Class of objects returned by fitting double generalized linear models.
Write m_i = E(y_i) for the expectation of the
Then Var(y_i) = s_iV(m_i) where V
is the variance function and s_i is the dispersion of the
(often denoted as the Greek character ‘phi’).
We assume the link linear models
g(m_i) = x_i^T b and
h(s_i) = z_i^T a,
where x_i and z_i are vectors of covariates,
and b and a are vectors of regression
cofficients affecting the mean and dispersion respectively.
dlink specifies h.
family for how to specify g.
The optional arguments
specify starting values for m_i, b
and s_i respectively.
The parameters b are estimated as for an ordinary glm. The parameters a are estimated by way of a dual glm in which the deviance components of the ordinary glm appear as responses. The estimation procedure alternates between one iteration for the mean submodel and one iteration for the dispersion submodel until overall convergence.
The output from
out say, consists of two
(that for the dispersion submodel is
out$dispersion.fit) with a few more
components for the outer iteration and overall likelihood.
anova functions have special methods for
Any generic function which has methods for
lms will work on
out, giving information about the mean submodel.
Information about the dispersion submodel can be obtained by using
out$dispersion.fit as argument rather than out itself.
drop1(out,scale=1) gives correct score statistics for
removing terms from the mean submodel,
drop1(out$dispersion.fit,scale=2) gives correct score
statistics for removing terms from the dispersion submodel.
The dispersion submodel is treated as a gamma family unless the original
reponses are gamma, in which case the dispersion submodel is digamma.
(Note that the digamma and trigamma functions are required to fit a digamma
family.) This is exact if the original glm family is
inverse.gaussian. In other cases it can be
justified by the saddle-point approximation to the density of the responses.
The results will therefore be close to exact ML or REML when the dispersions
are small compared to the means. In all cases the dispersion submodel as prior
weights 1, and has its own dispersion parameter which is 2.
This class of objects is returned by the
to represent a fitted double generalized linear model.
"dglm" inherits from class
since it consists of two coupled generalized linear models,
one for the mean and one for the dispersion.
it also inherits from
The object returned has all the components of a
The returned component
object$dispersion.fit is also a
glm object in its own right,
representing the result of modelling the dispersion.
Objects of this class have methods for the functions
step, amongst others.
Specific methods (not shared with
glm) exist for
dglm object consists of a
glm object with the following
dispersion.fitthe dispersion submodel: a
representing the fitted model for the dispersions.
The responses for this model are the deviance components from the original
generalized linear model.
The prior weights are 1 and the dispersion or scale of this model is 2.
iterthis component now represents the number of outer iterations used to fit the coupled mean-dispersion models. At each outer iteration, one IRLS is done for each of the mean and dispersion submodels.
methodfitting method used:
"ml" if maximum likelihood
was used or
"reml" if adjusted profile likelihood was used.
m2loglikminus twice the log-likelihood or adjusted profile likelihood of the fitted model.
The anova method is questionable when applied to an
dglm object with
method="reml" (stick to
Gordon Smyth, ported to R\ by Peter Dunn ([email protected])
Smyth, G. K. (1989). Generalized linear models with varying dispersion. J. R. Statist. Soc. B, 51, 47–60.
Smyth, G. K., and Verbyla, A. P. (1999). Adjusted likelihood methods for modelling dispersion in generalized linear models. Environmetrics, 10, 696-709.
Verbyla, A. P., and Smyth, G. K. (1998). Double generalized linear models: approximate residual maximum likelihood and diagnostics. Research Report, Department of Statistics, University of Adelaide.
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# Continuing the example from glm, but this time try # fitting a Gamma double generalized linear model also. clotting <- data.frame( u = c(5,10,15,20,30,40,60,80,100), lot1 = c(118,58,42,35,27,25,21,19,18), lot2 = c(69,35,26,21,18,16,13,12,12)) # The same example as in glm: the dispersion is modelled as constant out <- dglm(lot1 ~ log(u), ~1, data=clotting, family=Gamma) summary(out) # Try a double glm out2 <- dglm(lot1 ~ log(u), ~u, data=clotting, family=Gamma) summary(out2) anova(out2) # Summarize the mean model as for a glm summary.glm(out2) # Summarize the dispersion model as for a glm summary(out2$dispersion.fit) # Examine goodness of fit of dispersion model by plotting residuals plot(fitted(out2$dispersion.fit),residuals(out2$dispersion.fit))
Loading required package: statmod Call: dglm(formula = lot1 ~ log(u), dformula = ~1, family = Gamma, data = clotting) Mean Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -0.01655438 0.0009275491 -17.84744 4.279230e-07 log(u) 0.01534311 0.0004149596 36.97496 2.751191e-09 (Dispersion Parameters for Gamma family estimated as below ) Scaled Null Deviance: 1890.363 on 8 degrees of freedom Scaled Residual Deviance: 9.002787 on 7 degrees of freedom Dispersion Coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) -6.288103 0.4712586 -13.34321 1.297468e-40 (Dispersion parameter for Digamma family taken to be 2 ) Scaled Null Deviance: 8.90448 on 8 degrees of freedom Scaled Residual Deviance: 8.90448 on 8 degrees of freedom Minus Twice the Log-Likelihood: 31.98992 Number of Alternating Iterations: 4 Call: dglm(formula = lot1 ~ log(u), dformula = ~u, family = Gamma, data = clotting) Mean Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -0.01784797 0.0010062108 -17.73780 4.464149e-07 log(u) 0.01596262 0.0002301215 69.36604 3.402379e-11 (Dispersion Parameters for Gamma family estimated as below ) Scaled Null Deviance: 2313.573 on 8 degrees of freedom Scaled Residual Deviance: 9.003391 on 7 degrees of freedom Dispersion Coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) -4.59256962 0.76357166 -6.014589 1.803438e-09 u -0.06966577 0.01502817 -4.635680 3.557663e-06 (Dispersion parameter for Digamma family taken to be 2 ) Scaled Null Deviance: 16.75853 on 8 degrees of freedom Scaled Residual Deviance: 4.414477 on 7 degrees of freedom Minus Twice the Log-Likelihood: 22.17126 Number of Alternating Iterations: 5 Analysis of Deviance Table Gamma double generalized linear model Response: lot1 DF Seq.Chisq Seq.P Adj.Chisq Adj.P Mean model 1 48.686 0.0000000 47.403 0.0000000 Dispersion model 1 9.819 0.0017275 9.819 0.0017275 Call: dglm(formula = lot1 ~ log(u), dformula = ~u, family = Gamma, data = clotting) Deviance Residuals: Min 1Q Median 3Q Max -0.076478 -0.010122 0.001926 0.048233 0.093677 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -0.0178480 0.0010062 -17.74 4.46e-07 *** log(u) 0.0159626 0.0002301 69.37 3.40e-11 *** --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 (Dispersion parameter for Gamma family taken to be 1.307633) Null deviance: 2313.5733 on 8 degrees of freedom Residual deviance: 9.0034 on 7 degrees of freedom AIC: NA Number of Fisher Scoring iterations: 5 Call: dglm(formula = ~u, family = Digamma(link = "log"), data = clotting) Deviance Residuals: Min 1Q Median 3Q Max -2.18708 -0.81757 -0.07655 0.16104 1.16959 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -4.59257 0.53790 -8.538 6e-05 *** u -0.06967 0.01059 -6.581 0.00031 *** --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 (Dispersion parameter for Digamma family taken to be 0.9925192) Null deviance: 16.7585 on 7 degrees of freedom Residual deviance: 4.4145 on 7 degrees of freedom AIC: NA Number of Fisher Scoring iterations: 5
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