[s, dev, idf, b, irank, se, covar, v, ifail] = g02gd(link, mean_p, x, isx, ip, y, s, 'n', n, 'm', m, 'wt', wt, 'a', a, 'v', v, 'tol', tol, 'maxit', maxit, 'iprint', iprint, 'eps', eps)

[s, dev, idf, b, irank, se, covar, v, ifail] = nag_correg_glm_gamma(link, mean_p, x, isx, ip, y, s, '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 gamma errors consists of the following elements:

(a)

a set of

n

observations,

y_{i}

, from a gamma distribution with probability density function:

\frac{1}{Γ (ν)} {(\frac{ν y}{μ})}^{ν} \exp (- \frac{ν y}{μ}) \frac{1}{y}

ν

being constant for the sample.

(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, an adjusted deviance. This is a function related to the deviance, but defined for

y = 0

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

The linear arguments 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 \frac{d η}{d μ})}^{2},   where τ = \frac{1}{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

nag_correg_glm_gamma (g02gd) finds a

Q R

decomposition of

w^{\frac{1}{2}} X

, i.e.,

$w^{\frac{1}{2}} X = Q R$ where $R$ is a $p$ by $p$ triangular matrix and $Q$ is an $n$ by $p$ column orthogonal matrix.

R

is of full rank then

\hat{β}

is the solution to:

$R \hat{β} = Q^{T} w^{\frac{1}{2}} z$

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^{\frac{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^{\frac{1}{2}} z,

where

P_{1}

is 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 scale argument,

ν^{- 1}

is estimated by a moment estimator:

{\hat{ν}}^{- 1} = \sum_{i = 1}^{n} \frac{{[(y_{i} - {\hat{μ}}_{i}) / \hat{μ}]}^{2}}{(n - k)} .

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 or adjusted deviance between two nested models with known

ν

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 arguments estimates,

\hat{β}

, are asymptotically Normally distributed with variance-covariance matrix:

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

The residuals and influence statistics can also be examined.

The estimated linear predictor

\hat{η} = X \hat{β}

, can be written as

H w^{\frac{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})

nag_correg_glm_gamma (g02gd) also computes the Anscombe residuals,

r

r_{i} = \frac{3 (y_{i}^{\frac{1}{3}} - {\hat{μ}}_{i}^{\frac{1}{3}})}{{\hat{μ}}_{i}^{\frac{1}{3}}} .

An option allows the use of prior weights,

ω_{i}

. This gives a model with:

ν_{i} = ν ω_{i} .

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 arguments. These solutions can be obtained by using nag_correg_glm_constrain (g02gk) after using nag_correg_glm_gamma (g02gd). Only certain linear combinations of the arguments will have unique estimates, these are known as estimable functions, and 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^{*} = (\begin{matrix} D^{- 1} P_{1}^{T} \\ P_{0}^{T} \end{matrix}) .

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

Parameters

Compulsory Input Parameters

1: $link$ – string (length ≥ 1)

Indicates which link function is to be used.

$link ='E'$: An exponential 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'

2: $mean_p$ – string (length ≥ 1)

Indicates if a mean term is to be included.

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

Constraint:

mean_p ='M'

'Z'

3: $x (ldx, m)$ – double array

ldx, the first dimension of the array, must satisfy the constraint

ldx \geq n

x (i, j)

must contain the

i

th observation for the

j

th independent variable, for

i = 1, 2, \dots, n

and

j = 1, 2, \dots, m

4: $isx (m)$ – int64int32nag_int array

Indicates which independent variables are to be included in the model.

isx (j) > 0

, the variable contained in the

j

th column of x is included in the regression model.

Constraints:

$isx (j) \geq 0$ , for $i = 1, 2, \dots, m$ ;
if $mean_p ='M'$ , exactly $ip - 1$ values of isx must be $> 0$ ;
if $mean_p ='Z'$ , exactly ip values of isx must be $> 0$ .

5: $ip$ – int64int32nag_int scalar

The number of independent variables in the model, including the mean or intercept if present.

Constraint:

ip > 0

6: $y (n)$ – double array

y

, the dependent variable.

Constraint:

y (i) \geq 0.0

, for

i = 1, 2, \dots, n

7: $s$ – double scalar

The scale argument for the gamma model,

ν^{- 1}

$s = 0.0$: The scale argument is estimated with the function using the formula described in Description.

Constraint:

s \geq 0.0

Optional Input Parameters

1: $n$ – 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:

n \geq 2

2: $m$ – 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:

m \geq 1

3: $wt (:)$ – double array

The dimension of the array wt must be at least

n

weight ='W'

, and at least

1

otherwise

weight ='W'

, wt must contain the weights to be used in the weighted regression. If

wt (i) = 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.

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 = 1, 2, \dots, n

4: $a$ – double scalar

Default:

0

link ='E'

, a must contain the power of the exponential.

link \neq'E'

, a is not referenced.

Constraint: if

link ='E'

a \neq 0.0

5: $v (n, ip + 7)$ – double array

offset ='N'

, v need not be set.

offset ='Y'

v (i, 7)

, for

i = 1, 2, \dots, n

, must contain the offset values

o_{i}

. All other values need not be set.

6: $tol$ – 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

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.

0.0 \leq tol < machine precision

then the function will use

10 \times machine precision

instead.

Constraint:

tol \geq 0.0

7: $maxit$ – int64int32nag_int scalar

Default:

10

The maximum number of iterations for the iterative weighted least squares.

