Ordinal logistic regression models
Test of parallel regression assumption
All the significant explanatory variables (at 5% significance level) from uni-variable generalized ordered logistic regression models were included and assessed their significance in the proposed ordinal logistic regression models. The Brant test of parallel regression assumption from POM yielded a Chi-Square value of 90.27 with p-value=0.000, indicating that the proportional odds assumptions for the full-model was not upheld. This suggested that the effect of one or more of the explanatory variables was likely to differ across separate binary models fit to the cumulative cut points. This indicates that the null hypothesis that states the model parameters are equal across categories (i.e. parallel regression assumption) can be rejected. When the parallel regression assumptions are violated, models based on the parallel assumption for all the independent variables cannot be accurately applied to the whole population (the parameters of the model are said to be biased). Consequently, proportional odds were excluded from further analysis. Generalized ordered logit model and partial proportional odds models were fitted to the data and comparison of model was done.
Goodness of fit and model selection
Table 2: Log-likelihood and likelihood ratio estimates
Model
|
Obs
|
LL(null)
|
LL (model)
|
DF
|
LR chi2
|
Prob>chi2
|
GOLM
|
7910
|
-8049.262
|
-7269.128
|
84
|
1560.27
|
0.000
|
PPOM
|
7910
|
-8049.262
|
-7283.187
|
54
|
1532.15
|
0.000
|
From Table 2 one can conclude that both of the full model improves significantly over their null model (model only with intercept term) since there are evidences against the null hypothesis that all the coefficients of the predictors are zero (Prob>chi-square =0.000).
The model which represents the best fit according to AIC and BIC is PPOM as it has the smallest AIC (GOLM: 14710.26 and PPOM: 14678.37) and BIC (GOLM:15310.18 and PPOM: 15069.02). Partial proportional odds model is also more parsimonious than GOLM as it has fewer parameters. Thus, PPOM was used to identify significant determinants of undernutrition and parameter estimates of the PPOM are presented and interpreted for the significant predictors.
Results of Partial Proportional Odds Model
There are two result panels in Table 3 and 4. The first panel (Table 3) contrasts the moderately undernourished and severely undernourished categories with the nourished category. That is, the signs of the coefficients in the first panel imply how likely a child is nourished as opposed to the remaining two categories of undernutrition. Similarly, the second panel (Table 4) contrasts the severely undernourished category with nourished and moderately undernourished categories. Hence, positive coefficients indicate that higher category values on the explanatory variable make it more likely that the respondent will be in a higher category of Y than the current one, whereas negative coefficients indicate that higher category values on the explanatory variable increase the likelihood of being in the current or a lower category.
From the partial proportional odds model the categories Afar, Oromia, Somali, SNNPR, Gambela, Harari, Dire Dawa of the region; the category richest of wealth index; the category primary of husband’s education; the categories 2-3 and 4-5 of birth order and the variable sex were found to violate the parallel lines assumption. The partial proportional odds model therefore allows the coefficients of these variables to vary across the two equations. From PPOM results the predictors region, mother’s education, source of drinking water, number of children under five years, wealth index, anemia level, multiple birth, sex of a child, age of a child, fever, mother’s age at birth, body mass index of mother and husband’s education level were found to be significantly related to undernutrition.
