Application of ordinal logistic regression analysis in determining risk factors of child malnutrition in Bangladesh

Das, Sumonkanti; Rahman, Rajwanur M

doi:10.1186/1475-2891-10-124

Table 1 Functional form of BLR, POM and restricted and unrestricted PPOM

From: Application of ordinal logistic regression analysis in determining risk factors of child malnutrition in Bangladesh

Model	Functional form	Indication for use
Binary Logistic Model (BLM)	$λ_{} (\underset{\to}{x}) = ln \{\frac{Pr (Y = 1 \| \underset{\to}{x})}{Pr (Y = 0 \| \underset{\to}{x})}\} = α_{} + (β_{1} x_{1} + β_{2} x_{2} + . . . + β_{p} x_{p})$	Response variable with two categories (Y = 0,1)
Proportional Odds Model (POM)	$\begin{gathered} λ_{j} (\underset{\to}{x}) = ln \{\frac{Pr (Y = 1 \| \underset{\to}{x}) + . . . + Pr (Y = j \| \underset{\to}{x})}{Pr (Y = j + 1 \| \underset{\to}{x}) + . . . + Pr (Y = k \| \underset{\to}{x})}\} = ln \{\frac{\sum_{1}^{j} Pr (Y = j \| \underset{\to}{x})}{\sum_{j + 1}^{k} Pr (Y = j \| \underset{\to}{x})}\} \\ λ_{j} (\underset{\to}{x}) = α_{j} + (β_{1} x_{1} + β_{2} x_{2} + . . . + β_{p} x_{p}), j = 1,2, . . .,k - 1 \end{gathered}$	Originally continuous response variable, subsequently grouped, and valid proportional odds assumption
Unrestricted Partial Proportional Odds Model (PPOM-UR)*	$\begin{gathered} λ_{j} (\underset{\to}{x}) = ln \{\frac{Pr (Y = 1 \| \underset{\to}{x}) + . . . + Pr (Y = j \| \underset{\to}{x})}{Pr (Y = j + 1 \| \underset{\to}{x}) + . . . + Pr (Y = k \| \underset{\to}{x})}\} = ln \{\frac{\sum_{1}^{j} Pr (Y = j \| \underset{\to}{x})}{\sum_{j + 1}^{k} Pr (Y = j \| \underset{\to}{x})}\} \\ λ_{j} (\underset{\to}{x}) = α_{j} + [(β_{1} + γ_{j 1}) x_{1} + . . . + (β_{q} + γ_{j q}) x_{q} + (β_{q + 1}) x_{q + 1} . . . + β_{p} x_{p}], j = 1,2, . . .,k - 1 \end{gathered}$	Proportional odds assumption not valid
Restricted Partial Proportional Odds Model (PPOM-R)	$\begin{gathered} λ_{j} (\underset{\to}{x}) = ln \{\frac{Pr (Y = 1 \| \underset{\to}{x}) + . . . + Pr (Y = j \| \underset{\to}{x})}{Pr (Y = j + 1 \| \underset{\to}{x}) + . . . + Pr (Y = k \| \underset{\to}{x})}\} = ln \{\frac{\sum_{1}^{j} Pr (Y = j \| \underset{\to}{x})}{\sum_{j + 1}^{k} Pr (Y = j \| \underset{\to}{x})}\} \\ λ_{j} (\underset{\to}{x}) = α_{j} + [τ_{j} {(β_{1} + γ_{j 1}) x_{1} + . . . + (β_{q} + γ_{j q}) x_{q}} + (β_{q + 1}) x_{q + 1} . . . + β_{p} x_{p}], j = 1,2, . . .,k - 1 \end{gathered}$	Proportional odds assumption not valid, and linear relationship for odds ratio (OR) between a co-variable and the response variable

Note: Y = Response variable, $\underset{\to}{x}$ vector of explanatory variables = (x ₁, x ₂, ....., x _p)
*Stata uses P > j vs. < = j for the probability comparison in case of PPOM.

Back to article page

ISSN: 1475-2891

Contact us

General enquiries: journalsubmissions@springernature.com

Nutrition Journal

Contact us