g02gkc calculates the estimates of the arguments of a generalized linear model for given constraints from the singular value decomposition results.

2 Specification

#include <nag.h>

void	g02gkc (Integer ip, Integer nclin, const double v[], Integer tdv, const double c[], Integer tdc, double b[], double scale, double se[], double cov[], NagError *fail)

The function may be called by the names: g02gkc, nag_correg_glm_constrain or nag_glm_tran_model.

3 Description

g02gkc 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 g02gac, g02gbc, g02gcc or g02gdc.

In the case of a model not of full rank the functions use a singular value decomposition (SVD) to find the parameter estimates,

{\hat{β}}_{s v d}

, 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 g02gac, g02gbc, g02gcc and g02gdc.

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 \times n_{c}

matrix of constraints, such that

C^{T} β = 0,

then the new parameter estimates

{\hat{β}}_{c}

are given by:

\begin{matrix} {\hat{β}}_{c} & = & A {\hat{β}}_{s v d} \\ = & ({I - P}_{0} {(C^{T} P_{0})}^{−1}) {\hat{β}}_{s v d} \end{matrix},

where

I

is the identity matrix, 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: the number of terms in the linear model, $p$ .

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

Constraint: $0 < nclin < ip$ .
3: $v [ip \times tdv]$ – const double Input: Note: the $(i, j)$ th element of the matrix $V$ is stored in $v [(i - 1) \times tdv + j - 1]$ .

On entry: v as returned by g02gac, g02gbc, g02gcc or g02gdc.
4: $tdv$ – Integer Input: On entry: the stride separating matrix column elements in the array v.

Constraint: $tdv \geq ip + 6$ .
tdv should be as supplied to g02gac, g02gbc, g02gcc or g02gdc.
5: $c [ip \times tdc]$ – const double Input: Note: the $(i, j)$ th element of the matrix $C$ is stored in $c [(i - 1) \times tdc + j - 1]$ .

On entry: the nclin constraints stored by column, i.e., the $i$ th constraint is stored in the $i$ th column of c.
6: $tdc$ – Integer Input: On entry: the stride separating matrix column elements in the array c.

Constraint: $tdc \geq nclin$ .
7: $b [ip]$ – double Input/Output: On entry: the parameter estimates computed by using the singular value decomposition, ${\hat{β}}_{s v d}$ .

On exit: the parameter estimates of the arguments with the constraints imposed, ${\hat{β}}_{c}$ .
8: $scale$ – double Input: On entry: the estimate of the scale argument.
For results from g02gac and g02gdc then scale is the scale argument, for the model $σ^{2}$ and ${\hat{ν}}^{−1}$ respectively.

For results from g02gbc and g02gcc then scale should be set to 1.0.

Constraint: $scale > 0.0$ .
9: $se [ip]$ – double Output: On exit: the standard error of the parameter estimates in b.
10: $cov [(ip) \times (ip + 1) / 2]$ – double 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 $cov [j (j + 1) / 2 + i]$ , for $i = 0, 1, \dots, ip - 1$ and $j = i, \dots, ip - 1$ .
11: $fail$ – NagError * Input/Output: The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_2_INT_ARG_GE: On entry, $nclin = ⟨ value ⟩$ while $ip = ⟨ value ⟩$ . These arguments must satisfy $nclin < ip$ .
NE_2_INT_ARG_LT: On entry, $tdc = ⟨ value ⟩$ while $nclin = ⟨ value ⟩$ . These arguments must satisfy $tdc \geq nclin$ .

On entry, $tdv = ⟨ value ⟩$ while $ip = ⟨ value ⟩$ . These arguments must satisfy $tdv \geq ip + 6$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_INT_ARG_LE: On entry, $nclin = ⟨ value ⟩$ .
Constraint: $nclin > 0$ .
NE_INT_ARG_LT: On entry, $ip = ⟨ value ⟩$ .
Constraint: $ip \geq 1$ .
NE_MAT_NOT_FULL_RANK: Matrix c does not give a model of full rank.
NE_REAL_ARG_LE: On entry, scale must not be less than or equal to 0.0: $scale = ⟨ value ⟩$ .

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

Background information to multithreading can be found in the Multithreading documentation.

g02gkc is not threaded in any implementation.

9 Further Comments

g02gkc 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

A loglinear model is fitted to a 3 by 5 contingency table by g02gcc. 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 g02gkc and printed.

g02gk: FL CL CPP AD PY MB

NAG CL Interfaceg02gkc (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 CL Interface
g02gkc (glm_constrain)