On some extensions of generalized linear models with varying dispersion

Abstract

When dealing with exponential family distributions, a constant dispersion

is often assumed since it simplifies both model formulation and estimation.

In contrast, heteroscedasticity is a common feature of almost every empirical

data set. In this dissertation, the dispersion parameter is no longer considered

as constant throughout the entire sample, but defined as the expected

deviance of the individual response yi and its expected value _i such that it

will be expressed as a linear combination of some covariates and their coefficients.

At the same time, the dispersion regression is an essential part of a

double Generalized Linear Model in which mean and dispersion are modelled in two interlinked and pseudo-simultaneously estimated submodels.

In other words, the deviance is a function of the response mean which on the

other hand depends on the dispersion. Due to the mutual dependency, the

estimation algorithm will be iterated as long as the improvement of the one

parameter leads to significant changes of the other until it is not the case.

If appropriate covariates are chosen, the model’s goodness of fit should

be improved by the property that the dispersion is estimated by external

information instead of being a constant. In the following, the advantage

of dispersion modelling will be shown by its application on three different

types of data: a) zero-inflated data, b) non-linear time series data, and

c) clinical trials data. All these data follow distributions of the exponential

family for which the application of the Generalized Linear Model is justified,

but require certain extensions of modelling methodologies.

In this dissertation, The enhanced goodness of fit given that the constant

dispersion assumption is dropped will be shown in the above listed

examples. In fact, by formulating and carrying out score and Wald tests

on testing for the possible occurrence of varying dispersion, evidence of

heterogeneous dispersion could be found to be present in the data sets considered.

Furthermore, although model formulation, asymptotic properties

and computational effort are more extensive when dealing with the double

models, the benefits and advantages in terms of improved fitting results and

more efficient parameter estimates appear to justify the additional effort not

only for the types of data introduced, but also generally for empirical data

analysis, on different types of data as well.