A Bayesian approach to estimating the regression coefficients of a multinomial logit model

A Bayesian approach to estimation of the regression coefficients of a multinominal logit model with ordinal scale response categories is presented. A Monte Carlo method is used to construct the posterior distribution of the link function. The link function is treated as an arbitrary scalar function. Then the Gauss-Markov theorem is used to determine a function of the link which produces a random vector of coefficients. The posterior distribution of the random vector of coefficients is used to estimate the regression coefficients. The method described is referred to as a Bayesian generalized least square (BGLS) analysis. Two cases involving multinominal logit models are described. Case I involves a cumulative logit model and Case II involves a proportional-odds model. All inferences about the coefficients for both cases are described in terms of the posterior distribution of the regression coefficients. The results from the BGLS method are compared to maximum likelihood estimates of the regression coefficients. The BGLS method avoids the nonlinear problems encountered when estimating the regression coefficients of a generalized linear model. The method is not complex or computationally intensive. The BGLS method offers several advantages over Bayesian approaches. ^

The Effects of Experience on Preference Uncertainty: Theory and Empirics for Public and Quasi-Public Environmental Goods

This paper develop and estimates a model of demand estimation for environmental public goods which allows for consumers to learn about their preferences through consumption experiences. We develop a theoretical model of Bayesian updating, perform comparative statics over the model, and show how the theoretical model can be consistently incorporated into a reduced form econometric model. We then estimate the model using data collected for two environmental goods. We find that the predictions of the theoretical exercise that additional experience makes consumers more certain over their preferences in both mean and variance are supported in each case.

Statistiques appliquées en chirurgie cardiaque adulte : analyses de survie et applications du “propensity score”

L'objectif principal de ce travail est d’étudier en profondeur certaines techniques biostatistiques avancées en recherche évaluative en chirurgie cardiaque adulte. Les études ont été conçues pour intégrer les concepts d'analyse de survie, analyse de régression avec “propensity score”, et analyse de coûts. Le premier manuscrit évalue la survie après la réparation chirurgicale de la dissection aigüe de l’aorte ascendante. Les analyses statistiques utilisées comprennent : analyses de survie avec régression paramétrique des phases de risque et d'autres méthodes paramétriques (exponentielle, Weibull), semi-paramétriques (Cox) ou non-paramétriques (Kaplan-Meier) ; survie comparée à une cohorte appariée pour l’âge, le sexe et la race utilisant des tables de statistiques de survie gouvernementales ; modèles de régression avec “bootstrapping” et “multinomial logit model”. L'étude a démontrée que la survie s'est améliorée sur 25 ans en lien avec des changements dans les techniques chirurgicales et d’imagerie diagnostique. Le second manuscrit est axé sur les résultats des pontages coronariens isolés chez des patients ayant des antécédents d'intervention coronarienne percutanée. Les analyses statistiques utilisées comprennent : modèles de régression avec “propensity score” ; algorithme complexe d'appariement (1:3) ; analyses statistiques appropriées pour les groupes appariés (différences standardisées, “generalized estimating equations”, modèle de Cox stratifié). L'étude a démontrée que l’intervention coronarienne percutanée subie 14 jours ou plus avant la chirurgie de pontages coronariens n'est pas associée à des résultats négatifs à court ou long terme. Le troisième manuscrit évalue les conséquences financières et les changements démographiques survenant pour un centre hospitalier universitaire suite à la mise en place d'un programme de chirurgie cardiaque satellite. Les analyses statistiques utilisées comprennent : modèles de régression multivariée “two-way” ANOVA (logistique, linéaire ou ordinale) ; “propensity score” ; analyses de coûts avec modèles paramétriques Log-Normal. Des modèles d’analyse de « survie » ont également été explorés, utilisant les «coûts» au lieu du « temps » comme variable dépendante, et ont menés à des conclusions similaires. L'étude a démontrée que, après la mise en place du programme satellite, moins de patients de faible complexité étaient référés de la région du programme satellite au centre hospitalier universitaire, avec une augmentation de la charge de travail infirmier et des coûts.

