Monotonic regression based on Bayesian P-splines: an application to estimating price response functions from store-level scanner data

Generalized additive models have become a widely used instrument for flexible regression analysis. In many practical situations, however, it is desirable to restrict the flexibility of nonparametric estimation in order to accommodate a presumed monotonic relationship between a covariate and the response variable. For example, consumers usually will buy less of a brand if its price increases, and therefore one expects a brand's unit sales to be a decreasing function in own price. We follow a Bayesian approach using penalized B-splines and incorporate the assumption of monotonicity in a natural way by an appropriate specification of the respective prior distributions. We illustrate the methodology in an empirical application modeling demand for a brand of orange juice and show that imposing monotonicity constraints for own- and cross-item price effects improves the predictive validity of the estimated sales response function considerably

A Probabilistic One-Step Approach to the Optimal Product Line Design Problem Using Conjoint and Cost Data

Designing and pricing new products is one of the most critical activities for a firm, and it is well-known that taking into account consumer preferences for design decisions is essential for products later to be successful in a competitive environment (e.g., Urban and Hauser 1993). Consequently, measuring consumer preferences among multiattribute alternatives has been a primary concern in marketing research as well, and among many methodologies developed, conjoint analysis (Green and Rao 1971) has turned out to be one of the most widely used preference-based techniques for identifying and evaluating new product concepts. Moreover, a number of conjoint-based models with special focus on mathematical programming techniques for optimal product (line) design have been proposed (e.g., Zufryden 1977, 1982, Green and Krieger 1985, 1987b, 1992, Kohli and Krishnamurti 1987, Kohli and Sukumar 1990, Dobson and Kalish 1988, 1993, Balakrishnan and Jacob 1996, Chen and Hausman 2000). These models are directed at determining optimal product concepts using consumers' idiosyncratic or segment level part-worth preference functions estimated previously within a conjoint framework. Recently, Balakrishnan and Jacob (1996) have proposed the use of Genetic Algorithms (GA) to solve the problem of identifying a share maximizing single product design using conjoint data. In this paper, we follow Balakrishnan and Jacob's idea and employ and evaluate the GA approach with regard to the problem of optimal product line design. Similar to the approaches of Kohli and Sukumar (1990) and Nair et al. (1995), product lines are constructed directly from part-worths data obtained by conjoint analysis, which can be characterized as a one-step approach to product line design. In contrast, a two-step approach would start by first reducing the total set of feasible product profiles to a smaller set of promising items (reference set of candidate items) from which the products that constitute a product line are selected in a second step. Two-step approaches or partial models for either the first or second stage in this context have been proposed by Green and Krieger (1985, 1987a, 1987b, 1989), McBride and Zufryden (1988), Dobson and Kalish (1988, 1993) and, more recently, by Chen and Hausman (2000). Heretofore, with the only exception of Chen and Hausman's (2000) probabilistic model, all contributors to the literature on conjoint-based product line design have employed a deterministic, first-choice model of idiosyncratic preferences. Accordingly, a consumer is assumed to choose from her/his choice set the product with maximum perceived utility with certainty. However, the first choice rule seems to be an assumption too rigid for many product categories and individual choice situations, as the analyst often won't be in a position to control for all relevant variables influencing consumer behavior (e.g., situational factors). Therefore, in agreement with Chen and Hausman (2000), we incorporate a probabilistic choice rule to provide a more flexible representation of the consumer decision making process and start from segment-specific conjoint models of the conditional multinomial logit type. Favoring the multinomial logit model doesn't imply rejection of the widespread max-utility rule, as the MNL includes the option of mimicking this first choice rule. We further consider profit as a firm's economic criterion to evaluate decisions and introduce fixed and variable costs for each product profile. However, the proposed methodology is flexible enough to accomodate for other goals like market share (as well as for any other probabilistic choice rule). This model flexibility is provided by the implemented Genetic Algorithm as the underlying solver for the resulting nonlinear integer programming problem. Genetic Algorithms merely use objective function information (in the present context on expected profits of feasible product line solutions) and are easily adjustable to different objectives without the need for major algorithmic modifications. To assess the performance of the GA methodology for the product line design problem, we employ sensitivity analysis and Monte Carlo simulation. Sensitivity analysis is carried out to study the performance of the Genetic Algorithm w.r.t. varying GA parameter values (population size, crossover probability, mutation rate) and to finetune these values in order to provide near optimal solutions. Based on more than 1500 sensitivity runs applied to different problem sizes ranging from 12.650 to 10.586.800 feasible product line candidate solutions, we can recommend: (a) as expected, that a larger problem size be accompanied by a larger population size, with a minimum popsize of 130 for small problems and a minimum popsize of 250 for large problems, (b) a crossover probability of at least 0.9 and (c) an unexpectedly high mutation rate of 0.05 for small/medium-sized problems and a mutation rate in the order of 0.01 for large problem sizes. Following the results of the sensitivity analysis, we evaluated the GA performance for a large set of systematically varying market scenarios and associated problem sizes. We generated problems using a 4-factorial experimental design which varied by the number of attributes, number of levels in each attribute, number of items to be introduced by a new seller and number of competing firms except the new seller. The results of the Monte Carlo study with a total of 276 data sets that were analyzed show that the GA works efficiently in both providing near optimal product line solutions and CPU time. Particularly, (a) the worst-case performance ratio of the GA observed in a single run was 96.66%, indicating that the profit of the best product line solution found by the GA was never less than 96.66% of the profit of the optimal product line, (b) the hit ratio of identifying the optimal solution was 84.78% (234 out of 276 cases) and (c) it tooks at most 30 seconds for the GA to converge. Considering the option of Genetic Algorithms for repeated runs with (slightly) changed parameter settings and/or different initial populations (as opposed to many other heuristics) further improves the chances of finding the optimal solution.

