ProbitModelFit

ProbitModelFit[{y₁,y₂,…},{f₁,f₂,…},x]

constructs a binomial probit regression model of the form that fits the y_i for successive x values 1, 2, ….

ProbitModelFit[{{x₁₁,x₁₂,…,y₁},{x₂₁,x₂₂,…,y₂},…},{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.

Details and Options

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,…].
With data in the form {{x₁₁,x₁₂,…,y₁},{x₂₁,x₂₂,…,y₂},…}, the number of coordinates x_i1, x_i2, … should correspond to the number of variables x_i.
The y_i are probabilities between 0 and 1.
Data in the form {y₁,y₂,…} is equivalent to data in the form {{1,y₁},{2,y₂},…}.
ProbitModelFit produces a probit model under the assumption that the original are independent observations following binomial distributions with mean .
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.

Examples

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Basic Examples (1)

Define a dataset:

Fit a probit model to the data:

See the functional forms of the model:

Evaluate the model at a point:

Plot the data points and the models:

Compute the fitted values for the model:

Visualize the deviance residuals:

Scope (10)

Fit data with success probability responses:

Weight by the number of observations for each predictor value:

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

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:

Use Grid to add formatting:

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:

Add formatting using Grid:

Add formatting via TableForm:

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:

Use 99% intervals instead:

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:

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ProbitModelFit

Details and Options

Examples

Basic Examples (1)