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 ...
, ordered probit is a generalization of the widely used
probit
In probability theory and statistics, the probit function is the quantile function associated with the standard normal distribution. It has applications in data analysis and machine learning, in particular exploratory statistical graphics and s ...
analysis to the case of more than two outcomes of an
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 ...
(a dependent variable for which the potential values have a natural ordering, as in poor, fair, good, excellent). Similarly, the widely used
logit
In statistics, the logit ( ) function is the quantile function associated with the standard logistic distribution. It has many uses in data analysis and machine learning, especially in data transformations.
Mathematically, the logit is the ...
method also has a counterpart
ordered logit
In statistics, the ordered logit model (also ordered logistic regression or proportional odds model) is an ordinal regression model—that is, a regression model for ordinal dependent variables—first considered by Peter McCullagh. For exampl ...
. Ordered probit, like ordered logit, is a particular method of
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 ...
.
For example, in
clinical research
Clinical research is a branch of healthcare science that determines the safety and effectiveness ( efficacy) of medications, devices, diagnostic products and treatment regimens intended for human use. These may be used for prevention, treatm ...
, the effect a drug may have on a patient may be modeled with ordered probit regression. Independent variables may include the use or non-use of the drug as well as control variables such as age and details from medical history such as whether the patient suffers from high
blood pressure
Blood pressure (BP) is the pressure of circulating blood against the walls of blood vessels. Most of this pressure results from the heart pumping blood through the circulatory system. When used without qualification, the term "blood pressure" r ...
, heart disease, etc. The dependent variable would be ranked from the following list: complete cure, relieve symptoms, no effect, deteriorate condition, death.
Another example application are
Likert-type items commonly employed in survey research, where respondents rate their agreement on an ordered scale (e.g., "Strongly disagree" to "Strongly agree"). The ordered probit model provides an appropriate fit to these data, preserving the ordering of response options while making no assumptions of the interval distances between options.
Conceptual underpinnings
Suppose the underlying relationship to be characterized is
:
,
where
is the exact but unobserved dependent variable (perhaps the exact level of improvement by the patient);
is the vector of independent variables, 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:
: