# NAG Library Routine Document

## 1Purpose

g02ecf calculates ${R}^{2}$ and ${C}_{p}$-values from the residual sums of squares for a series of linear regression models.

## 2Specification

Fortran Interface
 Subroutine g02ecf ( mean, n, tss, nmod, rss, rsq, cp,
 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
#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)

## 3Description

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) ${R}^{2}$ 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 $\text{Total Sum of Squares}={\mathbf{tss}}=\sum {\left(y-\stackrel{-}{y}\right)}^{2}$ (if mean is fitted, otherwise ${\mathbf{tss}}=\sum {y}^{2}$) and $\text{Regression Sum of Squares}=\text{RegSS}={\mathbf{tss}}-{\mathbf{rss}}$, where ${\mathbf{rss}}=\text{residual sum of squares}=\sum {\left(y-\stackrel{^}{y}\right)}^{2}$.
The ${R}^{2}$-values can be examined to find a model with a high ${R}^{2}$-value but with small number of independent variables.
(b) ${C}_{p}$ statistic.
 $Cp=rssσ^2 -n-2p,$
where $p$ is the number of arguments (including the mean) in the model and ${\stackrel{^}{\sigma }}^{2}$ 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 ${C}_{p}\simeq p$. ${C}_{p}$ is often plotted against $p$ to see which models are closest to the ${C}_{p}=p$ line.
g02ecf may be called after g02eaf which calculates the residual sums of squares for all possible linear regression models.

## 4References

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

## 5Arguments

1:     $\mathbf{mean}$ – Character(1)Input
On entry: indicates if a mean term is to be included.
${\mathbf{mean}}=\text{'M'}$
A mean term, intercept, will be included in the model.
${\mathbf{mean}}=\text{'Z'}$
The model will pass through the origin, zero-point.
Constraint: ${\mathbf{mean}}=\text{'M'}$ or $\text{'Z'}$.
2:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of observations used in the regression model.
Constraint: ${\mathbf{n}}$ must be greater than $2×{p}_{\mathrm{max}}$, where ${p}_{\mathrm{max}}$ is the largest number of independent variables fitted (including the mean if fitted).
3:     $\mathbf{sigsq}$ – Real (Kind=nag_wp)Input
On entry: the best estimate of true variance of the errors, ${\stackrel{^}{\sigma }}^{2}$.
Constraint: ${\mathbf{sigsq}}>0.0$.
4:     $\mathbf{tss}$ – Real (Kind=nag_wp)Input
On entry: the total sum of squares for the regression model.
Constraint: ${\mathbf{tss}}>0.0$.
5:     $\mathbf{nmod}$ – IntegerInput
On entry: the number of regression models.
Constraint: ${\mathbf{nmod}}>0$.
6:     $\mathbf{nterms}\left({\mathbf{nmod}}\right)$ – Integer arrayInput
On entry: ${\mathbf{nterms}}\left(\mathit{i}\right)$ must contain the number of independent variables (not counting the mean) fitted to the $\mathit{i}$th model, for $\mathit{i}=1,2,\dots ,{\mathbf{nmod}}$.
7:     $\mathbf{rss}\left({\mathbf{nmod}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{rss}}\left(i\right)$ must contain the residual sum of squares for the $i$th model.
Constraint: ${\mathbf{rss}}\left(\mathit{i}\right)\le {\mathbf{tss}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{nmod}}$.
8:     $\mathbf{rsq}\left({\mathbf{nmod}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{rsq}}\left(\mathit{i}\right)$ contains the ${R}^{2}$-value for the $\mathit{i}$th model, for $\mathit{i}=1,2,\dots ,{\mathbf{nmod}}$.
9:     $\mathbf{cp}\left({\mathbf{nmod}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{cp}}\left(\mathit{i}\right)$ contains the ${C}_{p}$-value for the $\mathit{i}$th model, for $\mathit{i}=1,2,\dots ,{\mathbf{nmod}}$.
10:   $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. 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 $-1\text{​ or ​}1$ 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 $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{mean}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{mean}}=\text{'M'}$ or $\text{'Z'}$.
On entry, ${\mathbf{nmod}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nmod}}>0$.
On entry, ${\mathbf{sigsq}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{sigsq}}>0.0$.
On entry, ${\mathbf{tss}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{tss}}>0.0$.
${\mathbf{ifail}}=2$
On entry: the number of parameters, $p$, is $〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 2p$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{rss}}\left(〈\mathit{\text{value}}〉\right)=〈\mathit{\text{value}}〉$ and ${\mathbf{tss}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{rss}}\left(i\right)\le {\mathbf{tss}}$, for all $i$.
${\mathbf{ifail}}=4$
A value of ${C}_{p}$ is less than $0.0$. This may occur if sigsq is too large or if rss, n or IP are incorrect.
${\mathbf{ifail}}=-99$
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.
${\mathbf{ifail}}=-399$
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.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

Accuracy is sufficient for all practical purposes.

## 8Parallelism and Performance

g02ecf is not threaded in any implementation.

None.

## 10Example

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 ${R}^{2}$ and ${C}_{p}$ are then computed and printed along with the names of variables in the models.

### 10.1Program Text

Program Text (g02ecfe.f90)

### 10.2Program Data

Program Data (g02ecfe.d)

### 10.3Program Results

Program Results (g02ecfe.r)

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