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 )

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 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 )

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
)

F#
static member g02gc : link : string * mean : string * offset : string * weight : string * n : int * x : float[,] * m : int * isx : int[] * ip : int * y : float[] * wt : float[] * a : float * dev : float byref * idf : int byref * b : float[] * irank : int byref * se : float[] * cov : float[] * v : float[,] * tol : float * maxit : int * iprint : int * eps : float * ifail : int byref -> unit

static member g02gc : 
        link : string * 
        mean : string * 
        offset : string * 
        weight : string * 
        n : int * 
        x : float[,] * 
        m : int * 
        isx : int[] * 
        ip : int * 
        y : float[] * 
        wt : float[] * 
        a : float * 
        dev : float byref * 
        idf : int byref * 
        b : float[] * 
        irank : int byref * 
        se : float[] * 
        cov : float[] * 
        v : float[,] * 
        tol : float * 
        maxit : int * 
        iprint : int * 
        eps : float * 
        ifail : int byref -> unit

Parameters

link

Type: System..::..String

On entry: indicates which link function is to be used.

$link = "E"$: An exponent link is used.
$link = "I"$: An identity link is used.
$link = "L"$: A log link is used;
$link = "S"$: A square root link is used.
$link = "R"$: A reciprocal link is used.

Constraint:

link = "E"

"I"

"L"

"S"

"R"

mean

Type: System..::..String

On entry: indicates if a mean term is to be included.

$mean = "M"$: A mean term, intercept, will be included in the model.
$mean = "Z"$: The model will pass through the origin, zero-point.

Constraint:

mean = "M"

"Z"

offset

Type: System..::..String

On entry: indicates if an offset is required.

$offset = "Y"$: An offset is required and the offsets must be supplied in the seventh column of v.
$offset = "N"$: No offset is required.

Constraint:

offset = "N"

"Y"

weight

Type: System..::..String

On entry: indicates if prior weights are to be used.

$weight = "U"$: No prior weights are used.
$weight = "W"$: Prior weights are used and weights must be supplied in wt.

Constraint:

weight = "U"

"W"

n: Type: System..::..Int32
On entry: $n$ , the number of observations.

Constraint: $n \geq 2$ .

x: Type: array<System..::..Double,2>[,](,)[,][,]
An array of size [dim1, m]
Note: dim1 must satisfy the constraint: $dim1 \geq n$
On entry: the matrix of all possible independent variables. $x [i - 1, j - 1]$ must contain the $i j$ th element of x, for $i = 1, 2, \dots, n$ and $j = 1, 2, \dots, m$ .

m: Type: System..::..Int32
On entry: $m$ , the total number of independent variables.

Constraint: $m \geq 1$ .

isx

Type: array<System..::..Int32>[]()[][]

An array of size [m]

On entry: indicates which independent variables are to be included in the model.

$isx [j - 1] > 0$: The variable contained in the $j$ th column of x is included in the regression model.

Constraints:

$isx [j] \geq 0$ , for $j = 0, 1, \dots, m - 1$ ;
if $mean = "M"$ , exactly $ip - 1$ values of isx must be $> 0$ ;
if $mean = "Z"$ , exactly ip values of isx must be $> 0$ .

ip: Type: System..::..Int32
On entry: the number of independent variables in the model, including the mean or intercept if present.

Constraint: $ip > 0$ .

y: Type: array<System..::..Double>[]()[][]
An array of size [n]
On entry: $y$ , observations on the dependent variable.

Constraint: $y [i] \geq 0.0$ , for $i = 0, 1, \dots, n - 1$ .

wt: Type: array<System..::..Double>[]()[][]
An array of size [dim1]
Note: the dimension of the array wt must be at least $n$ if $weight = "W"$ , and at least $1$ otherwise.
On entry: if $weight = "W"$ >, wt must contain the weights to be used in the weighted regression.
If $wt [i - 1] = 0.0$ , 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 $weight = "U"$ , wt is not referenced and the effective number of observations is $n$ .

Constraint: if $weight = "W"$ , $wt [i] \geq 0.0$ , for $i = 0, 1, \dots, n - 1$ .

a: Type: System..::..Double
On entry: if $link = "E"$ , a must contain the power of the exponential.
If $link \neq "E"$ , a is not referenced.

Constraint: if $a \neq 0.0$ , $link = "E"$ .

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>[]()[][]
An array of size [ip]
On exit: the estimates of the parameters of the generalized linear model, $\hat{β}$ .
If $mean = "M"$ , the first element of b will contain the estimate of the mean parameter and $b [i]$ will contain the coefficient of the variable contained in column $j$ of $x$ , where $isx [j - 1]$ is the $i$ th positive value in the array isx.

If $mean = "Z"$ , $b [i - 1]$ will contain the coefficient of the variable contained in column $j$ of $x$ , where $isx [j - 1]$ 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, $irank = ip$ .

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 $eps \times$ (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>[]()[][]
An array of size [ip]
On exit: the standard errors of the linear parameters.
$se [i - 1]$ contains the standard error of the parameter estimate in $b [i - 1]$ , for $i = 1, 2, \dots, ip$ .

cov: Type: array<System..::..Double>[]()[][]
An array of size [ $ip \times (ip + 1) / 2$ ]
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 $b [i - 1]$ and the parameter estimate given in $b [j - 1]$ , $j \geq i$ , is stored in $cov [(j \times (j - 1) / 2 + i) - 1]$ .

v

Type: array<System..::..Double,2>[,](,)[,][,]

An array of size [dim1,

ip + 7

]

Note: dim1 must satisfy the constraint:

dim1 \geq n

On entry: if

offset = "N"

, v need not be set.

offset = "Y"

v [i - 1, 6]

, for

i = 1, 2, \dots, n

must contain the offset values

o_{i}

. All other values need not be set.

On exit: auxiliary information on the fitted model.

$v [i - 1, 0]$	contains the linear predictor value, $η_{i}$ , for $i = 1, 2, \dots, n$ .
$v [i - 1, 1]$	contains the fitted value, ${\hat{μ}}_{i}$ , for $i = 1, 2, \dots, n$ .
$v [i - 1, 2]$	contains the variance standardization, $\frac{1}{τ_{i}}$ , for $i = 1, 2, \dots, n$ .
$v [i - 1, 3]$	contains the square root of the working weight, $w_{i}^{\frac{1}{2}}$ , for $i = 1, 2, \dots, n$ .
$v [i - 1, 4]$	contains the deviance residual, $r_{i}$ , for $i = 1, 2, \dots, n$ .
$v [i - 1, 5]$	contains the leverage, $h_{i}$ , for $i = 1, 2, \dots, n$ .
$v [i - 1, 6]$	contains the offset, $o_{i}$ , for $i = 1, 2, \dots, n$ . If $offset = "N"$ , all values will be zero.
$v [i - 1, j - 1]$	for $j = 8, \dots, ip + 7$ , contains the results of the $Q R$ decomposition or the singular value decomposition.

If the model is not of full rank, i.e.,

irank < ip

, the first ip rows of columns

8

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 $tol \times (1.0 + Current Deviance)$ . This is approximately an absolute precision if the deviance is small and a relative precision if the deviance is large.

If $0.0 \leq tol < machine precision$ , the method will use $10 \times machine precision$ instead.

Constraint: $tol \geq 0.0$ .

maxit: Type: System..::..Int32
On entry: the maximum number of iterations for the iterative weighted least squares.
If $maxit = 0$ , a default value of $10$ is used.

Constraint: $maxit \geq 0$ .

iprint

Type: System..::..Int32

On entry: indicates if the printing of information on the iterations is required.

$iprint \leq 0$

There is no printing.

$iprint > 0$

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 $0.0 \leq eps < machine precision$ , the method will use machine precision instead.

Constraint: $eps \geq 0.0$ .

ifail: Type: System..::..Int32%
On exit: $ifail = 0$ 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:

\frac{μ^{y} e^{- 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 = \sum β_{j} x_{j} .

(d)

a link between the linear predictor,

η

, and the mean of the distribution,

μ

η = g (μ)

. The possible link functions are:

(i)	exponent link: $η = μ^{a}$ , for a constant $a$ ,
(ii)	identity link: $η = μ$ ,
(iii)	log link: $η = \log μ$ ,
(iv)	square root link: $η = \sqrt{μ}$ ,
(v)	reciprocal link: $η = \frac{1}{μ}$ .

(e)

a measure of fit, the deviance:

\sum_{i = 1}^{n} dev (y_{i}, {\hat{μ}}_{i}) = \sum_{i = 1}^{n} 2 (y_{i} \log (\frac{y_{i}}{{\hat{μ}}_{i}}) - (y_{i} - {\hat{μ}}_{i})) .

The linear parameters are estimated by iterative weighted least squares. An adjusted dependent variable,

z

, is formed:

z = η + (y - μ) \frac{d η}{d μ}

and a working weight,

w

w = {(τ d \frac{d η}{d μ})}^{2},

where

τ = \sqrt{μ}

At each iteration an approximation to the estimate of

β

\hat{β}

, is found by the weighted least squares regression of

z

X

with weights

w

g02gc finds a

Q R

decomposition of

w^{1 / 2} X

, i.e.,

w^{1 / 2} X = Q R

where

R

is a

p

p

triangular matrix and

Q

is an

n

p

column orthogonal matrix.

R

is of full rank, then

\hat{β}

is the solution to:

R \hat{β} = Q^{T} w^{1 / 2} z .

R

is not of full rank a solution is obtained by means of a singular value decomposition (SVD) of

R

R = Q_{*} (\begin{array}{l} D & 0 \\ 0 & 0 \end{array}) P^{T},

where

D

is a

k

k

diagonal matrix with nonzero diagonal elements,

k

being the rank of

R

and

w^{1 / 2} X

This gives the solution

\hat{β} = P_{1} D^{- 1} (\begin{array}{l} Q_{*} & 0 \\ 0 & I \end{array}) Q^{T} w^{1 / 2} z,

P_{1}

being the first

k

columns of

P

, i.e.,

P = (P_{1} P_{0})

The iterations are continued until there is only a small change in the deviance.

The initial values for the algorithm are obtained by taking

\hat{η} = g (y) .

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

χ^{2}

-distribution with degrees of freedom given by the difference in the degrees of freedom associated with the two deviances.

The parameters estimates,

\hat{β}

, are asymptotically Normally distributed with variance-covariance matrix

$C = R^{- 1} {R^{- 1}}^{T}$ in the full rank case, otherwise
$C = P_{1} D^{- 2} P_{1}^{T}$ .

The residuals and influence statistics can also be examined.

The estimated linear predictor

\hat{η} = X \hat{β}

, can be written as

H w^{1 / 2} z

for an

n

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{μ} = g^{- 1} (\hat{η})

g02gc also computes the deviance residuals,

r

r_{i} = sign (y_{i} - {\hat{μ}}_{i}) \sqrt{dev (y_{i}, {\hat{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 + \sum β_{j} x_{j} .

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^{*}

P^{*} = (\begin{matrix} D^{- 1} P_{1}^{T} \\ P_{0}^{T} \end{matrix}) .

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

Note: g02gc may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the method:

Some error messages may refer to parameters that are dropped from this interface (LDX, LDV) In these cases, an error in another parameter has usually caused an incorrect value to be inferred.

$ifail = 1$

On entry,	$n < 2$ ,
or	$m < 1$ ,
or	$ip < 1$ ,
or	$link \neq "E"$ , $"I"$ , $"L"$ , $"S"$ or $"R"$ ,
or	$link = "E"$ and $a = 0.0$ ,
or	$mean \neq "M"$ or $"Z"$ ,
or	$weight \neq "U"$ or $"W"$ ,
or	$offset \neq "N"$ or $"Y"$ ,
or	$maxit < 0$ ,
or	$tol < 0.0$ ,
or	$eps < 0.0$ .

$ifail = 2$

On entry,

weight = "W"

and a value of

wt < 0.0

$ifail = 3$

On entry,	a value of $isx < 0$ ,
or	the value of ip is incompatible with the values of mean and isx,
or	ip is greater than the effective number of observations.

$ifail = 4$

On entry,

y [i - 1] < 0.0

for some

i = 1, 2, \dots, n

$ifail = 5$: A fitted value is at the boundary, i.e., $\hat{μ} = 0.0$ . 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.

$ifail = 6$: The singular value decomposition has failed to converge. This is an unlikely error exit.

$ifail = 7$: 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.

$ifail = 8$: The rank of the model has changed during the weighted least squares iterations. The estimate for $β$ returned may be reasonable, but you should check how the deviance has changed during iterations.

$ifail = 9$: The degrees of freedom for error are $0$ . A saturated model has been fitted.

$ifail = -9000$: An error occured, see message report.
$ifail = -6000$: Invalid Parameters $〈value〉$
$ifail = -4000$: Invalid dimension for array $〈value〉$
$ifail = -8000$: Negative dimension for array $〈value〉$
$ifail = -6000$: Invalid Parameters $〈value〉$

Accuracy

The accuracy depends on the value of tol as described in [Parameters]. As the deviance is a function of

\log μ

the accuracy of the

\hat{β}

will only be a function of tol. tol should therefore be set smaller than the accuracy required for

\hat{β}

Parallelism and Performance

None.

Further Comments

None.

Example

3

5

contingency table given by Plackett (1974) is analysed by fitting 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} .

Example program (C#): g02gce.cs

Example program data: g02gce.d

Example program results: g02gce.r

Syntax

Parameters

Description

References

Error Indicators and Warnings

Accuracy

Parallelism and Performance

Further Comments

Example

See Also