@node REGRESSION
@comment  node-name,  next,  previous,  up
@section REGRESSION

@cindex regression
@cindex linear regression
The REGRESSION procedure fits linear models to data via least-squares
estimation. The procedure is appropriate for data which satisfy those
assumptions typical in linear regression:

@itemize @bullet
@item The data set contains @math{n} observations of a dependent variable, say
@math{Y_1,@dots{},Y_n}, and @math{n} observations of one or more explanatory
variables. Let @math{X_{11}, X_{12}}, @dots{}, @math{X_{1n}} denote the @math{n} observations of the
first explanatory variable; @math{X_{21}},@dots{},@math{X_{2n}} denote the @math{n} observations of the
second explanatory variable; @math{X_{k1}},@dots{},@math{X_{kn}} denote the @math{n} observations of the kth
explanatory variable.

@item The dependent variable @math{Y} has the following relationship to the 
explanatory variables:
@math{Y_i = b_0 + b_1 X_{1i} + ... + b_k X_{ki} + Z_i} 
where @math{b_0, b_1, @dots{}, b_k} are unknown
coefficients, and @math{Z_1,@dots{},Z_n} are independent, normally
distributed ``noise'' terms with mean zero and common variance. The noise, or
``error'' terms are unobserved. This relationship is called the
``linear model.''
@end itemize

The REGRESSION procedure estimates the coefficients
@math{b_0,@dots{},b_k} and produces output relevant to inferences for the
linear model. 

@c If you add any new commands, then don't forget to remove the entry in 
@c not-implemented.texi

@menu
* Syntax::                      Syntax definition.
* Examples::                    Using the REGRESSION procedure.
@end menu

@node Syntax
@subsection Syntax

@vindex REGRESSION
@display
REGRESSION
        /VARIABLES=var_list
        /DEPENDENT=var_list
        /STATISTICS=@{ALL, DEFAULTS, R, COEFF, ANOVA, BCOV@}
        /SAVE=@{PRED, RESID@}
@end display

The @cmd{REGRESSION} procedure reads the active file and outputs
statistics relevant to the linear model specified by the user.

The VARIABLES subcommand, which is required, specifies the list of
variables to be analyzed.  Keyword VARIABLES is required. The
DEPENDENT subcommand specifies the dependent variable of the linear
model. The DEPENDENT subcommand is required. All variables listed in
the VARIABLES subcommand, but not listed in the DEPENDENT subcommand,
are treated as explanatory variables in the linear model.

All other subcommands are optional:

The STATISTICS subcommand specifies the statistics to be displayed:

@table @code
@item ALL
All of the statistics below.
@item R
The ratio of the sums of squares due to the model to the total sums of
squares for the dependent variable.
@item COEFF
A table containing the estimated model coefficients and their standard errors.
@item ANOVA
Analysis of variance table for the model.
@item BCOV
The covariance matrix for the estimated model coefficients.
@end table

The SAVE subcommand causes PSPP to save the residuals or predicted
values from the fitted
model to the active file. PSPP will store the residuals in a variable
called RES1 if no such variable exists, RES2 if RES1 already exists,
RES3 if RES1 and RES2 already exist, etc. It will choose the name of
the variable for the predicted values similarly, but with PRED as a
prefix.

@node Examples
@subsection Examples
The following PSPP syntax will generate the default output and save the
predicted values and residuals to the active file.

@example
title 'Demonstrate REGRESSION procedure'.
data list / v0 1-2 (A) v1 v2 3-22 (10).
begin data.
b  7.735648 -23.97588
b  6.142625 -19.63854
a  7.651430 -25.26557
c  6.125125 -16.57090
a  8.245789 -25.80001
c  6.031540 -17.56743
a  9.832291 -28.35977
c  5.343832 -16.79548
a  8.838262 -29.25689
b  6.200189 -18.58219
end data.
list.
regression /variables=v0 v1 v2 /statistics defaults /dependent=v2 
           /save pred resid /method=enter.
@end example
@setfilename ignored
