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 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}

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 nag_correg_linregm_fit (g02da).

nag_correg_linregm_stat_resinf (g02fa) 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}})} .

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

Parameters

Compulsory Input Parameters

1: $n$ – int64int32nag_int scalar: $n$ , the number of observations included in the regression.

Constraint: $n > ip + 1$ .
2: $ip$ – int64int32nag_int scalar: $p$ , the number of linear arguments estimated in the regression model.

Constraint: $ip \geq 1$ .
3: $res (nres)$ – double array: The residuals, $r_{i}$ .
4: $h (nres)$ – double array: The diagonal elements of $H$ , $h_{i}$ , corresponding to the residuals in res.

Constraint: $0.0 < h (i) < 1.0$ , for $i = 1, 2, \dots, nres$ .
5: $rms$ – double scalar: The estimate of $σ^{2}$ based on all $n$ observations, $s^{2}$ , i.e., the residual mean square.

Constraint: $rms > 0.0$ .

Optional Input Parameters

1: $nres$ – int64int32nag_int scalar: Default: the dimension of the arrays res, h. (An error is raised if these dimensions are not equal.)
The number of residuals.

Constraint: $1 \leq nres \leq n$ .

Output Parameters

1: $sres (ldsres, 4)$ – double array

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, $T_{i}$ .

2: $ifail$ – int64int32nag_int scalar

ifail = 0

unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$

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.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

Accuracy is sufficient for all practical purposes.

Further Comments

None.

Example

A set of

24

residuals and

h_{i}

values from a

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.

Open in the MATLAB editor: g02fa_example

function g02fa_example


fprintf('g02fa example results\n\n');

res = [ 0.2660  -0.1387  -0.2971   0.5926  -0.4013   0.1396 ...
       -1.3173   1.1226   0.0321  -0.7111   0.3439  -0.4379 ...
        0.0633  -0.0936   0.9968   0.0209  -0.4056   0.1396 ...
        0.0327   0.2970  -0.2277   0.5180   0.5301  -1.0650 ];

h   = [ 0.5519   0.9746   0.6256   0.3144   0.4106   0.6268 ...
        0.5479   0.2325   0.4115   0.3577   0.3342   0.1673 ...
        0.3874   0.1705   0.3466   0.3743   0.7527   0.9069 ...
        0.2610   0.6256   0.2485   0.3072   0.5848   0.4794 ];

n  = int64(numel(res));
ip = int64(11);
rms  = 0.5798;

% Calculate standardised residuals
[sres, ifail] = g02fa( ...
                       n, ip, res, h, rms);

% Display results
fprintf('        Internally    Internally\n');
fprintf('Obs.   standardized  standardized   Cook''s D   Atkinson''s T\n');
fprintf('         residuals     residuals\n\n');
for j = 1:ip-1
  fprintf('%2d%13.3f%13.3f%13.3f%13.3f\n',j,sres(j,1:4))
end

g02fa example results

        Internally    Internally
Obs.   standardized  standardized   Cook's D   Atkinson's T
         residuals     residuals

 1        0.522        0.507        0.030        0.611
 2       -1.143       -1.158        4.557       -7.797
 3       -0.638       -0.622        0.062       -0.875
 4        0.940        0.935        0.037        0.689
 5       -0.686       -0.672        0.030       -0.610
 6        0.300        0.289        0.014        0.408
 7       -2.573       -3.529        0.729       -4.223
 8        1.683        1.828        0.078        1.094
 9        0.055        0.053        0.000        0.048
10       -1.165       -1.183        0.069       -0.960

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox