ESTIMATION OF DISCRETE CHOICE MODELS WITH MINIMAL VARIATION OF ALTERNATIVE-SPECIFIC VARIABLES

The paper states conditions for minimal variation within the explanatory variables such that the maximum likelihood estimate of the coefficient vector in the discrete choice logit model is unique. Special emphasis is given to the case that (almost) all individuals observe the same set of alternative-specific explanatory variables. The aspect of ‘experimental design’ in discrete choice models is discussed.     

ESTIMATION OF DISCRETE CHOICE MODELS WITH MINIMAL VARIATION OF ALTERNATIVE-SPECIFIC VARIABLES

The paper states conditions for minimal variation within the explanatory variables such that the maximum likelihood estimate of the coefficient vector in the discrete choice logit model is unique. Special emphasis is given to the case that (almost) all individuals observe the same set of alternative-specific explanatory variables. The aspect of 'experimental design' in discrete choice models is discussed.     

CONSTRUCTION OF CROSSOVER DESIGNS WITH CORRELATED ERRORS

Crossover experiments are widely used, particularly where a sequence of treatments is given to subjects. Correlations between observations on the same subject are therefore likely and should be considered in both the design and analysis of crossover experiments. This paper presents an algorithm for the generation of efficient crossover designs with autoregressive and linear variance structures. The algorithm has been implemented as a module in the experimental design generation package CycDesigN (Release 3.0; CycSoftware, Hamilton, New Zealand). Output from the algorithm is compared with earlier work. Some results are given from the analysis of a crossover experiment assuming correlated errors.     

Semiparametric multinomial logit models for analysing consumer choice behaviour

The multinomial logit model (MNL) is one of the most frequently used statistical models in marketing applications. It allows one to relate an unordered categorical response variable, for example representing the choice of a brand, to a vector of covariates such as the price of the brand or variables characterising the consumer. In its classical form, all covariates enter in strictly parametric, linear form into the utility function of the MNL model. In this paper, we introduce semiparametric extensions, where smooth effects of continuous covariates are modelled by penalised splines. A mixed model representation of these penalised splines is employed to obtain estimates of the corresponding smoothing parameters, leading to a fully automated estimation procedure. To validate semiparametric models against parametric models, we utilise different scoring rules as well as predicted market share and compare parametric and semiparametric approaches for a number of brand choice data sets.     

BAYESIAN ANALYSIS OF THE LINEAR REGRESSION MODEL WITH NON-NORMAL DISTURBANCES

This paper considers the Bayesian analysis of a linear regression model with identically independently distributed non-normal disturbances. The distribution of disturbances is approximated by an Edgeworth series distribution with cumulants, of order higher than fourth, negligible. The posterior distribution of the regression coefficients vector is obtained under the assumption of a g-prior distribution for the parameters of the model. The Bayes estimator and its Bayes risk of the estimator are derived under a quadratic loss structure.     

ON THE DESIGN OF FIELD EXPERIMENTS UNDER A ONE-DIMENSIONAL NEIGHBOUR MODEL

This paper examines the effect of randomisation restrictions, either to satisfy conditions for a balanced incomplete block design or to attain a higher level of partial neighbour balance, on the average variance of pair-wise treatment contrasts under a neighbour model discussed by Gleeson & Cullis (1987). Results suggest that smaller average pairwise variances can be obtained by ignoring requirements for incomplete block designs and concentrating on achieving a higher level of partial neighbour balance. Field layout of the design, although often determined by practical constraints, e.g. size, shape of site, minimum plot size and experimental husbandry, may markedly affect average pairwise variance. For the one-dimensional (row-wise) neighbour model considered here, investigation of three different layouts suggests that for a rectangular array of plots, smaller average pairwise variances can generally be obtained from layouts with fewer rows and more plots per row.     