Table 3: Maximum likelihood estimates of Partial proportional odds model
Predictors
|
Moderate and severe undernourished versus nourished
|
Coefficient
|
Std.Error
|
Z
|
P>|z|
|
Odds ratio
|
95% CI for OR
|
Region
|
Afar
|
-0.060
|
0.114
|
-0.53
|
0.597
|
0.942
|
(0.754, 1.177)
|
Amhara
|
0.328
|
0.101
|
3.25
|
0.001
|
1.389
|
(1.140, 1.693)
|
Oromia
|
-0.372
|
0.096
|
-3.86
|
0.000
|
0.689
|
(0.571, 0.833)
|
Somali
|
-0.670
|
0.108
|
-6.21
|
0.000
|
0.512
|
(0.414, 0.632)
|
Benishangul
|
0.117
|
0.112
|
1.05
|
0.295
|
1.124
|
(0.903, 1.401)
|
SNNPR
|
-0.355
|
0.100
|
-3.54
|
0.000
|
0.701
|
(0.576, 0.853)
|
Gambela
|
-0.715
|
0.127
|
-5.65
|
0.000
|
0.489
|
(0.382, 0.627)
|
Harari
|
-0.213
|
0.127
|
-1.68
|
0.092
|
0.808
|
(0.630, 1.036)
|
Addis Ababa
|
-0.812
|
0.173
|
-4.69
|
0.000
|
0.444
|
(0.316, 0.6236)
|
Dire Dawa
|
-0.097
|
0.138
|
-0.70
|
0.485
|
0.908
|
(0.692, 1.191)
|
Residence
|
Rural
|
-0.138
|
0.104
|
-1.32
|
0.186
|
0.871
|
(0.711, 1.069)
|
Mother’s education
|
Primary
|
-0.100
|
0.061
|
-1.62
|
0.105
|
0.905
|
(0.803, 1.021)
|
Secondary & >
|
-0.547
|
0.116
|
-4.72
|
0.000
|
0.579
|
(0.461, 0.726)
|
Drinking water
|
Unimproved
|
0.102
|
0.051
|
2.01
|
0.045
|
1.108
|
(1.002, 1.224)
|
House hold size
|
5-9
|
0.054
|
0.069
|
0.78
|
0.434
|
1.055
|
(0.922, 1.207)
|
10 and more
|
0.208
|
0.123
|
1.70
|
0.089
|
1.232
|
(0.969, 1.567)
|
No children <5 years
|
2
|
-0.144
|
0.076
|
-1.90
|
0.057
|
0.866
|
(0.746, 1.004)
|
3 and more
|
0.150
|
0.055
|
2.72
|
0.007
|
1.162
|
(1.043, 1.294)
|
Wealth index
|
Poorer
|
0.082
|
0.070
|
1.18
|
.238
|
1.085
|
(0.947, 1.244)
|
Middle
|
-0.204
|
0.076
|
-2.67
|
0.008
|
0.815
|
(0.702, 0.947)
|
Richer
|
-0.453
|
0.083
|
-5.44
|
0.000
|
0.636
|
(0.540, 0.748)
|
Richest
|
-0.447
|
0.113
|
-3.96
|
0.000
|
0.639
|
(0.512, 0.798)
|
Anemia
|
Anemic
|
0.197
|
0.050
|
3.95
|
0.000
|
1.218
|
(1.105, 1.3435)
|
Husband’s
Education
|
Primary
|
-0.087
|
0.059
|
-1.47
|
0.141
|
0.916
|
(0.816, 1.0296)
|
Secondary & >
|
-0.171
|
0.086
|
-2.00
|
0.045
|
0.924
|
(0.769, 1.109)
|
Birth order
|
2-3
|
-0.059
|
0.074
|
-0.79
|
0.428
|
0.943
|
(0.816, 1.090)
|
4-5
|
-0.035
|
0.091
|
-0.38
|
0.701
|
0.966
|
(0.809, 1.154)
|
6 and more
|
-0.079
|
0.093
|
-0.85
|
0.395
|
0.924
|
(0.769, 1.109)
|
Multiple birth
|
1st of multiple
|
0.725
|
0.210
|
3.45
|
0.001
|
2.066
|
(1.369, 3.117)
|
2nd of multiple
|
0.796
|
0.222
|
3.58
|
0.000
|
2.216
|
(1.430, 3.427)
|
Sex
|
Female
|
-0.110
|
0.049
|
-2.26
|
0.024
|
0.895
|
(0.814, 0.985)
|
Age of child in month
|
6-11
|
0.583
|
0.123
|
4.74
|
0.000
|
1.792
|
(1.408, 2.281)
|
12-23
|
1.646
|
0.107
|
15.37
|
0.000
|
5.185
|
(4.203, 6.395)
|
24-35
|
2.057
|
0.107
|
19.19
|
0.000
|
7.819
|
(6.338, 9.647)
|
36-47
|
1.980
|
0.107
|
18.44
|
0.000
|
7.244
|
(5.869, 8.941)
|
48-59
|
1.859
|
0.107
|
17.33
|
0.000
|
6.415
|
(5.199, 7.916)
|
Fever
|
Yes
|
0.237
|
0.077
|
3.06
|
0.002
|
1.267
|
(1.089, 1.475)
|
Cough
|
Yes
|
-0.019
|
0.074
|
-0.26
|
0.793
|
0.981
|
(0.849, 1.133)
|
Mother’s age at birth
|
20-34
|
-0.110
|
0.049
|
-2.24
|
0.025
|
0.895
|
(0.813, 0.986)
|
35-49
|
0.296
|
0.562
|
0.53
|
0.599
|
1.344
|
(0.447, 4.046)
|
BMI of mother
|
Normal
|
-0.349
|
0.055
|
-6.36
|
0.000
|
0.705
|
(0.633, 0.785)
|
Overweight
|
-0.939
|
0.106
|
-8.87
|
0.000
|
0.391
|
(0.318, 0.481)
|
Constant
|
-0.805
|
0.189
|
-4.26
|
0.000
|
0.447
|
(0.309, 0.647)
|
Table 4: Maximum likelihood estimates of Partial proportional odds model
Predictors
|
Severely undernourished versus nourished and moderately undernourished
|
Coefficient
|
Std.Error
|
Z
|
P>|z|
|
Odds ratio
|
95% CI for OR
|
Region
|
Afar
|
0.198
|
0.115
|
1.73
|
0.084
|
1.219
|
(0.974, 1.525)
|
Amhara
|
0.328
|
0.101
|
3.25
|
0.001
|
1.389
|
(1.139, 1.693)
|
Oromia
|
-0.152
|
0.102
|
-1.49
|
0.136
|
0.859
|
(0.704, 1.049)
|
Somali
|
-0.312
|
0.113
|
-2.76
|
0.006
|
0.732
|
(0.586, 0.914)
|
Benshangul
|
0.293
|
0.114
|
2.57
|
0.010
|
1.341
|
(1.073, 1.677)
|
SNNPR
|
-0.007
|
0.105
|
-0.07
|
0.946
|
0.993
|
(0.808, 1.220)
|
Gambela
|
-0.439
|
0.141
|
-3.12
|
0.002
|
0.645
|
(0.489, .849)
|
Harari
|
-0.213
|
0.127
|
-1.68
|
0.092
|
0.808
|
(0.630, 1.036)
|
Addis Ababa
|
-0.812
|
0.173
|
-4.69
|
0.000
|
0.444
|
(0.316, 0.624)
|
Dire Dawa
|
0.237
|
0.143
|
1.66
|
0.097
|
1.267
|
(0.958, 1.676)
|
Residence
|
Rural
|
-0.138
|
0.104
|
-1.32
|
0.186
|
0.871
|
(0.711, 1.069)
|
Mother’ education
|
Primary
|
-0.100
|
0.061
|
-1.62
|
0.105
|
0.905
|
(0.803, 1.021)
|
Secondary & >
|
-0.547
|
0.116
|
-4.72
|
0.000
|
0.579
|
(0.461, 0.726)
|
Drinking water
|
Unimproved
|
0.102
|
0.051
|
2.01
|
0.045
|
1.108
|
(1.002, 1.224)
|
House hold size
|
5-9
|
0.054
|
0.069
|
0.78
|
0.434
|
1.055
|
(0.922, 1.207)
|
10 & more
|
0.208
|
0.123
|
1.70
|
0.089
|
1.232
|
(0.969, 1.566)
|
No children
< 5 years
|
2
|
-0.144
|
0.076
|
-1.90
|
0.057
|
0.866
|
(0.746, 1.004)
|
3 and more
|
0.150
|
0.055
|
2.72
|
0.007
|
1.162
|
(1.043, 1.294)
|
Wealth index
|
Poorer
|
0.082
|
0.070
|
1.18
|
0.238
|
1.085
|
(0.947, 1.244)
|
Middle
|
-0.204
|
0.076
|
-2.67
|
0.008
|
0.815
|
(0.702, 0.947)
|
Richer
|
-0.453
|
0.083
|
-5.44
|
0.000
|
0.636
|
(0.540, 0.748)
|
Richest
|
-0.612
|
0.121
|
-5.06
|
0.000
|
0.542
|
(0.428, 0.688)
|
Anemia
|
Anemic
|
0.197
|
0.050
|
3.95
|
0.000
