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ProbitModelFit: Probit Regression—Wolfram Documentation

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• See Also
  • GeneralizedLinearModelFit
  • LogitModelFit
  • ExponentialFamily
  • LinkFunction
  • TimeSeriesModelFit
• Related Guides
  • Statistical Model Analysis
  • Supervised Machine Learning
• Tech Notes
  • Statistical Model Analysis

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

constructs a binomial probit regression model of the form that fits the y_i for each x_i.

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

constructs a binomial probit regression model of the form where the f_i depend on the variables x_k.

ProbitModelFit[{m,v}]

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

• 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 x_ij and the responses y_i {{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 responses {{x₁₁,…,y₁,…},…}n fit the n column of a matrix Tabular[…]name fit the column name in a tabular object
• With multivariate data such as , the number of coordinates x_i1, x_i2, … should equal the number of variables x_i.
• The y_i 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 f_i 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 f_i can be specified using the form ProbitModelFit[{m,v},{f₁,f₂,…}].
• ProbitModelFit is equivalent to GeneralizedLinearModelFit with ExponentialFamily->"Binomial" and LinkFunction->"ProbitLink".
• ProbitModelFit takes the same options as GeneralizedLinearModelFit, with the exception of ExponentialFamily and LinkFunction.

Define a dataset:

Fit a probit model to the data:

Evaluate the model at a point:

Plot the data points and the models:

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:

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:

Examine residuals for a fit:

Visualize the raw residuals:

Visualize Anscombe residuals and standardized Pearson residuals in stem plots:

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:

Obtain a formatted table of parameter information:

Extract the column of -statistic values:

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:

Fit a probit model:

Plot the predicted values against the observed values:

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:

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:

The default gives 95% confidence intervals:

Use 99% intervals instead:

Set the level to 90% within FittedModel:

Fit a probit model:

Compute the covariance matrix using the expected information matrix:

Use the observed information matrix instead:

Fit a probit model:

Compute the covariance matrix:

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

Fit a probit model:

Fit the model with zero constant term:

Fit data to a probit model:

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

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

Treat both variables as nominal:

Fit a model using equal weights:

Give explicit weights for the data points:

Use WorkingPrecision to get higher precision in parameter estimates:

Obtain the fitted function:

Reduce the precision in property computations after the fitting:

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

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