Optimal Designs for Stated Choice Experiments that Incorporate Position Effects

Davidson and Beaver (1977 Davidson , R. R. , Beaver , R. J. ( 1977 ). On extending the Bradley–Terry model to incorporate within-pair order effects . Biometrics 33 : 693 – 702 .[Crossref], [Web of Science ®] , [Google Scholar]) extended the Bradley–Terry model to incorporate the possible effect of position within a choice set on the choices made in paired comparisons experiments. We further extend the Davidson–Beaver result to choice sets of any size and show, under a mild restriction, that designs optimal for the multinomial logit model are still optimal. Designs balanced for carry-over effects of all orders can be used to construct designs with a diagonal information matrix for attribute effects. The theoretical results are derived assuming equal merits and we discuss the possible consequences of assuming unequal merits in an example.     

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.     

Cumulative or adjacent logits: Which choice for an ordinal logistic latent variable model?

With ordinal response items, a graded response model (GRM) is of cumulative logits type, while the polytomous Rasch model (PRM) is based on adjacent logits. In this work, we compare the two approaches. We show that the PRM is superior to the GRM, with interesting properties that we prove. Note S_ν the sum of item responses of individual ν and Θ_ν its latent parameter; we show i) S_ν is a sufficient statistic for θ_ν and ii) a property of “stochastic ordering” of the conditional distributions G_θ/S. The second property, less known, is, to our knowledge, nowhere satisfactorily demonstrated.     

Bayesian inference and model selection in latent class logit models with parameter constraints: an application to market segmentation

Latent class models have recently drawn considerable attention among many researchers and practitioners as a class of useful tools for capturing heterogeneity across different segments in a target market or population. In this paper, we consider a latent class logit model with parameter constraints and deal with two important issues in the latent class models--parameter estimation and selection of an appropriate number of classes--within a Bayesian framework. A simple Gibbs sampling algorithm is proposed for sample generation from the posterior distribution of unknown parameters. Using the Gibbs output, we propose a method for determining an appropriate number of the latent classes. A real-world marketing example as an application for market segmentation is provided to illustrate the proposed method.     

带异质线性趋势的动态二元选择面板模型估计

本文提出了带异质线性趋势的动态二元面板模型的极大似然偏误纠正估计量和近似条件Logit估计量。我们给出了通常极大似然估计量偏误的解析形式,并提供了相应的估计方法。小样本实验表明近似条件似然函数可以很好的消除异质性参数的影响,而偏误纠正估计量可以显著的修正极大似然估计量的偏误。最后我们将本文提出的方法应用到现金红利支付模型。     

Effect of choice complexity on design efficiency in conjoint choice experiments

Conjoint choice experiments have become a powerful tool to explore individual preferences. The consistency of respondents' choices depends on the choice complexity. For example, it is easier to make a choice between two alternatives with few attributes than between five alternatives with several attributes. In the latter case it will be much harder to choose the preferred alternative which is reflected in a higher response error. Several authors have dealt with this choice complexity in the estimation stage but very little attention has been paid to set up designs that take this complexity into account. The core issue of this paper is to find out whether it is worthwhile to take this complexity into account in the design stage. We construct efficient semi-Bayesian D-optimal designs for the heteroscedastic conditional logit model which is used to model the across respondent variability that occurs due to the choice complexity. The degree of complexity is measured by the entropy, as suggested by Swait and Adamowicz (2001). The proposed designs are compared with a semi-Bayesian D-optimal design constructed without taking the complexity into account. The simulation study shows that it is much better to take the choice complexity into account when constructing conjoint choice experiments.     

Rank-order choice-based conjoint experiments: Efficiency and design

In a rank-order choice-based conjoint experiment, the respondent is asked to rank a number of alternatives of a number of choice sets. In this paper, we study the efficiency of those experiments and propose a D-optimality criterion for rank-order experiments to find designs yielding the most precise parameter estimators. For that purpose, an expression of the Fisher information matrix for the rank-ordered conditional logit model is derived which clearly shows how much additional information is provided by each extra ranking step. A simulation study shows that, besides the Bayesian D-optimal ranking design, the Bayesian D-optimal choice design is also an appropriate design for this type of experiments. Finally, it is shown that considerable improvements in estimation and prediction accuracy are obtained by including extra ranking steps in an experiment.     

Optimal Designs for 2 k Choice Experiments

Abstract

In this article we establish the choice sets in the D-optimal design for a choice experiment for testing main effects and for testing main effects and two-factor interactions, when there are k attributes, each with two levels, for choice set size m. We also give a method to construct optimal and near-optimal designs with small numbers of choice sets.     

