NAG Library Routine Document

G02FAF

where	$y$ is a vector of length $n$ of the dependent variable,
	$X$ is an $n$ by $p$ matrix of the independent variables,
	$β$ is a vector of length $p$ of unknown parameters,
and	$ε$ is a vector of length $n$ of unknown random errors such that $var ε = σ^{2} I$ .

The residuals are given by

r = y - \hat{y} = y - X \hat{β}

and the fitted values,

\hat{y} = X \hat{β}

, can be written as

H y

for an

n

n

matrix

H

. The

i

th diagonal elements of

H

h_{i}

, give a measure of the influence of the

i

th values of the independent variables on the fitted regression model. The values of

r

and the

h_{i}

are returned by G02DAF.

G02FAF calculates statistics which help to indicate if an observation is extreme and having an undue influence on the fit of the regression model. Two types of standardized residual are calculated:

(i)

The

i

th residual is standardized by its variance when the estimate of

σ^{2}

s^{2}

, is calculated from all the data; this is known as internal Studentization.

R I_{i} = \frac{r_{i}}{s \sqrt{1 - h_{i}}} .

(ii)

The

i

th residual is standardized by its variance when the estimate of

σ^{2}

s_{- i}^{2}

is calculated from the data excluding the

i

th observation; this is known as external Studentization.

R E_{i} = \frac{r_{i}}{s_{- i} \sqrt{1 - h_{i}}} = r_{i} \sqrt{\frac{n - p - 1}{n - p - R I_{i}^{2}}} .

The two measures of influence are:

(i)

Cook's

D

D_{i} = \frac{1}{p} R E_{i}^{2} \frac{h_{i}}{1 - h_{i}} .

(ii)

Atkinson's

T

T_{i} = |R E_{i}| \sqrt{(\frac{n - p}{p}) (\frac{h_{i}}{1 - h_{i}})} .

4 References

Atkinson A C (1981) Two graphical displays for outlying and influential observations in regression Biometrika 68 13–20

Cook R D and Weisberg S (1982) Residuals and Influence in Regression Chapman and Hall

5 Parameters

1: N – INTEGERInput

On entry:

n

, the number of observations included in the regression.

Constraint:

N > IP + 1

2: IP – INTEGERInput

On entry:

p

, the number of linear parameters estimated in the regression model.

Constraint:

IP \geq 1

3: NRES – INTEGERInput

On entry: the number of residuals.

Constraint:

1 \leq NRES \leq N

4: RES(NRES) – REAL (KIND=nag_wp) arrayInput

On entry: the residuals,

r_{i}

5: H(NRES) – REAL (KIND=nag_wp) arrayInput

On entry: the diagonal elements of

H

h_{i}

, corresponding to the residuals in RES.

Constraint:

0.0 < H (i) < 1.0

, for

i = 1, 2, \dots, NRES

6: RMS – REAL (KIND=nag_wp)Input

On entry: the estimate of

σ^{2}

based on all

n

observations,

s^{2}

, i.e., the residual mean square.

Constraint:

RMS > 0.0

7: SRES(LDSRES, $4$ ) – REAL (KIND=nag_wp) arrayOutput

On exit: the standardized residuals and influence statistics.

For the observation with residual,

r_{i}

, given in

RES (i)

$SRES (i, 1)$: Is the internally standardized residual, ${RI}_{i}$ .
$SRES (i, 2)$: Is the externally standardized residual, ${RE}_{i}$ .
$SRES (i, 3)$: Is Cook's $D$ statistic, $D_{i}$ .
$SRES (i, 4)$: Is Atkinson's $T$ statistic, $T_{i}$ .

8: LDSRES – INTEGERInput

On entry: the first dimension of the array SRES as declared in the (sub)program from which G02FAF is called.

Constraint:

LDSRES \geq NRES

9: 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	$N \leq IP + 1$ ,
or	$NRES < 1$ ,
or	$NRES > N$ ,
or	$LDSRES < NRES$ ,
or	$RMS \leq 0.0$ .

$IFAIL = 2$

On entry,

H (i) \leq 0.0

\geq 1.0

, for some

i = 1, 2, \dots, NRES

$IFAIL = 3$

On entry,

the value of a residual is too large for the given value of RMS.

7 Accuracy

Accuracy is sufficient for all practical purposes.

8 Further Comments

None.

9 Example

A set of

24

residuals and

h_{i}

values from a

11

parameter model fitted to the cloud seeding data considered in Cook and Weisberg (1982) are input and the standardized residuals etc calculated and printed for the first

10

observations.

NAG Library Routine DocumentG02FAF

+− 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

G02FAF