Module Library

REGRESS Call

RUN REGRESS (x, y, name, <tval>, <l1>, <l2>, <l3>);

The REGRESS module performs ordinary least squares regression. It is primarily used for demonstration purposes.

The inputs to the REGRESS subroutine are as follows:

x

is an $n \times m$ numeric matrix, where m is the number of variables and n is the number of data points.

y

is an $n \times 1$ response vector.

name

is an $m \times 1$ matrix of variable names.

tval

is an optional t-value.

l1, l2, l3

are optional $1 \times m$ vectors that specify linear combinations of coefficients for hypothesis testing.

The design matrix is given by x, and y is the response vector. The name vector identifies each of the variables. If you specify a t-value, the module prints a table of observed and predicted values, residuals, hat diagonal, and confidence limits for the mean and predicted values. If you also specify linear combinations with l1, l2, and l3, the module performs the hypothesis test $\mb{H:} ~ l^{\prime }b = 0$, where b is the vector of parameter estimates. An example follows: 

/* U.S. Population for decades beginning 1790, in millions */
name = { "Intercept", "Decade", "Decade**2" };
x = { 1  1  1,   1  2  4,   1  3  9,   1  4 16,
      1  5 25,   1  6 36,   1  7 49,   1  8 64 };
y = {  3.929,    5.308,     7.239,     9.638,
      12.866,   17.069,    23.191,    31.443 };
/* 5 dof at 0.025 level to get 95% confidence interval */
tval = quantile("T", 1-0.025, 5);
l1 = { 0 1 0 };   /* test hypothesis lb=0 for linear coef */
l2 = { 0 1 0,     /* test hypothesis lb=0 for linear,quad */
       0 0 1 };
l3 = { 0 1 1 };   /* test hypothesis lb=0 for linear+quad */
run regress( x, y, name, tval, l1, l2, l3 );

Figure 25.2: Regression Analysis

name b stdb t probt
Intercept 5.0693393 0.9655939 5.2499702 0.0033263
Decade -1.109935 0.4923003 -2.254588 0.0738509
Decade**2 0.5396369 0.0533975 10.10604 0.0001625

Covariance of Estimates
  Intercept Decade Decade**2
Intercept 0.9324 -0.436 0.0428
Decade -0.436 0.2424 -0.026
Decade**2 0.0428 -0.026 0.0029

Correlation of Estimates
  Intercept Decade Decade**2
Intercept 1 -0.918 0.8295
Decade -0.918 1 -0.976
Decade**2 0.8295 -0.976 1

Predicted values, Residuals, and Limits

y yhat resid h lowerm upperm lower upper
3.929 4.499 -0.57 0.7083 3.0017 5.9964 2.1737 6.8244
5.308 5.008 0.3 0.2798 4.067 5.949 2.9954 7.0207
7.239 6.5963 0.6427 0.2321 5.7391 7.4535 4.6214 8.5711
9.638 9.2638 0.3742 0.2798 8.3228 10.205 7.2511 11.276
12.866 13.011 -0.145 0.2798 12.07 13.952 10.998 15.023
17.069 17.837 -0.768 0.2321 16.979 18.694 15.862 19.812
23.191 23.742 -0.551 0.2798 22.801 24.683 21.729 25.755
31.443 30.727 0.7164 0.7083 29.229 32.224 28.401 33.052

Test Hypothesis that l b = 0

  f dfn dfe prob
for Linear Coef 5.0831686 1 5 0.0739

  f dfn dfe prob
for Linear,Quad Coef 666.51095 2 5 <.0001

  f dfn dfe prob
for Linear+Quad Coef 1.6774629 1 5 0.2518