This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# LinearModelFit

 LinearModelFitconstructs a linear model of the form that fits the for successive x values 1, 2, .... LinearModelFitconstructs a linear model of the form where the depend on the variables . LinearModelFitconstructs a linear model from the design matrix m and response vector v.
• LinearModelFit returns a symbolic FittedModel object to represent the linear model it constructs. The properties and diagnostics of the model can be obtained from model["property"].
• The value of the best-fit function from LinearModelFit at a particular point , ... can be found from .
• With data in the form , the number of coordinates , , ... should equal the number of variables .
• Data in the form is equivalent to data in the form .
• LinearModelFit produces a linear model of the form under the assumption that the original are independent normally distributed with mean and common standard deviation.
 ConfidenceLevel 95/100 confidence level to use for parameters and predictions IncludeConstantBasis True whether to include a constant basis function LinearOffsetFunction None known offset in the linear predictor NominalVariables None variables considered as nominal or categorical VarianceEstimatorFunction Automatic function for estimating the error variance Weights Automatic weights for data elements WorkingPrecision Automatic precision used in internal computations
• With ConfidenceLevel->p, probability-p confidence intervals are computed for parameter and prediction intervals.
• With the setting Weights, the error variance for is assumed to be . By default, unit weights are used.
• With the setting VarianceEstimatorFunction->f, the variance is estimated by , where is the list of residuals and is the list of weights for the measurements .
• Using VarianceEstimatorFunction and Weights, is treated as the known uncertainty of measurement and parameter standard errors are effectively computed only from the weights.
• Properties related to data and the fitted function obtained using model["property"] include:
 "BasisFunctions" list of basis functions "BestFit" fitted function "BestFitParameters" parameter estimates "Data" the input data or design matrix and response vector "DesignMatrix" design matrix for the model "Function" best fit pure function "Response" response values in the input data
• Types of residuals include:
 "FitResiduals" difference between actual and predicted responses "StandardizedResiduals" fit residuals divided by the standard error for each residual "StudentizedResiduals" fit residuals divided by single deletion error estimates
• Properties related to the sum of squared errors include:
 "ANOVATable" analysis of variance table "ANOVATableDegreesOfFreedom" degrees of freedom from the ANOVA table "ANOVATableEntries" unformatted array of values from the table "ANOVATableFStatistics" F-statistics from the table "ANOVATableMeanSquares" mean square errors from the table "ANOVATablePValues" -values from the table "ANOVATableSumsOfSquares" sums of squares from the table "CoefficientOfVariation" response mean divided by the estimated standard deviation "EstimatedVariance" estimate of the error variance "PartialSumOfSquares" changes in model sum of squares as nonconstant basis functions are removed "SequentialSumOfSquares" the model sum of squares partitioned componentwise
• Properties and diagnostics for parameter estimates include:
 "CorrelationMatrix" parameter correlation matrix "CovarianceMatrix" parameter covariance matrix "EigenstructureTable" eigenstructure of the parameter correlation matrix "EigenstructureTableEigenvalues" eigenvalues from the table "EigenstructureTableEntries" unformatted array of values from the table "EigenstructureTableIndexes" index values from the table "EigenstructureTablePartitions" partitioning from the table "ParameterTable" table of fitted parameter information "ParameterConfidenceIntervals" parameter confidence intervals "ParameterConfidenceIntervalTable" table of confidence interval information for the fitted parameters "ParameterConfidenceIntervalTableEntries" unformatted array of values from the table "ParameterConfidenceRegion" ellipsoidal parameter confidence region "ParameterErrors" standard errors for parameter estimates "ParameterPValues" -values for parameter -statistics "ParameterTableEntries" unformatted array of values from the table "ParameterTStatistics" -statistics for parameter estimates "VarianceInflationFactors" list of inflation factors for the estimated parameters
• Properties related to influence measures include:
 "BetaDifferences" DFBETAS measures of influence on parameter values "CatcherMatrix" catcher matrix "CookDistances" list of Cook distances "CovarianceRatios" COVRATIO measures of observation influence "DurbinWatsonD" Durbin-Watson -statistic for autocorrelation "FitDifferences" DFFITS measures of influence on predicted values "FVarianceRatios" FVARATIO measures of observation influence "HatDiagonal" diagonal elements of the hat matrix "SingleDeletionVariances" list of variance estimates with the data point omitted
• Properties of predicted values include:
 "MeanPredictionBands" confidence bands for mean predictions "MeanPredictionConfidenceIntervals" confidence intervals for the mean predictions "MeanPredictionConfidenceIntervalTable" table of confidence intervals for the mean predictions "MeanPredictionConfidenceIntervalTableEntries" unformatted array of values from the table "MeanPredictionErrors" standard errors for mean predictions "PredictedResponse" fitted values for the data "SinglePredictionBands" confidence bands based on single observations "SinglePredictionConfidenceIntervals" confidence intervals for the predicted response of single observations "SinglePredictionConfidenceIntervalTable" table of confidence intervals for the predicted response of single observations "SinglePredictionConfidenceIntervalTableEntries" unformatted array of values from the table "SinglePredictionErrors" standard errors for the predicted response of single observations
• Properties that measure goodness of fit include:
 "AdjustedRSquared" adjusted for the number of model parameters "AIC" Akaike Information Criterion "BIC" Bayesian Information Criterion "RSquared" coefficient of determination
• In LinearModelFit, the design matrix m is formed from the values of basis functions at data points in the form . The response vector v is the list of responses .
• 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 can be specified using the form LinearModelFit.
Fit a linear model to some data:
Obtain the functional form:
Evaluate the model at a point:
Visualize the fitted function with the data:
Plot the residuals:
Fit a linear model to some data:
 Out[2]=
Obtain the functional form:
 Out[3]=
Evaluate the model at a point:
 Out[4]=
Visualize the fitted function with the data:
 Out[5]=
 Out[6]=
Plot the residuals:
 Out[7]=
 Scope   (11)
Fit a model of more than one variable:
Fit data to a linear combination of nonlinear functions of predictor variables:
Fit a model with categorical predictor variables:
Obtain an analysis of variance table for the model:
Extract the unformatted array of values:
Extract the mean squares values from the table:
Fit a model given a design matrix and response vector:
See the functional form:
Fit the model referring to the basis functions as x and y:
Obtain a list of available properties for a linear model:
Fit a linear 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 scaled residuals in stem plots:
Plot the absolute differences between the standardized and Studentized residuals:
Fit a linear model to some data:
Extract the estimated error variance and coefficient of variation:
Obtain an analysis of variance table for the model:
Get the F-statistics from the table:
Extract the numeric entries from the table:
Obtain a formatted table of parameter information:
Obtain an analysis of variance table for the model:
Get the unformatted array of values from the table:
Fit some data containing extreme values to a linear model:
Use single deletion variances to check the impact on the error variance of removing each point:
Check Cook distances to identify highly influential points:
Use DFFITS values to assess the influence of each point on the fitted values:
Use DFBETAS values to assess the influence of each point on each estimated parameter:
Fit a linear model:
Plot the predicted values against the observed values:
Obtain tabular results for mean and single prediction confidence intervals:
Get the single prediction intervals from the table:
Extract 99% mean prediction bands:
Obtain a table of goodness-of-fit measures for a linear model:
Compute goodness-of-fit measures for all possible linear submodels:
Rank the models by :
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   (7)
The default gives 95% confidence intervals:
Set the level to 90% within FittedModel:
Fit a simple linear regression model:
Fit the linear model with intercept zero:
Fit data to a linear model:
Fit data to a linear model with a known Sqrt[x] term:
Fit data treating the first variable as a nominal variable:
Treat both variables as nominal:
Use the default unbiased estimate of error variance:
Assume a known error variance:
Estimate the variance by the mean squared error:
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:
 Applications   (6)
Fit the first 100 primes to a linear model:
Visualize the fit:
The systematic trend in the residuals violates the assumption of independent normal errors:
Fit a linear model of multiple variables:
Visually inspect the residuals by data point:
Plot the residuals against each predictor variable:
Plot Cook's distances to diagnose leverage:
Find the positions of distances above a given cutoff value:
Extract the associated data points:
Use - plots to check the assumption of normal errors:
Compare standardized residuals to standard normal values:
Do the comparison with Studentized residuals:
Simulate some data with a continuous and a nominal variable:
Fit an analysis of covariance model to the data:
Obtain an analysis of variance table for the model:
Group the data by :
Visualize the grouped data and associated curves:
Use properties to compute additional results:
Extract the design matrix and residuals:
Compute White's heteroskedasticity-consistent covariance estimate:
Compare with the covariance assuming homoskedasticity:
Compare standard errors based on the two covariance estimates:
Perform a Breusch-Pagan test:
Fit a model:
Fit the squared errors to a model with the same predictors:
Compute the Breusch-Pagan test statistic:
Compute the -value:
DesignMatrix constructs the design matrix used by LinearModelFit:
By default LinearModelFit and GeneralizedLinearModelFit fit equivalent models:
LinearModelFit fits linear models assuming normally distributed errors:
NonlinearModelFit fits nonlinear models assuming normally distributed errors:
Fit and LinearModelFit fit equivalent models: