where	$y$	is a vector of length $n$ of the dependent variable,
	$X$	is an $n \times p$ matrix of the independent variables,
	$β$	is a vector of length $p$ of unknown arguments,
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} .

The fitted values,

\hat{y} = X \hat{β}

, can be written as

H y

for an

n \times n

matrix

H

. The

i

th diagonal element of

H

h_{i}

, gives a measure of the influence of the

i

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

r

and the

h_{i}

are returned by g02dac.

g02fac 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:

(a)The $i$ th residual is standardized by its variance when the estimate of $σ^{2}$ , $s^{2}$ , is calculated from all the data; known as internal studentization.

${R I}_{i} = \frac{r_{i}}{s \sqrt{1 - h_{i}}} .$

(b)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; 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:

(a)Cook's $D$

$D_{i} = \frac{1}{p} {R E}_{i}^{2} \frac{h_{i}}{1 - h_{i}}$
(b)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 Arguments

1: $n$ – Integer Input

On entry: number of observations included in the regression,

n

Constraint:

n > ip + 1

2: $ip$ – Integer Input

On entry: the number of linear arguments estimated in the regression model,

p

Constraint:

ip \geq 1

3: $nres$ – Integer Input

On entry: the number of residuals.

Constraint:

1 \leq nres \leq n

4: $res [nres]$ – const double Input

On entry: the residuals,

r_{i}

5: $h [nres]$ – const double Input

On entry: the diagonal elements of

H

h_{i}

, corresponding to the residuals in res.

Constraint:

0.0 < h [i] < 1.0

, for

i = 0, 1, \dots, nres - 1

6: $rms$ – double 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: $sres [nres \times 4]$ – double Output

On exit: the standardized residuals and influence statistics.

For the observation with residual given in

res [i]

$sres [(i) \times 4]$ is the internally studentized residual
$sres [(i) \times 4 + 1]$ is the externally studentized residual
$sres [(i) \times 4 + 2]$ is Cook's $D$ statistic
$sres [(i) \times 4 + 3]$ is Atkinson's $T$ statistic.

8: $fail$ – 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_2_INT_ARG_GT: On entry, $nres = ⟨ value ⟩$ while $n = ⟨ value ⟩$ . These arguments must satisfy $nres \geq n$ .
NE_2_INT_ARG_LE: On entry, $n = ⟨ value ⟩$ while $ip + 1 = ⟨ value ⟩$ . These arguments must satisfy $n > ip + 1$ .
NE_INT_ARG_LT: On entry, $ip = ⟨ value ⟩$ .
Constraint: $ip \geq 1$ .

On entry, $nres = ⟨ value ⟩$ .
Constraint: $nres \geq 1$ .
NE_REAL_ARG_GE: On entry, $h [⟨ value ⟩]$ must not be greater than or equal to 1.0: $h [⟨ value ⟩] = ⟨ value ⟩$ .
NE_REAL_ARG_LE: On entry, $h [⟨ value ⟩]$ must not be less than or equal to 0.0: $h [⟨ value ⟩] = ⟨ value ⟩$ .

On entry, rms must not be less than or equal to 0.0: $rms = ⟨ value ⟩$ .
NE_RESID_LARG: On entry, the value of a residual is too large for the given value of rms: $res [⟨ value ⟩] = ⟨ value ⟩$ , $rms = ⟨ value ⟩$ .

7 Accuracy

Accuracy is sufficient for all practical purposes.

8 Parallelism and Performance

g02fac is not threaded in any implementation.

9 Further Comments

None.

10 Example

A set of 24 residuals and

h_{i}

values from an 11 argument 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.

g02fa: FL CL CPP AD

NAG CL Interfaceg02fac (linregm_​stat_​resinf)

▸▿ 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
g02fac (linregm_stat_resinf)