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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_correg_glm_gamma (g02gd)

Purpose

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

Syntax

[s, dev, idf, b, irank, se, cov, v, ifail] = g02gd(link, mean, 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, cov, v, ifail] = nag_correg_glm_gamma(link, mean, 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:
Mark 23: offset & weight omitted; v, wt, tol, maxit, iprint, eps, a optional
.

Description

A generalized linear model with gamma errors consists of the following elements:
(a) a set of nn observations, yiyi, from a gamma distribution with probability density function:
1/(Γ(ν)) ((νy)/μ)νexp((νy)/μ) 1/y
1Γ(ν) (νy μ ) νexp(-νy μ ) 1y
νν being constant for the sample.
(b) XX, a set of pp independent variables for each observation, x1,x2,,xpx1,x2,,xp.
(c) a linear model:
η = βjxj.
η=βjxj.
(d) a link between the linear predictor, ηη, and the mean of the distribution, μμ, η = g(μ)η=g(μ). The possible link functions are:
(i) exponent link: η = μaη=μa, for a constant aa,
(ii) identity link: η = μη=μ,
(iii) log link: η = logμη=logμ,
(iv) square root link: η = sqrt(μ)η=μ,
(v) reciprocal link: η = 1/μ η= 1μ .
(e) a measure of fit, an adjusted deviance. This is a function related to the deviance, but defined for y = 0y=0:
n n
dev*(yi,μ̂i) = 2(log(μ̂i) + ((yi)/(μ̂i))).
i = 1 i = 1
i=1ndev*(yi,μ^i)=i=1n2 (log(μ^i)+(yiμ^i) ) .
The linear parameters are estimated by iterative weighted least squares. An adjusted dependent variable, zz, is formed:
z = η + (yμ)(dη)/(dμ)
z=η+(y-μ)dη dμ
and a working weight, ww,
w = (τ(dη)/(dμ))2,   where  τ = 1/μ.
w= (τdη dμ ) 2 ,   where  τ=1μ.
At each iteration an approximation to the estimate of ββ, β̂β^ is found by the weighted least squares regression of zz on XX with weights ww.
nag_correg_glm_gamma (g02gd) finds a QRQR decomposition of w(1/2)Xw12X, i.e.,
If RR is of full rank then β̂β^ is the solution to:
If RR is not of full rank a solution is obtained by means of a singular value decomposition (SVD) of RR.
R = Q*
(D0)
0 0
PT.
R=Q* D 0 0 0 PT.
where DD is a kk by kk diagonal matrix with nonzero diagonal elements, kk being the rank of RR and w(1/2)Xw12X.
This gives the solution
β̂ = P1D1
(Q*0)
0 I
QTw(1/2)z,
β^=P1D-1 Q* 0 0 I QTw12z,
where P1P1 is the first kk columns of PP, i.e., P = (P1P0)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
η̂ = g(y).
η^=g(y).
The scale parameter, ν1ν-1 is estimated by a moment estimator:
n
ν̂1 = ( [(yiμ̂i) / μ̂] 2 )/((nk)).
i = 1
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 νν has, asymptotically, a χ2χ2-distribution with degrees of freedom given by the difference in the degrees of freedom associated with the two deviances.
The parameters estimates, β̂β^, are asymptotically Normally distributed with variance-covariance matrix:
The residuals and influence statistics can also be examined.
The estimated linear predictor η̂ = Xβ̂η^=Xβ^, can be written as Hw(1/2)zHw12z for an nn by nn matrix HH. The iith diagonal elements of HH, hihi, give a measure of the influence of the iith values of the independent variables on the fitted regression model. These are known as leverages.
The fitted values are given by μ̂ = g1(η̂)μ^=g-1(η^).
nag_correg_glm_gamma (g02gd) also computes the Anscombe residuals, rr:
ri = ( 3 (yi(1/3)μ̂i(1/3)) )/( μ̂i(1/3) ) .
ri = 3 ( y i 13 - μ^ i 13 ) μ^ i 13 .
An option allows the use of prior weights, ωiωi. This gives a model with:
νi = νωi .
νi = νωi .
In many linear regression models the first term is taken as a mean term or an intercept, i.e., xi,1 = 1 xi,1 = 1 , for i = 1,2,,n 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, oo:
η = o + βj xj .
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 parameters. 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 parameters 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*:
P* =
( D1 P1T )
P0T
.
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'link='E'
An exponential link is used.
link = 'I'link='I'
An identity link is used.
link = 'L'link='L'
A log link is used.
link = 'S'link='S'
A square root link is used.
link = 'R'link='R'
A reciprocal link is used.
Constraint: link = 'E'link='E', 'I''I', 'L''L', 'S''S' or 'R''R'.
2:     mean – string (length ≥ 1)
Indicates if a mean term is to be included.
mean = 'M'mean='M'
A mean term, intercept, will be included in the model.
mean = 'Z'mean='Z'
The model will pass through the origin, zero-point.
Constraint: mean = 'M'mean='M' or 'Z''Z'.
3:     x(ldx,m) – double array
ldx, the first dimension of the array, must satisfy the constraint ldxnldxn.
x(i,j)xij must contain the iith observation for the jjth independent variable, for i = 1,2,,ni=1,2,,n and j = 1,2,,mj=1,2,,m.
4:     isx(m) – int64int32nag_int array
m, the dimension of the array, must satisfy the constraint m1m1.
Indicates which independent variables are to be included in the model.
If isx(j) > 0isxj>0, the variable contained in the jjth column of x is included in the regression model.
Constraints:
  • isx(j)0isxj0, for i = 1,2,,mi=1,2,,m;
  • if mean = 'M'mean='M', exactly ip1ip-1 values of isx must be > 0>0;
  • if mean = 'Z'mean='Z', exactly ip values of isx must be > 0>0.
5:     ip – int64int32nag_int scalar
The number of independent variables in the model, including the mean or intercept if present.
Constraint: ip > 0ip>0.
6:     y(n) – double array
n, the dimension of the array, must satisfy the constraint n2n2.
yy, the dependent variable.
Constraint: y(i)0.0yi0.0, for i = 1,2,,ni=1,2,,n.
7:     s – double scalar
The scale parameter for the gamma model, ν1ν-1.
s = 0.0s=0.0
The scale parameter is estimated with the function using the formula described in Section [Description].
Constraint: s0.0s0.0.

