g02gkf calculates the estimates of the parameters of a generalized linear model for given constraints from the singular value decomposition results.

2 Specification

Fortran Interface

Subroutine g02gkf (

ip, iconst, v, ldv, c, ldc, b, s, se, cov, wk, ifail)

Integer, Intent (In)	::	ip, iconst, ldv, ldc
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	v(ldv,ip+7), c(ldc,iconst), s
Real (Kind=nag_wp), Intent (Inout)	::	b(ip)
Real (Kind=nag_wp), Intent (Out)	::	se(ip), cov(ip(ip+1)/2), wk(2ipip+ipiconst+2iconsticonst+4*iconst)

C Header Interface

#include <nag.h>

void	g02gkf_ (const Integer ip, const Integer iconst, const double v[], const Integer ldv, const double c[], const Integer ldc, double b[], const double s, double se[], double cov[], double wk[], Integer ifail)

The routine may be called by the names g02gkf or nagf_correg_glm_constrain.

3 Description

g02gkf computes the estimates given a set of linear constraints for a generalized linear model which is not of full rank. It is intended for use after a call to g02gaf, g02gbf, g02gcf or g02gdf.

In the case of a model not of full rank the routines use a singular value decomposition to find the parameter estimates,

{\hat{β}}_{svd}

, and their variance-covariance matrix. Details of the SVD are made available in the form of the matrix

P^{*}

P^{*} = (\begin{matrix} D^{- 1} P_{1}^{T} \\ P_{0}^{T} \end{matrix})

as described by g02gaf, g02gbf, g02gcf and g02gdf. Alternative solutions can be formed by imposing constraints on the parameters. If there are

p

parameters 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 \times n_{c}

matrix of constraints, such that

C^{T} β = 0,

then the new parameter estimates

{\hat{β}}_{c}

are given by:

\begin{array}{l} {\hat{β}}_{c} & = A {\hat{β}}_{svd} \\ = (I - P_{0} {(C^{T} P_{0})}^{- 1}) {\hat{β}}_{svd},   where ​ I ​ is the identity matrix, \end{array}

and the variance-covariance matrix is given by

A P_{1} D^{- 2} P_{1}^{T} A^{T}

provided

{(C^{T} P_{0})}^{- 1}

exists.

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: $ip$ – Integer Input: On entry: $p$ , the number of terms in the linear model.

Constraint: $ip \geq 1$ .
2: $iconst$ – Integer Input: On entry: the number of constraints to be imposed on the parameters, $n_{c}$ .

Constraint: $0 < iconst < ip$ .
3: $v (ldv, ip + 7)$ – Real (Kind=nag_wp) array Input: On entry: the array v as returned by g02gaf, g02gbf, g02gcf or g02gdf.
4: $ldv$ – Integer Input: On entry: the first dimension of the array v as declared in the (sub)program from which g02gkf is called.

Constraint: $ldv \geq ip$ .
ldv should be as supplied to g02gaf, g02gbf, g02gcf or g02gdf
5: $c (ldc, iconst)$ – Real (Kind=nag_wp) array Input: On entry: contains the iconst constraints stored by column, i.e., the $i$ th constraint is stored in the $i$ th column of c.
6: $ldc$ – Integer Input: On entry: the first dimension of the array c as declared in the (sub)program from which g02gkf is called.

Constraint: $ldc \geq ip$ .
7: $b (ip)$ – Real (Kind=nag_wp) array Input/Output: On entry: the parameter estimates computed by using the singular value decomposition, ${\hat{β}}_{svd}$ .
On exit: the parameter estimates of the parameters with the constraints imposed, ${\hat{β}}_{c}$ .
8: $s$ – Real (Kind=nag_wp) Input: On entry: the estimate of the scale parameter.
For results from g02gaf and g02gdf then s is the scale parameter for the model.

For results from g02gbf and g02gcf then s should be set to $1.0$ .

Constraint: $s > 0.0$ .
9: $se (ip)$ – Real (Kind=nag_wp) array Output: On exit: the standard error of the parameter estimates in b.
10: $cov (ip \times (ip + 1) / 2)$ – Real (Kind=nag_wp) array Output: 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 $b (j)$ , $j \geq i$ , is stored in $cov ((j \times (j - 1) / 2 + i))$ .
11: $wk (2 \times ip \times ip + ip \times iconst + 2 \times iconst \times iconst + 4 \times iconst)$ – Real (Kind=nag_wp) array Workspace: Note: a simple upper bound for the size of the workspace is $5 \times ip \times ip + 4 \times ip$ .
12: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $- 1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $- 1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $- 1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit: $ifail = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $iconst = ⟨ value ⟩$ .
Constraint: $iconst > 0$ .

On entry, $iconst = ⟨ value ⟩$ and $ip = ⟨ value ⟩$ .
Constraint: $iconst < ip$ .

On entry, $ip = ⟨ value ⟩$ .
Constraint: $ip \geq 1$ .

On entry, $ldc = ⟨ value ⟩$ and $ip = ⟨ value ⟩$ .
Constraint: $ldc \geq ip$ .

On entry, $ldv = ⟨ value ⟩$ and $ip = ⟨ value ⟩$ .
Constraint: $ldv \geq ip$ .

On entry, $s = ⟨ value ⟩$ .
Constraint: $s > 0.0$ .

$ifail = 2$: c does not give a model of full rank.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

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

g02gkf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

g02gkf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

g02gkf 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 parameters of the model to give a full rank model. The routine is not intended for situations in which the relationships between the independent variables are only approximate.

10 Example

A loglinear model is fitted to a

3 \times 5

contingency table by g02gcf. The model consists of terms for rows and columns. The table is

\begin{array}{r} 141 & 67 & 114 & 79 & 39 \\ 131 & 66 & 143 & 72 & 35 \\ 36 & 14 & 38 & 28 & 16 \end{array} .

The constraints that the sum of row effects and the sum of column effects are zero are then read in and the parameter estimates with these constraints imposed are computed by g02gkf and printed.

g02gk: FL CL CPP AD

NAG FL Interfaceg02gkf (glm_​constrain)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
g02gkf (glm_constrain)