g02gc fits a generalized linear model with Poisson errors.
Syntax
C# 

public static void g02gc(
string link,
string mean,
string offset,
string weight,
int n,
double[,] x,
int m,
int[] isx,
int ip,
double[] y,
double[] wt,
double a,
out double dev,
out int idf,
double[] b,
out int irank,
double[] se,
double[] cov,
double[,] v,
double tol,
int maxit,
int iprint,
double eps,
out int ifail
) 
Visual Basic 

Public Shared Sub g02gc ( _
link As String, _
mean As String, _
offset As String, _
weight As String, _
n As Integer, _
x As Double(,), _
m As Integer, _
isx As Integer(), _
ip As Integer, _
y As Double(), _
wt As Double(), _
a As Double, _
<OutAttribute> ByRef dev As Double, _
<OutAttribute> ByRef idf As Integer, _
b As Double(), _
<OutAttribute> ByRef irank As Integer, _
se As Double(), _
cov As Double(), _
v As Double(,), _
tol As Double, _
maxit As Integer, _
iprint As Integer, _
eps As Double, _
<OutAttribute> ByRef ifail As Integer _
) 
Visual C++ 

public:
static void g02gc(
String^ link,
String^ mean,
String^ offset,
String^ weight,
int n,
array<double,2>^ x,
int m,
array<int>^ isx,
int ip,
array<double>^ y,
array<double>^ wt,
double a,
[OutAttribute] double% dev,
[OutAttribute] int% idf,
array<double>^ b,
[OutAttribute] int% irank,
array<double>^ se,
array<double>^ cov,
array<double,2>^ v,
double tol,
int maxit,
int iprint,
double eps,
[OutAttribute] int% ifail
) 
Parameters
 link
 Type: System..::..String
On entry: indicates which link function is to be used.
 ${\mathbf{link}}=\text{"E"}$
 An exponent link is used.
 ${\mathbf{link}}=\text{"I"}$
 An identity link is used.
 ${\mathbf{link}}=\text{"L"}$
 A log link is used;
 ${\mathbf{link}}=\text{"S"}$
 A square root link is used.
 ${\mathbf{link}}=\text{"R"}$
 A reciprocal link is used.
Constraint:
${\mathbf{link}}=\text{"E"}$, $\text{"I"}$, $\text{"L"}$, $\text{"S"}$ or $\text{"R"}$.
 mean
 Type: System..::..String
On entry: indicates if a mean term is to be included.
 ${\mathbf{mean}}=\text{"M"}$
 A mean term, intercept, will be included in the model.
 ${\mathbf{mean}}=\text{"Z"}$
 The model will pass through the origin, zeropoint.
Constraint:
${\mathbf{mean}}=\text{"M"}$ or $\text{"Z"}$.
 offset
 Type: System..::..String
On entry: indicates if an offset is required.
 ${\mathbf{offset}}=\text{"Y"}$
 An offset is required and the offsets must be supplied in the seventh column of v.
 ${\mathbf{offset}}=\text{"N"}$
 No offset is required.
Constraint:
${\mathbf{offset}}=\text{"N"}$ or $\text{"Y"}$.
 weight
 Type: System..::..String
On entry: indicates if prior weights are to be used.
 ${\mathbf{weight}}=\text{"U"}$
 No prior weights are used.
 ${\mathbf{weight}}=\text{"W"}$
 Prior weights are used and weights must be supplied in wt.
Constraint:
${\mathbf{weight}}=\text{"U"}$ or $\text{"W"}$.
 n
 Type: System..::..Int32
On entry: $n$, the number of observations.
Constraint:
${\mathbf{n}}\ge 2$.
 x
 Type: array<System..::..Double,2>[,](,)[,][,]
An array of size [
dim1,
m]
Note: dim1 must satisfy the constraint:
$\mathrm{dim1}\ge {\mathbf{n}}$
On entry: the matrix of all possible independent variables.
${\mathbf{x}}[\mathit{i}1,\mathit{j}1]$ must contain the
$\mathit{i}\mathit{j}$th element of
x, for
$\mathit{i}=1,2,\dots ,n$ and
$\mathit{j}=1,2,\dots ,{\mathbf{m}}$.
 m
 Type: System..::..Int32
On entry: $m$, the total number of independent variables.
Constraint:
${\mathbf{m}}\ge 1$.
 isx
 Type: array<System..::..Int32>[]()[][]
On entry: indicates which independent variables are to be included in the model.
 ${\mathbf{isx}}\left[j1\right]>0$
 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 $\mathit{j}=0,1,\dots ,{\mathbf{m}}1$;
 if ${\mathbf{mean}}=\text{"M"}$, exactly ${\mathbf{ip}}1$ values of isx must be $\text{}>0$;
 if ${\mathbf{mean}}=\text{"Z"}$, exactly ip values of isx must be $\text{}>0$.
 ip
 Type: System..::..Int32
On entry: the number of independent variables in the model, including the mean or intercept if present.
Constraint:
${\mathbf{ip}}>0$.
 y
 Type: array<System..::..Double>[]()[][]
On entry: $y$, observations on the dependent variable.
Constraint:
${\mathbf{y}}\left[\mathit{i}\right]\ge 0.0$, for $\mathit{i}=0,1,\dots ,n1$.
 wt
 Type: array<System..::..Double>[]()[][]
An array of size [dim1]
Note: the dimension of the array
wt
must be at least
${\mathbf{n}}$ if
${\mathbf{weight}}=\text{"W"}$, and at least
$1$ otherwise.
On entry: if
${\mathbf{weight}}=\text{"W"}$ >,
wt must contain the weights to be used in the weighted regression.
If ${\mathbf{wt}}\left[i1\right]=0.0$, 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
${\mathbf{weight}}=\text{"U"}$,
wt is not referenced and the effective number of observations is
$n$.
Constraint:
