Estimate of an estimable function for a general linear regression model

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NAG[g02dnc] NAG[nag_regsn_mult_linear_est_func] - Estimate of an estimable function for a general linear regression model

Calling Sequence

g02dnc(rank, b, cov, p, f, est, stat, sestat, t, tol, 'ip'=ip, 'fail'=fail)

nag_regsn_mult_linear_est_func(. . .)

Parameters

rank - integer;

On entry: the rank of the independent variables, .

Constraint: . .

b - Vector(1..ip, datatype=float[8]);

On entry: the ip values of the estimates of the arguments of the model, .

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

Note: the dimension, dim, of the array cov must be at least .

On entry: the upper triangular part of the variance-covariance matrix of the ip argument estimates given in b. They are stored packed by column, i.e., the covariance between the argument estimate given in and the argument estimate given in , , is stored in , for and .

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

Note: the dimension, dim, of the array p must be at least .

On entry: p as returned by g02dac (nag_regsn_mult_linear) or g02ddc (nag_regsn_mult_linear_upd_model).

f - Vector(1..ip, datatype=float[8]);

On entry: the linear function to be estimated, .

est - assignable;

Note: On exit the variable est will have a value of type boolean.

On exit: est indicates if the function was estimable.

The function is estimable.

The function is not estimable and stat, sestat and t are not set.

stat - assignable;

Note: On exit the variable stat will have a value of type float.

On exit: if , stat contains the estimate of the function, .

sestat - assignable;

Note: On exit the variable sestat will have a value of type float.

On exit: if , sestat contains the standard error of the estimate of the function, .

t - assignable;

Note: On exit the variable t will have a value of type float.

On exit: if , t contains the -statistic for the test of the function being equal to zero.

tol - float;

On entry: tol is the tolerance value used in the check for estimability, . If , is used instead.

'ip'=ip - integer; (optional)

Default value: the first dimension of the arrays b, f.

On entry: the number of terms in the linear model, .

Constraint: . .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_regsn_mult_linear_est_func (g02dnc) gives the estimate of an estimable function along with its standard error.

Description

This function computes the estimates of an estimable function for a general linear regression model which is not of full rank. It is intended for use after a call to g02dac (nag_regsn_mult_linear) or g02ddc (nag_regsn_mult_linear_upd_model). An estimable function is a linear combination of the arguments such that it has a unique estimate. For a full rank model all linear combinations of arguments are estimable.

In the case of a model not of full rank the functions use a singular value decomposition (SVD) to find the argument estimates, , and their variance-covariance matrix. Given the upper triangular matrix obtained from the decomposition of the independent variables the SVD gives:

R=Q[*]([[D,0],[0,0]])P^T

where is a by diagonal matrix with non-zero diagonal elements, being the rank of , and and are by orthogonal matrices. This leads to a solution:

being the first columns of , i.e., , being the first columns of and being the first elements of .

Details of the SVD are made available, in the form of the matrix :

P^(*)=([[D^(-1)P[1]^T],[P[0]^T]])

as given by g02dac (nag_regsn_mult_linear) and g02ddc (nag_regsn_mult_linear_upd_model).

A linear function of the arguments, , can be tested to see if it is estimable by computing . If is zero, then the function is estimable, if not, the function is not estimable. In practice is tested against some small quantity .

Given that is estimable it can be estimated by and its standard error calculated from the variance-covariance matrix of , , as

Also a -statistic:

can be computed. The -statistic will have a Student's -distribution with degrees of freedom as given by the degrees of freedom for the residual sum of squares for the model.

Error Indicators and Warnings

"NE_2_INT_ARG_GT"

On entry, while . These arguments must satisfy .

"NE_ALLOC_FAIL"

Dynamic memory allocation failed.

"NE_INT_ARG_LT"

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

"NE_RANK_EQ_IP"

On entry, . In this case, the boolean variable est is returned as true and all statistics are calculated.

"NE_STDES_ZERO"

se probably due to rounding error or due to incorrectly specified inputs cov and f.

	Accuracy
	The computations are believed to be stable.

Further Comments

The value of estimable functions is independent of the solution chosen from the many possible solutions. While nag_regsn_mult_linear_est_func (g02dnc) may be used to estimate functions of the arguments of the model as computed by g02dkc (nag_regsn_mult_linear_tran_model), , these must be expressed in terms of the original arguments, . The relation between the two sets of arguments may not be straightforward.

Examples

ip := 5:
rank := 4:
tol := 1e-05:
b := Vector([30.55666666666667, 5.446666666666667, 6.743333333333339, 11.04666666666667, 7.320000000000003], datatype=float[8]):
cov := Vector([0.1481786666666665, 0.03704466666666668, 0.703848666666666, 0.0370446666666667, -0.222268, 0.7038486666666662, 0.03704466666666655, -0.2222679999999998, -0.2222679999999997, 0.7038486666666665, 0.03704466666666659, -0.2222679999999998, -0.2222679999999999, -0.2222680000000002, 0.7038486666666667], datatype=float[8]):
p := Vector([1.135198279858991, 1.13078348304906, 1.234006904805161, 1.228003039942231, 1.148419798367432, 3.872983346207417, 1.732050807568878, 1.732050807568877, 1.732050807568877, 3.166264259090703e-16, 0.2309401076758502, 0.05773502691896256, 0.05773502691896257, 0.0577350269189626, 0.05773502691896259, -8.986850838355147e-17, -0.2180297316413536, -0.3448196554213736, 0.3466016006011757, 0.2162477864615517, 3.472144456476812e-18, 0.4492089773025322, -0.3565253366776346, -0.03799964639604183, -0.05468399422885582, 4.280983223691876e-18, -0.02596788076090121, -0.06315923956148034, -0.3583620478433494, 0.4474891681657309, -0.447213595499958, 0.4472135954999579, 0.447213595499958, 0.4472135954999578, 0.4472135954999579], datatype=float[8]):
f := Vector([1, 1, 0, 0, 0], datatype=float[8]):
NAG:-g02dnc(rank, b, cov, p, f, est, stat, sestat, t, tol, 'ip' = ip):

est;

stat;

sestat;

	See Also
	Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore Hammarling S (1985) The singular value decomposition in multivariate statistics SIGNUM Newsl. 20 (3) 2–25 Searle S R (1971) Linear Models Wiley g02 Chapter Introduction. NAG Toolbox Overview. NAG Web Site.