probit analysis

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1. an investigation of the component parts of a whole and their relations in making up the whole
2. the abstract separation of a whole into its constituent parts in order to study the parts and their relations (同)analytic thinking
3. a branch of mathematics involving calculus and the theory of limits; sequences and series and integration and differentiation
4. a form of literary criticism in which the structure of a piece of writing is analyzed
5. the use of closed-class words instead of inflections: e.g., `the father of the bride instead of `the brides father

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1. (内容・状況などの)『分析』,分解;(詳細な)検討 / (化学・物理で)分析;《米》(心理学で)[精神]分析;(数学で)解析

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出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2013/02/12 22:28:02」(JST)

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Regression analysis
Models
Linear regression Simple regression Ordinary least squares Polynomial regression General linear model
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Errors-in-variables
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Background
Regression model validation Mean and predicted response Errors and residuals Goodness of fit Studentized residual Gauss–Markov theorem

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 name is from probability + unit.

A probit model is a popular specification for an ordinal or a binary response model that employs a probit link function. As such it treats the same set of problems as does logistic regression using similar techniques. The probit model is most often estimated using the standard maximum likelihood procedure, such an estimation being called a probit regression.

Probit models were introduced by Chester Bliss in 1934, and a fast method for computing maximum likelihood estimates for them was proposed by Ronald Fisher in an appendix to Bliss 1935.

Introduction

Suppose response variable Y is binary, that is it can have only two possible outcomes which we will denote as 1 and 0. For example Y may represent presence/absence of a certain condition, success/failure of some device, answer yes/no on a survey, etc. We also have a vector of regressors X, which are assumed to influence the outcome Y. Specifically, we assume that the model takes form

$\Pr(Y=1 \mid X) = \Phi(X'\beta),$

where Pr denotes probability, and Φ is the Cumulative Distribution Function (CDF) of the standard normal distribution. The parameters β are typically estimated by maximum likelihood.

It is also possible to motivate the probit model as a latent variable model. Suppose there exists an auxiliary random variable

where ε ~ N(0, 1). Then Y can be viewed as an indicator for whether this latent variable is positive:

$Y = \begin{cases} 1 & \text{if }Y^\ast > 0 \ \text{ i.e. } - \varepsilon < X'\beta, \\ 0 &\text{otherwise.} \end{cases}$

The use of the standard normal distribution causes no loss of generality compared with using an arbitrary mean and standard deviation because adding a fixed amount to the mean can be compensated by subtracting the same amount from the intercept, and multiplying the standard deviation by a fixed amount can be compensated by multiplying the weights by the same amount.

To see that the two models are equivalent, note that

$\begin{align} \Pr(Y = 1 \mid X) &= \Pr(Y^\ast > 0) = \Pr(X'\beta + \varepsilon > 0) \\ &= \Pr(\varepsilon > -X'\beta) \\ &= \Pr(\varepsilon < X'\beta) \quad \text{(by symmetry of the normal dist)}\\ &= \Phi(X'\beta) \end{align}$

Maximum likelihood estimation

Suppose data set contains n independent statistical units corresponding to the model above. Then their joint log-likelihood function is

The estimator which maximizes this function will be consistent, asymptotically normal and efficient provided that E[XX'] exists and is not singular. It can be shown that this log-likelihood function is globally concave in β, and therefore standard numerical algorithms for optimization will converge rapidly to the unique maximum.

Asymptotic distribution for is given by

where

$\Omega = \operatorname{E}\bigg[ \frac{\varphi^2(X'\beta)}{\Phi(X'\beta)(1-\Phi(X'\beta))}XX' \bigg], \qquad \hat\Omega = \frac{1}{n}\sum_{i=1}^n \frac{\varphi^2(x'_i\hat\beta)}{\Phi(x'_i\hat\beta)(1-\Phi(x'_i\hat\beta))}x_ix'_i$

and φ = Φ' is the Probability Density Function (PDF) of standard normal distribution.

Berkson's minimum chi-square method

This method can be applied only when there are many observations of response variable having the same value of the vector of regressors (such situation may be referred to as "many observations per cell"). More specifically, the model can be formulated as follows.

Suppose among n observations there are only T distinct values of the regressors, which can be denoted as . Let be the number of observations with and the number of such observations with . We assume that there are indeed "many" observations per each "cell": for each .

Denote

Then Berkson's minimum chi-square estimator is a generalized least squares estimator in a regression of on with weights :

It can be shown that this estimator is consistent (as n→∞ and T fixed), asymptotically normal and efficient.^{[citation needed]} Its advantage is the presence of a closed-form formula for the estimator. However, it is only meaningful to carry out this analysis when individual observations are not available, only their aggregated counts , , and (for example in the analysis of voting behavior).

Gibbs sampling

Gibbs sampling of a probit model is possible because regression models typically use normal prior distributions over the weights, and this distribution is conjugate with the normal distribution of the errors (and hence of the latent variablesY^*). The model can be described as

$\begin{array}{lcl} \boldsymbol\beta &\sim& \mathcal{N}(\mathbf{b}_0, \mathbf{B}_0) \\ [3pt] y_i^\ast|\mathbf{x}_i,\boldsymbol\beta &\sim& \mathcal{N}(\mathbf{x}'_i\boldsymbol\beta, 1) \\ [3pt] y_i &=& \begin{cases} 1 & \text{if } y_i^\ast > 0 \\ 0 & \text{otherwise} \end{cases} \end{array}$

From this, we can determine the full conditional densities needed:

$\begin{array}{lcl} \mathbf{B} &=& (\mathbf{B}_0^{-1} + \mathbf{X}'\mathbf{X})^{-1} \\ [3pt] \boldsymbol\beta|\mathbf{y}^\ast &\sim& \mathcal{N}(\mathbf{B}(\mathbf{B}_0^{-1}\mathbf{b}_0 + \mathbf{X}'\mathbf{y}^\ast), \mathbf{B}) \\ [3pt] y_i^\ast|y_i=0,\mathbf{x}_i,\boldsymbol\beta &\sim& \mathcal{N}(\mathbf{x}'_i\boldsymbol\beta, 1)[y_i^\ast < 0] \\ [3pt] y_i^\ast|y_i=1,\mathbf{x}_i,\boldsymbol\beta &\sim& \mathcal{N}(\mathbf{x}'_i\boldsymbol\beta, 1)[y_i^\ast \ge 0] \end{array}$

The result for β is given in the article on Bayesian linear regression, although specified with different notation.

The only trickiness is in the last two equations. The notation is the Iverson bracket, sometimes written or similar. It indicates that the distribution must be truncated within the given range, and rescaled appropriately. In this particular case, a truncated normal distribution arises. Sampling from this distribution depends on how much is truncated. If a large fraction of the original mass remains, sampling can be easily done with rejection sampling — simply sample a number from the non-truncated distribution, and reject it if it falls outside the restriction imposed by the truncation. If sampling from only a small fraction of the original mass, however (e.g. if sampling from one of the tails of the normal distribution — for example if is around 3 or more, and a negative sample is desired), then this will be inefficient and it becomes necessary to fall back on other sampling algorithms. General sampling from the truncated normal can be achieved using approximations to the normal CDF and the probit function, and R has a function rtnorm() for generating truncated-normal samples.

See also

• Generalized linear model
• Limited dependent variable
• Multivariate probit models
• Ordered probit and Ordered logit model
• Separation (statistics)

References

• Bliss, C.I. (1935). "The calculation of the dosage-mortality curve". Annals of Applied Biology (22)134–167. doi:10.1111/j.1744-7348.1935.tb07713.x
• Bliss, C.I (1938). "The determination of the dosage-mortality curve from small numbers". Quarterly Journal of Pharmacology (11)192–216.
• McCullagh, Peter; John Nelder (1989). Generalized Linear Models. London: Chapman and Hall. ISBN 0-412-31760-5.
• Albert, J.H., and Chib, S. (1993). "Bayesian Analysis of Binary and Polychotomous Response Data." Journal of the American Statistical Association (88)422: pp. 669-679. http://www.jstor.org/stable/2290350

Notes

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English Journal

• Control of human filarial vector, Culex quinquefasciatus Say 1823 (Diptera: Culicidae) through bioactive fraction of Cayratia trifolia leaf.