Methods for Categorical Longitudinal Survey Data: Understanding Employment Status of Australian Women

Many variables that are of interest in social science research are nominal variables with two or more categories, such as employment status, occupation, political preference, or self-reported health status. With longitudinal survey data it is possible to analyse the transitions of individuals between different employment states or occupations (for example). In the statistical literature, models for analysing categorical dependent variables with repeated observations belong to the family of models known as generalized linear mixed models (GLMMs). The specific GLMM for a dependent variable with three or more categories is the multinomial logit random effects model. For these models, the marginal distribution of the response does not have a closed form solution and hence numerical integration must be used to obtain maximum likelihood estimates for the model parameters. Techniques for implementing the numerical integration are available but are computationally intensive requiring a large amount of computer processing time that increases with the number of clusters (or individuals) in the data and are not always readily accessible to the practitioner in standard software. For the purposes of analysing categorical response data from a longitudinal social survey, there is clearly a need to evaluate the existing procedures for estimating multinomial logit random effects model in terms of accuracy, efficiency and computing time. The computational time will have significant implications as to the preferred approach by researchers. In this paper we evaluate statistical software procedures that utilise adaptive Gaussian quadrature and MCMC methods, with specific application to modeling employment status of women using a GLMM, over three waves of the HILDA survey.

Bayesian Inference in Large-scale Problems

Many modern applications fall into the category of "large-scale" statistical problems, in which both the number of observations n and the number of features or parameters p may be large. Many existing methods focus on point estimation, despite the continued relevance of uncertainty quantification in the sciences, where the number of parameters to estimate often exceeds the sample size, despite huge increases in the value of n typically seen in many fields. Thus, the tendency in some areas of industry to dispense with traditional statistical analysis on the basis that "n=all" is of little relevance outside of certain narrow applications. The main result of the Big Data revolution in most fields has instead been to make computation much harder without reducing the importance of uncertainty quantification. Bayesian methods excel at uncertainty quantification, but often scale poorly relative to alternatives. This conflict between the statistical advantages of Bayesian procedures and their substantial computational disadvantages is perhaps the greatest challenge facing modern Bayesian statistics, and is the primary motivation for the work presented here.

Two general strategies for scaling Bayesian inference are considered. The first is the development of methods that lend themselves to faster computation, and the second is design and characterization of computational algorithms that scale better in n or p. In the first instance, the focus is on joint inference outside of the standard problem of multivariate continuous data that has been a major focus of previous theoretical work in this area. In the second area, we pursue strategies for improving the speed of Markov chain Monte Carlo algorithms, and characterizing their performance in large-scale settings. Throughout, the focus is on rigorous theoretical evaluation combined with empirical demonstrations of performance and concordance with the theory.

One topic we consider is modeling the joint distribution of multivariate categorical data, often summarized in a contingency table. Contingency table analysis routinely relies on log-linear models, with latent structure analysis providing a common alternative. Latent structure models lead to a reduced rank tensor factorization of the probability mass function for multivariate categorical data, while log-linear models achieve dimensionality reduction through sparsity. Little is known about the relationship between these notions of dimensionality reduction in the two paradigms. In Chapter 2, we derive several results relating the support of a log-linear model to nonnegative ranks of the associated probability tensor. Motivated by these findings, we propose a new collapsed Tucker class of tensor decompositions, which bridge existing PARAFAC and Tucker decompositions, providing a more flexible framework for parsimoniously characterizing multivariate categorical data. Taking a Bayesian approach to inference, we illustrate empirical advantages of the new decompositions.

Latent class models for the joint distribution of multivariate categorical, such as the PARAFAC decomposition, data play an important role in the analysis of population structure. In this context, the number of latent classes is interpreted as the number of genetically distinct subpopulations of an organism, an important factor in the analysis of evolutionary processes and conservation status. Existing methods focus on point estimates of the number of subpopulations, and lack robust uncertainty quantification. Moreover, whether the number of latent classes in these models is even an identified parameter is an open question. In Chapter 3, we show that when the model is properly specified, the correct number of subpopulations can be recovered almost surely. We then propose an alternative method for estimating the number of latent subpopulations that provides good quantification of uncertainty, and provide a simple procedure for verifying that the proposed method is consistent for the number of subpopulations. The performance of the model in estimating the number of subpopulations and other common population structure inference problems is assessed in simulations and a real data application.

In contingency table analysis, sparse data is frequently encountered for even modest numbers of variables, resulting in non-existence of maximum likelihood estimates. A common solution is to obtain regularized estimates of the parameters of a log-linear model. Bayesian methods provide a coherent approach to regularization, but are often computationally intensive. Conjugate priors ease computational demands, but the conjugate Diaconis--Ylvisaker priors for the parameters of log-linear models do not give rise to closed form credible regions, complicating posterior inference. In Chapter 4 we derive the optimal Gaussian approximation to the posterior for log-linear models with Diaconis--Ylvisaker priors, and provide convergence rate and finite-sample bounds for the Kullback-Leibler divergence between the exact posterior and the optimal Gaussian approximation. We demonstrate empirically in simulations and a real data application that the approximation is highly accurate, even in relatively small samples. The proposed approximation provides a computationally scalable and principled approach to regularized estimation and approximate Bayesian inference for log-linear models.

Another challenging and somewhat non-standard joint modeling problem is inference on tail dependence in stochastic processes. In applications where extreme dependence is of interest, data are almost always time-indexed. Existing methods for inference and modeling in this setting often cluster extreme events or choose window sizes with the goal of preserving temporal information. In Chapter 5, we propose an alternative paradigm for inference on tail dependence in stochastic processes with arbitrary temporal dependence structure in the extremes, based on the idea that the information on strength of tail dependence and the temporal structure in this dependence are both encoded in waiting times between exceedances of high thresholds. We construct a class of time-indexed stochastic processes with tail dependence obtained by endowing the support points in de Haan's spectral representation of max-stable processes with velocities and lifetimes. We extend Smith's model to these max-stable velocity processes and obtain the distribution of waiting times between extreme events at multiple locations. Motivated by this result, a new definition of tail dependence is proposed that is a function of the distribution of waiting times between threshold exceedances, and an inferential framework is constructed for estimating the strength of extremal dependence and quantifying uncertainty in this paradigm. The method is applied to climatological, financial, and electrophysiology data.

The remainder of this thesis focuses on posterior computation by Markov chain Monte Carlo. The Markov Chain Monte Carlo method is the dominant paradigm for posterior computation in Bayesian analysis. It has long been common to control computation time by making approximations to the Markov transition kernel. Comparatively little attention has been paid to convergence and estimation error in these approximating Markov Chains. In Chapter 6, we propose a framework for assessing when to use approximations in MCMC algorithms, and how much error in the transition kernel should be tolerated to obtain optimal estimation performance with respect to a specified loss function and computational budget. The results require only ergodicity of the exact kernel and control of the kernel approximation accuracy. The theoretical framework is applied to approximations based on random subsets of data, low-rank approximations of Gaussian processes, and a novel approximating Markov chain for discrete mixture models.

Data augmentation Gibbs samplers are arguably the most popular class of algorithm for approximately sampling from the posterior distribution for the parameters of generalized linear models. The truncated Normal and Polya-Gamma data augmentation samplers are standard examples for probit and logit links, respectively. Motivated by an important problem in quantitative advertising, in Chapter 7 we consider the application of these algorithms to modeling rare events. We show that when the sample size is large but the observed number of successes is small, these data augmentation samplers mix very slowly, with a spectral gap that converges to zero at a rate at least proportional to the reciprocal of the square root of the sample size up to a log factor. In simulation studies, moderate sample sizes result in high autocorrelations and small effective sample sizes. Similar empirical results are observed for related data augmentation samplers for multinomial logit and probit models. When applied to a real quantitative advertising dataset, the data augmentation samplers mix very poorly. Conversely, Hamiltonian Monte Carlo and a type of independence chain Metropolis algorithm show good mixing on the same dataset.

