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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_correg_glm_gamma (g02gd)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_correg_glm_gamma (g02gd) fits a generalized linear model with gamma errors.

Syntax

[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, yi, from a gamma distribution with probability density function:
1Γν νy μ νexp-νy μ 1y  
ν being constant for the sample.
(b) X, a set of p independent variables for each observation, x1,x2,,xp.
(c) a linear model:
η=βjxj.  
(d) a link between the linear predictor, η, and the mean of the distribution, μ, η=gμ. The possible link functions are:
(i) exponent link: η=μa, for a constant a,
(ii) identity link: η=μ,
(iii) log link: η=logμ,
(iv) square root link: η=μ,
(v) reciprocal link: η= 1μ .
(e) a measure of fit, an adjusted deviance. This is a function related to the deviance, but defined for y=0:
i=1ndev*yi,μ^i=i=1n2 logμ^i+yiμ^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= τdη dμ 2 ,   where  τ=1μ.  
At each iteration an approximation to the estimate of β, β^ is found by the weighted least squares regression of z on X with weights w.
nag_correg_glm_gamma (g02gd) finds a QR decomposition of w12X, i.e.,
If R is of full rank then β^ 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.
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 w12X.
This gives the solution
β^=P1D-1 Q* 0 0 I QTw12z,  
where P1 is the first k columns of P, i.e., P=P1P0.
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 scale argument, ν-1 is estimated by a moment estimator:
ν^ -1 = i=1 n yi - μ^i / 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 χ2-distribution with degrees of freedom given by the difference in the degrees of freedom associated with the two deviances.
The arguments estimates, β^, are asymptotically Normally distributed with variance-covariance matrix:
The residuals and influence statistics can also be examined.
The estimated linear predictor η^=Xβ^, can be written as Hw12z for an n by n matrix H. The ith diagonal elements of H, hi, give a measure of the influence of the ith values of the independent variables on the fitted regression model. These are known as leverages.
The fitted values are given by μ^=g-1η^.
nag_correg_glm_gamma (g02gd) also computes the Anscombe residuals, r:
ri = 3 y i 13 - μ^ i 13 μ^ i 13 .  
An option allows the use of prior weights, ωi. This gives a model with:
νi = νωi .  
In many linear regression models the first term is taken as a mean term or an intercept, i.e., xi,1 = 1 , for i=1,2,,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 + βj xj .  
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* = D-1 P1T P0T .  

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

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' or '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' or 'Z'.
3:     xldxm – double array
ldx, the first dimension of the array, must satisfy the constraint ldxn.
xij must contain the ith observation for the jth independent variable, for i=1,2,,n and j=1,2,,m.
4:     isxm int64int32nag_int array
Indicates which independent variables are to be included in the model.
If isxj>0, the variable contained in the jth column of x is included in the regression model.
Constraints:
  • isxj0, for i=1,2,,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:     yn – double array
y, the dependent variable.
Constraint: yi0.0, for i=1,2,,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: s0.0.

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: n2.
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: m1.
3:     wt: – double array
The dimension of the array wt must be at least n if weight='W', and at least 1 otherwise
If weight='W', wt must contain the weights to be used in the weighted regression. If wti=0.0, the ith 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', wti0.0, for i=1,2,,n.
4:     a – double scalar
Default: 0
If link='E', a must contain the power of the exponential.
If link'E', a is not referenced.
Constraint: if link='E', a0.0.
5:     vnip+7 – double array
If offset='N', v need not be set.
If offset='Y', vi7, for i=1,2,,n, must contain the offset values oi. 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×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 0.0tol<machine precision then the function will use 10×machine precision instead.
Constraint: tol0.0.
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: maxit0.
8:     iprint int64int32nag_int scalar
Default: 0
Indicates if the printing of information on the iterations is required.
iprint0
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.
If 0.0eps<machine precision then the function will use machine precision instead.
Constraint: eps0.0.

Output Parameters

1:     s – double scalar
If on input s=0.0, s contains the estimated value of the scale argument, ν^-1.
If on input s0.0, 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:     bip – double array
The estimates of the parameters of the generalized linear model, β^.
If mean_p='M', the first element of b will contain the estimate of the mean parameter and bi+1 will contain the coefficient of the variable contained in column j of x, where isxj is the ith positive value in the array isx.
If mean_p='Z', bi will contain the coefficient of the variable contained in column j of x, where isxj is the ith 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×(largest singular value). It is possible for the SVD to be carried out but for irank to be returned as ip.
6:     seip – double array
The standard errors of the linear parameters.
sei contains the standard error of the parameter estimate in bi, for i=1,2,,ip.
7:     covarip×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 bi and the parameter estimate given in bj, ji, is stored in covarj×j-1/2+i.
8:     vnip+7 – double array
Auxiliary information on the fitted model.
vi1 contains the linear predictor value, ηi, for i=1,2,,n.
vi2 contains the fitted value, μ^i, for i=1,2,,n.
vi3 contains the variance standardization, 1τi, for i=1,2,,n.
vi4 contains the square root of the working weight, wi12, for i=1,2,,n.
vi5 contains the Anscombe residual, ri, for i=1,2,,n.
vi6 contains the leverage, hi, for i=1,2,,n.
vi7 contains the offset, oi, for i=1,2,,n. If offset='N', all values will be zero.
vij, for j=8,,ip+7, contains the results of the QR 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 to 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,
orm<1,
orldx<n,
orldv<n,
orip<1,
orlink'E','I','L','S' or 'R',
ors<0.0,
orlink='E' and a=0.0,
ormean_p'M' or 'Z',
orweight'U' or 'W',
oroffset'N' or 'Y',
ormaxit<0,
ortol<0.0,
oreps<0.0.
   ifail=2
On entry, weight='W' and a value of wt<0.0.
   ifail=3
On entry,a value of isx<0,
orthe value of ip is incompatible with the values of mean_p and isx,
orip is greater than the effective number of observations.
   ifail=4
On entry,yi<0.0 for some i=1,2,,n.
   ifail=5
A fitted value is at the boundary, i.e., 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 logμ, the accuracy of the β^s will be a function of tol, so tol should be set to a smaller value than the accuracy required for β^.

Further Comments

None.

Example

A set of 10 observations from two groups is input and a model for the two groups is fitted.
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

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