$maxit = 0$: A default value of $10$ is used.

Constraint:

maxit \geq 0

8: $iprint$ – int64int32nag_int scalar

Default:

0

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 nag_file_set_unit_advisory (x04ab)).

9: $eps$ – 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.

0.0 \leq eps < machine precision

then the function will use machine precision instead.

Constraint:

eps \geq 0.0

Output Parameters

1: $s$ – double scalar

If on input

s = 0.0

, s contains the estimated value of the scale argument,

{\hat{ν}}^{- 1}

If on input

s \neq 0.0

, s is unchanged on exit.

2: $dev$ – double scalar

The adjusted deviance for the fitted model.

3: $idf$ – int64int32nag_int scalar

The degrees of freedom asociated with the deviance for the fitted model.

4: $b (ip)$ – double array

The estimates of the parameters of the generalized linear model,

\hat{β}

mean_p ='M'

, the first element of b will contain the estimate of the mean parameter and

b (i + 1)

will contain the coefficient of the variable contained in column

j

x

, where

isx (j)

is the

i

th positive value in the array isx.

mean_p ='Z'

b (i)

will contain the coefficient of the variable contained in column

j

x

, where

isx (j)

is the

i

th positive value in the array isx.

5: $irank$ – int64int32nag_int scalar

The rank of the independent variables.

If the model is of full rank then

irank = ip

If the model is not of full rank then 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.

6: $se (ip)$ – double array

The standard errors of the linear parameters.

se (i)

contains the standard error of the parameter estimate in

b (i)

, for

i = 1, 2, \dots, ip

7: $covar (ip \times (ip + 1) / 2)$ – double array

The upper triangular part of the variance-covariance matrix of the ip parameter estimates given in b. They are stored in packed form by column, i.e., the covariance between the parameter estimate given in

b (i)

and the parameter estimate given in

b (j)

j \geq i

, is stored in

covar ((j \times (j - 1) / 2 + i))

8: $v (n, ip + 7)$ – double array

Auxiliary information on the fitted model.

$v (i, 1)$	contains the linear predictor value, $η_{i}$ , for $i = 1, 2, \dots, n$ .
$v (i, 2)$	contains the fitted value, ${\hat{μ}}_{i}$ , for $i = 1, 2, \dots, n$ .
$v (i, 3)$	contains the variance standardization, $\frac{1}{τ_{i}}$ , for $i = 1, 2, \dots, n$ .
$v (i, 4)$	contains the square root of the working weight, $w_{i}^{\frac{1}{2}}$ , for $i = 1, 2, \dots, n$ .
$v (i, 5)$	contains the Anscombe residual, $r_{i}$ , for $i = 1, 2, \dots, n$ .
$v (i, 6)$	contains the leverage, $h_{i}$ , for $i = 1, 2, \dots, n$ .
$v (i, 7)$	contains the offset, $o_{i}$ , for $i = 1, 2, \dots, n$ . If $offset ='N'$ , all values will be zero.
$v (i, j)$ ,	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.

9: $ifail$ – int64int32nag_int scalar

ifail = 0

unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Note: nag_correg_glm_gamma (g02gd) 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.

$ifail = 1$

On entry,	$n < 2$ ,
or	$m < 1$ ,
or	$ldx < n$ ,
or	$ldv < n$ ,
or	$ip < 1$ ,
or	$link \neq'E','I','L','S'$ or 'R',
or	$s < 0.0$ ,
or	$link ='E'$ and $a = 0.0$ ,
or	$mean_p \neq'M'$ 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_p and isx,
or	ip is greater than the effective number of observations.

$ifail = 4$

On entry,

y (i) < 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 small values of $y$ and the model is not suitable 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.

W $ifail = 8$: 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.

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

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

The accuracy depends on tol as described in Arguments. As the adjusted deviance is a function of

\log μ

, the accuracy of the

\hat{β}

s will be a function of tol, so tol should be set to a smaller value than the accuracy required for

\hat{β}

Further Comments

None.

Example

A set of

10

observations from two groups is input and a model for the two groups is fitted.

Open in the MATLAB editor: g02gd_example

function g02gd_example


fprintf('g02gd example results\n\n'); 

x = [1;  1;    1;    1;    1;    0;     0;     0;     0;    0   ];
y = [1;  0.3; 10.5;  9.7; 10.9;  0.62;  0.12;  0.09;  0.5;  2.14];

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

link   = 'R';
mean_p = 'M';
s      = 0;
tol    = 5e-5;

% Fit generalized linear model with Gamma errors
[s, dev, idf, b, irank, se, covar, v, ifail] = ...
  g02gd( ...
         link, mean_p, x, isx, ip, y, s, 'tol', tol);

  %  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

g02gd example results

Deviance           =   3.5034e+01
Degrees of freedom =  8

Variable   Parameter estimate   Standard error

     1          1.4408              0.6678
     2         -1.2865              0.6717

     y        fv     residual       h

    1.0      6.48     -1.3909     0.200
    0.3      6.48     -1.9228     0.200
   10.5      6.48      0.5236     0.200
    9.7      6.48      0.4318     0.200
   10.9      6.48      0.5678     0.200
    0.6      0.69     -0.1107     0.200
    0.1      0.69     -1.3287     0.200
    0.1      0.69     -1.4815     0.200
    0.5      0.69     -0.3106     0.200
    2.1      0.69      1.3665     0.200

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox