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NAG Toolbox

NAG Toolbox: nag_correg_glm_estfunc (g02gn)

Purpose

nag_correg_glm_estfunc (g02gn) gives the estimate of an estimable function along with its standard error from the results from fitting a generalized linear model.

Syntax

[est, stat, sestat, z, ifail] = g02gn(irank, b, covar, v, f, tol, 'ip', ip)
[est, stat, sestat, z, ifail] = nag_correg_glm_estfunc(irank, b, covar, v, f, tol, 'ip', ip)

Description

nag_correg_glm_estfunc (g02gn) 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 nag_correg_glm_normal (g02ga), nag_correg_glm_binomial (g02gb), nag_correg_glm_poisson (g02gc) or nag_correg_glm_gamma (g02gd). 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, $\stackrel{^}{\beta }$, 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 nag_correg_glm_normal (g02ga), nag_correg_glm_binomial (g02gb), nag_correg_glm_poisson (g02gc) and nag_correg_glm_gamma (g02gd).
A linear function of the arguments, $F={f}^{\mathrm{T}}\beta$, 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.

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

Parameters

Compulsory Input Parameters

1:     $\mathrm{irank}$int64int32nag_int scalar
$k$, the rank of the dependent variables.
Constraint: $1\le {\mathbf{irank}}\le {\mathbf{ip}}$.
2:     $\mathrm{b}\left({\mathbf{ip}}\right)$ – double array
The ip values of the estimates of the arguments of the model, $\stackrel{^}{\beta }$.
3:     $\mathrm{covar}\left({\mathbf{ip}}×\left({\mathbf{ip}}+1\right)/2\right)$ – double array
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 ${\mathbf{covar}}\left(\left(j×\left(j-1\right)/2+i\right)\right)$.
4:     $\mathrm{v}\left(\mathit{ldv},{\mathbf{ip}}+7\right)$ – double array
ldv, the first dimension of the array, must satisfy the constraint $\mathit{ldv}\ge {\mathbf{ip}}$.
5:     $\mathrm{f}\left({\mathbf{ip}}\right)$ – double array
$f$, the linear function to be estimated.
6:     $\mathrm{tol}$ – double scalar
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.

Optional Input Parameters

1:     $\mathrm{ip}$int64int32nag_int scalar
Default: the dimension of the arrays b, f and the first dimension of the array v. (An error is raised if these dimensions are not equal.)
$p$, the number of terms in the linear model.
Constraint: ${\mathbf{ip}}\ge 1$.

Output Parameters

1:     $\mathrm{est}$ – logical scalar
Indicates if the function was estimable.
${\mathbf{est}}=\mathit{true}$
The function is estimable.
${\mathbf{est}}=\mathit{false}$
The function is not estimable and stat, sestat and z are not set.
2:     $\mathrm{stat}$ – double scalar
If ${\mathbf{est}}=\mathit{true}$, stat contains the estimate of the function, ${f}^{\mathrm{T}}\stackrel{^}{\beta }$
3:     $\mathrm{sestat}$ – double scalar
If ${\mathbf{est}}=\mathit{true}$, sestat contains the standard error of the estimate of the function, $\mathrm{se}\left(F\right)$.
4:     $\mathrm{z}$ – double scalar
If ${\mathbf{est}}=\mathit{true}$, z contains the $z$ statistic for the test of the function being equal to zero.
5:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Note: nag_correg_glm_estfunc (g02gn) may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

${\mathbf{ifail}}=1$
 On entry, ${\mathbf{ip}}<1$, or ${\mathbf{irank}}<1$, or ${\mathbf{irank}}>{\mathbf{ip}}$, or $\mathit{ldv}<{\mathbf{ip}}$.
W  ${\mathbf{ifail}}=2$
 On entry, ${\mathbf{irank}}={\mathbf{ip}}$. In this case est is returned as true and all statistics are calculated.
W  ${\mathbf{ifail}}=3$
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 covar and f.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

Accuracy

The computations are believed to be stable.

Further Comments

The value of estimable functions is independent of the solution chosen from the many possible solutions. While nag_correg_glm_estfunc (g02gn) may be used to estimate functions of the arguments of the model as computed by nag_correg_glm_constrain (g02gk), ${\beta }_{\mathrm{c}}$, these must be expressed in terms of the original arguments, $\beta$. The relation between the two sets of arguments may not be straightforward.

Example

A loglinear model is fitted to a $3$ by $5$ contingency table by nag_correg_glm_poisson (g02gc). 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 nag_correg_glm_estfunc (g02gn). The results of nag_correg_glm_estfunc (g02gn) are printed.
```function g02gn_example

fprintf('g02gn example results\n\n');

x = [
1  0  0  1  0  0  0  0;
1  0  0  0  1  0  0  0;
1  0  0  0  0  1  0  0;
1  0  0  0  0  0  1  0;
1  0  0  0  0  0  0  1;
0  1  0  1  0  0  0  0;
0  1  0  0  1  0  0  0;
0  1  0  0  0  1  0  0;
0  1  0  0  0  0  1  0;
0  1  0  0  0  0  0  1;
0  0  1  1  0  0  0  0;
0  0  1  0  1  0  0  0;
0  0  1  0  0  1  0  0;
0  0  1  0  0  0  1  0;
0  0  1  0  0  0  0  1];

y = [141  67 114  79  39 131  66 143  72  35  36  14  38  28  16];

[n,m] = size(x);
isx = ones(m,1,'int64');
ip = int64(m+1);

link   = 'L';
mean_p = 'M';
eps = 1e-6;
tol = 5e-5;

% Fit generalized linear model with Poisson errors
[dev, idf, b, irank, se, covar, v, ifail] = ...
g02gc( ...
link, mean_p, x, isx, ip, y, 'eps', eps, 'tol', tol);

% Display initial results
fprintf('Deviance           = %12.4e\n', dev);
fprintf('Degrees of freedom = %2d\n', idf);
fprintf('\nVariable   Parameter estimate   Standard error\n\n');
ivar = double([1:ip]');
fprintf('%6d%16.4f%20.4f\n',[ivar b se]');

f = [1  0  0;
1  1  1;
0 -1  0;
0  0  0;
1  0  0;
0  0  0;
0  0  0;
0  0  0;
0  0  0];
tol = 5e-05;

% Estimate the estimable functions
for j = 1:size(f,2)
[est, stat, sestat, z, ifail] = ...
g02gn( ...
irank, b, covar, v, f(:,j), tol, 'ip', ip);

% Display results
fprintf('\nFunction %2d\n\n', j);
fprintf('%6.1f', f(:,j)');
if est
fprintf('\n\nstat = %10.4f, se = %10.4f, z = %10.4f\n', stat, sestat, z);
else
fprintf('\n\nFunction not estimable\n');
end
end

```
```g02gn example results

Deviance           =   9.0379e+00
Degrees of freedom =  8

Variable   Parameter estimate   Standard error

1          2.5977              0.0258
2          1.2619              0.0438
3          1.2777              0.0436
4          0.0580              0.0668
5          1.0307              0.0551
6          0.2910              0.0732
7          0.9876              0.0559
8          0.4880              0.0675
9         -0.1996              0.0904

Function  1

1.0   1.0   0.0   0.0   1.0   0.0   0.0   0.0   0.0

stat =     4.8903, se =     0.0674, z =    72.5934

Function  2

0.0   1.0  -1.0   0.0   0.0   0.0   0.0   0.0   0.0

stat =    -0.0158, se =     0.0672, z =    -0.2350

Function  3

0.0   1.0   0.0   0.0   0.0   0.0   0.0   0.0   0.0

Function not estimable
```

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