|
In probability theory and statistics, the probit function is the quantile function associated with the standard normal distribution, which is commonly denoted as N(0,1). Mathematically, it is the inverse of the cumulative distribution function of the standard normal distribution, which is denoted as , so the probit is denoted as . It has applications in exploratory statistical graphics and specialized regression modeling of binary response variables. Largely because of the central limit theorem, the standard normal distribution plays a fundamental role in probability theory and statistics. If we consider the familiar fact that the standard normal 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. Continuing the example, :. In general, : :and : ==Conceptual development== The idea of the probit function was published by Chester Ittner Bliss (1899–1979) in a 1934 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.〔Finney, D.J. (1947), ''Probit Analysis''. (1st edition) Cambridge University Press, Cambridge, UK.〕 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 (5 added ) was 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Probit」の詳細全文を読む スポンサード リンク
|