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In
statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, the ordered logit model (also ordered logistic regression or proportional odds model) is an
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 dif ...
model—that is, a regression model for ordinal
dependent variable Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or demand ...
s—first considered by
Peter McCullagh Peter McCullagh (born 8 January 1952) is a Northern Irish-born American statistician and John D. MacArthur Distinguished Service Professor in the Department of Statistics at the University of Chicago. Education McCullagh is from Plumbridge ...
. For example, if one question on a survey is to be answered by a choice among "poor", "fair", "good", "very good" and "excellent", and the purpose of the analysis is to see how well that response can be predicted by the responses to other questions, some of which may be quantitative, then ordered logistic regression may be used. It can be thought of as an extension of the
logistic regression In statistics, the logistic model (or logit model) is a statistical model that models the probability of an event taking place by having the log-odds for the event be a linear function (calculus), linear combination of one or more independent var ...
model that applies to
dichotomous A dichotomy is a partition of a whole (or a set) into two parts (subsets). In other words, this couple of parts must be * jointly exhaustive: everything must belong to one part or the other, and * mutually exclusive: nothing can belong simultan ...
dependent variables, allowing for more than two (ordered) response categories.


The model and the proportional odds assumption

The model only applies to data that meet the ''proportional odds assumption'', the meaning of which can be exemplified as follows. Suppose there are five outcomes: "poor", "fair", "good", "very good", and "excellent". We assume that the probabilities of these outcomes are given by ''p''1(''x''), ''p''2(''x''), ''p''3(''x''), ''p''4(''x''), ''p''5(''x''), all of which are functions of some independent variable(s) ''x''. Then, for a fixed value of ''x,'' the logarithms of the
odds Odds provide a measure of the likelihood of a particular outcome. They are calculated as the ratio of the number of events that produce that outcome to the number that do not. Odds are commonly used in gambling and statistics. Odds also have ...
(not the logarithms of the probabilities) of answering in certain ways are: : \begin \text: & \log\frac, \\ pt\text: & \log\frac, \\ pt\text: & \log\frac, \\ pt\text: & \log\frac \end The proportional odds assumption states that the numbers added to each of these logarithms to get the next are the same regardless of ''x''. In other words, the difference between the logarithm of the odds of having poor or fair health minus the logarithm of having poor health is the same regardless of ''x''; similarly, the logarithm of the odds of having poor, fair, or good health minus the logarithm of having poor or fair health is the same regardless of ''x''; etc. Examples of multiple-ordered response categories include bond ratings, opinion surveys with responses ranging from "strongly agree" to "strongly disagree," levels of state spending on government programs (high, medium, or low), the level of insurance coverage chosen (none, partial, or full), and employment status (not employed, employed part-time, or fully employed). Ordered logit can be derived from a latent-variable model, similar to the one from which binary logistic regression can be derived. Suppose the underlying process to be characterized is :y^ = \mathbf^ \beta + \varepsilon, \, where y^ is an unobserved dependent variable (perhaps the exact level of agreement with the statement proposed by the pollster); \mathbf is the vector of independent variables; \varepsilon is the
error term In mathematics and statistics, an error term is an additive type of error. Common examples include: * errors and residuals in statistics, e.g. in linear regression * the error term in numerical integration In analysis, numerical integration ...
, assumed to follow a standard logistic distribution; and \beta is the vector of regression coefficients which we wish to estimate. Further suppose that while we cannot observe y^, we instead can only observe the categories of response : y= \begin 0 & \text y^* \le \mu_1, \\ 1 & \text \mu_1 where the parameters \mu_i are the externally imposed endpoints of the observable categories. Then the ordered logit technique will use the observations on ''y'', which are a form of censored data on ''y*'', to fit the parameter vector \beta.


Estimation

For details on how the equation is estimated, see the article
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 dif ...
.


See also

*
Multinomial logit In statistics, multinomial logistic regression is a classification method that generalizes logistic regression to multiclass problems, i.e. with more than two possible discrete outcomes. That is, it is a model that is used to predict the prob ...
*
Multinomial probit In statistics and econometrics, the multinomial probit model is a generalization of the probit model used when there are several possible categories that the dependent variable can fall into. As such, it is an alternative to the multinomial log ...
*
Ordered probit In statistics, ordered probit is a generalization of the widely used probit analysis to the case of more than two outcomes of an ordinal dependent variable (a dependent variable for which the potential values have a natural ordering, as in poor, f ...


References


Further reading

* * * *


External links

* * {{cite web , first=Germán , last=Rodríguez , title=Ordered Logit Models , work=Princeton University , url=http://data.princeton.edu/wws509/stata/c6s5.html Logistic regression