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

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

LogitModelFit [data,{f₁,…},{x₁,x₂,…}]

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

LogitModelFit [{m,v}]

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

Details and Options

Details and Options Details and Options

Examples

Basic Examples

Scope

Data

Properties

Data & Fitted Functions

Residuals

Dispersion and Deviances

Parameter Estimation Diagnostics

Influence Measures

Prediction Values

Goodness-of-Fit Measures

Generalizations & Extensions

Options

ConfidenceLevel

CovarianceEstimatorFunction

DispersionEstimatorFunction

Show More Show More

IncludeConstantBasis

LinearOffsetFunction

NominalVariables

Weights

WorkingPrecision

Properties & Relations

Possible Issues

LogitModelFit

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

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

LogitModelFit [data,{f₁,…},{x₁,x₂,…}]

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

LogitModelFit [{m,v}]

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

Details and Options

LogitModelFit attempts to model the data using a linear combination of basis functions composed with a logistic sigmoid.
LogitModelFit is typically used in classification to model probability values.
LogitModelFit 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 LogisticSigmoid .
LogitModelFit returns a symbolic FittedModel object to represent the logistic model it constructs. The properties and diagnostics of the model can be obtained from model["property"].
The value of the best-fit function from LogitModelFit 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^(th) column of a matrix

Tabular […]name fit the column name in a tabular object
With multivariate data such as {{x_(11),x_(12),... ,y_(1)},{x_(21),x_(22),... ,y_(2)},...}, 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 LogitModelFit [{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 LogitModelFit [{m,v},{f₁,f₂,…}].
LogitModelFit is equivalent to GeneralizedLinearModelFit with ExponentialFamily ->"Binomial" and LinkFunction ->Automatic .
LogitModelFit takes the same options as GeneralizedLinearModelFit , with the exception of ExponentialFamily and LinkFunction .

Examples

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Basic Examples (1)

Define a dataset:

Wolfram Language code:

data = {{1, 0}, {2, 0}, {2, 0}, {2, 1}, {2, 0}, {3, 0}, {3, 0}, {3, 0}, {3, 1}, {3, 1}, {3, 1}, {4, 1}, {4, 1}, {5, 0}, {6, 1}, {7, 1}};

Fit a logistic model to the data:

Wolfram Language code: logit = LogitModelFit[data, x, x]

Evaluate the model at a point:

Wolfram Language code: logit[1.5]

Plot the data points and the models:

Wolfram Language code: Show[ListPlot[data, PlotStyle -> PointSize[Medium]], Plot[logit[x], {x, 0, 8}]]

Scope (13)

Data (6)

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

Wolfram Language code: LogitModelFit[{1 / 3, 2 / 3, 5 / 5}, x, x]//Normal

This is equivalent to:

Wolfram Language code: LogitModelFit[{{1, 1 / 3}, {2, 2 / 3}, {3, 5 / 5}}, x, x]//Normal

Weight by the number of observations for each predictor value:

Wolfram Language code: LogitModelFit[{{1, 1 / 3}, {2, 2 / 3}, {3, 5 / 5}}, x, x, Weights -> {3, 6, 5}]//Normal

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

Wolfram Language code:

sfdata = {{1, 0}, {1, 1}, {1, 0}, {2, 0}, {2, 0}, {2, 1}, {2, 1}, {2, 1}, {2, 1}, {3, 1}, {3, 1}, {3, 1}, {3, 1}, {3, 1}};
GroupBy[sfdata, First -> Last]

Wolfram Language code: LogitModelFit[sfdata, x, x] //Normal

Fit a list of rules:

Wolfram Language code: LogitModelFit[{1 -> 1 / 3, 2 -> 2 / 3, 3 -> 3 / 3}, x, x]

Fit a rule of input values and responses:

Wolfram Language code: LogitModelFit[{1, 2, 3} -> {1 / 3, 2 / 3, 3 / 3}, x, x]

Specify a column as the response:

Wolfram Language code: data = {...};

Wolfram Language code: fit = LogitModelFit[data -> 1, {1, x, y}, {x, y}]

