Statistical Modelling

We wish to evaluate and compare models that are non-nested and fit to data using different fitting criteria. We first estimate parameters in all models by optimizing goodness-of-fit to a dataset. Then, to assess a candidate model, we simulate a population of datasets from it and evaluate the goodness-of-fit of all the models, without re-estimating parameter values. Finally, we see whether the vector of goodness-of-fit criteria for the original data is compatible with the multivariate distribution of these criteria for the simulated datasets. By simulating from each model in turn, we determine whether any, or several, models are consistent with the data. We apply the method to compare three models, fit at different temporal resolutions to binary time series of animal behaviour data, concluding that a semi-Markov model gives a better fit than latent Gaussian and hidden Markov models.

Reduced-rank regression is a method with great potential for dimension reduction but has found few applications in applied statistics. To address this, reduced-rank regression is proposed for the class of vector generalized linear models (VGLMs), which is very large. The resulting class, which we call reduced-rank VGLMs (RR-VGLMs), enables the benefits of reduced-rank regression to be conveyed to a wide range of data types, including categorical data. RR-VGLMs are illustrated by focussing on models for categorical data, and especially the multinomial logit model. General algorithmic details are provided and software written by the first author is described. The reduced-rank multinomial logit model is illustrated with real data in two contexts: a regression analysis of workforce data and a classification problem.

The paper describes the analysis of data originating from the German Deep Drill Program. The amount of ‘cataclastic rocks’ is modelled with data resulting from a series of measurements taken from deep drill samples ranging from 1000 up to 5000 m depth. The measurements thereby describe the amount of strongly deformed rock particles and serve as indicator for the occurrence of cataclastic shear zones, which are areas of severely‘ground’ stones due to movements of different layers in the earth crust. The data represent a ‘depth series’ as analogue to a ‘time series’, with mean, dispersion and correlation structure varying in depth. The general smooth structure is thereby disturbed by peaks and outliers so that robust procedures have to be applied for estimation. In terms of statistical modelling technology three different peculiarities of the data have to be tackled simultaneously, that is estimation of the correlation structure, local bandwidth selection and robust smoothing. To do so, existing routines are adapted and combined in new ‘two-stage’ estimation procedures.

A statistical model for ranks is presented, and some results on its parameter are discussed. In particular, maximum likelihood inference is developed, with and without covariates; thus, a statistical model for rank data is introduced in order to link the expressed ranks to the main features of the raters. Some empirical evidence from a marketing survey confirms the usefulness of the proposal in the study of the preferences.