Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
In mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa. i.e., ƒ=y, and g=x...
In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...
In probability and statistics, the quantile function of the probability distribution of a random variable specifies, for a given probability, the value which the random variable will be at, or below, with that probability...
In statistics, a Q-Q plot is a probability plot, which is a graphical method for comparing two probability distributions by plotting their quantiles against each other. First, the set of intervals for the quantiles are chosen...
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
In mathematics, a monotonic function is a function that preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory....
Many natural processes, including those of complex system learning curves, exhibit a progression from small beginnings that accelerates and approaches a climax over time. When a detailed description is lacking, a sigmoid function is often used. A sigmoid curve is produced by a mathematical...
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, .
Chester Ittner Bliss was primarily a biologist, who is best known for his contributions to statistics. He was born in Springfield, Ohio in 1899 and died in 1979.-Academic qualifications:*Bachelor of Arts in Entomology from Ohio State University, 1921...
Science is the academic journal of the American Association for the Advancement of Science and is one of the world's top scientific journals....
on how to treat data such as the percentage of a pest killed by a pesticide
Pesticide
Pesticides are substances or mixture of substances intended for preventing, destroying, repelling or mitigating any pest.A pesticide may be a chemical unicycle, biological agent , antimicrobial, disinfectant or device used against any pest...
. 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
Probit model
In statistics, a probit model is a type of regression where the dependent variable can only take two values, for example married or not married....
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 an important text on toxicological applications by D. J. Finney
D. J. Finney
David John Finney CBE is Professor Emeritus of Statistics at the University of Edinburgh. He was Director of the Agricultural Research Council's Unit of Statistics from 1954 to 1984 and a former President of the Royal Statistical Society and of the Biometric Society...
. 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 [with 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.
Diagnosing deviation of a distribution from normality
In addition to providing a basis for important types of regression, the probit function is useful in statistical analysis for diagnosing deviation from normality, according to the method of Q-Q plotting. If a set of data is actually a sample
Sample (statistics)
In statistics, a sample is a subset of a population. Typically, the population is very large, making a census or a complete enumeration of all the values in the population impractical or impossible. The sample represents a subset of manageable size...
of a normal distribution, a plot of the values against their probit scores will be approximately linear. Specific deviations from normality such as asymmetry
Skewness
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. The skewness value can be positive or negative, or even undefined...
In statistics, a bimodal distribution is a continuous probability distribution with two different modes. These appear as distinct peaks in the probability density function, as shown in Figure 1....
can be diagnosed based on detection of specific deviations from linearity. While the Q-Q plot can be used for comparison to any distribution family (not only the normal), the normal Q-Q plot is a relatively standard exploratory data analysis procedure because the assumption of normality is often a starting point for analysis.
Computation
The normal distribution CDF and its inverse are not available in closed form
Closed form
-Maths:* Closed-form expression, a finitary expression* Closed differential form, a differential form \alpha with the property that d\alpha = 0-Poetry:* In poetry analysis, a type of poetry that exhibits regular structure, such as meter or a rhyming pattern;...
, and computation requires careful use of numerical procedures. However, the functions are widely available in software for statistics and probability modeling, and in spreadsheets. In Microsoft Excel
Microsoft Excel
Microsoft Excel is a proprietary commercial spreadsheet application written and distributed by Microsoft for Microsoft Windows and Mac OS X. It features calculation, graphing tools, pivot tables, and a macro programming language called Visual Basic for Applications...
, for example, the probit function is available as normsinv(p). In computing environments where numerical implementations of the inverse error function
Error function
In mathematics, the error function is a special function of sigmoid shape which occurs in probability, statistics and partial differential equations...
are available, the probit function may be obtained as
An example is MATLAB
MATLAB
MATLAB is a numerical computing environment and fourth-generation programming language. Developed by MathWorks, MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages,...
, where an 'erfinv' function is available. The language Mathematica
Mathematica
Mathematica is a computational software program used in scientific, engineering, and mathematical fields and other areas of technical computing...
implements 'InverseErf'. Other environments directly implement the probit function as is shown in the following session in the R programming language.
> qnorm(0.025)
[1] -1.959964
> pnorm(-1.96)
[1] 0.02499790
An ordinary differential equation for the probit function
Another means of computation is based on forming a non-linear ordinary differential equation for probit. Abbreviating the probit function as , the ODE is
with the centre (boundary) conditions
This equation may be solved by several methods, including the classical power series approach. From this, solutions of arbitrarily high accuracy may be developed based on Steinbrecher's approach to the series for the inverse error function. The power series solution is given by
where the coefficients satisfy the non-linear recurrence
with . In this form the ratio as .
Related topics
Closely related to the probit function (and probit model
Probit model
In statistics, a probit model is a type of regression where the dependent variable can only take two values, for example married or not married....
The logit function is the inverse of the sigmoidal "logistic" function used in mathematics, especially in statistics.Log-odds and logit are synonyms.-Definition:The logit of a number p between 0 and 1 is given by the formula:...
function and logit model. The inverse of the logistic function is given by
Analogously to the probit model, we may assume that such a quantity is related linearly to a set of predictors, resulting in the logit model, the basis in particular of logistic regression
Logistic regression
In statistics, logistic regression is used for prediction of the probability of occurrence of an event by fitting data to a logit function logistic curve. It is a generalized linear model used for binomial regression...
In statistics, regression analysis includes many techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables...
for binary response data. In current statistical practice, probit and logit regression models are often handled as cases of the generalized linear model
Generalized linear model
In statistics, the generalized linear model is a flexible generalization of ordinary linear regression. The GLM generalizes linear regression by allowing the linear model to be related to the response variable via a link function and by allowing the magnitude of the variance of each measurement to...
In statistics, logistic regression is used for prediction of the probability of occurrence of an event by fitting data to a logit function logistic curve. It is a generalized linear model used for binomial regression...
The logit function is the inverse of the sigmoidal "logistic" function used in mathematics, especially in statistics.Log-odds and logit are synonyms.-Definition:The logit of a number p between 0 and 1 is given by the formula:...
In statistics, a Q-Q plot is a probability plot, which is a graphical method for comparing two probability distributions by plotting their quantiles against each other. First, the set of intervals for the quantiles are chosen...
In probability and statistics, the quantile function of the probability distribution of a random variable specifies, for a given probability, the value which the random variable will be at, or below, with that probability...
In statistics, rankits of a set of data are the expected values of the order statistics of a sample from the standard normal distribution the same size as the data. They are primarily used in the normal probability plot, a graphical technique for normality testing.-Example:This is perhaps most...
In econometrics, ridit scoring is a statistical method used to analyze ordered qualitative measurements.The tools of ridit analysis were developed and first applied by Bross, who coined the term "ridit" by analogy with other statistical transformations such as probit and logit.-Choosing a reference...
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. Function 
is a continuous
, denoted 
. Continuing the example,
.
and 
The idea of probit was published in 1934 by Chester Ittner Bliss
, the ODE is
satisfy the non-linear recurrence
. In this form the ratio 
as 
.