLR models was assessed by "Hosmer and Lemeshow test" [33, 44, 45].
Though POM is suitable for analyzing ordinal variables arising from a continuous variable, the proportional odds assumption is satisfied seldom in practice. When this assumption is violated, a legal alternative is to develop a PPOM which allows some covariables with proportional odds assumption to be modeled, but for the covariables failed to perform the proportional odds assumption, it is augmented by a coefficient (γ), which is the effect linked with each jth cumulative logit, adjusted by the other covariables [33]. Thus, PPOM releases the constraint of having a common parameter across the response logits for all the predictors considered in the model [17]. Since both PPOM and separate binary logistic regression approaches are based on cumulative logit, the PPOM is directly comparable with separate BLR models [37]. In the same way, the formulation of the logit functions in POM and PPOM are identical (i.e. nourish vs. moderately & severely undernourish; nourish and moderately undernourish vs. severely undernourish), so overall fit of these two models are comparable [28]. So the study compared the results of the separate BLR models with that of the PPOM, and also compared POM with PPOM. The study fitted unrestricted PPOM model. STATA procedure OLOGIT and SPSS procedure PLUM with TPARALLEL option for POM, SPSS procedure LOGISTIC REGRESSION for separate BLR models [46], STATA procedure GOLOGIT2 with AUTOFIT option for PPOM [47] were employed in the study. For graphical tests of proportional odds assumption, PROC LOGISTIC procedure of SAS is used to obtain the estimated logits for at least moderate undernutrition (logit {P[Y≤1]/P[Y = 2]}) and for at least severe undernutrition (logit {P[Y = 0]/P[Y≥1]}). To see the parallel regression assumptions, estimated logits are plotted against all categories of each explanatory variable. SPSS 17.0, STATA 11.1, and SAS 9.2 are utilized for the complex statistical analysis.
Results
Children's nutrition status according to selected independent variables
Covariables  Nutrition Status according to WeightforAge Zscore (WAZ)  Pearson Chisquare (pvalue)  

Severe Malnourish (WAZ < 3.00)  Moderate Malnourish (3.00≤WAZ≤2.01)  Nourish (WAZ≥2.00)  Total  
Children age (in months)  
011  5.2  14.4  80.5  1140  455.986 (0.000) 
1223  17.8  41.0  41.2  1170  
24^{+}  13.6  38.6  47.7  3695  
Birth interval (months)  
48+ months  10.6  31.0  58.4  1787  36.924 (0.000) 
2447 months  14.3  36.4  49.3  2586  
< 24 months  13.0  35.2  51.8  1523  
Mother's education  
Higher  3.4  17.4  79.1  422  204.092 (0.000) 
Secondary  10.4  33.1  56.5  1987  
Primary  12.6  35.9  51.5  1292  
No education  17.1  38.1  44.8  2217  
Household wealth status  
Richest  5.9  24.3  69.7  987  251.337 (0.000) 
Richer  10.5  32.1  57.4  1091  
Middle  12.0  32.5  55.6  1179  
Poor  14.4  38.5  47.1  1237  
Poorest  18.4  41.2  40.3  1512  
Child feeding status  
High (1012)  8.5  26.0  65.5  1148  99.780 (0.000) 
Medium (79)  13.2  36.8  50.0  3367  
Low (06)  15.5  35.7  48.7  1381  
Mothers' antenatalpostnatal care status  
Sufficient (1618)  3.6  17.5  78.8  290  136.482 (0.000) 
Merely sufficient (1115)  10.1  27.1  62.9  796  
Less sufficient (610)  11.3  33.0  55.8  1396  
Least sufficient (15)  13.8  36.1  50.1  1102  
No care (0)  16.5  36.8  46.7  1137  
Mother's BMI  
Normal (≥18.5)  9.5  31.0  59.4  3664  198.723 (0.000) 
Thinness (< 18.5)  18.3  40.1  41.7  2232  
Incidence of ARI in the last two weeks  
No  12.2  33.8  54.0  4654  15.833 (0.000) 
Yes  15.1  36.9  48.0  1242  
Incidence of fever in the last two weeks  
No  11.2  33.3  55.5  3499  34.490 (0.000) 
Yes  15.2  36.1  48.6  2397  
Incidence of diarrhoea in the last two weeks  
No  12.5  34.2  53.3  5445  13.350 (0.001) 
Yes  16.9  38.0  45.1  451  
Total  12.8  34.5  52.7  6005  N/A 
To identify the risk factors of child malnutrition, the study fitted POM, separate BLR models, and PPOM. At first competence of the models are described and then the results of the models are interpreted.
Proportional Odds Model
Results of the multiple POM using nutrition status as response three ordered categories ^{♣}
Covariable  Regression coefficient  Standard error  pvalue  Odds ration  95% CI of OR  Single score test (pvalue) 

Intercept _{ 1 }  3.888  .209  .000       
Intercept _{ 2 }  5.888  .218  .000      
Children age (in months) [011 months as Reference]  
1223  1.877  .099  .000  6.534  5.3817.934  0.005 
24^{+}  1.638  .091  .000  5.147  4.3026.157  
Birth interval (months) [48+ months as reference]  
2447  .403  .071  .000  1.496  1.3011.719  0.928 
< 24  .465  .083  .000  1.591  1.3531.871  
Mother's education [Higher education as reference]  
Secondary  .820  .156  .000  2.270  1.6723.080  0.549 
Primary  .760  .167  .000  2.139  1.5422.967  
No education  .982  .166  .000  2.670  1.9293.694  
Household wealth status [Richest as reference]  
Richer  .273  .112  .015  1.314  1.0541.638  0.799 
Middle  .359  .115  .002  1.432  1.1441.792  
Poorer  .534  .116  .000  1.705  1.3592.139  
Poorest  .695  .118  .000  2.005  1.5902.527  
Child feeding status [High (1012) as reference]  
Medium (79)  .045  .081  .578  1.046  0.8931.226  0.180 
Low (06)  .248  .095  .009  1.281  1.0631.544  
Mothers' antenatalpostnatal care status [Sufficient (1618) as reference]  
Merely sufficient (1115)  .338  .177  .057  1.402  0.9901.984  0.678 
Less sufficient (610)  .318  .177  .073  1.375  0.9711.946  
Least sufficient (15)  .453  .182  .013  1.573  1.1012.246  
No care (0)  .472  .185  .011  1.603  1.1162.301  
Mother's BMI [Normal (≥18.5) as reference]  
Thinness (< 18.5)  .553  .063  .000  1.738  1.5371.965  0.665 
Incidence of ARI in the last two weeks [No as reference]  
Yes  .195  .078  .012  1.215  1.0441.415  0.678 
Incidence of fever in the last two weeks [No as reference]  
Yes  .234  .066  .000  1.263  1.1111.436  0.292 
Incidence of diarrhoea in the last two weeks [No as reference]  
Yes  .245  .107  .021  1.278  1.0371.575  0.876 
Score test for the proportional odds assumption: Chisquare = 27.83, df = 21, pvalue = 0.144  
Goodnessoffit test of overall model (Likelihood Ratio): Chisquare = 926.52, df = 21, pvalue = 0.000, Pseudo R^{2} = 0.1029 
Separate Binary Logistic Regressions
Results of two separate multiple binary logistic regression models using child nutrition status as binary response ^{♣}
Comparisons  

Covariable  Nourish vs. (moderately & severely malnourished) ^{1}  Nourish & moderately malnourished vs. (severely malnourished) ^{2}  
β_{1}  OR_{1}  CI (Pvalue)  β_{2}  OR_{2}  CI (Pvalue)  
Coefficient  3.976    .000  
Children age (in months) [011 months as Reference]  
1223  1.943  6.977  5.698.55 (.000)  1.445  4.242  3.085.84 (.000) 
24^{+}  1.683  5.381  4.476.48 (.000)  1.207  3.343  2.464.55 (.000) 
Birth interval (months) [48+ months as reference]  
2447  .433  1.542  1.331.79 (.000)  .284  1.329  1.071.65 (.009) 
< 24  .514  1.671  1.411.99 (.000)  .316  1.371  1.071.76 (.014) 
Mother's education [Higher education as reference]  
Secondary  .827  2.286  1.683.12 (.000)  .823  2.276  1.224.25 (.010) 
Primary  .776  2.172  1.553.04 (.000)  .797  2.219  1.164.24 (.016) 
No education  .954  2.596  1.863.62 (.000)  1.141  3.131  1.655.94 (.000) 
Household wealth status [Richest as reference]  
Richer  .260  1.297  1.031.63 (.025)  .338  1.402  0.952.07 (.088) 
Middle  .352  1.421  1.131.80 (.003)  .438  1.549  1.052.28 (.027) 
Poorer  .594  1.812  1.432.30 (.000)  .426  1.531  1.042.26 (.032) 
Poorest  .698  2.011  1.582.57 (.000)  .687  1.988  1.352.93 (.000) 
Child feeding status [High (1012) as reference]  
Medium (79)  .072  1.075  0.911.27 (.398)  .005  1.005  0.781.30 (.970) 
Low (06)  .240  1.271  1.041.55 (.019)  .305  1.357  1.021.81 (.038) 
Mothers' antenatalpostnatal care status [Sufficient (1618) as reference]  
Merely sufficient (1115)  .305  1.357  0.951.94 (.092)  .510  1.665  0.833.34 (.150) 
Less sufficient (610)  .330  1.390  0.981.98 (.069)  .339  1.403  0.702.81 (.339) 
Least sufficient (15)  .462  1.587  1.102.29 (.013)  .459  1.582  0.783.19 (.200) 
No care (0)  .455  1.576  1.092.29 (.017)  .510  1.665  0.823.37 (.157) 
Mother's BMI [Normal (≥18.5) as reference]  
Thinness (< 18.5)  .581  1.789  1.572.04 (.000)  .541  1.717  1.432.06 (.000) 
Incidence of ARI in the last two weeks [No as reference]  
Yes  .242  1.274  1.081.50 (.004)  .117  1.124  0.901.41 (.308) 
Incidence of fever in the last two weeks [No as reference]  
Yes  .211  1.234  1.081.42 (.003)  .286  1.331  1.101.62 (.004) 
Incidence of diarrhoea in the last two weeks [No as reference]  
Yes  .271  1.311  1.041.65 (.023)  .150  1.161  0.861.57 (.327) 
HosmerLemeshow goodnessoffit test  pvalue = 0.589  pvalue = 0.610 
Partial Proportional Odds Model
Results of multiple PPOM using child nutrition status as response with three ordered categories ^{♣}
Comparisons  

