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The ordered logit model (also known as ordered/ordinal logistic regression) is an extension of the logistic regression from binary dependent variable to an ordinal dependent variable. A widespread special case is proportional odds model.

These models are a class of generalized linear models (GLMs) using cumulative links to borrow information across several possible polytomous groups when the usual probability model for such an outcome has a restrictive support (e.g. 0/1 responses).

Ordinal logistic regression is a popular method for analyzing ordinal response variables. Ordinal responses are variables that can be ranked in terms of the relative severity or intensity of associated response, but there may be no associated units or conventional scale to relate differences meaningfully across groups. Examples of ordinal variables include Likert responses about inclination toward voting for a political candidate, pain severity scales, or ECOG status.

Ordinal logistic regression employs a cumulative link where a separate likelihood is generated comparing thresholds at each possible value of the ordinal response and jointly fitting logistic regression models for each threshold, having common odds ratios but differing intercepts for each likelihood. The odds ratio(s) is/are then interpreted as the ratio of odds for an incrementally higher ordinal response for a unit difference in the fitted variable(s).

Valid and alternative approaches to analyzing data from a similar setting include a linear regression model treating ordinal outcomes as a direct numeric value. The slope parameter(s) are then interpreted as the expected difference in ordinal responses comparing groups differing by 1 unit in the covariate(s). Alternately, a single thresholded binary outcome can be estimates from a standard logistic regression model for this purpose.

Literature references;

• Wooldridge, J. M. (2002). Econometric analysis of cross section and panel data. Cambridge, Mass: MIT Press. p. 504-509.

• McCullagh, P., & Nelder, J. A. (1989). Generalized linear models. London: Chapman and Hall. Chapter 4.

• McCullagh, Peter (1980). "Regression Models for Ordinal Data". Journal of the Royal Statistical Society. Series B (Methodological). 42 (2): 109–142. JSTOR 2984952.

• Jackman, S. (2000). Models for Ordered Outcomes. Political Science 200C. Stanford University

• Greene, William H. (2012). Econometric Analysis (Seventh ed.). Boston: Pearson Education. pp. 827–831/726-740. ISBN 978-0-273-75356-8.

• http://www.pmean.com/04/OrdinalLogistic.html

• https://www.analyticsvidhya.com/blog/2016/02/multinomial-ordinal-logistic-regression/

• Gelman, Andrew; Hill, Jennifer (2007). Data Analysis Using Regression and Multilevel/Hierarchical Models. New York: Cambridge University Press. pp. 119–124. ISBN 978-0-521-68689-1.

• Hardin, James; Hilbe, Joseph (2007). Generalized Linear Models and Extensions (2nd ed.). College Station: Stata Press. ISBN 978-1-59718-014-6.

• Woodward, Mark (2005). Epidemiology: Study Design and Data Analysis (2nd ed.). Chapman & Hall/CRC. ISBN 978-1-58488-415-6.

Software implementation