allvc produce objects of type
if Gaussian quadrature (Hinde, 1982,
was applied for computation, and objects of class
parameter estimation was carried out by nonparametric maximum likelihood
random.distribution="np" ). The functions presented here
give predictions from those objects.
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a fitted object of class
a data frame with covariates from which prediction is desired. If omitted, empirical Bayes predictions for the original data will be given.
if set to
further arguments which will mostly not have any effect (and are
included only to ensure compatibility
with the generic
The predicted values are obtained by
Empirical Bayes (Aitkin, 1996b), if
newdata has not been specified.
That is, the prediction on the linear predictor scale is given by
sum(eta_ik * w_ik),
whereby eta_ik are the fitted linear predictors,
w_ik are the weights in the final iteration of the EM algorithm
(corresponding to the posterior probability for observation i
to come from component k ), and the sum is taken over the number
of components k for fixed i.
the marginal model, if object is of class
newdata has been specified. That is, computation is identical as above, but
with w_ik replaced by the masses pi_k of the fitted model.
the analytical expression for the marginal mean of the responses,
if object is of class
newdata has been specified.
See Aitkin et al. (2009), p. 481, for the formula. This method is only
supported for the logarithmic link function, as otherwise no analytical
expression for the marginal mean of the responses exists.
It is sufficient to call
predict instead of
predict.glmmGQ, since the generic predict function provided in R automatically selects the right
A vector of predicted values.
The results of the generic
predict(object, type="response"). Note that, as we are
working with random effects, fitted values are never really ‘fitted’ but rather
Jochen Einbeck and John Hinde (2007).
Aitkin, M. (1996a). A general maximum likelihood analysis of overdispersion in generalized linear models. Statistics and Computing 6, 251-262.
Aitkin, M. (1996b). Empirical Bayes shrinkage using posterior random effect means from nonparametric maximum likelihood estimation in general random effect models. Statistical Modelling: Proceedings of the 11th IWSM 1996, 87-94.
Aitkin, M., Francis, B. and Hinde, J. (2009). Statistical Modelling in R. Oxford Statistical Science Series, Oxford, UK.
Hinde, J. (1982). Compound Poisson regression models. Lecture Notes in Statistics 14, 109-121.
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# Toxoplasmosis data: data(rainfall, package="forward") rainfall$x<-rainfall$Rain/1000 toxo.0.3x<- alldist(cbind(Cases,Total-Cases)~1, random=~x, data=rainfall, k=3, family=binomial(link=logit)) toxo.1.3x<- alldist(cbind(Cases,Total-Cases)~x, random=~x, data=rainfall, k=3, family=binomial(link=logit)) predict(toxo.0.3x, type="response", newdata=data.frame(x=2)) #  0.4608 predict(toxo.1.3x, type="response", newdata=data.frame(x=2)) #  0.4608 # gives the same result, as both models are equivalent and only differ # by a parameter transformation. # Fabric faults data: data(fabric) names(fabric) #  "leng" "y" "x" faults.g2<- alldist(y ~ x, family=poisson(link=log), random=~1, data= fabric,k=2, random.distribution="gq") predict(faults.g2, type="response",newdata=fabric[1:6,]) #  8.715805 10.354556 13.341242 5.856821 11.407828 13.938013 # is not the same as predict(faults.g2, type="response")[1:6] #  6.557786 7.046213 17.020242 7.288989 13.992591 9.533823 # since in the first case prediction is done using the analytical # mean of the marginal distribution, and in the second case using the # individual posterior probabilities in an empirical Bayes approach.