Calculates standardized residuals and influence statistics

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NAG[g02fac] NAG[nag_regsn_std_resid_influence] - Calculates standardized residuals and influence statistics

Calling Sequence

g02fac(n, ip, res, h, rms, sres, 'nres'=nres, 'fail'=fail)

nag_regsn_std_resid_influence(. . .)

Parameters

n - integer;

On entry: number of observations included in the regression, .

Constraint: . .

ip - integer;

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

Constraint: . .

res - Vector(1..nres, datatype=float[8]);

On entry: the residuals, .

h - Vector(1..nres, datatype=float[8]);

On entry: the diagonal elements of , , corresponding to the residuals in res.

Constraint: , for . .

rms - float;

On entry: the estimate of based on all observations, , i.e., the residual mean square.

Constraint: . .

sres - Matrix(1..nres, 1.., datatype=float[8], order=C_order);

On exit: the standardized residuals and influence statistics.

For the observation with residual given in :

is the internally studentized residual

is the externally studentized residual

is Cook's statistic

is Atkinson's statistic.

'nres'=nres - integer; (optional)

Default value: the first dimension of the arrays res, h, sres.

On entry: the number of residuals.

Constraint: . .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_regsn_std_resid_influence (g02fac) calculates two types of standardized residuals and two measures of influence for a linear regression.

Description

For the general linear regression model is defined by

where		is a vector of length of the dependent variable,
		is an by matrix of the independent variables,
		is a vector of length of unknown arguments,
and		is a vector of length of unknown random errors such that .

The residuals are given by

The fitted values, , can be written as for an by matrix . The th diagonal element of , , gives a measure of the influence of the th value of the independent variables on the fitted regression model. The values of and the are returned by g02dac (nag_regsn_mult_linear).

nag_regsn_std_resid_influence (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 th residual is standardized by its variance when the estimate of , , is calculated from all the data; known as internal studentization.

I*R*[i] = r[i]/(s*sqrt(1-h[i]))

b. The th residual is standardized by its variance when the estimate of , is calculated from the data excluding the th observation; known as external studentization.

R*E[i] = r[i]/(s[-i]*sqrt(1-h[i])) and r[i]/(s[-i]*sqrt(1-h[i])) = r[i]*sqrt((n-p-1)/(n-p-I*R*[i]^2))

The two measures of influence are:

a. Cook's

D[i] = R*E[i]^2*h[i]/(p*(1-h[i]))

b. Atkinson's

T[i]=|(RE)[i]|sqrt(((n-p)/p)((h[i])/(1-h[i])))

Error Indicators and Warnings

"NE_2_INT_ARG_GT"

On entry, while . These arguments must satisfy .

"NE_2_INT_ARG_LE"

On entry, while . These arguments must satisfy .

"NE_INT_ARG_LT"

On entry, ip must not be less than 1: .

"NE_REAL_ARG_GE"

On entry, must not be greater than or equal to 1.0: .

"NE_REAL_ARG_LE"

On entry, rms must not be less than or equal to 0.0: .

"NE_RESID_LARG"

On entry, the value of a residual is too large for the given value of rms: , .

	Accuracy
	Accuracy is sufficient for all practical purposes.

	Further Comments
	None.

Examples

n := 24:
ip := 11:
nres := 10:
rms := 0.5798:
res := Vector([0.266, -0.1387, -0.2971, 0.5926, -0.4013, 0.1396, -1.3173, 1.1226, 0.0321, -0.7111], datatype=float[8]):
h := Vector([0.5518999999999999, 0.9746, 0.6256, 0.3144, 0.4106, 0.6268, 0.5479000000000001, 0.2325, 0.4115, 0.3577], datatype=float[8]):
sres := Matrix(10, 4, datatype=float[8], order='C_order'):
NAG:-g02fac(n, ip, res, h, rms, sres, 'nres' = nres):

sres;

	See Also
	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 g02 Chapter Introduction. NAG Toolbox Overview. NAG Web Site.