Wolfram Language code:

Show[
	Plot3D[fit[x, y], {x, -5, 5}, {y, -5, 5}], 
	ListPointPlot3D[data[[All, {2, 3, 1}]]], 
	PlotRange -> All
	]

Fit a model given a design matrix and response vector:

Wolfram Language code:

dm = {{1, 1}, {1, 2}, {1, 3}, {1, 4}};
resp = {0, 1, 0, 1};LogitModelFit[{dm, resp}]

See the functional form:

Wolfram Language code: Normal[%]

Fit the model referring to the basis functions as x and y:

Wolfram Language code: LogitModelFit[{dm, resp}, {x, y}]//Normal

Obtain a list of available properties:

Wolfram Language code: logit = LogitModelFit[RandomReal[1, 10], x, x]

Wolfram Language code: logit["Properties"]

Properties (7)

Data & Fitted Functions (1)

Fit a logit model:

Wolfram Language code: data = Join[RandomReal[10, {10, 2}], Transpose[{RandomReal[1, 10]}], 2];

Wolfram Language code: logit = LogitModelFit[data, {x, y}, {x, y}]

Extract the original data:

Wolfram Language code: logit["Data"]

Obtain and plot the best fit:

Wolfram Language code: fit = logit["BestFit"]

Wolfram Language code: Show[Plot3D[fit, {x, 0, 10}, {y, 0, 10}], Graphics3D[{PointSize[0.025], Point[logit["Data"]]}]]

Obtain the fitted function as a pure function:

Wolfram Language code: logit["Function"]

Get the design matrix and response vector for the fitting:

Wolfram Language code: MatrixForm /@ logit[{"DesignMatrix", "Response"}]

Residuals (1)

Examine residuals for a fit:

Wolfram Language code: data = Join[RandomReal[10, {100, 2}], Transpose[{RandomReal[1, 100]}], 2];

Wolfram Language code: logit = LogitModelFit[data, {x, y}, {x, y}]

Wolfram Language code: {fr, ar, spr} = logit[{"FitResiduals", "StandardizedPearsonResiduals", "AnscombeResiduals"}];

Visualize the raw residuals:

Wolfram Language code: ListPlot[fr]

Visualize Anscombe residuals and standardized Pearson residuals in stem plots:

Wolfram Language code: Map[ListPlot[#, Filling -> 0]&, {ar, spr}]

Dispersion and Deviances (1)

Fit a logit model to some data:

Wolfram Language code:

SeedRandom[1];
data = Flatten[Table[{x, y, 1 / (1 + Exp[.2 + .3x + .1y + RandomReal[{-1, 1}]])}, {x, RandomReal[5, 10]}, {y, RandomReal[5, 10]}], 1];logit = LogitModelFit[data, {x, y}, {x, y}]

The estimated dispersion is 1 by default:

Wolfram Language code: logit["EstimatedDispersion"]

Use Pearson's as the dispersion estimator instead:

Wolfram Language code: logit["EstimatedDispersion", DispersionEstimatorFunction -> "PearsonChiSquare"]

Plot the deviances for each point:

Wolfram Language code: ListPlot[logit["Deviances"], Filling -> 0]

Obtain the analysis of deviance table:

Wolfram Language code: logit["DevianceData"]

Get the residual deviances from the table:

Wolfram Language code: logit["DevianceData"][All, "ResidualDeviance"]//Normal

Parameter Estimation Diagnostics (1)

Obtain a formatted table of parameter information:

Wolfram Language code:

SeedRandom[1];
data = Table[{i, RandomInteger[BinomialDistribution[20, .05 + i / 30 + Sin[i] / 10]] / 20}, {i, 1, 20, .2}];
logit = LogitModelFit[data, {x, Sin[x], Cos[x]}, x];

Wolfram Language code: logit["ParameterEstimates"]

Extract the column of -statistic values:

Wolfram Language code: logit["ParameterEstimates"][All, "ZStatistic"]//Normal

Influence Measures (1)

Fit some data containing extreme values to a logit model:

Wolfram Language code:

SeedRandom[3];data = Table[{i, 1 / (1 + Exp[2 + 4i - Log[i] + RandomReal[]])}, {i, RandomReal[{1, 10}, 20]}];
data[[{3, 8}, -1]] = 1 - data[[{3, 8}, -1]];

Wolfram Language code: ListPlot[data]

Wolfram Language code: logit = LogitModelFit[data, {x, Log[x]}, x]

Check Cook distances to identify highly influential points:

Wolfram Language code: ListPlot[logit["CookDistances"], PlotRange -> {0, All}, Filling -> 0]

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

Wolfram Language code: ListPlot[logit["HatDiagonal"], PlotRange -> {0, All}, Filling -> 0]

Prediction Values (1)

Fit a logit model:

Wolfram Language code:

SeedRandom[3];data = Flatten[Table[{x, y, 1 / (1 + Exp[.3x - .5y + RandomReal[{-1, 1}]])}, {x, RandomReal[5, 3]}, {y, RandomReal[5, 3]}], 1];logit = LogitModelFit[data, {x, y}, {x, y}]

Plot the predicted values against the observed values:

Wolfram Language code:

ListPlot[Transpose[logit[{"Response", "PredictedResponse"}]], FrameLabel -> {"observed", "predicted"}, Frame -> True, Axes -> False]

Goodness-of-Fit Measures (1)

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

Wolfram Language code:

SeedRandom[3];data = Flatten[Table[{i, j, RandomReal[10], 1 / (1 + Exp[30i - 10j + RandomReal[5]])}, {i, RandomReal[10, 5]}, {j, RandomReal[10, 5]}], 1];

Wolfram Language code: logit = LogitModelFit[data, {x, y, z}, {x, y, z}]

Wolfram Language code: Grid[Transpose[{#, logit[#]}&[{"AIC", "BIC", "LikelihoodRatioIndex", "PearsonChiSquare"}]], Alignment -> Left]

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

Wolfram Language code:

sub = Table[Join[{i}, LogitModelFit[data, i, {x, y, z}][{"AIC", "BIC", "LikelihoodRatioIndex", "PearsonChiSquare"}]], {i, Subsets[{x, y, z}]}]

Rank the models by AIC:

Wolfram Language code: Grid[Join[{{"Model", "AIC", "BIC", "LikelihoodRatioIndex", "PearsonChiSquare"}}, SortBy[sub, -#[[2]]&]]]

Generalizations & Extensions (1)

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

Wolfram Language code: logit = LogitModelFit[Table[{i, RandomReal[{i - 1, i}] / 10}, {i, 10}], x, x]

Integrate symbolically and numerically:

Wolfram Language code: Integrate[logit[x], x]

Wolfram Language code: NIntegrate[logit[x], {x, 1, 5}]

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

Wolfram Language code: FindRoot[logit[x] == .5, {x, 5}]

Options (8)

ConfidenceLevel (1)

The default gives 95% confidence intervals:

Wolfram Language code: data = {{0, 1}, {1, 0}, {3, 1}, {5, 0}};

Wolfram Language code: logit = LogitModelFit[data, x, x]

Wolfram Language code: logit["ParameterEstimates"][All, "ConfidenceInterval"]//Normal

Use 99% intervals instead:

Wolfram Language code: logit = LogitModelFit[data, x, x, ConfidenceLevel -> .99]

Wolfram Language code: logit["ParameterEstimates"][All, "ConfidenceInterval"]//Normal

Set the level to 90% within FittedModel :

Wolfram Language code: logit["ParameterEstimates", ConfidenceLevel -> .9][All, "ConfidenceInterval"]//Normal

CovarianceEstimatorFunction (1)

Fit a logit model:

Wolfram Language code: data = {{0, 1}, {1, 0}, {3, 1}, {5, 0}};

Wolfram Language code: logit = LogitModelFit[data, x, x]

Compute the covariance matrix using the expected information matrix:

Wolfram Language code: logit["CovarianceMatrix"]

Use the observed information matrix instead:

Wolfram Language code: logit["CovarianceMatrix", CovarianceEstimatorFunction -> "ObservedInformation"]

