NAG Toolbox: nag_correg_glm_poisson (g02gc)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{link}$ – string (length ≥ 1)

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 $\text{'R'}$.

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 $>0$;
• if $\mathbf{mean\_p}=\text{'Z'}$, exactly ip values of isx must be $>0$.

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

If $0.0\le \mathbf{tol}<\mathit{machine precision}$, the function will use $10\times \mathit{machine precision}$ 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 $0.0\le \mathbf{eps}<\mathit{machine precision}$, 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, $\hat{\beta }$.

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 $\mathbf{eps}\times$(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:     $\mathrm{covar}\left(\mathbf{ip}\times \left(\mathbf{ip}+1\right)/2\right)$ – 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 $\mathbf{covar}\left(\left(j\times \left(j-1\right)/2+i\right)\right)$.

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, ${\hat{\mu }}_{\mathit{i}}$, 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

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., $\hat{\mu }=0.0$. 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.

   $\mathbf{ifail}=-99$

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

Further Comments

Example

Open in the MATLAB editor: g02gc_example

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);

link = 'L';
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