• Chakraborty S, Singha S, Bhattacharya K, Chandra G.SourceMosquito and Microbiology Research Units, Parasitology Laboratory, Department of Zoology, The University of Burdwan, Burdwan-713104, West Bengal, India.
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• OBJECTIVE: To investigate the mosquito larvicidal activity of Cayratia trifolia (L.) Domin (Vitaceae: Vitales) (C. trifolia) which is distributed in many parts of India with medicinal properties as vector control is facing threat due to the emergence of resistance to synthetic insecticides.METHODS:
• PMID 24093790

• Cross cultural analysis of factors associated with age at natural menopause among latin-american immigrants to madrid and their spanish neighbors.

• Pérez-Alcalá I, Sievert LL, Obermeyer CM, Reher DS.SourceDepartment of Sociology II (Human Ecology and Population Studies), Universidad Complutense de Madrid, Madrid, Spain.
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• OBJECTIVES: In this study, age at menopause was examined in relation to demographic and life style factors among Latin-American immigrants to Madrid and their Spanish counterparts.METHODS: Respondents were drawn from the Decisions at Menopause Study (2002-2003) and from a recent sample of Latin-Amer
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• Larvicidal efficacy of Catharanthus roseus Linn. (Family: Apocynaceae) leaf extract and bacterial insecticide Bacillus thuringiensis against Anopheles stephensi Liston.

• Panneerselvam C, Murugan K, Kovendan K, Kumar PM, Ponarulselvam S, Amerasan D, Subramaniam J, Hwang JS.SourceDRDO-BU Center for Life Sciences, Bharathiar University, Coimbatore 641046, Tamil Nadu, India; Division of Entomology, Department of Zoology, School of Life Sciences, Bharathiar University, Coimbatore 641046, Tamil Nadu, India.
• Asian Pacific journal of tropical medicine.Asian Pac J Trop Med.2013 Nov;6(11):847-53. doi: 10.1016/S1995-7645(13)60151-0.
• OBJECTIVE: To explore the larvicidal activity of Catharanthus roseus (C. roseus) leaf extract and Bacillus thuringiensis (B. thuringiensis) against the malarial vector Anopheles stephensi (An. stephensi), when being used alone or together.METHODS: The larvicidal activity was assayed at various conce
• PMID 24083578

Japanese Journal

• Chronic and Transient Poverty in Indonesia : A Spatial Perspective with the 2008-2010 Susenas Panel Data

• Akita Takahiro,Dariwardani Ni Made Inna,秋田 隆裕
• Economics & management series (EMS-2013-19), 1-26, 2013-09-00
• … It also conducts a probit analysis to explore the determinants of poverty based on the 2008 consumption module Susenas. …
• NAID 120005328946

• Chronic and Transient Poverty in Indonesia: A Spatial Perspective with the 2008-2010 Susenas Panel Data

• Akita Takahiro,Dariwardani Ni Made Inna
• Economics & management series EMS-2013-19, 1-26, 2013-09-00
• … It also conducts a probit analysis to explore the determinants of poverty based on the 2008 consumption module Susenas. …
• NAID 110009613809

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• 宇山 満,吉野 章,北野 慎一
• 生駒経済論叢 11(1), 155-167, 2013-08-00
• … This study employed an ordered logit model and an ordered probit model to clarify what kinds of people would participate in opposition activities against imaginary plans for the construction of a waste incineration facility based on a questionnaire survey. …
• NAID 120005328852

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