NAG Library Routine Document

g02ecf (linregm_rssq_stat)


    1  Purpose
    7  Accuracy


g02ecf calculates R2 and Cp-values from the residual sums of squares for a series of linear regression models.


Fortran Interface
Subroutine g02ecf ( mean, n, sigsq, tss, nmod, nterms, rss, rsq, cp, ifail)
Integer, Intent (In):: n, nmod, nterms(nmod)
Integer, Intent (Inout):: ifail
Real (Kind=nag_wp), Intent (In):: sigsq, tss, rss(nmod)
Real (Kind=nag_wp), Intent (Out):: rsq(nmod), cp(nmod)
Character (1), Intent (In):: mean
C Header Interface
#include nagmk26.h
void  g02ecf_ (const char *mean, const Integer *n, const double *sigsq, const double *tss, const Integer *nmod, const Integer nterms[], const double rss[], double rsq[], double cp[], Integer *ifail, const Charlen length_mean)


When selecting a linear regression model for a set of n observations a balance has to be found between the number of independent variables in the model and fit as measured by the residual sum of squares. The more variables included the smaller will be the residual sum of squares. Two statistics can help in selecting the best model.
(a) R2 represents the proportion of variation in the dependent variable that is explained by the independent variables.
R2=Regression Sum of SquaresTotal Sum of Squares,  
where Total Sum of Squares=tss= y-y- 2 (if mean is fitted, otherwise tss=y2) and
Regression Sum of Squares=RegSS=tss-rss, where
rss=residual sum of squares= y-y^ 2.
The R2-values can be examined to find a model with a high R2-value but with small number of independent variables.
(b) Cp statistic.
Cp=rssσ^2 -n-2p,  
where p is the number of arguments (including the mean) in the model and σ^2 is an estimate of the true variance of the errors. This can often be obtained from fitting the full model.
A well fitting model will have Cpp. Cp is often plotted against p to see which models are closest to the Cp=p line.
g02ecf may be called after g02eaf which calculates the residual sums of squares for all possible linear regression models.


Draper N R and Smith H (1985) Applied Regression Analysis (2nd Edition) Wiley
Weisberg S (1985) Applied Linear Regression Wiley


1:     mean – Character(1)Input
On entry: indicates if a mean term is to be included.
A mean term, intercept, will be included in the model.
The model will pass through the origin, zero-point.
Constraint: mean='M' or 'Z'.
2:     n – IntegerInput
On entry: n, the number of observations used in the regression model.
Constraint: n must be greater than 2×pmax, where pmax is the largest number of independent variables fitted (including the mean if fitted).
3:     sigsq – Real (Kind=nag_wp)Input
On entry: the best estimate of true variance of the errors, σ^2.
Constraint: sigsq>0.0.
4:     tss – Real (Kind=nag_wp)Input
On entry: the total sum of squares for the regression model.
Constraint: tss>0.0.
5:     nmod – IntegerInput
On entry: the number of regression models.
Constraint: nmod>0.
6:     ntermsnmod – Integer arrayInput
On entry: ntermsi must contain the number of independent variables (not counting the mean) fitted to the ith model, for i=1,2,,nmod.
7:     rssnmod – Real (Kind=nag_wp) arrayInput
On entry: rssi must contain the residual sum of squares for the ith model.
Constraint: rssitss, for i=1,2,,nmod.
8:     rsqnmod – Real (Kind=nag_wp) arrayOutput
On exit: rsqi contains the R2-value for the ith model, for i=1,2,,nmod.
9:     cpnmod – Real (Kind=nag_wp) arrayOutput
On exit: cpi contains the Cp-value for the ith model, for i=1,2,,nmod.
10:   ifail – IntegerInput/Output
On entry: ifail must be set to 0, -1​ or ​1. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1​ or ​1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this argument, the recommended value is 0. When the value -1​ or ​1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

Error Indicators and Warnings

If on entry ifail=0 or -1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, mean=value.
Constraint: mean='M' or 'Z'.
On entry, nmod=value.
Constraint: nmod>0.
On entry, sigsq=value.
Constraint: sigsq>0.0.
On entry, tss=value.
Constraint: tss>0.0.
On entry: the number of parameters, p, is value and n=value.
Constraint: n2p.
On entry, rssvalue=value and tss=value.
Constraint: rssitss, for all i.
A value of Cp is less than 0.0. This may occur if sigsq is too large or if rss, n or IP are incorrect.
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.


Accuracy is sufficient for all practical purposes.

Parallelism and Performance

g02ecf is not threaded in any implementation.

Further Comments



The data, from an oxygen uptake experiment, is given by Weisberg (1985). The independent and dependent variables are read and the residual sums of squares for all possible models computed using g02eaf. The values of R2 and Cp are then computed and printed along with the names of variables in the models.

Program Text

Program Text (g02ecfe.f90)

Program Data

Program Data (g02ecfe.d)

Program Results

Program Results (g02ecfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017