nag_regsn_mult_linear_tran_model (g02dkc) (PDF version)
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NAG Library Manual

NAG Library Function Document

nag_regsn_mult_linear_tran_model (g02dkc)


    1  Purpose
    7  Accuracy

1  Purpose

nag_regsn_mult_linear_tran_model (g02dkc) calculates the estimates of the arguments of a general linear regression model for given constraints from the singular value decomposition results.

2  Specification

#include <nag.h>
#include <nagg02.h>
void  nag_regsn_mult_linear_tran_model (Integer ip, Integer iconst, const double p[], const double c[], Integer tdc, double b[], double rss, double df, double se[], double cov[], NagError *fail)

3  Description

nag_regsn_mult_linear_tran_model (g02dkc) computes the estimates given a set of linear constraints for a general linear regression model which is not of full rank. It is intended for use after a call to nag_regsn_mult_linear (g02dac) or nag_regsn_mult_linear_upd_model (g02ddc).
In the case of a model not of full rank the functions use a singular value decomposition (SVD) to find the parameter estimates, β ^ svd , and their variance-covariance matrix. Details of the SVD are made available, in the form of the matrix P * :
P * = D -1 P1T P0T  
as described by nag_regsn_mult_linear (g02dac) and nag_regsn_mult_linear_upd_model (g02ddc).
Alternative solutions can be formed by imposing constraints on the arguments. If there are p  arguments and the rank of the model is k , then n c = p - k  constraints will have to be imposed to obtain a unique solution.
Let C  be a p  by n c  matrix of constraints, such that
CT β = 0 ,  
then the new parameter estimates β ^ c  are given by:
β ^ c = A β ^ svd = I-P 0 CT P 0 -1 β ^ svd ,  
where I  is the identity matrix, and the variance-covariance matrix is given by:
A P 1 D -2 P1T AT  
provided CT P 0 -1  exists.

4  References

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

5  Arguments

1:     ip IntegerInput
On entry: the number of terms in the linear model, p .
Constraint: ip1 .
2:     iconst IntegerInput
On entry: the number of constraints to be imposed on the arguments, n c .
Constraint: 0 < iconst < ip .
3:     p[ip×ip+2×ip] const doubleInput
4:     c[ip×tdc] const doubleInput
Note: the i,jth element of the matrix C is stored in c[i-1×tdc+j-1].
On entry: the iconst constraints stored by column, i.e., the i th constraint is stored in the i th column of c.
5:     tdc IntegerInput
On entry: the stride separating matrix column elements in the array c.
Constraint: tdciconst .
6:     b[ip] doubleInput/Output
On entry: the parameter estimates computed by using the singular value decomposition, β ^ svd .
On exit: the parameter estimates of the arguments with the constraints imposed, β ^ c .
7:     rss doubleInput
On entry: the residual sum of squares as returned by nag_regsn_mult_linear (g02dac) or nag_regsn_mult_linear_upd_model (g02ddc).
Constraint: rss>0.0 .
8:     df doubleInput
On entry: the degrees of freedom associated with the residual sum of squares as returned by nag_regsn_mult_linear (g02dac) or nag_regsn_mult_linear_upd_model (g02ddc).
Constraint: df>0.0 .
9:     se[ip] doubleOutput
On exit: the standard error of the parameter estimates in b.
10:   cov[ip×ip+1/2] doubleOutput
On exit: the upper triangular part of the variance-covariance matrix of the ip parameter estimates given in b. They are stored packed by column, i.e., the covariance between the parameter estimate given in b[i]  and the parameter estimate given in b[j] , ji , is stored in cov[ j j+1 / 2 + i ] , for i=0,1,,ip - 1 and j=i,,ip - 1.
11:   fail NagError *Input/Output
The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

6  Error Indicators and Warnings

On entry, iconst=value  while ip=value . These arguments must satisfy iconst<ip .
On entry, tdc=value  while iconst=value . These arguments must satisfy tdciconst .
Dynamic memory allocation failed.
On entry, iconst=value.
Constraint: iconst>0.
On entry, ip=value.
Constraint: ip1.
Matrix c does not give a model of full rank.
On entry, df must not be less than or equal to 0.0: df=value .
On entry, rss must not be less than or equal to 0.0: rss=value .

7  Accuracy

It should be noted that due to rounding errors an argument that should be zero when the constraints have been imposed may be returned as a value of order machine precision.

8  Parallelism and Performance

nag_regsn_mult_linear_tran_model (g02dkc) is not threaded in any implementation.

9  Further Comments

nag_regsn_mult_linear_tran_model (g02dkc) is intended for use in situations in which dummy (0-1) variables have been used such as in the analysis of designed experiments when you do not wish to change the arguments of the model to give a full rank model. The function is not intended for situations in which the relationships between the independent variables are only approximate.

10  Example

Data from an experiment with four treatments and three observations per treatment are read in. A model, including the mean term, is fitted by nag_regsn_mult_linear (g02dac) and the results printed. The constraint that the sum of treatment effects is zero is then read in and the parameter estimates with this constraint imposed are computed by nag_regsn_mult_linear_tran_model (g02dkc) and printed.

10.1  Program Text

Program Text (g02dkce.c)

10.2  Program Data

Program Data (g02dkce.d)

10.3  Program Results

Program Results (g02dkce.r)

nag_regsn_mult_linear_tran_model (g02dkc) (PDF version)
g02 Chapter Contents
g02 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2016