void	g02ecc (Nag_IncludeMean mean, Integer n, double sigsq, double tss, Integer nmod, const Integer nterms[], const double rss[], double rsq[], double cp[], NagError *fail)

The function may be called by the names: g02ecc, nag_correg_linregm_rssq_stat or nag_cp_stat.

3 Description

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.

$R^{2} = \frac{Regression Sum of Squares}{Total Sum of Squares},$

where $Total Sum of Squares = tss = \sum {(y - \bar{y})}^{2}$ (if mean is fitted, otherwise $tss = \sum y^{2}$ ) and

$Regression Sum of Squares = RegSS = tss - rss$ , where
$rss = residual sum of squares = \sum {(y - \hat{y})}^{2}$ .

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.

$C_{p} = \frac{rss}{{\hat{σ}}^{2}} - (n - 2 p),$

where $p$ is the number of parameters (including the mean) in the model and ${\hat{σ}}^{2}$ 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 $C_{p} ≃ p$ . $C_{p}$ is often plotted against $p$ to see which models are closest to the $C_{p} = p$ line.

g02ecc may be called after g02eac which calculates the residual sums of squares for all possible linear regression models.

4 References

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

Weisberg S (1985) Applied Linear Regression Wiley

5 Arguments

1: $mean$ – Nag_IncludeMean Input

On entry: indicates if a mean term is to be included.

$mean = Nag_MeanInclude$: A mean term, intercept, will be included in the model.
$mean = Nag_MeanZero$: The model will pass through the origin, zero-point.

Constraint:

mean = Nag_MeanInclude

Nag_MeanZero

2: $n$ – Integer Input

On entry:

n

, the number of observations used in the regression model.

Constraint:

n

must be greater than

2 \times p_{\max}

, where

p_{\max}

is the largest number of independent variables fitted (including the mean if fitted).

3: $sigsq$ – double Input

On entry: the best estimate of true variance of the errors,

{\hat{σ}}^{2}

Constraint:

sigsq > 0.0

4: $tss$ – double Input

On entry: the total sum of squares for the regression model.

Constraint:

tss > 0.0

5: $nmod$ – Integer Input

On entry: the number of regression models.

Constraint:

nmod > 0

6: $nterms [nmod]$ – const Integer Input

On entry:

nterms [i - 1]

must contain the number of independent variables (not counting the mean) fitted to the

i

th model, for

i = 1, 2, \dots, nmod

7: $rss [nmod]$ – const double Input

On entry:

rss [i - 1]

must contain the residual sum of squares for the

i

th model.

Constraint:

rss [i - 1] \leq tss

, for

i = 1, 2, \dots, nmod

8: $rsq [nmod]$ – double Output

On exit:

rsq [i - 1]

contains the

R^{2}

-value for the

i

th model, for

i = 1, 2, \dots, nmod

9: $cp [nmod]$ – double Output

On exit:

cp [i - 1]

contains the

C_{p}

-value for the

i

th model, for

i = 1, 2, \dots, nmod

10: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_INT: On entry, $nmod = ⟨ value ⟩$ .
Constraint: $nmod > 0$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_MODEL_PARAMETERS: On entry: the number of parameters, $p$ , is $⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $n \geq 2 p$ .
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL: On entry, $sigsq = ⟨ value ⟩$ .
Constraint: $sigsq > 0.0$ .

On entry, $tss = ⟨ value ⟩$ .
Constraint: $tss > 0.0$ .
NE_REAL_ARRAY_ELEM_CONS: 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.

On entry, $rss [⟨ value ⟩] = ⟨ value ⟩$ and $tss = ⟨ value ⟩$ .
Constraint: $rss [i] \leq tss$ , for all $i$ .

7 Accuracy

Accuracy is sufficient for all practical purposes.

8 Parallelism and Performance

g02ecc is not threaded in any implementation.

9 Further Comments

None.

10 Example

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 g02eac. The values of

R^{2}

and

C_{p}

are then computed and printed along with the names of variables in the models.

g02ec: FL CL CPP AD

where	$Total Sum of Squares = tss = \sum {(y - \bar{y})}^{2}$ (if mean is fitted, otherwise $tss = \sum y^{2}$ ) and
	$Regression Sum of Squares = RegSS = tss - rss$ , where $rss = residual sum of squares = \sum {(y - \hat{y})}^{2}$ .

NAG CL Interfaceg02ecc (linregm_​rssq_​stat)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
g02ecc (linregm_rssq_stat)