# NAG Library Routine Document

## 1Purpose

g02faf calculates two types of standardized residuals and two measures of influence for a linear regression.

## 2Specification

Fortran Interface
 Subroutine g02faf ( n, ip, nres, res, h, rms, sres,
 Integer, Intent (In) :: n, ip, nres, ldsres Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: res(nres), h(nres), rms Real (Kind=nag_wp), Intent (Inout) :: sres(ldsres,4)
#include nagmk26.h
 void g02faf_ (const Integer *n, const Integer *ip, const Integer *nres, const double res[], const double h[], const double *rms, double sres[], const Integer *ldsres, Integer *ifail)

## 3Description

For the general linear regression model
 $y=Xβ+ε,$
 where $y$ is a vector of length $n$ of the dependent variable, $X$ is an $n$ by $p$ matrix of the independent variables, $\beta$ is a vector of length $p$ of unknown arguments, and $\epsilon$ is a vector of length $n$ of unknown random errors such that $\mathrm{var}\epsilon ={\sigma }^{2}I$.
The residuals are given by
 $r=y-y^=y-Xβ^$
and the fitted values, $\stackrel{^}{y}=X\stackrel{^}{\beta }$, can be written as $Hy$ for an $n$ by $n$ 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 ${\sigma }^{2}$, ${s}^{2}$, is calculated from all the data; this is known as internal Studentization.
 $RIi=ris⁢1-hi .$
(ii) The $i$th residual is standardized by its variance when the estimate of ${\sigma }^{2}$, ${s}_{-i}^{2}$ is calculated from the data excluding the $i$th observation; this is known as external Studentization.
 $REi=ris-i1-hi =rin-p-1 n-p-RIi2 .$
The two measures of influence are:
(i) Cook's $D$
 $Di=1pREi2hi1-hi .$
(ii) Atkinson's $T$
 $Ti=REi n-pp hi1-hi .$

## 4References

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

## 5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of observations included in the regression.
Constraint: ${\mathbf{n}}>{\mathbf{ip}}+1$.
2:     $\mathbf{ip}$ – IntegerInput
On entry: $p$, the number of linear arguments estimated in the regression model.
Constraint: ${\mathbf{ip}}\ge 1$.
3:     $\mathbf{nres}$ – IntegerInput
On entry: the number of residuals.
Constraint: $1\le {\mathbf{nres}}\le {\mathbf{n}}$.
4:     $\mathbf{res}\left({\mathbf{nres}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the residuals, ${r}_{i}$.
5:     $\mathbf{h}\left({\mathbf{nres}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the diagonal elements of $H$, ${h}_{i}$, corresponding to the residuals in res.
Constraint: $0.0<{\mathbf{h}}\left(\mathit{i}\right)<1.0$, for $\mathit{i}=1,2,\dots ,{\mathbf{nres}}$.
6:     $\mathbf{rms}$ – Real (Kind=nag_wp)Input
On entry: the estimate of ${\sigma }^{2}$ based on all $n$ observations, ${s}^{2}$, i.e., the residual mean square.
Constraint: ${\mathbf{rms}}>0.0$.
7:     $\mathbf{sres}\left({\mathbf{ldsres}},4\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the standardized residuals and influence statistics.
For the observation with residual, ${r}_{i}$, given in ${\mathbf{res}}\left(i\right)$.
${\mathbf{sres}}\left(i,1\right)$
Is the internally standardized residual, ${\mathrm{RI}}_{i}$.
${\mathbf{sres}}\left(i,2\right)$
Is the externally standardized residual, ${\mathrm{RE}}_{i}$.
${\mathbf{sres}}\left(i,3\right)$
Is Cook's $D$ statistic, ${D}_{i}$.
${\mathbf{sres}}\left(i,4\right)$
Is Atkinson's $T$ statistic, ${T}_{i}$.
8:     $\mathbf{ldsres}$ – IntegerInput
On entry: the first dimension of the array sres as declared in the (sub)program from which g02faf is called.
Constraint: ${\mathbf{ldsres}}\ge {\mathbf{nres}}$.
9:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{ip}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ip}}\ge 1$.
On entry, ${\mathbf{ip}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}-1>{\mathbf{ip}}$.
On entry, ${\mathbf{ldsres}}=〈\mathit{\text{value}}〉$ and ${\mathbf{nres}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldsres}}\ge {\mathbf{nres}}$.
On entry, ${\mathbf{nres}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nres}}\ge 1$.
On entry, ${\mathbf{nres}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nres}}\le {\mathbf{n}}$.
On entry, ${\mathbf{rms}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{rms}}>0.0$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{h}}\left(〈\mathit{\text{value}}〉\right)=〈\mathit{\text{value}}〉$.
Constraint: $0.0<{\mathbf{h}}\left(i\right)<1.0$, for all $i$.
${\mathbf{ifail}}=3$
On entry, a value in res is too large for given rms. ${\mathbf{res}}\left(〈\mathit{\text{value}}〉\right)=〈\mathit{\text{value}}〉$ and ${\mathbf{rms}}=〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

Accuracy is sufficient for all practical purposes.

## 8Parallelism and Performance

g02faf is not threaded in any implementation.

None.

## 10Example

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.

### 10.1Program Text

Program Text (g02fafe.f90)

### 10.2Program Data

Program Data (g02fafe.d)

### 10.3Program Results

Program Results (g02fafe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017