# ProbitModelFit

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

constructs a binomial probit regression model of the form that fits the yi 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 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 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 xi1, xi2, should correspond to the number of variables xi.
• The yi 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 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.

# 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:

#### 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:  