SMALL AREA ESTIMATION USING SURVEY WEIGHTS WITH FUNCTIONAL MEASUREMENT ERROR IN THE COVARIATE

Nested error linear regression models using survey weights have been studied in small area estimation to obtain efficient model‐based and design‐consistent estimators of small area means. The covariates in these nested error linear regression models are not subject to measurement errors. In practical applications, however, there are many situations in which the covariates are subject to measurement errors. In this paper, we develop a nested error linear regression model with an area‐level covariate subject to functional measurement error. In particular, we propose a pseudo‐empirical Bayes (PEB) predictor to estimate small area means. This predictor borrows strength across areas through the model and makes use of the survey weights to preserve the design consistency as the area sample size increases. We also employ a jackknife method to estimate the mean squared prediction error (MSPE) of the PEB predictor. Finally, we report the results of a simulation study on the performance of our PEB predictor and associated jackknife MSPE estimator.     

BOOTSTRAP TESTS FOR THE ERROR DISTRIBUTION IN LINEAR AND NONPARAMETRIC REGRESSION MODELS

In this paper we investigate several tests for the hypothesis of a parametric form of the error distribution in the common linear and non‐parametric regression model, which are based on empirical processes of residuals. It is well known that tests in this context are not asymptotically distribution‐free and the parametric bootstrap is applied to deal with this problem. The performance of the resulting bootstrap test is investigated from an asymptotic point of view and by means of a simulation study. The results demonstrate that even for moderate sample sizes the parametric bootstrap provides a reliable and easy accessible solution to the problem of goodness‐of‐fit testing of assumptions regarding the error distribution in linear and non‐parametric regression models.     

THE REDUCED-RANK GROWTH CURVE MODEL FOR DISCRIMINANT ANALYSIS OF LONGITUDINAL DATA

This paper presents a method of discriminant analysis especially suited to longitudinal data. The approach is in the spirit of canonical variate analysis (CVA) and is similarly intended to reduce the dimensionality of multivariate data while retaining information about group differences. A drawback of CVA is that it does not take advantage of special structures that may be anticipated in certain types of data. For longitudinal data, it is often appropriate to specify a growth curve structure (as given, for example, in the model of Potthoff & Roy, 1964). The present paper focuses on this growth curve structure, utilizing it in a model-based approach to discriminant analysis. For this purpose the paper presents an extension of the reduced-rank regression model, referred to as the reduced-rank growth curve (RRGC) model. It estimates discriminant functions via maximum likelihood and gives a procedure for determining dimensionality. This methodology is exploratory only, and is illustrated by a well-known dataset from Grizzle & Allen (1969).     

MULTILEVEL DISCRETE‐TIME EVENT HISTORY MODELS WITH APPLICATIONS TO THE ANALYSIS OF RECURRENT EMPLOYMENT TRANSITIONS

Data on the timing of events such as births, residential moves and changes in employment status are collected in many longitudinal surveys. These data often have a highly complex structure, with events of several types occurring repeatedly over time to an individual and interdependences between different event processes (e.g. births and employment transitions). The aim of this paper is to review a general class of multilevel discrete‐time event history models for handling recurrent events and transitions between multiple states. It is also shown how standard methods can be extended to allow for time‐varying covariates that are outcomes of an event process that is jointly determined with the process of interest. The considerable potential of these methods for studying transitions through the life course is illustrated in analyses of the effect of the presence and age of children on women's employment transitions, using data from the British Household Panel Survey.     

MODELING THE LIFETIME OF LONGITUDINAL ELEMENTS

In the study of the stochastic behaviour of the lifetime of an element as a function of its length, it is often observed that the failure time (or lifetime) decreases as the length increases. In probabilistic terms, such an idea can be expressed as follows. Let T be the lifetime of a specimen of length x, so the survival function, which denotes the probability that an element of length x survives till time t, will be given by S_T (t, x) = P(T > t/α(x), where α(x) is a monotonically decreasing function. In particular, it is often assumed that T has a Weibull distribution. In this paper, we propose a generalization of this Weibull model by assuming that the distribution of T is Generalized gamma (GG). Since the GG model contains the Weibull, Gamma and Lognormal models as special and limiting cases, a GG regression model is an appropriate tool for describing the size effect on the lifetime and for selecting among the embedded models. Maximum likelihood estimates are obtained for the GG regression model with α(x) = cx^b . As a special case this provide an alternative to the usual approach to estimation for the GG distribution which involves reparametrization. Related parametric inference issues are addressed and illustrated using two experimental data sets. Some discussion of censored data is also provided.