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:
:
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
:
where
is an unobserved dependent variable (perhaps the exact level of agreement with the statement proposed by the pollster);
is the vector of independent variables;
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
is the vector of regression coefficients which we wish to estimate. Further suppose that while we cannot observe
, we instead can only observe the categories of response
: