nag_glm_est_func (g02gnc) (PDF version)
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nag_glm_est_func (g02gnc)


    1  Purpose
    7  Accuracy

1  Purpose

nag_glm_est_func (g02gnc) gives the estimate of an estimable function along with its standard error from the results from fitting a generalized linear model.

2  Specification

#include <nag.h>
#include <nagg02.h>
void  nag_glm_est_func (Integer ip, Integer rank, const double b[], const double cov[], const double v[], Integer tdv, const double f[], Nag_Boolean *est, double *stat, double *sestat, double *z, double tol, NagError *fail)

3  Description

nag_glm_est_func (g02gnc) 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 nag_glm_normal (g02gac), nag_glm_binomial (g02gbc), nag_glm_poisson (g02gcc) or nag_glm_gamma (g02gdc). 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 parameter estimates, β ^ , and their variance-covariance matrix. Given the upper triangular matrix R  obtained from the QR  decomposition of the independent variables the SVD gives:
R = Q * D 0 0 0 PT  
where D  is a k  by k  diagonal matrix with nonzero diagonal elements, k  being the rank of R , and Q *  and P  are p  by p  orthogonal matrices. This leads to a solution:
β ^ = P 1 D -1 Q * 1 T c 1  
P 1  being the first k  columns of P , i.e., P = P 1 P 0 ; Q * 1  being the first k  columns of Q *  and c 1  being the first p  elements of c .
Details of the SVD are made available, in the form of the matrix P * :
P * = D -1 P1T P0T  
as given by nag_glm_normal (g02gac), nag_glm_binomial (g02gbc), nag_glm_poisson (g02gcc) and nag_glm_gamma (g02gdc).
A linear function of the arguments, F = fT β , can be tested to see if it is estimable by computing ζ = P0T f . If ζ  is zero, then the function is estimable, if not, the function is not estimable. In practice ζ  is tested against some small quantity η .
Given that F  is estimable it can be estimated by fT β ^  and its standard error calculated from the variance-covariance matrix of β ^ , C β , as
seF = fT C β f  
Also a z  statistic:
z = fT β ^ seF ,  
can be computed. The distribution of z  will be approximately Normal.

4  References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
McCullagh P and Nelder J A (1983) Generalized Linear Models Chapman and Hall
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:     rank IntegerInput
On entry: the rank of the independent variables, k .
Constraint: 1 rank ip .
3:     b[ip] const doubleInput
On entry: the ip values of the estimates of the arguments of the model, β ^ .
4:     cov[ip×ip+1/2] const doubleInput
On entry: 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.
5:     v[ip×tdv] const doubleInput
Note: the i,jth element of the matrix V is stored in v[i-1×tdv+j-1].
6:     tdv IntegerInput
On entry: the stride separating matrix column elements in the array v.
7:     f[ip] const doubleInput
On entry: the linear function to be estimated, f .
8:     est Nag_Boolean *Output
On exit: est indicates if the function was estimable.
The function is estimable.
The function is not estimable and stat, sestat and z are not set.
9:     stat double *Output
On exit: if est=Nag_TRUE , stat contains the estimate of the function, fT β ^ .
10:   sestat double *Output
On exit: if est=Nag_TRUE , sestat contains the standard error of the estimate of the function, seF .
11:   z double *Output
On exit: if est=Nag_TRUE , z contains the z  statistic for the test of the function being equal to zero.
12:   tol doubleInput
On entry: tol is the tolerance value used in the check for estimability, η .
If tol0.0 , then machine precision  is used instead.
13:   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, ip=value  while rank=value . These arguments must satisfy rankip .
On entry, tdv=value  while ip=value . These arguments must satisfy tdv ip + 6 .
Dynamic memory allocation failed.
On entry, ip=value.
Constraint: ip1.
On entry, rank=value.
Constraint: rank1.
On entry, rank=ip . In this case, the boolean variable est is returned as Nag_TRUE and all statistics are calculated.
sestat, the standard error of the estimate of the function, se F = 0.0 ; probably due to rounding error or due to incorrectly specified input values of cov and f.

7  Accuracy

The computations are believed to be stable.

8  Parallelism and Performance

nag_glm_est_func (g02gnc) is not threaded in any implementation.

9  Further Comments

The value of estimable functions is independent of the solution chosen from the many possible solutions. While nag_glm_est_func (g02gnc) may be used to estimate functions of the arguments of the model as computed by nag_glm_tran_model (g02gkc), β c , these must be expressed in terms of the original arguments, β . The relation between the two sets of arguments may not be straightforward.

10  Example

A loglinear model is fitted to a 3 by 5 contingency table by nag_glm_poisson (g02gcc). The model consists of terms for for rows and columns. The table is:
141 67 114 79 39 131 66 143 72 35 36 14 38 28 16  
The number of functions to be tested is read in, then the linear functions themselves are read in and tested with nag_glm_est_func (g02gnc). The results of nag_glm_est_func (g02gnc) are printed.

10.1  Program Text

Program Text (g02gnce.c)

10.2  Program Data

Program Data (g02gnce.d)

10.3  Program Results

Program Results (g02gnce.r)

nag_glm_est_func (g02gnc) (PDF version)
g02 Chapter Contents
g02 Chapter Introduction
NAG Library Manual

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