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:

  • The data set contains \(n\) observations of a dependent variable, say \(y_1,...,y_n\), and \(n\) observations of one or more explanatory variables. Let \(x_{11}, x_{12}, ..., x_{1n}\) denote the \(n\) observations of the first explanatory variable; \(x_{21},...,x_{2n}\) denote the \(n\) observations of the second explanatory variable; \(x_{k1},...,x_{kn}\) denote the \(n\) observations of the kth explanatory variable.

  • The dependent variable \(y\) has the following relationship to the explanatory variables: \(y_i = b_0 + b_1 x_{1i} + ... + b_k x_{ki} + z_i\) where \(b_0, b_1, ..., b_k\) are unknown coefficients, and \(z_1,...,z_n\) 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".

    The REGRESSION procedure estimates the coefficients \(b_0,...,b_k\) and produces output relevant to inferences for the linear model.

Syntax

REGRESSION
        /VARIABLES=VAR_LIST
        /DEPENDENT=VAR_LIST
        /STATISTICS={ALL, DEFAULTS, R, COEFF, ANOVA, BCOV, CI[CONF, TOL]}
        { /ORIGIN | /NOORIGIN }
        /SAVE={PRED, RESID}

The REGRESSION procedure reads the active dataset 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 which statistics are to be displayed. The following keywords are accepted:

  • ALL
    All of the statistics below.
  • R
    The ratio of the sums of squares due to the model to the total sums of squares for the dependent variable.
  • COEFF
    A table containing the estimated model coefficients and their standard errors.
  • CI (CONF)
    This item is only relevant if COEFF has also been selected. It specifies that the confidence interval for the coefficients should be printed. The optional value CONF, which must be in parentheses, is the desired confidence level expressed as a percentage.
  • ANOVA
    Analysis of variance table for the model.
  • BCOV
    The covariance matrix for the estimated model coefficients.
  • TOL
    The variance inflation factor and its reciprocal. This has no effect unless COEFF is also given.
  • DEFAULT
    The same as if R, COEFF, and ANOVA had been selected. This is what you get if the /STATISTICS command is not specified, or if it is specified without any parameters.

The ORIGIN and NOORIGIN subcommands are mutually exclusive. ORIGIN indicates that the regression should be performed through the origin. You should use this option if, and only if you have reason to believe that the regression does indeed pass through the origin -- that is to say, the value b_0 above, is zero. The default is NOORIGIN.

The 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 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. When SAVE is used, PSPP ignores TEMPORARY, treating temporary transformations as permanent.

Example

The following PSPP syntax will generate the default output and save the predicted values and residuals to the active dataset.

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.