NAG Library Routine Document

G02DFF

When selecting a linear regression model it is sometimes useful to drop independent variables from the model and to examine the resulting sub-model. G02DFF updates the

Q R

decomposition used in the computation of the linear regression model. The

Q R

decomposition may come from G02DAF or G02DEF, or a previous call to G02DFF.

For the general linear regression model with

p

independent variables fitted G02DAF or G02DEF compute a

Q R

decomposition of the (weighted) independent variables and form an upper triangular matrix

R

and a vector

c

. To remove an independent variable

R

and

c

have to be updated. The column of

R

corresponding to the variable to be dropped is removed and the matrix is then restored to upper triangular form by applying a series of Givens rotations. The rotations are then applied to

c

. Note only the first

p

elements of

c

are affected.

The method used means that while the updated values of

R

and

c

are computed an updated value of

Q

from the

Q R

decomposition is not available so a call to G02DEF cannot be made after a call to G02DFF.

G02DDF can be used to calculate the parameter estimates,

\hat{β}

, from the information provided by G02DFF.

4 References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Hammarling S (1985) The singular value decomposition in multivariate statistics SIGNUM Newsl. 20(3) 2–25

5 Parameters

1: IP – INTEGERInput: On entry: $p$ , the number of independent variables already in the model.
Constraint: $IP \geq 1$ .
2: Q(LDQ, $IP + 1$ ) – REAL (KIND=nag_wp) arrayInput/Output: On entry: the results of the $Q R$ decomposition as returned by routines G02DAF, G02DCF, G02DEF or G02EEF, or previous calls to G02DFF.

On exit: the updated $Q R$ decomposition.
3: LDQ – INTEGERInput: On entry: the first dimension of the array Q as declared in the (sub)program from which G02DFF is called.

Constraint: $LDQ \geq IP$ .
4: INDX – INTEGERInput: On entry: indicates which independent variable is to be deleted from the model.
Constraint: $1 \leq INDX \leq IP$ .
5: RSS – REAL (KIND=nag_wp)Input/Output: On entry: the residual sum of squares for the full regression.
Constraint: $RSS \geq 0.0$ .

On exit: the residual sum of squares with the (INDX)th variable removed. Note that the residual sum of squares will only be valid if the regression is of full rank, otherwise the residual sum of squares should be obtained using G02DDF.
6: WK( $2 \times IP$ ) – REAL (KIND=nag_wp) arrayWorkspace
7: 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,	$IP < 1$ ,
or	$LDQ < IP$ ,
or	$INDX < 1$ ,
or	$INDX > IP$ ,
or	$RSS < 0.0$ .

$IFAIL = 2$

On entry,

a diagonal element of

R

is zero.

7 Accuracy

There will inevitably be some loss in accuracy in fitting a model by dropping terms from a more complex model rather than fitting it afresh using G02DAF.

8 Further Comments

None.

9 Example

A dataset consisting of

12

observations on four independent variables and one dependent variable is read in. The full model, including a mean term, is fitted using G02DAF. The value of INDX is read in and that variable dropped from the regression. The parameter estimates are calculated by G02DDF and printed. This process is repeated until INDX is

0

NAG Library Routine DocumentG02DFF

+− Contents

1 Purpose

2 Specification

3 Description