if ${\mathbf{weight}}=\text{"W"}$, ${\mathbf{wt}}\left[\mathit{i}\right]\ge 0.0$, for $\mathit{i}=0,1,\dots ,n1$.
 a
 Type: System..::..Double
On entry: 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 ${\mathbf{a}}\ne 0.0$, ${\mathbf{link}}=\text{"E"}$.
 dev
 Type: System..::..Double%
On exit: the deviance for the fitted model.
 idf
 Type: System..::..Int32%
On exit: the degrees of freedom asociated with the deviance for the fitted model.
 b
 Type: array<System..::..Double>[]()[][]
On exit: the estimates of the parameters of the generalized linear model,
$\hat{\beta}$.
If
${\mathbf{mean}}=\text{"M"}$, the first element of
b will contain the estimate of the mean parameter and
${\mathbf{b}}\left[i\right]$ will contain the coefficient of the variable contained in column
$j$ of
${\mathbf{x}}$, where
${\mathbf{isx}}\left[j1\right]$ is the
$i$th positive value in the array
isx.
If
${\mathbf{mean}}=\text{"Z"}$,
${\mathbf{b}}\left[i1\right]$ will contain the coefficient of the variable contained in column
$j$ of
${\mathbf{x}}$, where
${\mathbf{isx}}\left[j1\right]$ is the
$i$th positive value in the array
isx.
 irank
 Type: System..::..Int32%
On exit: 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
${\mathbf{eps}}\times \text{}$(largest singular value). It is possible for the SVD to be carried out but for
irank to be returned as
ip.
 se
 Type: array<System..::..Double>[]()[][]
On exit: the standard errors of the linear parameters.
${\mathbf{se}}\left[\mathit{i}1\right]$ contains the standard error of the parameter estimate in ${\mathbf{b}}\left[\mathit{i}1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{ip}}$.
 cov
 Type: array<System..::..Double>[]()[][]
An array of size [${\mathbf{ip}}\times \left({\mathbf{ip}}+1\right)/2$]
On exit: the upper triangular part of the variancecovariance 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
${\mathbf{b}}\left[i1\right]$ and the parameter estimate given in
${\mathbf{b}}\left[j1\right]$,
$j\ge i$, is stored in
${\mathbf{cov}}\left[\left(j\times \left(j1\right)/2+i\right)1\right]$.
 v
 Type: array<System..::..Double,2>[,](,)[,][,]
An array of size [dim1, ${\mathbf{ip}}+7$]
Note: dim1 must satisfy the constraint:
$\mathrm{dim1}\ge {\mathbf{n}}$
On entry: if
${\mathbf{offset}}=\text{"N"}$,
v need not be set.
If ${\mathbf{offset}}=\text{"Y"}$,
${\mathbf{v}}[\mathit{i}1,6]$, for $\mathit{i}=1,2,\dots ,n$ must contain the offset values ${o}_{\mathit{i}}$. All other values need not be set.
On exit: auxiliary information on the fitted model.
${\mathbf{v}}[i1,0]$ 
contains the linear predictor value,
${\eta}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$. 
${\mathbf{v}}[i1,1]$ 
contains the fitted value,
${\hat{\mu}}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$. 
${\mathbf{v}}[i1,2]$ 
contains the variance standardization,
$\frac{1}{{\tau}_{\mathit{i}}}$, for $\mathit{i}=1,2,\dots ,n$. 
${\mathbf{v}}[i1,3]$ 
contains the square root of the working weight,
${w}_{\mathit{i}}^{\frac{1}{2}}$, for $\mathit{i}=1,2,\dots ,n$. 
${\mathbf{v}}[i1,4]$ 
contains the deviance residual,
${r}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$. 
${\mathbf{v}}[i1,5]$ 
contains the leverage,
${h}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$. 
${\mathbf{v}}[i1,6]$ 
contains the offset,
${o}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$. If ${\mathbf{offset}}=\text{"N"}$, all values will be zero. 
${\mathbf{v}}[i1,j1]$ 
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.
 tol
 Type: System..::..Double
On entry: 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 ${\mathbf{tol}}\times \left(1.0+\text{Current Deviance}\right)$. This is approximately an absolute precision if the deviance is small and a relative precision if the deviance is large.
If $0.0\le {\mathbf{tol}}<\mathit{machineprecision}$, the method will use $10\times \mathit{machineprecision}$ instead.
Constraint:
${\mathbf{tol}}\ge 0.0$.
 maxit
 Type: System..::..Int32
On entry: the maximum number of iterations for the iterative weighted least squares.
If ${\mathbf{maxit}}=0$, a default value of $10$ is used.
Constraint:
${\mathbf{maxit}}\ge 0$.
 iprint
 Type: System..::..Int32
On entry: 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 (X04ABF not in this release)).
 eps
 Type: System..::..Double
On entry: 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 $0.0\le {\mathbf{eps}}<\mathit{machineprecision}$, the method will use machine precision instead.
Constraint:
${\mathbf{eps}}\ge 0.0$.
 ifail
 Type: System..::..Int32%
On exit:
${\mathbf{ifail}}={0}$ unless the method detects an error or a warning has been flagged (see
[Error Indicators and Warnings]).
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:

(b) 
$X$, a set of $p$ independent variables for each observation, ${x}_{1},{x}_{2},\dots ,{x}_{p}$. 
(c) 
a linear model:

(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}\u200a\mu $, 
(iv) 
square root link: $\eta =\sqrt{\mu}$, 
(v) 
reciprocal link: $\eta =\frac{1}{\mu}$. 

(e) 
a measure of fit, the deviance:

The linear parameters are estimated by iterative weighted least squares. An adjusted dependent variable,
$z$, is formed:
and a working weight,
$w$,
where
$\tau =\sqrt{\mu}$.
At each iteration an approximation to the estimate of $\beta $, $\hat{\beta}$, is found by the weighted least squares regression of $z$ on $X$ with weights $w$.
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
$\hat{\beta}$ is the solution to:
If
$R$ is not of full rank a solution is obtained by means of a singular value decomposition (SVD) of
$R$.
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
${P}_{1}$ 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
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 submodel 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 parameters estimates,
$\hat{\beta}$, are asymptotically Normally distributed with variancecovariance 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 $\hat{\eta}=X\hat{\beta}$, 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 $\hat{\mu}={g}^{1}\left(\hat{\eta}\right)$.
g02gc also computes the deviance residuals,
$r$:
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,
$o$:
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 parameters. These solutions can be obtained by using
g02gk after using
g02gc. Only certain linear combinations of the parameters will have unique estimates, these are known as estimable functions, these can be estimated and tested using
g02gn.
Details of the SVD are made available in the form of the matrix
${P}^{*}$:
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
Error Indicators and Warnings
Accuracy
The accuracy depends on the value of
tol as described in
[Parameters]. As the deviance is a function of
$\mathrm{log}\u200a\mu $ the accuracy of the
$\hat{\beta}$ will only be a function of
tol.
tol should therefore be set smaller than the accuracy required for
$\hat{\beta}$.
Parallelism and Performance
Further Comments
Example
See Also