# ProbitModelFit

ProbitModelFit[{{x1,y1},{x2,y2},},{f1,f2,},x]

constructs a binomial probit regression model of the form that fits the yi for each xi.

ProbitModelFit[data,{f1,f2,},{x1,x2,}]

constructs a binomial probit regression model of the form where the fi depend on the variables xk.

ProbitModelFit[{m,v}]

constructs a binomial probit regression model from the design matrix m and response vector v.

# Details and Options

• ProbitModelFit attempts to model the data using a linear combination of basis functions composed with the inverse of the probit function ().
• LogitModelFit is typically used in classification to model probability values.
• ProbitModelFit produces a generalized linear model of the form under the assumption that the original are independent realizations of Bernoulli trials with probabilities .
• The function is the CDF of the standard NormalDistribution.
• ProbitModelFit returns a symbolic FittedModel object to represent the probit model it constructs. The properties and diagnostics of the model can be obtained from model["property"].
• The value of the best-fit function from ProbitModelFit at a particular point x1, can be found from model[x1,].
• Possible forms of data are:
•  {y1,y2,…} equivalent to the form {{1,y1},{2,y2},…} {{x11,x12,…,y1},…} a list of independent values xij and the responses yi {{x11,x12,…}y1,…} a list of rules between input values and response {{x11,x12,…},…}{y1,y2,…} a rule between a list of input values and of responses {{x11,…,y1,…},…}n fit the nth column of a matrix
• With multivariate data such as , the number of coordinates xi1, xi2, should equal the number of variables xi.
• The yi are probabilities between 0 and 1.
• Additionally, data can be specified using a design matrix without specifying functions and variables:
•  {m,v} a design matrix m and response vector v
• In ProbitModelFit[{m,v}], the design matrix m is formed from the values of basis functions fi at data points in the form {{f1,f2,},{f1,f2,},}. The response vector v is the list of responses {y1,y2,}.
• For a design matrix m and response vector v, the model is where is the vector of parameters to be estimated.
• When a design matrix is used, the basis functions fi can be specified using the form ProbitModelFit[{m,v},{f1,f2,}].
• ProbitModelFit takes the same options as GeneralizedLinearModelFit, with the exception of ExponentialFamily and LinkFunction.
• ## List of all options

•  AccuracyGoal Automatic the accuracy sought ConfidenceLevel 95/100 confidence level for parameters and predictions CovarianceEstimatorFunction "ExpectedInformation" estimation method for the parameter covariance matrix DispersionEstimatorFunction Automatic function for estimating the dispersion parameter ExponentialFamily Automatic exponential family distribution for y IncludeConstantBasis True whether to include a constant basis function LinearOffsetFunction None known offset in the linear predictor LinkFunction Automatic link function for the model MaxIterations Automatic maximum number of iterations to use NominalVariables None variables considered as nominal PrecisionGoal Automatic the precision sought Weights Automatic weights for data elements WorkingPrecision Automatic the precision for internal computations

# Examples

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## Basic Examples(1)

Define a dataset:

Fit a probit model to the data:

Evaluate the model at a point:

Plot the data points and the models:

## Scope(13)

### Data(6)

Fit data with success probability responses, assuming increasing integer independent values:

This is equivalent to:

Weight by the number of observations for each predictor value:

This gives the same best fit function as success failure data:

Fit a list of rules:

Fit a rule of input values and responses:

Specify a column as the response:

Fit a model given a design matrix and response vector:

See the functional form:

Fit the model referring to the basis functions as and :

Obtain a list of available properties:

### Properties(7)

#### Data & Fitted Functions(1)

Fit a probit model:

Extract the original data:

Obtain and plot the best fit:

Obtain the fitted function as a pure function:

Get the design matrix and response vector for the fitting:

#### Residuals(1)

Examine residuals for a fit:

Visualize the raw residuals:

Visualize Anscombe residuals and standardized Pearson residuals in stem plots:

#### Dispersion and Deviances(1)

Fit a probit model to some data:

The estimated dispersion is 1 by default:

Use Pearson's as the dispersion estimator instead:

Plot the deviances for each point:

Obtain the analysis of deviance table:

Get the residual deviances from the table:

Extract the numeric entries from the table:

#### Parameter Estimation Diagnostics(1)

Obtain a formatted table of parameter information:

Extract the column of -statistic values:

Get the unformatted array of values from the table:

#### Influence Measures(1)

Fit some data containing extreme values to a probit model:

Check Cook distances to identify highly influential points:

Check the diagonal elements of the hat matrix to assess influence of points on the fitting:

#### Prediction Values(1)

Fit a probit model:

Plot the predicted values against the observed values:

#### Goodness-of-Fit Measures(1)

Obtain a table of goodness-of-fit measures for a probit model:

Compute goodness-of-fit measures for all subsets of predictor variables:

Rank the models by AIC:

## Generalizations & Extensions(1)

Perform other mathematical operations on the functional form of the model:

Integrate symbolically and numerically:

Find a predictor value that gives a particular value for the model:

## Options(8)

### ConfidenceLevel(1)

The default gives 95% confidence intervals:

Set the level to 90% within FittedModel:

### CovarianceEstimatorFunction(1)

Fit a probit model:

Compute the covariance matrix using the expected information matrix:

Use the observed information matrix instead:

### DispersionEstimatorFunction(1)

Fit a probit model:

Compute the covariance matrix:

Compute the covariance matrix estimating the dispersion by Pearson's :

### IncludeConstantBasis(1)

Fit a probit model:

Fit the model with zero constant term:

### LinearOffsetFunction(1)

Fit data to a probit model:

Fit data to a model with a known Sqrt[x] term:

### NominalVariables(1)

Fit the data, treating the first variable as a nominal variable:

Treat both variables as nominal:

### Weights(1)

Fit a model using equal weights:

Give explicit weights for the data points:

### WorkingPrecision(1)

Use WorkingPrecision to get higher precision in parameter estimates:

Obtain the fitted function:

Reduce the precision in property computations after the fitting:

## Properties & Relations(4)

ProbitModelFit is equivalent to a "Binomial" model from GeneralizedLinearModelFit with "ProbitLink":

LogitModelFit is a "Binomial" model from GeneralizedLinearModelFit with default "LogitLink":

ProbitModelFit assumes binomially distributed responses:

NonlinearModelFit assumes normally distributed responses:

The fits are not identical:

ProbitModelFit will use the time stamps of a TimeSeries as variables:

Rescale the time stamps and fit again:

Find fit for the values:

ProbitModelFit acts pathwise on a multipath TemporalData:

## Possible Issues(1)

Responses outside the interval from 0 to 1 are not valid for probit models:

Wolfram Research (2008), ProbitModelFit, Wolfram Language function, https://reference.wolfram.com/language/ref/ProbitModelFit.html.

#### Text

Wolfram Research (2008), ProbitModelFit, Wolfram Language function, https://reference.wolfram.com/language/ref/ProbitModelFit.html.

#### CMS

Wolfram Language. 2008. "ProbitModelFit." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ProbitModelFit.html.

#### APA

Wolfram Language. (2008). ProbitModelFit. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ProbitModelFit.html

#### BibTeX

@misc{reference.wolfram_2023_probitmodelfit, author="Wolfram Research", title="{ProbitModelFit}", year="2008", howpublished="\url{https://reference.wolfram.com/language/ref/ProbitModelFit.html}", note=[Accessed: 01-December-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2023_probitmodelfit, organization={Wolfram Research}, title={ProbitModelFit}, year={2008}, url={https://reference.wolfram.com/language/ref/ProbitModelFit.html}, note=[Accessed: 01-December-2023 ]}