Covariable  Nourish vs. (moderately & severely malnourished)  Nourish & moderately malnourished vs. (severely malnourished)  
β_{1}  OR_{1}  Pvalue  β_{2}  OR_{2}  Pvalue  
Coefficient  3.9269    0.000  5.576    0.000 
Children age (in months) [011 months as Reference]  
1223  1.9264  6.8645  0.000  1.4328  4.1904  0.000 
2435  1.6772  5.3508  0.000  1.1986  3.3156  0.000 
Birth interval (months) [48+ months as reference]  
2447  0.4035  1.4971  0.000  0.4035  1.4971  0.000 
< 24  0.4683  1.5973  0.000  0.4683  1.5973  0.000 
Mother's education [Higher education as reference]  
Secondary  0.8323  2.2987  0.000  0.8323  2.2987  0.000 
Primary  0.7759  2.1726  0.000  0.7759  2.1726  0.000 
No education  0. 9467  2.5772  0.000  0. 9467  2.5772  0.000 
Household wealth status [Richest as reference]  
Richer  0.2786  1.3213  0.014  0.2786  1.3213  0.014 
Middle  0.368  1.4449  0.001  0.368  1.4449  0.001 
Poorer  0.5437  1.7224  0.000  0.5437  1.7224  0.000 
Poorest  0.7068  2.0276  0.000  0.7068  2.0276  0.000 
Child feeding status [High as reference]  
Medium  0.0407  1.0415  0.617  0.0407  1.0415  0.617 
Low  0.2424  1.2744  0.012  0.2424  1.2744  0.012 
Mothers' antenatalpostnatal care status [Sufficient as reference]  
Merely sufficient  0.3418  1.4075  0.055  0.3418  1.4075  0.055 
Less sufficient  0.3238  1.3823  0.068  0.3238  1.3823  0.068 
Least sufficient  0.4588  1.5822  0.012  0.4588  1.5822  0.012 
No care  0.4796  1.6155  0.009  0.4796  1.6155  0.009 
Mother's BMI [Normal (≥18.5) as reference]  
Thinness (< 18.5)  0.5557  1.7432  0.000  0.5557  1.7432  0.000 
Incidence of ARI in the last two weeks [No as reference]  
Yes  0.1961  1.2167  0.011  0.1961  1.2167  0.011 
Incidence of fever in the last two weeks [No as reference]  
Yes  0.2363  1.2665  0.000  0.243  1.2751  0.022 
Incidence of diarrhoea in the last two weeks [No as reference]  
Yes  0.2430  1.2751  0.022  0.2363  1.2665  0.000 
Score test for the proportional odds assumption: Chisquare = 12.95, df = 19, pvalue = 0.7943  
Goodnessoffit test of overall model (Likelihood Ratio): Chisquare = 941.55, df = 23, pvalue = 0.000, Pseudo R^{2} = 0.1046 
Graphical Test of Proportional Odds Assumption
Determinants of Child Undernutrition
In POM and PPOM, all the considered variables are found as significant predictors of child malnutrition as in previous studies. The covariates were also found significant in both the separate BLR models except antenatalpostnatal care status, incidence of diarrhoea and ARI in the 2^{nd} BLR model with the response variable "at least severe undernutrition". These results support the use of POM and PPOM instead of BLR models to determine the predictors of child undernutrition as well as severe undernutrition.
The results of POM reveal that the risk of having worse nutrition status were 6.53 and 5.15 times higher among the children belonging to the age group 1223 and 24+ months respectively, when compared with the infants (Table 3). Since this variable violated the proportional odds assumption, this interpretation may be invalid. However, from separate BLR models and PPOM it is clear that the odds ratios for the children aged 1223 months and 24^{+} months compared to infants were about 6.9 and 5.4 respectively when no undernutrition state is compared with moderate and severe undernutrition states (Table 4 &5). When no undernutrition and moderate undernutrition states are compared with severe undernutrition state, the odds ratios were found about 4.2 and 3.3 respectively for children belonging to age group 1223 and 24^{+} months compared to infants (Table 4 &5). Since all other covariates did not violate the proportional odds assumption and PPOM performed better than POM as well as separate BLR models, the results for other covariates are described from Table 5.
Children having birth interval < 24 and 2447 months had 1.6 and 1.5 times greater risk of having worse nutrition status compared with the children having 48^{+} months birth interval (Table 5). The risk of having worse nutrition status was found highest for the children having mothers with no education (about 3.0 times) when compared with highly educated mothers' children. Compared to the children of the richest households, the chances of having worse nutrition status was found to increase with decrease of household wealth condition (2.03 for the children of poorest household and 1.32 for those of richer household). The risk of having poor nutrition condition was found significantly higher for the children with poor feeding practices compared to those having better feeding practices. Mothers who received no antenatalpostnatal care had 