Solutions Returned by the Regression Commands
This help page describes the forms of solutions returned by the regression commands in the Statistics package. See the Statistics/Regression help page for an overview of the regression commands.
The Default Solution
The output Option
The Solution Module
Each data-fitting command in the Statistics package attempts to fit a model function to a given set of data by determining values of the model parameters that minimize the least-squares error. The model function may be specified in algebraic, operator or Matrix forms, with algebraic form being the most commonly used form. (See Statistics/Regression/InputForms for more details about the input forms.)
When the model function is given in algebraic form, then an algebraic expression, representing the model function with the final parameter values, is returned by default. Otherwise, the model parameters are returned in a Vector.
The output option can be used to control the form of the returned solution. If the output=solutionmodule option is provided, then a solution module, as described in the following section, is returned. The output option can also take as value a single name (or string) or a list of names (or strings), and the associated results are returned. The acceptable names are described in the Results section below.
When the output=solutionmodule option is provided to one of the fitting commands, a module containing two exports, Settings and Results, is returned. Each export is a procedure that accepts a name (or string) or a list of names (or strings) and returns the associated values. The acceptable names are described in the Settings and Results sections below. For example, if the module is assigned to variable m, then the call m:-Results(residualsumofsquares) returns the sum of squares of the final residuals.
If no argument is provided to the Settings or Results export, then all available information is returned as a list of equations nm=val, where nm is a name and val is the associated value.
The following names can be given to the Settings export of the solution module.
These settings are available for the linear fitting routines only.
confidencelevel -- Confidence level.
svdtolerance -- Tolerance controlling when a singular-value decomposition takes place.
These settings are available for the nonlinear fitting routines only.
initialpoint -- Initial point used by the nonlinear optimization solver.
parameternames -- The names of the parameters, if they are available.
The following names can be given to the Results export of the solution module or used as a value of the output option.
These settings are available for both the linear and nonlinear fitting routines.
leastsquaresfunction -- The model function containing the computed least-squares parameters. This result is available only when the input is in algebraic form.
parametervalues -- The values of the computed parameters. If the input is in algebraic form, then a list of equations of the form nm=val is returned, where nm is the name of the parameter and val is its value. Otherwise, a Vector of parameter values is returned.
parametervector -- A Vector containing the values of the computed parameters. If the model function is given in algebraic form, then the parameternames option must be provided so that the order of the values in the parameter Vector is specified.
residualsumofsquares -- The sum of squares of the residuals.
residuals -- The residual Vector.
degreesoffreedom -- The degrees of freedom in the least-squares problem.
residualmeansquare -- The residual mean square (residual sum of squares divided by degrees of freedom).
residualstandarddeviation -- The residual standarddeviation (square root of residual mean square).
standarderrors -- Standard errors for the parameters.
confidenceintervals -- Confidence intervals for the parameters using the confidencelevel setting.
leverages -- Leverages.
variancecovariancematrix -- Variance-covariance Matrix.
The following are available only if certain conditions are met by the solution of the least-squares problem.
internallystandardizedresiduals -- Internally standardized residuals.
externallystandardizedresiduals -- Externally standardized residuals.
CookDstatistic -- Cook's D statistic (also known as Cook's distance).
AtkinsonTstatistic -- Atkinson's T statistic.
rsquared -- The coefficient of determination, often used to indicate how well data fits a model.
rsquaredadjusted -- The adjusted coefficient of determination that accounts for the number of variables in the model.
tprobability -- The p-value for the hypothesis test for which the t value is the test statistic.
tvalue -- The value of the t statistic for testing whether the corresponding regression coefficient is different than 0.
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