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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_correg_glm_poisson (g02gc)

## Purpose

nag_correg_glm_poisson (g02gc) fits a generalized linear model with Poisson errors.

## Syntax

[dev, idf, b, irank, se, covar, v, ifail] = g02gc(link, mean_p, x, isx, ip, y, 'n', n, 'm', m, 'wt', wt, 'a', a, 'v', v, 'tol', tol, 'maxit', maxit, 'iprint', iprint, 'eps', eps)
[dev, idf, b, irank, se, covar, v, ifail] = nag_correg_glm_poisson(link, mean_p, x, isx, ip, y, 'n', n, 'm', m, 'wt', wt, 'a', a, 'v', v, 'tol', tol, 'maxit', maxit, 'iprint', iprint, 'eps', eps)
Note: the interface to this routine has changed since earlier releases of the toolbox:
 At Mark 23: offset and weight were removed from the interface; v, wt, tol, maxit, iprint, eps and a were made optional

## Description

A generalized linear model with Poisson errors consists of the following elements:
(a) a set of $n$ observations, ${y}_{i}$, from a Poisson distribution:
 $μye-μ y! .$
(b) $X$, a set of $p$ independent variables for each observation, ${x}_{1},{x}_{2},\dots ,{x}_{p}$.
(c) a linear model:
 $η=∑βjxj.$
(d) a link between the linear predictor, $\eta$, and the mean of the distribution, $\mu$, $\eta =g\left(\mu \right)$. The possible link functions are:
 (i) exponent link: $\eta ={\mu }^{a}$, for a constant $a$, (ii) identity link: $\eta =\mu$, (iii) log link: $\eta =\mathrm{log}\mu$, (iv) square root link: $\eta =\sqrt{\mu }$, (v) reciprocal link: $\eta =\frac{1}{\mu }$.
(e) a measure of fit, the deviance:
 $∑i=1ndevyi,μ^i=∑i=1n2 yilogyiμ^i-yi-μ^i .$
The linear arguments are estimated by iterative weighted least squares. An adjusted dependent variable, $z$, is formed:
 $z=η+y-μdη dμ$
and a working weight, $w$,
 $w= τddη dμ 2,$
where $\tau =\sqrt{\mu }$.
At each iteration an approximation to the estimate of $\beta$, $\stackrel{^}{\beta }$, is found by the weighted least squares regression of $z$ on $X$ with weights $w$.
nag_correg_glm_poisson (g02gc) finds a $QR$ decomposition of ${w}^{1/2}X$, i.e., ${w}^{1/2}X=QR$ where $R$ is a $p$ by $p$ triangular matrix and $Q$ is an $n$ by $p$ column orthogonal matrix.
If $R$ is of full rank, then $\stackrel{^}{\beta }$ is the solution to:
 $Rβ^=QTw1/2z.$
If $R$ is not of full rank a solution is obtained by means of a singular value decomposition (SVD) of $R$.
 $R=Q* D 0 0 0 PT,$
where $D$ is a $k$ by $k$ diagonal matrix with nonzero diagonal elements, $k$ being the rank of $R$ and ${w}^{1/2}X$.
This gives the solution
 $β^=P1D-1 Q* 0 0 I QTw1/2z,$
${P}_{1}$ being the first $k$ columns of $P$, i.e., $P=\left({P}_{1}{P}_{0}\right)$.
The iterations are continued until there is only a small change in the deviance.
The initial values for the algorithm are obtained by taking
 $η^=gy.$
The fit of the model can be assessed by examining and testing the deviance, in particular by comparing the difference in deviance between nested models, i.e., when one model is a sub-model of the other. The difference in deviance between two nested models has, asymptotically, a ${\chi }^{2}$-distribution with degrees of freedom given by the difference in the degrees of freedom associated with the two deviances.
The arguments estimates, $\stackrel{^}{\beta }$, are asymptotically Normally distributed with variance-covariance matrix
• $C={R}^{-1}{{R}^{-1}}^{\mathrm{T}}$ in the full rank case, otherwise
• $C={P}_{1}{D}^{-2}{P}_{1}^{\mathrm{T}}$.
The residuals and influence statistics can also be examined.
The estimated linear predictor $\stackrel{^}{\eta }=X\stackrel{^}{\beta }$, can be written as $H{w}^{1/2}z$ for an $n$ by $n$ matrix $H$. The $i$th diagonal elements of $H$, ${h}_{i}$, give a measure of the influence of the $i$th values of the independent variables on the fitted regression model. These are known as leverages.
The fitted values are given by $\stackrel{^}{\mu }={g}^{-1}\left(\stackrel{^}{\eta }\right)$.
nag_correg_glm_poisson (g02gc) also computes the deviance residuals, $r$:
 $ri=signyi-μ^idevyi,μ^i.$
An option allows prior weights to be used with the model.
In many linear regression models the first term is taken as a mean term or an intercept, i.e., ${x}_{i,1}=1$, for $i=1,2,\dots ,n$. This is provided as an option.
Often only some of the possible independent variables are included in a model; the facility to select variables to be included in the model is provided.
If part of the linear predictor can be represented by a variables with a known coefficient then this can be included in the model by using an offset, $o$:
 $η=o+∑βjxj.$
If the model is not of full rank the solution given will be only one of the possible solutions. Other estimates may be obtained by applying constraints to the arguments. These solutions can be obtained by using nag_correg_glm_constrain (g02gk) after using nag_correg_glm_poisson (g02gc). Only certain linear combinations of the arguments will have unique estimates, these are known as estimable functions, these can be estimated and tested using nag_correg_glm_estfunc (g02gn).
Details of the SVD are made available in the form of the matrix ${P}^{*}$:
 $P*= D-1 P1T P0T .$
The generalized linear model with Poisson errors can be used to model contingency table data; see Cook and Weisberg (1982) and McCullagh and Nelder (1983).

## References

Cook R D and Weisberg S (1982) Residuals and Influence in Regression Chapman and Hall
McCullagh P and Nelder J A (1983) Generalized Linear Models Chapman and Hall
Plackett R L (1974) The Analysis of Categorical Data Griffin

## Parameters

### Compulsory Input Parameters

Indicates which link function is to be used.
${\mathbf{link}}=\text{'E'}$
${\mathbf{link}}=\text{'I'}$
${\mathbf{link}}=\text{'L'}$
${\mathbf{link}}=\text{'S'}$
A square root link is used.
${\mathbf{link}}=\text{'R'}$
Constraint: ${\mathbf{link}}=\text{'E'}$, $\text{'I'}$, $\text{'L'}$, $\text{'S'}$ or $\text{'R'}$.
2:     $\mathrm{mean_p}$ – string (length ≥ 1)
Indicates if a mean term is to be included.
${\mathbf{mean_p}}=\text{'M'}$
A mean term, intercept, will be included in the model.
${\mathbf{mean_p}}=\text{'Z'}$
The model will pass through the origin, zero-point.
Constraint: ${\mathbf{mean_p}}=\text{'M'}$ or $\text{'Z'}$.
3:     $\mathrm{x}\left(\mathit{ldx},{\mathbf{m}}\right)$ – double array
ldx, the first dimension of the array, must satisfy the constraint $\mathit{ldx}\ge {\mathbf{n}}$.
The matrix of all possible independent variables. ${\mathbf{x}}\left(\mathit{i},\mathit{j}\right)$ must contain the $\mathit{i}\mathit{j}$th element of x, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,{\mathbf{m}}$.
4:     $\mathrm{isx}\left({\mathbf{m}}\right)$int64int32nag_int array
Indicates which independent variables are to be included in the model.
${\mathbf{isx}}\left(j\right)>0$
The variable contained in the $j$th column of x is included in the regression model.
Constraints:
• ${\mathbf{isx}}\left(\mathit{j}\right)\ge 0$, for $\mathit{j}=1,2,\dots ,{\mathbf{m}}$;
• if ${\mathbf{mean_p}}=\text{'M'}$, exactly ${\mathbf{ip}}-1$ values of isx must be $\text{}>0$;
• if ${\mathbf{mean_p}}=\text{'Z'}$, exactly ip values of isx must be $\text{}>0$.
5:     $\mathrm{ip}$int64int32nag_int scalar
The number of independent variables in the model, including the mean or intercept if present.
Constraint: ${\mathbf{ip}}>0$.
6:     $\mathrm{y}\left({\mathbf{n}}\right)$ – double array
$y$, observations on the dependent variable.
Constraint: ${\mathbf{y}}\left(\mathit{i}\right)\ge 0.0$, for $\mathit{i}=1,2,\dots ,n$.

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the dimension of the array y and the first dimension of the arrays x, v. (An error is raised if these dimensions are not equal.)
$n$, the number of observations.
Constraint: ${\mathbf{n}}\ge 2$.
2:     $\mathrm{m}$int64int32nag_int scalar
Default: the dimension of the array isx and the second dimension of the array x. (An error is raised if these dimensions are not equal.)
$m$, the total number of independent variables.
Constraint: ${\mathbf{m}}\ge 1$.
3:     $\mathrm{wt}\left(:\right)$ – double array
The dimension of the array wt must be at least ${\mathbf{n}}$ if $\mathit{weight}=\text{'W'}$, and at least $1$ otherwise
If provided>, wt must contain the weights to be used in the weighted regression.
If ${\mathbf{wt}}\left(i\right)=0.0$, the $i$th observation is not included in the model, in which case the effective number of observations is the number of observations with nonzero weights.
If wt is not provided the effective number of observations is $n$.
Constraint: if $\mathit{weight}=\text{'W'}$, ${\mathbf{wt}}\left(\mathit{i}\right)\ge 0.0$, for $\mathit{i}=1,2,\dots ,n$.
4:     $\mathrm{a}$ – double scalar
Default: $0$
If ${\mathbf{link}}=\text{'E'}$, a must contain the power of the exponential.
If ${\mathbf{link}}\ne \text{'E'}$, a is not referenced.
Constraint: if ${\mathbf{a}}\ne 0.0$, ${\mathbf{link}}=\text{'E'}$.
5:     $\mathrm{v}\left({\mathbf{n}},{\mathbf{ip}}+7\right)$ – double array
If $\mathit{offset}=\text{'N'}$, v need not be set.
If $\mathit{offset}=\text{'Y'}$, ${\mathbf{v}}\left(\mathit{i},7\right)$, for $\mathit{i}=1,2,\dots ,n$ must contain the offset values ${o}_{\mathit{i}}$. All other values need not be set.
6:     $\mathrm{tol}$ – double scalar
Default: $0$
Indicates the accuracy required for the fit of the model.
The iterative weighted least squares procedure is deemed to have converged if the absolute change in deviance between iterations is less than ${\mathbf{tol}}×\left(1.0+\text{Current Deviance}\right)$. This is approximately an absolute precision if the deviance is small and a relative precision if the deviance is large.
If , the function will use  instead.
Constraint: ${\mathbf{tol}}\ge 0.0$.
7:     $\mathrm{maxit}$int64int32nag_int scalar
Default: $10$
The maximum number of iterations for the iterative weighted least squares.
If ${\mathbf{maxit}}=0$, a default value of $10$ is used.
Constraint: ${\mathbf{maxit}}\ge 0$.
8:     $\mathrm{iprint}$int64int32nag_int scalar
Default: $0$
Indicates if the printing of information on the iterations is required.
${\mathbf{iprint}}\le 0$
There is no printing.
${\mathbf{iprint}}>0$
Every iprint iteration, the following are printed:
• the deviance;
• the current estimates;
• and if the weighted least squares equations are singular then this is indicated.
When printing occurs the output is directed to the current advisory message unit (see nag_file_set_unit_advisory (x04ab)).
9:     $\mathrm{eps}$ – double scalar
Default: $0$
The value of eps is used to decide if the independent variables are of full rank and, if not, what is the rank of the independent variables. The smaller the value of eps the stricter the criterion for selecting the singular value decomposition.
If , the function will use machine precision instead.
Constraint: ${\mathbf{eps}}\ge 0.0$.