DispersionEstimatorFunction (1)

Fit a logit model:

Wolfram Language code: data = {{1, 0}, {1, 1}, {1, 1}, {2, 0}, {2, 0}, {2, 1}, {2, 1}, {2, 1}, {2, 1}};

Wolfram Language code: logit = LogitModelFit[data, x, x]

Compute the covariance matrix:

Wolfram Language code: logit["CovarianceMatrix"]

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

Wolfram Language code: logit["CovarianceMatrix", DispersionEstimatorFunction -> "PearsonChiSquare"]

IncludeConstantBasis (1)

Fit a logit model:

Wolfram Language code: data = {{0, 1}, {1, 0}, {3, 1}, {5, 0}};

Wolfram Language code: LogitModelFit[data, x, x]//Normal

Fit the model with no constant term:

Wolfram Language code: LogitModelFit[data, x, x, IncludeConstantBasis -> False]//Normal

LinearOffsetFunction (1)

Fit data to a logit model:

Wolfram Language code: data = {{0, 1}, {1, 0}, {3, 1}, {5, 0}};

Wolfram Language code: LogitModelFit[data, x, x]//Normal

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

Wolfram Language code: LogitModelFit[data, x, x, LinearOffsetFunction -> (Sqrt[#]&)]//Normal

NominalVariables (1)

Wolfram Language code: data = {{a, 0, 0}, {b, 2, 1}, {a, 2, 0}, {b, 0, 1}, {a, 2, 1}, {b, 0, 0}};

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

Wolfram Language code: nom = LogitModelFit[data, {x, y}, {x, y}, NominalVariables -> x]

Wolfram Language code: Normal[nom]

Treat both variables as nominal:

Wolfram Language code: LogitModelFit[data, {x, y}, {x, y}, NominalVariables -> All]//Normal

Weights (1)

Fit a model using equal weights:

Wolfram Language code: data = {{0, 1}, {1, 0}, {3, 1}, {5, 0}};

Wolfram Language code: LogitModelFit[data, x, x]//Normal

Give explicit weights for the data points:

Wolfram Language code: LogitModelFit[data, x, x, Weights -> {1, (1/2), (1/3), (1/4)}]//Normal

WorkingPrecision (1)

Use WorkingPrecision to get higher precision in parameter estimates:

Wolfram Language code: data = Table[{x, (10 - x) / 10}, {x, 10}]

Wolfram Language code: logit = LogitModelFit[data, x, x, WorkingPrecision -> 30]

Obtain the fitted function:

Wolfram Language code: logit["BestFit"]

Reduce the precision in property computations after the fitting:

Wolfram Language code: logit["BestFit", WorkingPrecision -> MachinePrecision]

Properties & Relations (4)

A default "Binomial" model from GeneralizedLinearModelFit is equivalent to the model for LogitModelFit :

Wolfram Language code: data = Table[{i, RandomReal[{i - 1, i}] / 10}, {i, 10}];

Wolfram Language code: GeneralizedLinearModelFit[data, x, x, ExponentialFamily -> "Binomial"]//Normal

Wolfram Language code: LogitModelFit[data, x, x]//Normal

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

Wolfram Language code: GeneralizedLinearModelFit[data, x, x, ExponentialFamily -> "Binomial", LinkFunction -> "ProbitLink"]//Normal

Wolfram Language code: ProbitModelFit[data, x, x]//Normal

LogitModelFit assumes binomially distributed responses:

Wolfram Language code: data = Table[{i, RandomReal[{i - 1, i}] / 10}, {i, 10}];

Wolfram Language code: lm = LogitModelFit[data, x, x]

NonlinearModelFit assumes normally distributed responses:

Wolfram Language code: nm = NonlinearModelFit[data, 1 / (1 + Exp[a + b x]), {a, b}, x]

The fits are not identical:

Wolfram Language code: {Normal[lm], Normal[nm]}

Wolfram Language code: Plot[{lm[x], nm[x]}, {x, 1, 5}]

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

Wolfram Language code:

ts1 = TemporalData[TimeSeries, {{{0.7758532198866042, 0.20426632015371096, 0.2353690565215641, 
 0.4448009174525732, 0.5081575045542254, 0.5371908040104083, 0.6223323745794831, 
 0.669522447912184, 0.7204685251554224, 0.7633871517003976}}, {{0, 9, 1}}, 1, {"Continuous", 1}, 
 {"Discrete", 1}, 1, {ResamplingMethod -> {"Interpolation", InterpolationOrder -> 1}}}, False, 
 10.1];

Wolfram Language code: ts1["Times"]

Wolfram Language code: LogitModelFit[ts1, x, x]

Rescale the time stamps and fit again:

Wolfram Language code: ts2 = TimeSeriesRescale[ts1, {.1, 2}]

Wolfram Language code: ts2["Times"]

Wolfram Language code: LogitModelFit[ts2, x, x]

Find fit for the values:

Wolfram Language code: LogitModelFit[ts1["Values"], x, x]

LogitModelFit acts pathwise on a multipath TemporalData :

Wolfram Language code:

LogitModelFit[TemporalData[Automatic, {{{0.009753347177267509, 0.18455899093641098, 0.20260886672737863, 
 0.37977270854029854, 0.45927203330240407, 0.5010683896315415, 0.6468842072138713, 
 0.7452632634563319, 0.8683334166368966, 0.9876532168951402}, 
 ... 2138713, 0.7452632634563319, 
 0.8683334166368966, 0.9876532168951402}}, {{0, 9, 1}, {0.1, 2., 0.2111111111111111}}, 2, 
 {"Continuous", 2}, {"Discrete", 2}, 1, 
 {ResamplingMethod -> {"Interpolation", InterpolationOrder -> 1}}}, False, 10.1], x, x]

Possible Issues (1)

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

Wolfram Language code: LogitModelFit[Range[5], x, x]

Wolfram Language code: LogitModelFit[Table[{i, 2i}, {i, 10}], x, x]

Tech Notes

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Statistical Model Analysis

Related Guides

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Statistical Model Analysis

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Life Sciences & Medicine: Data & Computation

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Supervised Machine Learning

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Matrix-Based Minimization

History

Introduced in 2008 (7.0) | Updated in 2023 (13.3) ▪ 2025 (14.2)

Wolfram Research (2008), LogitModelFit, Wolfram Language function, https://reference.wolfram.com/language/ref/LogitModelFit.html (updated 2025).

Text

Wolfram Research (2008), LogitModelFit, Wolfram Language function, https://reference.wolfram.com/language/ref/LogitModelFit.html (updated 2025).

CMS

Wolfram Language. 2008. "LogitModelFit." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2025. https://reference.wolfram.com/language/ref/LogitModelFit.html.

APA

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

BibTeX

@misc{reference.wolfram_2026_logitmodelfit, author="Wolfram Research", title="{LogitModelFit}", year="2025", howpublished="\url{https://reference.wolfram.com/language/ref/LogitModelFit.html}", note=[Accessed: 13-August-2026]}

BibLaTeX

@online{reference.wolfram_2026_logitmodelfit, organization={Wolfram Research}, title={LogitModelFit}, year={2025}, url={https://reference.wolfram.com/language/ref/LogitModelFit.html}, note=[Accessed: 13-August-2026]}

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LogitModelFit

Details and Options

Examples

Basic Examples (1)

Scope (13)

Data (6)

Properties (7)

Data & Fitted Functions (1)

Residuals (1)

Dispersion and Deviances (1)

Parameter Estimation Diagnostics (1)

Influence Measures (1)

Prediction Values (1)

Goodness-of-Fit Measures (1)

Generalizations & Extensions (1)

Options (8)

ConfidenceLevel (1)

CovarianceEstimatorFunction (1)

DispersionEstimatorFunction (1)

IncludeConstantBasis (1)

LinearOffsetFunction (1)

NominalVariables (1)

Weights (1)

WorkingPrecision (1)

Properties & Relations (4)

Possible Issues (1)

See Also

Tech Notes

Related Guides

History

Text

CMS

APA

BibTeX

BibLaTeX