1.62 times greater risk of having malnourished children compared to those received sufficient care. The risk reduced with the increase of mothers' antenatalpostnatal care. Children belonging to acutely malnourished mothers, compared to those of nourished mothers, were 1.74 times (95% CI: 1.541.97) as likely to be malnourished moderately or severely. Table 5 also shows that children of acutely malnourished mothers had 1.74 times greater risk of being undernourished compared to those of nourished mothers. Children experienced with ARI, fever, and diarrhoea within last two weeks of the survey had 1.22, 1.27 and 1.28 times higher risk of being undernourished respectively when comparison is made with the children having no such problems (Table 5).
Discussion
At first sight the POM seems to be an appropriate model for analyzing the considered data since the pvalue of chisquared score test for overall model is insignificant at 5% level of significance indicating proportional odds assumption is not violated. All of the considered variables were found significant in the POM. However, the pvalue of the score test for overall model was very much small which compels to conduct single score test for each covariate. These tests show that only 'age of children' violates the vital assumption of POM which may lead invalid results. Separate BLR models also indicate the coefficients and the odds ratios for the each age categories varied in the models. Graphical test of proportional odds assumption reveals the same result. In case of all other variables, coefficients and odds ratios are not identical but almost closer. In PPOM, coefficients and odds ratios for the variable 'age of children' are almost same with the result of BLR models. However, the coefficients and odds ratios for other covariates in PPOM are slightly different compared to separate binary logistic regression models, but almost identical with those of POM. Moreover, all the variables are significant in PPOM but in separate binary logistic regression models few are insignificant.
Conclusion
Despite some differences in the results of the fitted models, the results of POM and PPOM are reasonably comparable with those of BLR models. The POM and PPOM have proved adequate for data analysis of child nutritional status, due to the nature of the response variable (grouped continuous variable), in addition, the parsimony and ease of interpretation. Furthermore, PPOM is fitted better for the data than POM. From the results of POM and PPOM it is clear that all the considered variables in the study are significant predictors of child malnutrition as previous research works. Moreover, these findings clearly justify that OLR models (POM & PPOM) are appropriate to find predictors of malnutrition as well as sever undernutrition instead of using two separate binary logistic regression models.
Author's information
SD, Faculty, Department of Statistics, Shahjalal University of Science & Technology (SUST), Sylhet, Bangladesh, obtained his graduation and M.S. in Statistics from the same university. His research and publications focus on the following areas: Time Series Analysis, Stochastic Model Building, BioStatistics, Child Health & Nutrition, and Analytic Hierarchy Process. He carried out his M.S. Thesis on Child Health and Nutrition.
RMR is working as Statistical Programmer in Shafi Consultancy Bangladesh, Sylhet, Bangladesh. He completed his graduation and M.S. in Statistics from Department of Statistics, SUST, Sylhet, Bangladesh. His Interested areas of research are Data Management, Biostatistics, Time Series Modeling, and Child Health & Nutrition.
List of Abbreviations
 ARI :

Acute Respiratory Infections
 BDHS :

Bangladesh Demographic and Health Survey
 BLR :

Binary Logistic Regression
 BMI :

Body Mass Index
 CI :

Confidence Interval
 CRM :

Continuous Ratio Model
 NIPORT :

National Institute of Population Research and Training
 OLR :

Ordinal Logistic Regression
 OR :

Odds Ratio
 POM :

Proportional Odds Model
 PPOM :

Partial Proportional Odds Model
 PPOMR :

Partial Proportional Odds ModelWith Restrictions
 PPOMUR :

Partial Proportional Odds ModelWithout Restrictions
 SM :

Stereotype Model
 WAZ :

WeightforAge Zscore
 WHO :

World Health Organization.
Declarations
Acknowledgements
The authors wish to thank DHS [ICF Macro] and National Institute of Population Research and Training (NIPORT) [Bangladesh] providing the permission of using the nationwide data of BDHS 2004. Special thanks to the reviewers for their valuable comments and suggestions that enrich the paper.
Authors’ Affiliations
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