### Output Parameters

1:     $\mathrm{dev}$ – double scalar
The deviance for the fitted model.
2:     $\mathrm{idf}$int64int32nag_int scalar
The degrees of freedom asociated with the deviance for the fitted model.
3:     $\mathrm{b}\left({\mathbf{ip}}\right)$ – double array
The estimates of the parameters of the generalized linear model, $\stackrel{^}{\beta }$.
If ${\mathbf{mean_p}}=\text{'M'}$, the first element of b will contain the estimate of the mean parameter and ${\mathbf{b}}\left(i+1\right)$ will contain the coefficient of the variable contained in column $j$ of ${\mathbf{x}}$, where ${\mathbf{isx}}\left(j\right)$ is the $i$th positive value in the array isx.
If ${\mathbf{mean_p}}=\text{'Z'}$, ${\mathbf{b}}\left(i\right)$ will contain the coefficient of the variable contained in column $j$ of ${\mathbf{x}}$, where ${\mathbf{isx}}\left(j\right)$ is the $i$th positive value in the array isx.
4:     $\mathrm{irank}$int64int32nag_int scalar
The rank of the independent variables.
If the model is of full rank, ${\mathbf{irank}}={\mathbf{ip}}$.
If the model is not of full rank, irank is an estimate of the rank of the independent variables. irank is calculated as the number of singular values greater that ${\mathbf{eps}}×\text{}$(largest singular value). It is possible for the SVD to be carried out but for irank to be returned as ip.
5:     $\mathrm{se}\left({\mathbf{ip}}\right)$ – double array
The standard errors of the linear parameters.
${\mathbf{se}}\left(\mathit{i}\right)$ contains the standard error of the parameter estimate in ${\mathbf{b}}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{ip}}$.
6:     $\mathrm{covar}\left({\mathbf{ip}}×\left({\mathbf{ip}}+1\right)/2\right)$ – 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 ${\mathbf{covar}}\left(\left(j×\left(j-1\right)/2+i\right)\right)$.
7:     $\mathrm{v}\left({\mathbf{n}},{\mathbf{ip}}+7\right)$ – double array
Auxiliary information on the fitted model.
 ${\mathbf{v}}\left(i,1\right)$ contains the linear predictor value, ${\eta }_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$. ${\mathbf{v}}\left(i,2\right)$ contains the fitted value, ${\stackrel{^}{\mu }}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$. ${\mathbf{v}}\left(i,3\right)$ contains the variance standardization, $\frac{1}{{\tau }_{\mathit{i}}}$, for $\mathit{i}=1,2,\dots ,n$. ${\mathbf{v}}\left(i,4\right)$ contains the square root of the working weight, ${w}_{\mathit{i}}^{\frac{1}{2}}$, for $\mathit{i}=1,2,\dots ,n$. ${\mathbf{v}}\left(i,5\right)$ contains the deviance residual, ${r}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$. ${\mathbf{v}}\left(i,6\right)$ contains the leverage, ${h}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$. ${\mathbf{v}}\left(i,7\right)$ contains the offset, ${o}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$. If $\mathit{offset}=\text{'N'}$, all values will be zero. ${\mathbf{v}}\left(i,j\right)$ for $j=8,\dots ,{\mathbf{ip}}+7$, contains the results of the $QR$ decomposition or the singular value decomposition.
If the model is not of full rank, i.e., ${\mathbf{irank}}<{\mathbf{ip}}$, the first ip rows of columns $8$ to ${\mathbf{ip}}+7$ contain the ${P}^{*}$ matrix.
8:     $\mathrm{ifail}$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_poisson (g02gc) 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, ${\mathbf{n}}<2$, or ${\mathbf{m}}<1$, or $\mathit{ldx}<{\mathbf{n}}$, or $\mathit{ldv}<{\mathbf{n}}$, or ${\mathbf{ip}}<1$, or ${\mathbf{link}}\ne \text{'E'}$, $\text{'I'}$, $\text{'L'}$, $\text{'S'}$ or $\text{'R'}$, or ${\mathbf{link}}=\text{'E'}$ and ${\mathbf{a}}=0.0$, or ${\mathbf{mean_p}}\ne \text{'M'}$ or $\text{'Z'}$, or $\mathit{weight}\ne \text{'U'}$ or $\text{'W'}$, or $\mathit{offset}\ne \text{'N'}$ or $\text{'Y'}$, or ${\mathbf{maxit}}<0$, or ${\mathbf{tol}}<0.0$, or ${\mathbf{eps}}<0.0$.
${\mathbf{ifail}}=2$
 On entry, $\mathit{weight}=\text{'W'}$ and a value of ${\mathbf{wt}}<0.0$.
${\mathbf{ifail}}=3$
 On entry, a value of ${\mathbf{isx}}<0$, or the value of ip is incompatible with the values of mean_p and isx, or ip is greater than the effective number of observations.
${\mathbf{ifail}}=4$
 On entry, ${\mathbf{y}}\left(i\right)<0.0$ for some $i=1,2,\dots ,n$.
${\mathbf{ifail}}=5$
A fitted value is at the boundary, i.e., $\stackrel{^}{\mu }=0.0$. This may occur if there are $y$ values of $0.0$ and the model is too complex for the data. The model should be reformulated with, perhaps, some observations dropped.
${\mathbf{ifail}}=6$
The singular value decomposition has failed to converge. This is an unlikely error exit.
${\mathbf{ifail}}=7$
The iterative weighted least squares has failed to converge in maxit (or default $10$) iterations. The value of maxit could be increased but it may be advantageous to examine the convergence using the iprint option. This may indicate that the convergence is slow because the solution is at a boundary in which case it may be better to reformulate the model.
W  ${\mathbf{ifail}}=8$
The rank of the model has changed during the weighted least squares iterations. The estimate for $\beta$ returned may be reasonable, but you should check how the deviance has changed during iterations.
W  ${\mathbf{ifail}}=9$
The degrees of freedom for error are $0$. A saturated model has been fitted.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