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.)
nn, the number of observations.
Constraint: n2n2.
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.)
mm, the total number of independent variables.
Constraint: m1m1.
3:     wt( : :) – double array
Note: the dimension of the array wt must be at least nn if weight = 'W'weight='W', and at least 11 otherwise.
If weight = 'W'weight='W', wt must contain the weights to be used in the weighted regression. If wt(i) = 0.0wti=0.0, the iith 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'weight='U', wt is not referenced and the effective number of observations is nn.
Constraint: if weight = 'W'weight='W', wt(i)0.0wti0.0, for i = 1,2,,ni=1,2,,n.
4:     a – double scalar
If link = 'E'link='E', a must contain the power of the exponential.
If link'E'link'E', a is not referenced.
Default: 00
Constraint: if link = 'E'link='E', a0.0a0.0.
5:     v(n,ip + 7ip+7) – double array
If offset = 'N'offset='N', v need not be set.
If offset = 'Y'offset='Y', v(i,7)vi7, for i = 1,2,,ni=1,2,,n, must contain the offset values oioi. All other values need not be set.
6:     tol – double scalar
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)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 precision0.0tol<machine precision then the function will use 10 × machine precision10×machine precision instead.
Default: 00
Constraint: tol0.0tol0.0.
7:     maxit – int64int32nag_int scalar
The maximum number of iterations for the iterative weighted least squares.
maxit = 0maxit=0
A default value of 1010 is used.
Default: 1010
Constraint: maxit0maxit0.
8:     iprint – int64int32nag_int scalar
Indicates if the printing of information on the iterations is required.
iprint0iprint0
There is no printing.
iprint > 0iprint>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)).
Default: 00
9:     eps – double scalar
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 precision0.0eps<machine precision then the function will use machine precision instead.
Default: 00
Constraint: eps0.0eps0.0.

Input Parameters Omitted from the MATLAB Interface

offset weight ldx ldv wk

Output Parameters

1:     s – double scalar
If on input s = 0.0s=0.0, s contains the estimated value of the scale parameter, ν̂1ν^-1.
If on input s0.0s0.0, 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, β̂β^.
If mean = 'M'mean='M', the first element of b will contain the estimate of the mean parameter and b(i + 1)bi+1 will contain the coefficient of the variable contained in column jj of xx, where isx(j)isxj is the iith positive value in the array isx.
If mean = 'Z'mean='Z', b(i)bi will contain the coefficient of the variable contained in column jj of xx, where isx(j)isxj is the iith 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 = ipirank=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 × eps×(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)sei contains the standard error of the parameter estimate in b(i)bi, for i = 1,2,,ipi=1,2,,ip.
7:     cov(ip × (ip + 1) / 2ip×(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)bi and the parameter estimate given in b(j)bj, jiji, is stored in cov((j × (j1) / 2 + i))cov(j×(j-1)/2+i).
8:     v(n,ip + 7ip+7) – double array
Auxiliary information on the fitted model.
v(i,1)vi1 contains the linear predictor value, ηiηi, for i = 1,2,,ni=1,2,,n.
v(i,2)vi2 contains the fitted value, μ̂iμ^i, for i = 1,2,,ni=1,2,,n.
v(i,3)vi3 contains the variance standardization, 1/(τi)1τi, for i = 1,2,,ni=1,2,,n.
v(i,4)vi4 contains the square root of the working weight, wi(1/2)wi12, for i = 1,2,,ni=1,2,,n.
v(i,5)vi5 contains the Anscombe residual, riri, for i = 1,2,,ni=1,2,,n.
v(i,6)vi6 contains the leverage, hihi, for i = 1,2,,ni=1,2,,n.
v(i,7)vi7 contains the offset, oioi, for i = 1,2,,ni=1,2,,n. If offset = 'N'offset='N', all values will be zero.
v(i,j)vij, for j = 8,,ip + 7j=8,,ip+7, contains the results of the QRQR decomposition or the singular value decomposition.
If the model is not of full rank, i.e., irank < ipirank<ip, the first ip rows of columns 88 to ip + 7ip+7 contain the P*P* matrix.
9:     ifail – int64int32nag_int scalar
ifail = 0ifail=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 = 1ifail=1
On entry,n < 2n<2,
orm < 1m<1,
orldx < nldx<n,
orldv < nldv<n,
orip < 1ip<1,
orlink'E','I','L','S'link'E','I','L','S' or 'R',
ors < 0.0s<0.0,
orlink = 'E'link='E' and a = 0.0a=0.0,
ormean'M'mean'M' or 'Z''Z',
orweight'U'weight'U' or 'W''W',
oroffset'N'offset'N' or 'Y',
ormaxit < 0maxit<0,
ortol < 0.0tol<0.0,
oreps < 0.0eps<0.0.
  ifail = 2ifail=2
On entry, weight = 'W'weight='W' and a value of wt < 0.0wt<0.0.
  ifail = 3ifail=3
On entry,a value of isx < 0isx<0,
orthe value of ip is incompatible with the values of mean and isx,
orip is greater than the effective number of observations.
  ifail = 4ifail=4
On entry,y(i) < 0.0yi<0.0 for some i = 1,2,,ni=1,2,,n.
  ifail = 5ifail=5
A fitted value is at the boundary, i.e., μ̂ = 0.0μ^=0.0. This may occur if there are small values of yy and the model is not suitable for the data. The model should be reformulated with, perhaps, some observations dropped.
  ifail = 6ifail=6
The singular value decomposition has failed to converge. This is an unlikely error exit.
  ifail = 7ifail=7
The iterative weighted least squares has failed to converge in maxit (or default 1010) 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 = 8ifail=8
The rank of the model has changed during the weighted least squares iterations. The estimate for ββ returned may be reasonable, but you should check how the deviance has changed during iterations.
W ifail = 9ifail=9
The degrees of freedom for error are 00. A saturated model has been fitted.

Accuracy

The accuracy depends on tol as described in Section [Parameters]. As the adjusted deviance is a function of logμlogμ, the accuracy of the β̂β^s will be a function of tol, so tol should be set to a smaller value than the accuracy required for β̂β^.

Further Comments

None.

Example

function nag_correg_glm_gamma_example
link = 'R';
mean_p = 'M';
x = [1;
     1;
     1;
     1;
     1;
     0;
     0;
     0;
     0;
     0];
isx = [int64(1)];
ip = int64(2);
y = [1;
     0.3;
     10.5;
     9.7;
     10.9;
     0.62;
     0.12;
     0.09;
     0.5;
     2.14];
s = 0;
[sOut, dev, idf, b, irank, se, covar, vOut, ifail] = ...
     nag_correg_glm_gamma(link, mean_p, x, isx, ip, y, s, 'tol', 5e-5)
 

sOut =

    1.0742


dev =

   35.0344


idf =

                    8


b =

    1.4408
   -1.2865


irank =

                    2


se =

    0.6678
    0.6717


covar =

    0.4460
   -0.4460
    0.4511


vOut =

    0.1543    6.4800    0.1543   -6.4800   -1.3909    0.2000         0   14.5726   14.4073
    0.1543    6.4800    0.1543   -6.4800   -1.9228    0.2000         0    0.3700    1.5431
    0.1543    6.4800    0.1543   -6.4800    0.5236    0.2000         0    0.3700    0.0324
    0.1543    6.4800    0.1543   -6.4800    0.4318    0.2000         0    0.3700    0.0324
    0.1543    6.4800    0.1543   -6.4800    0.5678    0.2000         0    0.3700    0.0324
    1.4408    0.6940    1.4408   -0.6940   -0.1107    0.2000         0    0.0396   -0.4391
    1.4408    0.6940    1.4408   -0.6940   -1.3287    0.2000         0    0.0396   -0.4391
    1.4408    0.6940    1.4408   -0.6940   -1.4815    0.2000         0    0.0396   -0.4391
    1.4408    0.6940    1.4408   -0.6940   -0.3106    0.2000         0    0.0396   -0.4391
    1.4408    0.6940    1.4408   -0.6940    1.3665    0.2000         0    0.0396   -0.4391


ifail =

                    0


function g02gd_example
link = 'R';
mean_p = 'M';
x = [1;
     1;
     1;
     1;
     1;
     0;
     0;
     0;
     0;
     0];
isx = [int64(1)];
ip = int64(2);
y = [1;
     0.3;
     10.5;
     9.7;
     10.9;
     0.62;
     0.12;
     0.09;
     0.5;
     2.14];
s = 0;
[sOut, dev, idf, b, irank, se, covar, vOut, ifail] = ...
     g02gd(link, mean_p, x, isx, ip, y, s, 'tol', 5e-5)
 

sOut =

    1.0742


dev =

   35.0344


idf =

                    8


b =

    1.4408
   -1.2865


irank =

                    2


se =

    0.6678
    0.6717


covar =

    0.4460
   -0.4460
    0.4511


vOut =

    0.1543    6.4800    0.1543   -6.4800   -1.3909    0.2000         0   14.5726   14.4073
    0.1543    6.4800    0.1543   -6.4800   -1.9228    0.2000         0    0.3700    1.5431
    0.1543    6.4800    0.1543   -6.4800    0.5236    0.2000         0    0.3700    0.0324
    0.1543    6.4800    0.1543   -6.4800    0.4318    0.2000         0    0.3700    0.0324
    0.1543    6.4800    0.1543   -6.4800    0.5678    0.2000         0    0.3700    0.0324
    1.4408    0.6940    1.4408   -0.6940   -0.1107    0.2000         0    0.0396   -0.4391
    1.4408    0.6940    1.4408   -0.6940   -1.3287    0.2000         0    0.0396   -0.4391
    1.4408    0.6940    1.4408   -0.6940   -1.4815    0.2000         0    0.0396   -0.4391
    1.4408    0.6940    1.4408   -0.6940   -0.3106    0.2000         0    0.0396   -0.4391
    1.4408    0.6940    1.4408   -0.6940    1.3665    0.2000         0    0.0396   -0.4391


ifail =

                    0



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Chapter Contents
Chapter Introduction
NAG Toolbox

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