Testing for state dependence in binary panel data with individual covariates by a modified quadratic exponential model

We propose a test for state dependence in binary panel data with individual covariates. For this aim, we rely on a quadratic exponential model in which the association between the response variables is accounted for in a different way with respect to more standard formulations. The level of association is measured by a single parameter that may be estimated by a Conditional Maximum Likelihood (CML) approach. Under the dynamic logit model, the conditional estimator of this parameter converges to zero when the hypothesis of absence of state dependence is true. Therefore, it is possible to implement a t-test for this hypothesis which may be very simply performed and attains the nominal significance level under several structures of the individual covariates. Through an extensive simulation study, we find that our test has good finite sample properties and it is more robust to the presence of (autocorrelated) covariates in the model specification in comparison with other existing testing procedures for state dependence. The proposed approach is illustrated by two empirical applications: the first is based on data coming from the Panel Study of Income Dynamics and concerns employment and fertility; the second is based on the Health and Retirement Study and concerns the self reported health status.     

Optimal designs for estimating a choice hierarchy by a general nested multinomial logit model

Abstract

In choice experiments the process of decision-making can be more complex than the proposed by the Multinomial Logit Model (MNL). In these scenarios, models such as the Nested Multinomial Logit Model (NMNL) are often employed to model a more complex decision-making. Understanding the decision-making process is important in some fields such as marketing. Achieving a precise estimation of the models is crucial to the understanding of this process. To do this, optimal experimental designs are required. To construct an optimal design, information matrix is key. A previous research by others has developed the expression for the information matrix of the two-level NMNL model with two nests: Alternatives nest (J alternatives) and No-Choice nest (1 alternative). In this paper, we developed the likelihood function for a two-stage NMNL model for M nests and we present the expression for the information matrix for 2 nests with any amount of alternatives in them. We also show alternative D-optimal designs for No-Choice scenarios with similar relative efficiency but with less complex alternatives which can help to obtain more reliable answers and one application of these designs.     

Model-Robust Design of Conjoint Choice Experiments

Within the context of choice experimental designs, most authors have proposed designs for the multinomial logit model under the assumption that only the main effects matter. Very little attention has been paid to designs for attribute interaction models. In this article, three types of Bayesian D-optimal designs for the multinomial logit model are studied: main-effects designs, interaction-effects designs, and composite designs. Simulation studies are used to show that in situations where a researcher is not sure whether or not attribute interaction effects are present, it is best to take into account interactions in the design stage. In particular, it is shown that a composite design constructed by including an interaction-effects model and a main-effects model in the design criterion is most robust against misspecification of the underlying model when it comes to making precise predictions.     

An Empirical Investigation of the “Dynamic McFadden” Model of Purchase Timing and Brand Choice: Implications for Market Structure

Purchase timing and brand-choice decisions of households are jointly investigated using the “dynamic McFadden” model of Heckman and Singer. The hazard of brand purchase is decomposed into the category purchase hazard and the probability of brand choice conditional on category purchase. The former is modeled using the hazard-function approach and the latter using a logit model. Unobserved heterogeneity in brand preferences, marketing effects, and baseline hazard parameters is accounted for in the empirical analysis. The distribution of preference heterogeneity identifies the locations of brands in multiattribute perceptual space and the distribution of attribute importance weights across households.     

Bayesian analysis of binary survey data

First a comprehensive treatment of the hierarchical-conjugate Bayesian predictive approach to binary survey data is presented, encompassing simple random, stratified, cluster, and two-stage sampling, as well as two-stage sampling within strata. For the case of two-stage sampling within strata when there is more than one variable of stratification, analysis using an unsaturated logit linear model on the prior means is proposed. This allows there to be cells containing no sampled clusters. Formulas for posterior predictive means, variances, and covariances of numbers of successes in unsampled portions of clusters are presented in terms of posterior expectations of certain functions of hyperparameters; these may be evaluated by existing methods. The technique is illustrated using a small subset of Canada Youth & AIDS Study data. A sample of students within each of various selected school boards was chosen and interviewed via questionnaire. The boards were stratified/poststratified in two dimensions, but some of the resulting cells contained no data. The additive logit linear model on the prior means produced estimates and posterior variances for boards in all cells. Data showed the additive model to be plausible.     

