# NAG Library Routine Document

## 1Purpose

g02gnf gives the estimate of an estimable function along with its standard error from the results from fitting a generalized linear model.

## 2Specification

Fortran Interface
 Subroutine g02gnf ( ip, b, cov, v, ldv, f, est, stat, z, tol, wk,
 Integer, Intent (In) :: ip, irank, ldv Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: b(ip), cov(ip*(ip+1)/2), v(ldv,ip+7), f(ip), tol Real (Kind=nag_wp), Intent (Out) :: stat, sestat, z, wk(ip) Logical, Intent (Out) :: est
#include <nagmk26.h>
 void g02gnf_ (const Integer *ip, const Integer *irank, const double b[], const double cov[], const double v[], const Integer *ldv, const double f[], logical *est, double *stat, double *sestat, double *z, const double *tol, double wk[], Integer *ifail)

## 3Description

g02gnf computes the estimates of an estimable function 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. An estimable function is a linear combination of the parameters such that it has a unique estimate. For a full rank model all linear combinations of parameters are estimable.
In the case of a model not of full rank the routines use a singular value decomposition (SVD) to find the parameter estimates, $\stackrel{^}{\beta }$, 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:
 $β^=P1D-1Q*1Tc1,$
${P}_{1}$ being the first $k$ columns of $P$, i.e., $P=\left({P}_{1}{P}_{0}\right)$; ${Q}_{{*}_{1}}$ 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 described by g02gaf, g02gbf, g02gcf and g02gdf.
A linear function of the parameters, $F={f}^{\mathrm{T}}\beta$, can be tested to see if it is estimable by computing $\zeta ={P}_{0}^{\mathrm{T}}f$. If $\zeta$ is zero, then the function is estimable, if not; the function is not estimable. In practice $\left|\zeta \right|$ is tested against some small quantity $\eta$.
Given that $F$ is estimable it can be estimated by ${f}^{\mathrm{T}}\stackrel{^}{\beta }$ and its standard error calculated from the variance-covariance matrix of $\stackrel{^}{\beta }$, ${C}_{\beta }$, as
 $seF=fTCβf.$
Also a $z$ statistic
 $z=fTβ^ seF ,$
can be computed. The distribution of $z$ will be approximately Normal.

## 4References

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

## 5Arguments

1:     $\mathbf{ip}$ – IntegerInput
On entry: $p$, the number of terms in the linear model.
Constraint: ${\mathbf{ip}}\ge 1$.
2:     $\mathbf{irank}$ – IntegerInput
On entry: $k$, the rank of the dependent variables.
Constraint: $1\le {\mathbf{irank}}\le {\mathbf{ip}}$.
3:     $\mathbf{b}\left({\mathbf{ip}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the ip values of the estimates of the parameters of the model, $\stackrel{^}{\beta }$.
4:     $\mathbf{cov}\left({\mathbf{ip}}×\left({\mathbf{ip}}+1\right)/2\right)$ – Real (Kind=nag_wp) arrayInput
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 ${\mathbf{b}}\left(i\right)$ and the parameter estimate given in ${\mathbf{b}}\left(j\right)$, $j\ge i$, is stored in ${\mathbf{cov}}\left(\left(j×\left(j-1\right)/2+i\right)\right)$.
5:     $\mathbf{v}\left({\mathbf{ldv}},{\mathbf{ip}}+7\right)$ – Real (Kind=nag_wp) arrayInput
On entry: as returned by g02gaf, g02gbf, g02gcf and g02gdf.
6:     $\mathbf{ldv}$ – IntegerInput
On entry: the first dimension of the array v as declared in the (sub)program from which g02gnf is called.
Constraint: ${\mathbf{ldv}}\ge {\mathbf{ip}}$.
7:     $\mathbf{f}\left({\mathbf{ip}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: $f$, the linear function to be estimated.
8:     $\mathbf{est}$ – LogicalOutput
On exit: indicates if the function was estimable.
${\mathbf{est}}=\mathrm{.TRUE.}$
The function is estimable.
${\mathbf{est}}=\mathrm{.FALSE.}$
The function is not estimable and stat, sestat and z are not set.
9:     $\mathbf{stat}$ – Real (Kind=nag_wp)Output
On exit: if ${\mathbf{est}}=\mathrm{.TRUE.}$, stat contains the estimate of the function, ${f}^{\mathrm{T}}\stackrel{^}{\beta }$
10:   $\mathbf{sestat}$ – Real (Kind=nag_wp)Output
On exit: if ${\mathbf{est}}=\mathrm{.TRUE.}$, sestat contains the standard error of the estimate of the function, $\mathrm{se}\left(F\right)$.
11:   $\mathbf{z}$ – Real (Kind=nag_wp)Output
On exit: if ${\mathbf{est}}=\mathrm{.TRUE.}$, z contains the $z$ statistic for the test of the function being equal to zero.
12:   $\mathbf{tol}$ – Real (Kind=nag_wp)Input
On entry: the tolerance value used in the check for estimability, $\eta$.
If ${\mathbf{tol}}\le 0.0$ then $\sqrt{\epsilon }$, where $\epsilon$ is the machine precision, is used instead.
13:   $\mathbf{wk}\left({\mathbf{ip}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
14:   $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value  is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, because for this routine the values of the output arguments may be useful even if ${\mathbf{ifail}}\ne {\mathbf{0}}$ on exit, the recommended value is $-1$. When the value  is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Note: g02gnf may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{ip}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ip}}\ge 1$.
On entry, ${\mathbf{irank}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{irank}}\ge 1$.
On entry, ${\mathbf{irank}}=〈\mathit{\text{value}}〉$ and ${\mathbf{ip}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{irank}}\le {\mathbf{ip}}$.
On entry, ${\mathbf{ldv}}=〈\mathit{\text{value}}〉$ and ${\mathbf{ip}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldv}}\ge {\mathbf{ip}}$.
${\mathbf{ifail}}=2$
${\mathbf{irank}}={\mathbf{ip}}$. In this case est is returned as true and all statistics are calculated.
${\mathbf{ifail}}=3$
Standard error of statistic $\text{}=0.0$; this may be due to rounding errors if the standard error is very small or due to mis-specified inputs cov and f.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

The computations are believed to be stable.

## 8Parallelism and Performance

g02gnf 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.

The value of estimable functions is independent of the solution chosen from the many possible solutions. While g02gnf may be used to estimate functions of the parameters of the model as computed by g02gkf, ${\beta }_{\mathrm{c}}$, these must be expressed in terms of the original parameters, $\beta$. The relation between the two sets of parameters may not be straightforward.

## 10Example

A loglinear model is fitted to a $3$ by $5$ contingency table by g02gcf. The model consists of terms 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 g02gnf. The results of g02gnf are printed.

### 10.1Program Text

Program Text (g02gnfe.f90)

### 10.2Program Data

Program Data (g02gnfe.d)

### 10.3Program Results

Program Results (g02gnfe.r)