Ordinal regression
In statistics, ordinal regression (also called "ordinal classification") is a type of regression analysis used for predicting an ordinal variable, i.e. a variable whose value exists on an arbitrary scale where only the relative ordering between different values is significant. It can be considered an intermediate problem between regression and classification.[1][2] Examples of ordinal regression are ordered logit and ordered probit. Ordinal regression turns up often in the social sciences, for example in the modeling of human levels of preference (on a scale from, say, 1–5 for "very poor" through "excellent"), as well as in information retrieval. In machine learning, ordinal regression may also be called ranking learning.[3][loweralpha 1]
Part of a series on Statistics 
Regression analysis 

Models 
Estimation 
Background 

Linear models for ordinal regression
Ordinal regression can be performed using a generalized linear model (GLM) that fits both a coefficient vector and a set of thresholds to a dataset. Suppose one has a set of observations, represented by lengthp vectors x_{1} through x_{n}, with associated responses y_{1} through y_{n}, where each y_{i} is an ordinal variable on a scale 1, ..., K. For simplicity, and without loss of generality, we assume y is a nondecreasing vector, that is, y_{i} y_{i+1}. To this data, one fits a lengthp coefficient vector w and a set of thresholds θ_{1}, ..., θ_{K−1} with the property that θ_{1} < θ_{2} < ... < θ_{K−1}. This set of thresholds divides the real number line into K disjoint segments, corresponding to the K response levels.
The model can now be formulated as
or, the cumulative probability of the response y being at most i is given by a function σ (the inverse link function) applied to a linear function of x. Several choices exist for σ; the logistic function
gives the ordered logit model, while using the probit function gives the ordered probit model. A third option is to use an exponential function
which gives the proportional hazards model.[4]
Latent variable model
The probit version of the above model can be justified by assuming the existence of a realvalued latent variable (unobserved quantity) y*, determined by[5]
where ε is normally distributed with zero mean and unit variance, conditioned on x. The response variable y results from an "incomplete measurement" of y*, where one only determines the interval into which y* falls:
Defining θ_{0} = ∞ and θ_{K} = ∞, the above can be summarized as y = k if and only if θ_{k−1} < y* ≤ θ_{k}.
From these assumptions, one can derive the conditional distribution of y as[5]
where Φ is the cumulative distribution function of the standard normal distribution, and takes on the role of the inverse link function σ. The loglikelihood of the model for a single training example x_{i}, y_{i} can now be stated as[5]
(using the Iverson bracket [y_{i} = k].) The loglikelihood of the ordered logit model is analogous, using the logistic function instead of Φ.[6]
Alternative models
In machine learning, alternatives to the latentvariable models of ordinal regression have been proposed. An early result was PRank, a variant of the perceptron algorithm that found multiple parallel hyperplanes separating the various ranks; its output is a weight vector w and a sorted vector of K−1 thresholds θ, as in the ordered logit/probit models. The prediction rule for this model is to output the smallest rank k such that wx < θ_{k}.[7]
Other methods rely on the principle of largemargin learning that also underlies support vector machines.[8][9]
Another approach is given by Rennie and Srebro, who, realizing that "even just evaluating the likelihood of a predictor is not straightforward" in the ordered logit and ordered probit models, propose fitting ordinal regression models by adapting common loss functions from classification (such as the hinge loss and log loss) to the ordinal case.[10]
Software
ORCA (Ordinal Regression and Classification Algorithms) is an Octave/MATLAB framework including a wide set of ordinal regression methods.[11]
See also
Notes
 Not to be confused with learning to rank.
References
 Winship, Christopher; Mare, Robert D. (1984). "Regression Models with Ordinal Variables" (PDF). American Sociological Review. 49 (4): 512–525. doi:10.2307/2095465. JSTOR 2095465.
 Gutiérrez, P. A.; PérezOrtiz, M.; SánchezMonedero, J.; FernándezNavarro, F.; HervásMartínez, C. (January 2016). "Ordinal Regression Methods: Survey and Experimental Study". IEEE Transactions on Knowledge and Data Engineering. 28 (1): 127–146. doi:10.1109/TKDE.2015.2457911. hdl:10396/14494. ISSN 10414347.
 Shashua, Amnon; Levin, Anat (2002). Ranking with large margin principle: Two approaches. NIPS.
 McCullagh, Peter (1980). "Regression models for ordinal data". Journal of the Royal Statistical Society. Series B (Methodological). 42 (2): 109–142.
 Wooldridge, Jeffrey M. (2010). Econometric Analysis of Cross Section and Panel Data. MIT Press. pp. 655–657. ISBN 9780262232586.
 Agresti, Alan (23 October 2010). "Modeling Ordinal Categorical Data" (PDF). Retrieved 23 July 2015.
 Crammer, Koby; Singer, Yoram (2001). Pranking with ranking. NIPS.
 Chu, Wei; Keerthi, S. Sathiya (2007). "Support vector ordinal regression". Neural Computation. 19 (3): 792–815. CiteSeerX 10.1.1.297.3637. doi:10.1162/neco.2007.19.3.792. PMID 17298234.
 Herbrich, Ralf; Graepel, Thore; Obermayer, Klaus (2000). "Large Margin Rank Boundaries for Ordinal Regression". Advances in Large Margin Classifiers. MIT Press. pp. 115–132.
 Rennie, Jason D. M.; Srebro, Nathan (2005). Loss Functions for Preference Levels: Regression with Discrete Ordered Labels (PDF). Proc. IJCAI Multidisciplinary Workshop on Advances in Preference Handling.
 orca: Ordinal Regression and Classification Algorithms, AYRNA, 20171121, retrieved 20171121
Further reading
 Agresti, Alan (2010). Analysis of ordinal categorical data. Hoboken, N.J: Wiley. ISBN 9780470082898.
 Greene, William H. (2012). Econometric Analysis (Seventh ed.). Boston: Pearson Education. pp. 824–842. ISBN 9780273753568.
 Hardin, James; Hilbe, Joseph (2007). Generalized Linear Models and Extensions (2nd ed.). College Station: Stata Press. ISBN 9781597180146.