Analytical closed-form solution for binary logit regression by categorical predictors

In contrast to the common belief that the logit model has no analytical presentation, it is possible to find such a solution in the case of categorical predictors. This paper shows that a binary logistic regression by categorical explanatory variables can be constructed in a closed-form solution. No special software and no iterative procedures of nonlinear estimation are needed to obtain a model with all its parameters and characteristics, including coefficients of regression, their standard errors and t-statistics, as well as the residual and null deviances. The derivation is performed for logistic models with one binary or categorical predictor, and several binary or categorical predictors. The analytical formulae can be used for arithmetical calculation of all the parameters of the logit regression. The explicit expressions for the characteristics of logit regression are convenient for the analysis and interpretation of the results of logistic modeling.     

Book Reviews

Aggregate forecasts using McFadden's conditional logit model of discrete choice harbor the unrealistic implicit assumption of a random-utility distribution that is homogeneous across both alternatives and individuals. This article relaxes that assumption. A choice model is developed that describes the random-utility component as a probability-mixture model. Some numerical results illustrate that the derived model is not constrained by the independence-of-irrelevant-alternatives property. An experimental test of visual perceptions demonstrates the potential superiority of the model.     

BINARY REGRESSION WITH A CLASS OF SKEWED t LINK MODELS

ABSTRACT

In this paper we propose a class of skewed t link models for analyzing binary response data with covariates. It is a class of asymmetric link models designed to improve the overall fit when commonly used symmetric links, such as the logit and probit links, do not provide the best fit available for a given binary response dataset. Introducing a skewed t distribution for the underlying latent variable, we develop the class of models. For the analysis of the models, a Bayesian and non-Bayesian methods are pursued using a Markov chain Monte Carlo (MCMC) sampling based approach. Necessary theories involved in modelling and computation are provided. Finally, a simulation study and a real data example are used to illustrate the proposed methodology.     

A Graphical Tool for Interpreting Regression Coefficients of Trinomial Logit Models

Multinomial logit (also termed multi-logit) models permit the analysis of the statistical relation between a categorical response variable and a set of explicative variables (called covariates or regressors). Although multinomial logit is widely used in both the social and economic sciences, the interpretation of regression coefficients may be tricky, as the effect of covariates on the probability distribution of the response variable is nonconstant and difficult to quantify. The ternary plots illustrated in this article aim at facilitating the interpretation of regression coefficients and permit the effect of covariates (either singularly or jointly considered) on the probability distribution of the dependent variable to be quantified. Ternary plots can be drawn both for ordered and for unordered categorical dependent variables, when the number of possible outcomes equals three (trinomial response variable); these plots allow not only to represent the covariate effects over the whole parameter space of the dependent variable but also to compare the covariate effects of any given individual profile. The method is illustrated and discussed through analysis of a dataset concerning the transition of master’s graduates of the University of Trento (Italy) from university to employment.     

THE ANALYSIS OF DISCRETE CHOICE EXPERIMENTS WITH CORRELATED ERROR STRUCTURE

In a stated preference discrete choice experiment each subject is typically presented with several choice sets, and each choice set contains a number of alternatives. The alternatives are defined in terms of their name (brand) and their attributes at specified levels. The task for the subject is to choose from each choice set the alternative with highest utility for them. The multinomial is an appropriate distribution for the responses to each choice set since each subject chooses one alternative, and the multinomial logit is a common model. If the responses to the several choice sets are independent, the likelihood function is simply the product of multinomials. The most common and generally preferred method of estimating the parameters of the model is maximum likelihood (that is, selecting as estimates those values that maximize the likelihood function). If the assumption of within-subject independence to successive choice tasks is violated (it is almost surely violated), the likelihood function is incorrect and maximum likelihood estimation is inappropriate. The most serious errors involve the estimation of the variance-covariance matrix of the model parameter estimates, and the corresponding variances of market shares and changes in market shares.

In this paper we present an alternative method of estimation of the model parameter coefficients that incorporates a first-order within-subject covariance structure. The method involves the familiar log-odds transformation and application of the multivariate delta method. Estimation of the model coefficients after the transformation is a straightforward generalized least squares regression, and the corresponding improved estimate of the variance-covariance matrix is in closed form. Estimates of market share (and change in market share) follow from a second application of the multivariate delta method. The method and comparison with maximum likelihood estimation are illustrated with several simulated and actual data examples.

Advantages of the proposed method are: 1) it incorporates the within-subject covariance structure; 2) it is completely data driven; 3) it requires no additional model assumptions; 4) assuming asymptotic normality, it provides a simple procedure for computing confidence regions on market shares and changes in market shares; and 5) it produces results that are asymptotically equivalent to those produced by maximum likelihood when the data are independent.