Predictors of non-response to a questionnaire survey of a volunteer twin panel: Findings from the Australian 1989 twin cohort

Questionnaire surveys, while more economical, typically achieve poorer response rates than interview surveys. We used data from a national volunteer cohort of young adult twins, who were scheduled for assessment by questionnaire in 1989 and by interview in 1996-2000, to identify predictors of questionnaire non-response. Out of a total of 8536 twins, 5058 completed the questionnaire survey (59% response rate), and 6255 completed a telephone interview survey conducted a decade later (73% response rate). Multinomial logit models were fitted to the interview data to identify socioeconomic, psychiatric and health behavior correlates of non-response in the earlier questionnaire survey. Male gender, education below University level, and being a dizygotic rather than monozygotic twin, all predicted reduced likelihood of participating in the questionnaire survey. Associations between questionnaire response status and psychiatric history and health behavior variables were modest, with history of alcohol dependence and childhood conduct disorder predicting decreased probability of returning a questionnaire, and history of smoking and heavy drinking more weakly associated with non-response. Body-mass index showed no association with questionnaire non-response. Despite a poor response rate to the self-report questionnaire survey, we found only limited sampling biases for most variables. While not appropriate for studies where socioeconomic variables are critical, it appears that survey by questionnaire, with questionnaire administration by telephone to non-responders, will represent a viable strategy for gene-mapping studies requiring that large numbers of relatives be screened.

A produção de educação nos Açores: a nota de matemática

Tese de Doutoramento, Ciências Económicas e Empresariais (Desenvolvimento Económico e Social e Economia Pública), 16 de Janeiro de 2014, Universidade dos Açores.

Firm Size and Geographical Aggregation: An Empirical Appraisal in Industrial Location

This paper assesses empirically the importance of size discrimination and disaggregate data for deciding where to locate a start-up concern. We compare three econometric specifications using Catalan data: a multinomial logit with 4 and 41 alternatives (provinces and comarques, respectively) in which firm size is the main covariate; a conditional logit with 4 and 41 alternatives including attributes of the sites as well as size-site interactions; and a Poisson model on the comarques and the full spatial choice set (942 municipalities) with site-specific variables. Our results suggest that if these two issues are ignored, conclusions may be misleading. We provide evidence that large and small firms behave differently and conclude that Catalan firms tend to choose between comarques rather than between municipalities. Moreover, labour-intensive firms seem more likely to be located in the city of Barcelona. Keywords: Catalonia, industrial location, multinomial response model. JEL: C250, E30, R00, R12

Economic development and changes in car ownership patterns

The contributions of this paper are twofold: On the one hand, the paper analyses the factors determining the growth in car ownership in Spain over the last two decades, and, on the other, the paper provides empirical evidence for a controversial methodological issue. From a methodological point of view, the paper compares the two alternative decision mechanisms used for modelling car ownership: ordered-response versus unordered-response mechanisms. A discrete choice model is estimated at three points in time: 1980, 1990 and 2000. The study concludes that on the basis of forecasting performance, the multinomial logit model and the ordered probit model are almost undistinguishable. As for the empirical results, it can be emphasised that income elasticity is not constant and declines as car ownership increases. Besides, households living in rural areas are less sensitive than those living in urban areas. Car ownership is also sensitive to the quality of public transport for those living in the largest cities. The results also confirmed the existence of a generation effect, which will vanish around the year 2020, a weak life-cycle effect, and a positive effect of employment on the number of cars per household. Finally, the change in the estimated coefficients over time reflects an increase in mobility needs and, consequently, an increase in car ownership.

Choice of Contracts in the British National Health Service: An Empirical Study

Following major reforms of the British National Health Service (NHS) in 1990, the roles of purchasing and providing health services were separated, with the relationship between purchasers and providers governed by contracts. Using a mixed multinomial logit analysis, we show how this policy shift led to a selection of contracts that is consistent with the predictions of a simple model, based on contract theory, in which the characteristics of the health services being purchased and of the contracting parties influence the choice of contract form. The paper thus provides evidence in support of the practical relevance of theory in understanding health care market reform.

A tale of two logits, compositional data analysis and zero observations

The application of compositional data analysis through log ratio trans-formations corresponds to a multinomial logit model for the shares themselves.This model is characterized by the property of Independence of Irrelevant Alter-natives (IIA). IIA states that the odds ratio in this case the ratio of shares is invariant to the addition or deletion of outcomes to the problem. It is exactlythis invariance of the ratio that underlies the commonly used zero replacementprocedure in compositional data analysis. In this paper we investigate using thenested logit model that does not embody IIA and an associated zero replacementprocedure and compare its performance with that of the more usual approach ofusing the multinomial logit model. Our comparisons exploit a data set that com-bines voting data by electoral division with corresponding census data for eachdivision for the 2001 Federal election in Australia

The Impacts of social networks on immigrants' employment prospect: the spanish case 1997-2007

This paper studies the extent to which social networks influence the employment stability and wages of immigrants in Spain. By doing so, I consider an aspect that has not been previously addressed in the empirical literature, namely the connection between immigrants' social networks and labor market outcomes in Spain. For this purpose, I use micro-data from the National Immigrant Survey carried out in 2007. The analysis is conducted in two stages. First, the impact of social networks on the probability of keeping the first job obtained in Spain is studied through a multinomial logit regression. Second, quantile regressions are used to estimate a wage equation. The empirical results suggest that once the endogeneity problem has been accounted for, immigrants' social networks influence their labor market outcomes. On arrival, immigrants experience a mismatch in the labor market. In addition, different effects of social networks on wages by gender and wage distribution are found. While contacts on arrival and informal job access mechanisms positively influence women's wages, a wage penalty is observed for men.

Unstable equilibrium: young adult women in family and career formation in Southern Europe

The paper examines the relationship between family formation (i.e., living with a partner and having children) and women’s occupational career in southern Europe (i.e., Greece, Italy, Portugal and Spain). The relationship is explored by analysing the impact that different family structures and male [nvolvement in caring activities have on women’s early occupational trajectories (i.e., remaining in the same occupational status, experiencing downward or upward mobility, or withdrawing from paid work). This research shows that male involvement in caring activities does not really push women ahead in their career, but the absolute lack of male support seems to negatively affect women’s permanence in paid work. These results apply to all southern European countries except Portugal, where the absolute absence of the partners’ support in caring activities does not seem to alter women’s determination to remain in paid work. The methodology applied consists of the estimation of multinomial logit regression models and the analysis is based on eight waves (1994-2001) of the European Community Household Panel (ECHP).

A new compact linear programming formulation for choice network revenue management

The choice network revenue management model incorporates customer purchase behavioras a function of the offered products, and is the appropriate model for airline and hotel networkrevenue management, dynamic sales of bundles, and dynamic assortment optimization.The optimization problem is a stochastic dynamic program and is intractable. A certainty-equivalencerelaxation of the dynamic program, called the choice deterministic linear program(CDLP) is usually used to generate dyamic controls. Recently, a compact linear programmingformulation of this linear program was given for the multi-segment multinomial-logit (MNL)model of customer choice with non-overlapping consideration sets. Our objective is to obtaina tighter bound than this formulation while retaining the appealing properties of a compactlinear programming representation. To this end, it is natural to consider the affine relaxationof the dynamic program. We first show that the affine relaxation is NP-complete even for asingle-segment MNL model. Nevertheless, by analyzing the affine relaxation we derive a newcompact linear program that approximates the dynamic programming value function betterthan CDLP, provably between the CDLP value and the affine relaxation, and often comingclose to the latter in our numerical experiments. When the segment consideration sets overlap,we show that some strong equalities called product cuts developed for the CDLP remain validfor our new formulation. Finally we perform extensive numerical comparisons on the variousbounds to evaluate their performance.

When to carry eccentric products? Optimal retail assortment under consumer returns

To understand whether retailers should consider consumer returns when merchandising, we study howthe optimal assortment of a price-taking retailer is influenced by its return policy. The retailer selects itsassortment from an exogenous set of horizontally differentiated products. Consumers make purchase andkeep/return decisions in nested multinomial logit fashion. Our main finding is that the optimal assortmenthas a counterintuitive structure for relatively strict return policies: It is optimal to offer a mix of the mostpopular and most eccentric products when the refund amount is sufficiently low, which can be viewed asa form of risk sharing between the retailer and consumers. In contrast, if the refund is sufficiently high, orwhen returns are disallowed, optimal assortment is composed of only the most popular products (a commonfinding in the literature). We provide preliminary empirical evidence for one of the key drivers of our results:more eccentric products have higher probability of return conditional on purchase. In light of our analyticalfindings and managerial insights, we conclude that retailers should take their return policies into accountwhen merchandising.