G02ECF : NAG Library, Mark 24

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.

G02ECF may be called after G02EAF which calculates the residual sums of squares for all possible linear regression models.

5 Parameters

1: MEAN – CHARACTER(1)Input

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

$MEAN ='M'$: A mean term, intercept, will be included in the model.
$MEAN ='Z'$: The model will pass through the origin, zero-point.

Constraint:

MEAN ='M'

'Z'

2: N – INTEGERInput

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 – REAL (KIND=nag_wp)Input

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

{\hat{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: NTERMS(NMOD) – INTEGER arrayInput

On entry:

NTERMS (i)

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) – REAL (KIND=nag_wp) arrayInput

On entry:

RSS (i)

must contain the residual sum of squares for the

i

th model.

Constraint:

RSS (i) \leq TSS

, for

i = 1, 2, \dots, NMOD

8: RSQ(NMOD) – REAL (KIND=nag_wp) arrayOutput

On exit:

RSQ (i)

contains the

R^{2}

-value for the

i

th model, for

i = 1, 2, \dots, NMOD

9: CP(NMOD) – REAL (KIND=nag_wp) arrayOutput

On exit:

CP (i)

contains the

C_{p}

-value for the

i

th model, for

i = 1, 2, \dots, NMOD

10: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction 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 parameter, 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).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

IFAIL = 1

On entry,	$NMOD < 1$ ,
or	$SIGSQ \leq 0.0$ ,
or	$TSS \leq 0.0$ .
or	$MEAN \neq'M'$ or $'Z'$ .

IFAIL = 2

On entry, the number of parameters for a model is too large for the number of observations, i.e.,

2 \times p \geq n

IFAIL = 3

On entry,

RSS (i) > TSS

, for some

i = 1, 2, \dots, NMOD

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.

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.

NAG Library Routine Document

G02ECF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine DocumentG02ECF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

G02ECF