BUILT-IN MATHEMATICA SYMBOL
GeneralizedLinearModelFit
GeneralizedLinearModelFit[{y1, y2, ...}, {f1, f2, ...}, x]
constructs a generalized linear model of the form 
that fits the 
for successive x values 1, 2, ....
GeneralizedLinearModelFit[{{x11, x12, ..., y1}, {x21, x22, ..., y2}, ...}, {f1, f2, ...}, {x1, x2, ...}]
constructs a generalized linear model of the form 
where the 
depend on the variables 
.
- GeneralizedLinearModelFit returns a symbolic FittedModel object to represent the generalized 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 GeneralizedLinearModelFit at a particular point
, ... can be found from 
.
- With data in the form
, the number of coordinates 
, 
, ... should correspond to the number of variables 
.
- Data in the form
is equivalent to data in the form 
.
- GeneralizedLinearModelFit produces a generalized linear model of the form
under the assumption that the original 
are independent observations following an exponential family distribution with mean 
and the function 
being an invertible link function.
- GeneralizedLinearModelFit takes the following options:
-
- With the setting IncludeConstantBasis->False, a model of the form
is fitted.
- With the setting LinearOffsetFunction->h, a model of the form
is fitted.
- With ConfidenceLevel->p, probability-p confidence intervals are computed for parameter and prediction intervals.
- With the setting DispersionEstimatorFunction->f, the common dispersion is estimated by
where 
is the list of observations, 
is the list of predicted values, and 
is the list of weights for the measurements 
.
- Possible settings for ExponentialFamily include:
, 
, 
, 
, 
, or 
.
- 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 |
| "LinearPredictor" | fitted linear combination |
| "Response" | response values in the input data |
- Properties related to dispersion and model deviances include:
-
| "Deviances" | deviances |
| "DevianceTable" | deviance table |
| "DevianceTableDegreesOfFreedom" | degrees of freedom differences from the table |
| "DevianceTableDeviances" | deviance differences from the table |
| "DevianceTableEntries" | unformatted array of values from the table |
| "DevianceTableResidualDegreesOfFreedom" | residual degrees of freedom from the table |
| "DevianceTableResidualDeviances" | residual deviances from the table |
| "EstimatedDispersion" | estimated dispersion parameter |
| "NullDeviance" | deviance for the null model |
| "NullDegreesOfFreedom" | degrees of freedom for the null model |
| "ResidualDeviance" | difference between the deviance for the fitted model and the deviance for the full model |
| "ResidualDegreesOfFreedom" | difference between the model degrees of freedom and null degrees of freedom |
- Types of residuals include:
-
| "AnscombeResiduals" | Anscombe residuals |
| "DevianceResiduals" | deviance residuals |
| "FitResiduals" | difference between actual and predicted responses |
| "LikelihoodResiduals" | likelihood residuals |
| "PearsonResiduals" | Pearson residuals |
| "StandardizedDevianceResiduals" | standardized deviance residuals |
| "StandardizedPearsonResiduals" | standardized Pearson residuals |
| "WorkingResiduals" | working residuals |
- Properties and diagnostics for parameter estimates include:
-
| "CorrelationMatrix" | asymptotic parameter correlation matrix |
| "CovarianceMatrix" | asymptotic parameter covariance matrix |
| "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 |
| "ParameterTableEntries" | unformatted array of values from the table |
| "ParameterErrors" | standard errors for parameter estimates |
| "ParameterPValues" | p-values for parameter z-statistics |
| "ParameterTable" | table of fitted parameter information |
| "ParameterZStatistics" | z-statistics for parameter estimates |
- Properties related to influence measures include:
-
| "CookDistances" | list of Cook distances |
| "HatDiagonal" | diagonal elements of the hat matrix |
- Properties of predicted values include:
-
| "PredictedResponse" | fitted values for the data |
- Properties that measure goodness of fit include:
-
| "AdjustedLikelihoodRatioIndex" | Ben-Akiva and Lerman's adjusted likelihood ratio index |
| "AIC" | Akaike Information Criterion |
| "BIC" | Bayesian Information Criterion |
| "CoxSnellPseudoRSquared" | Cox and Snell's pseudo  |
| "CraggUhlerPseudoRSquared" | Cragg and Uhler's pseudo  |
| "EfronPseudoRSquared" | Efron's pseudo  |
| "LikelihoodRatioIndex" | McFadden's likelihood ratio index |
| "LikelihoodRatioStatistic" | likelihood ratio |
| "LogLikelihood" | log likelihood for the fitted model |
| "PearsonChiSquare" | Pearson's  statistic |
- In GeneralizedLinearModelFit[m, v], 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 GeneralizedLinearModelFit[{m, v}, {f1, f2, ...}].
Define a dataset:
Fit a log-linear Poisson model to the data:
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See the functional forms of the model:
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Evaluate the model at a point:
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Plot the data points and the models:
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Compute and plot the deviance residuals for the model:
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