The accuracy depends on the value of tol as described in Arguments. As the deviance is a function of $\mathrm{log}\mu$ the accuracy of the $\stackrel{^}{\beta }$ will only be a function of tol. tol should therefore be set smaller than the accuracy required for $\stackrel{^}{\beta }$.

None.

## Example

A $3$ by $5$ contingency table given by Plackett (1974) is analysed by fitting terms for rows and columns. The table is:
 $141 67 114 79 39 131 66 143 72 35 36 14 38 28 16 .$
function g02gc_example

fprintf('g02gc 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);

mean_p = 'M';
eps = 1e-6;
maxit = int64(20);

% 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, 'maxit', maxit);

%  Display 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]');
fprintf('\n     y        fv     residual       h\n\n');
for j=1:n
fprintf('%7.1f%10.2f%12.4f%10.3f\n',y(j),v(j,2),v(j,5),v(j,6));
end

g02gc 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

y        fv     residual       h

141.0    132.99      0.6875     0.604
67.0     63.47      0.4386     0.514
114.0    127.38     -1.2072     0.596
79.0     77.29      0.1936     0.532
39.0     38.86      0.0222     0.482
131.0    135.11     -0.3553     0.608
66.0     64.48      0.1881     0.520
143.0    129.41      1.1749     0.601
72.0     78.52     -0.7465     0.537
35.0     39.48     -0.7271     0.488
36.0     39.90     -0.6276     0.393
14.0     19.04     -1.2131     0.255
38.0     38.21     -0.0346     0.382
28.0     23.19      0.9675     0.282
16.0     11.66      1.2028     0.206