Probit

In probability theory and statistics, the probit function is the quantile function, i.e., the inverse cumulative distribution function (CDF), associated with the standard normal distribution. It has applications in exploratory statistical graphics and specialized regression modeling of binary response variables.

The standard normal distribution is commonly denoted as N(0,1) and its CDF as . Function is a continuous, monotone increasing sigmoid function whose domain is the real line and range is (0,1). As an example, consider the familiar fact that the N(0,1) distribution places 95% of probability between −1.96 and 1.96, and is symmetric around zero. It follows that

The probit function gives the 'inverse' computation, generating a value of an N(0,1) random variable, associated with specified cumulative probability. Formally, the probit function is the inverse of, denoted . Continuing the example,

.

In general,

and

The idea of probit was published in 1934 by Chester Ittner Bliss (1899–1979) in an article in Science on how to treat data such as the percentage of a pest killed by a pesticide. Bliss proposed transforming the percentage killed into a "probability unit" (or "probit") which was linearly related to the modern definition (he defined it arbitrarily as equal to 0 for 0.0001 and 10 for 0.9999). He included a table to aid other researchers to convert their kill percentages to his probit, which they could then plot against the logarithm of the dose and thereby, it was hoped, obtain a more or less straight line. Such a so-called probit model is still important in toxicology, as well as other fields. The approach is justified in particular if response variation can be rationalized as a lognormal distribution of tolerances among subjects on test, where the tolerance of a particular subject is the dose just sufficient for the response of interest.

The method introduced by Bliss was carried forward in Probit Analysis, an important text on toxicological applications by D. J. Finney. Values tabled by Finney can be derived from probits as defined here by adding a value of 5. This distinction is summarized by Collett (p. 55): "The original definition of a probit primarily to avoid having to work with negative probits; ... This definition is still used in some quarters, but in the major statistical software packages for what is referred to as probit analysis, probits are defined without the addition of 5." It should be observed that probit methodology, including numerical optimization for fitting of probit functions, was introduced before widespread availability of electronic computing. When using tables, it was convenient to have probits uniformly positive. Common areas of application do not require positive probits.