@node REGRESSION
@section REGRESSION

@cindex regression
@cindex linear regression
The @cmd{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 @math{k}th 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 @dfn{noise} terms with mean zero and common variance.
The noise, or @dfn{error} terms are unobserved.
This relationship is called the @dfn{linear model}.
@end itemize

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

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

@node Syntax
@subsection Syntax

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

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

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

All other subcommands are optional:

The @subcmd{STATISTICS} subcommand specifies the statistics to be displayed:

@table @subcmd
@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 @subcmd{SAVE} subcommand causes @pspp{} to save the residuals or predicted
values from the fitted
model to the active dataset. @pspp{} will store the residuals in a variable
called @samp{RES1} if no such variable exists, @samp{RES2} if @samp{RES1} 
already exists,
@samp{RES3} if @samp{RES1} and @samp{RES2} already exist, etc. It will
choose the name of
the variable for the predicted values similarly, but with @samp{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 dataset.

@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
