The Nonlinear Estimation module allows the user to fit essentially any type of nonlinear model.
One of the unique features of this module is that (unlike traditional nonlinear estimation
programs) it does not impose any limits on the size of data files that it can process.
Estimation Methods.
The models can be fit using least squares or maximum-likelihood estimation, or any user-specified
loss function. When using the least-squares criterion, the very efficient Levenberg-Marquardt and
Gauss-Newton algorithms can be used to estimate the parameters for arbitrary linear and nonlinear
regression problems.
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For large datasets or for difficult nonlinear regression problems, when using the
least-squares criterion, this is the recommended method for computing precise parameter estimates.
When using arbitrary loss functions, the user can choose from among four very different, powerful
estimation procedures (quasi-Newton, Simplex, Hooke-Jeeves pattern moves, and Rosenbrock pattern
search method of rotating coordinates) so that stable parameter estimates can be obtained in
practically all cases, and even in extremely numerically-demanding conditions.
Models
The user can specify any type of model by typing in the respective equation into an equation editor.
The equations may include logical operators; thus, discontinuous (piecewise) regression models and
models including indicator variables can also be estimated. The equations may also include a wide
selection of distribution functions and cumulative distribution functions (Beta, Binomial, Cauchy,
Chi-square, Exponential, Extreme value, F, Gamma, Geometric, Laplace, Logistic, Normal, Log-Normal,
Pareto, Poisson, Rayleigh, t (Student), or Weibull distribution). The user has full control over all
aspects of the estimation procedure (e.g., starting values, step sizes, convergence criteria, etc.).
The most common nonlinear regression models are predefined in the Nonlinear Estimation module, and
can be chosen simply as menu options. Those regression models include stepwise Probit and Logit
regression, the exponential regression model, and linear piecewise (break point) regression. Note
that STATISTICA also includes implementations of powerful algorithms for fitting generalized linear
models, including probit and multinomial logit models, and generalized additive models; see the
respective descriptions for additional details.
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Results
In addition to various descriptive statistics, standard results of the nonlinear estimation include
the parameter estimates and their standard errors (computed independently of the estimation itself,
via finite differencing to optimize precision; see the Validation Benchmarks ); the variance/
covariance matrix of parameter estimates, the predicted values, residuals, and appropriate measures
of goodness-of-fit (e.g., log-likelihood of estimated/null models and Chi-square test of difference,
proportion of variance accounted for, classification of cases and odds-ratios for Logit and Probit
models, etc.). Predicted and residual values can be appended to the data file for further analyses.
For Probit and Logit models, the incremental fit is also automatically computed when adding or
deleting parameters from the regression model.
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Graphs
All output is integrated with extensive selections of graphs, including interactively-adjustable 2D
and 3D (surface) arbitrary function fitting graphs which allow the user to visualize the quality of
the fit and identify outliers or ranges of discrepancy between the model and the data; the user can
interactively adjust the equation of the fitted function (as shown in the graph) without re-processing
the data and visualize practically all aspects of the nonlinear fitting process). Many other
specialized graphs are provided to evaluate the fitting process and visualize the results, such as
histograms of all selected variables and residual values, scatterplots of observed versus predicted
values and predicted versus residual values, normal and half-normal probability plots of residuals,
and many others.
This module offers a complete implementation of log-linear modeling procedures for multi-way frequency
tables. Note that Statistica also includes the Generalized Linear Models module, which provides options
for analyzing binomial and multinomial logit models with coded ANOVA/ANCOVA-like designs. In the
Log-Linear Analysis module, the user can analyze up to 7-way tables in a single run. Both complete
and incomplete tables (with structural zeros) can be analyzed. Frequency tables can be computed from
raw data, or may be entered directly into the program. The Log-Linear Analysis module provides a
comprehensive selection of advanced modeling procedures in an interactive and flexible environment
that greatly facilitates exploratory and confirmatory analyses of complex tables. The user may at
all times review the complete observed table as well as marginal tables, and fitted (expected) values,
and may evaluate the fit of all partial and marginal association models or select specific models
(marginal tables) to be fitted to the observed data. The program also offers an intelligent automatic
model selection procedure that first determines the necessary order of interaction terms required for
a model to fit the data, and then, through backwards elimination, determines the best sufficient model
to satisfactorily fit the data . The standard output includes G-square, the standard Pearson Chi-square
with the appropriate degrees of freedom and significance levels, the observed and expected tables, marginal
tables, and other statistics. Graphics options available in the Log-linear module include a variety of
2D and 3D graphs designed to visualize 2-way and multi-way frequency tables (including interactive,
user-controlled cascades of categorized histograms and 3D histograms revealing "slices" of multi-way
tables), plots of observed and fitted frequencies, plots of various residuals (standardized, components of
Maximum-Likelihood Chi-square, Freeman-Tukey deviates, etc.), and many others.
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