|
1.218
|
(1.105, 1.343)
|
Husband’s education
|
Primary
|
-0.188
|
0.063
|
-3.01
|
0.003
|
0.828
|
(0.732, 0.936)
|
Secondary & >
|
-0.171
|
0.086
|
-2.00
|
0.045
|
0.843
|
(0.712, 0.996)
|
Birth order
|
2-3
|
-0.059
|
0.074
|
-0.79
|
0.428
|
0.943
|
(0.816, 1.090)
|
4-5
|
0.096
|
0.093
|
1.03
|
0.302
|
1.101
|
(0.917, 1.320)
|
6 and more
|
0.050
|
0.095
|
0.53
|
0.598
|
1.052
|
(0.872, 1.268)
|
Multiple birth
|
1st of multiple
|
0.725
|
0.210
|
3.45
|
0.001
|
2.066
|
(1.369, 3.112)
|
2nd of multiple
|
0.796
|
0.222
|
3.58
|
0.000
|
2.216
|
(1.433, 3.427)
|
Sex
|
Female
|
-0.214
|
0.053
|
-4.07
|
0.000
|
0.808
|
(0.729, .895)
|
Age of child in month
|
6-11
|
0.583
|
0.123
|
4.74
|
0.000
|
1.792
|
(1.408, 2.281)
|
12-23
|
1.646
|
0.107
|
15.37
|
0.000
|
5.185
|
(4.203, 6.395)
|
24-35
|
2.057
|
0.107
|
19.19
|
0.000
|
7.819
|
(6.338, 9.647)
|
36-34
|
1.980
|
0.107
|
0.000
|
0.000
|
7.244
|
(5.869, 8.941)
|
48-59
|
1.859
|
0.107
|
17.33
|
0.000
|
6.415
|
(5.199, 7.916)
|
Fever
|
Yes
|
0.237
|
0.077
|
3.06
|
0.002
|
1.267
|
(1.089, 1.475)
|
Cough
|
Yes
|
-0.019
|
0.074
|
-0.26
|
0.793
|
0.981
|
(0.849, 1.133)
|
Mother’s age at birth
|
20-34
|
-0.110
|
0.049
|
-2.24
|
0.025
|
0.895
|
(0.813, 0.986)
|
35-49
|
0.296
|
0.562
|
0.53
|
0.599
|
1.344
|
(0.447, 4.046)
|
BMI of mother
|
Normal
|
-0.349
|
0.055
|
-6.36
|
0.000
|
0.705
|
(0.633, 0.785)
|
Overweight
|
-0.939
|
0.106
|
-8.87
|
0.000
|
0.391
|
(0.318, 0.481)
|
Constant
|
-1.929
|
0.191
|
10.09
|
0.000
|
0.145
|
(0.100, 0.211)
|
Key: the reference category for predictors is: Region (Tigray), Residence (urban), mother’s education (no education), source of drinking water (improved source), house hold size (1-4), number of children <5 years (1), wealth index (poorest), anemia (no), husband’s education (no education), birth order (1), multiple birth (single), sex (male), age of child (0-6), diarrhea (no), cough (no), fever (no), mother’s age at birth (<20), BMI (thin).
Predictors that do not violate the parallel line assumption
The results of PPOM revealed a child who lived in Amhara was 1.4 (OR=1.4; CI: 1.14 - 1.69) times more likely to be in moderate or severe undernutrition status than nourished status as opposed to a child in Tigray, holding all variables constant. Similarly, a child born in Amhara was 1.4 (p-value =0.0001) times more likely to be in severe undernutrition status than moderate or nourished status compared to a child in Tigray. The odds of being worse undernourished was 2.3(OR=0.44; CI: 0.32-0.62) times in Tigray’s children as opposed to children of Addis Ababa, holding all other variables constant.
The fitted model showed the risk of having worse undernutrition status was 1.7= (0.58)-1 (OR=0.58; CI: 0.46 - 0.73) times higher for children born to mother without education compared to children born to mother with secondary or higher education. Children with secondary or higher educated fathers were around 8% (OR=0.92; p-value =0.045) less likely to be in the worst nutrition status, compared to the children with illiterate fathers. The risk of being in worse undernutrition status was decreased by 11% (OR=0.89; p-value =0.025) in a child born to a mother aged 20-34 years as compared to a child born to a mother aged <20 years. A child with mother of BMI<18.5 was 1.4 (OR=0.71; CI: 0.63 - 0 .79) and 2.6 (OR=0.39; CI: 0.31 - 0.65) times more likely to be in worse undernutrition status as opposed to a child who had normal and obese mother respectively, keeping all other variables constant.
The results of this study revealed that the odds of being in worse undernutrition status were 1.2 (OR=1.2; p-value =0.007) times higher for children from families having 3 or more under five years children compared to children from families having one child aged under five years. The risk of being worse undernourished was decreased by 18% (OR=0.82; p-value =0.008) in children from families with middle wealth index and 36% (OR=0.64; p-value=0.000) in children from families with richer wealth index, respectively as opposed to children from households with poorest wealth index. The children born who were first of multiple and second of multiple births were 2.1 (OR=2.1; p-value =0.001) and 2.2 (OR=2.2; CI: 1.43-3.43) times more likely to be in worse undernutrition status respectively as compared to single birth. The children aged 6-11, 12-23, 24-35, 36-47 and 48-59 months were 1.8 (OR=1.8; CI: 1.4-2.3), 5.2 (CI: 4.2-6.4), 7.8 (CI: 6.3-9.6), 7.2 (CI: 5.9-8.9) and 6.4 (CI: 5.2-7.9) times more likely to be in worse undernutrition status respectively as opposed to children aged 0-6 months, holding all other variables constant.
Holding all variables constant, the fitted model indicated the risk of having worse undernutrition status was 1.2 (OR=1.2; CI: 1.1-1.3) times higher among anemic children when compared to the non-anemic children. The risk of being in worse undernutrition status was 1.3 (OR=1.3; p-value=0.002) times higher for children who had fever in the last two weeks before the survey as compared to children who had no fever. The odds of being undernourished was increased by 10% (OR=1.1; p-value=0.045) in children from household who have not consumed water from improved source compared to children from household who have consumed water from improved source.
Predictors that violate the parallel regression assumption
The results of PPOM showed a child from Tigray region was 1.4 (OR=0.69; CI: 0.58-0.83), 2 (OR=0.50; CI: 0.41-0.63), 1.4 (OR=0.70; CI: 0.58-0.85) and 2 (OR=0.49; CI: 0.38-0.63) times more likely to be in moderate or severe undernutrition status than nourished status as opposed to a child from Oromia, Somali, SNNP and Gambella respectively. A child who lived in Tigray was 1.4 (p-value =0.006) and 1.5 (p-value =0.002) times more likely to be in severe undernutrition status than nourished or moderate undernutrition status as compared to a child in Somali and Gambella respectively. A child born in Benshangul was 1.34 (p-value =0.01) times more likely to be in severe undernutrition status than nourished or moderate undernutrition status as opposed to a child in Tigray. The children from families with poorest wealth index were found 1.5 (OR=0.64; CI: 0.51-0.80) times more likely to be in moderate or severe undernutrition status than nourished status when compared to children who had richest wealth index households. The children who had families with poorest wealth index were found 1.8 (OR=0.54; CI: 0.43-0.69) times more likely to be in severe undernutrition status than nourished or moderate undernutrition status as opposed to children from families with richest wealth index, keeping all other variables constant.
The fitted model had showed that male children were 1.1 (p-value =0.024) times more likely to be in moderate or severe undernutrition status than nourished status when compared to female. The risk of male child being in severe undernutrition status was 1.2 (OR=0.81; CI: 0.73-0.89) times higher than being in nourished or moderate status as opposed to female child. The risk of children born to husband without education being in severe undernutrition status were 1.2 (p-value =0.003) times higher than being in nourished or moderate status as compared to children born to husband with secondary or higher education, holding all other variables constant.