On the Complexity of Digraph Colourings and Vertex Arboricity

It has been shown by Bokal et al. that deciding 2-colourability of digraphs is an NP-complete problem. This result was later on extended by Feder et al. to prove that deciding whether a digraph has a circular $p$-colouring is NP-complete for all rational $p>1$. In this paper, we consider the complexity of corresponding decision problems for related notions of fractional colourings for digraphs and graphs, including the star dichromatic number, the fractional dichromatic number and the circular vertex arboricity. We prove the following results: Deciding if the star dichromatic number of a digraph is at most $p$ is NP-complete for every rational $p>1$. Deciding if the fractional dichromatic number of a digraph is at most $p$ is NP-complete for every $p>1, p \neq 2$. Deciding if the circular vertex arboricity of a graph is at most $p$ is NP-complete for every rational $p>1$. To show these results, different techniques are required in each case. In order to prove the first result, we relate the star dichromatic number to a new notion of homomorphisms between digraphs, called circular homomorphisms, which might be of independent interest. We provide a classification of the computational complexities of the corresponding homomorphism colouring problems similar to the one derived by Feder et al. for acyclic homomorphisms.Comment: 21 pages, 1 figur

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 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 proper scoring rules and compare parametric and semiparametric approaches for a number of brand choice data sets

Semiparametric Stepwise Regression to Estimate Sales Promotion Effects

Kalyanam and Shively (1998) and van Heerde et al. (2001) have proposed semiparametric models to estimate the influence of price promotions on brand sales, and both obtained superior performance for their models compared to strictly parametric modeling. Following these researchers, we suggest another semiparametric framework which is based on penalized B-splines to analyze sales promotion effects flexibly. Unlike these researchers, we introduce a stepwise procedure with simultaneous smoothing parameter choice for variable selection. Applying this stepwise routine enables us to deal with product categories with many competitive items without imposing restrictions on the competitive market structure in advance. We illustrate the new methodology in an empirical application using weekly store-level scanner data

Complete Acyclic Colorings

We study two parameters that arise from the dichromatic number and the vertex-arboricity in the same way that the achromatic number comes from the chromatic number. The adichromatic number of a digraph is the largest number of colors its vertices can be colored with such that every color induces an acyclic subdigraph but merging any two colors yields a monochromatic directed cycle. Similarly, the a-vertex arboricity of an undirected graph is the largest number of colors that can be used such that every color induces a forest but merging any two yields a monochromatic cycle. We study the relation between these parameters and their behavior with respect to other classical parameters such as degeneracy and most importantly feedback vertex sets.Comment: 17 pages, no figure

Bayesian Geoadditive Seemingly Unrelated Regression

Parametric seemingly unrelated regression (SUR) models are a common tool for multivariate regression analysis when error variables are reasonably correlated, so that separate univariate analysis may result in inefficient estimates of covariate effects. A weakness of parametric models is that they require strong assumptions on the functional form of possibly nonlinear effects of metrical covariates. In this paper, we develop a Bayesian semiparametric SUR model, where the usual linear predictors are replaced by more flexible additive predictors allowing for simultaneous nonparametric estimation of such covariate effects and of spatial effects. The approach is based on appropriate smoothness priors which allow different forms and degrees of smoothness in a general framework. Inference is fully Bayesian and uses recent Markov chain Monte Carlo techniques

A semiparametric approach to estimating reference price effects in sales response models

It is well known that store-level brand sales may not only depend on contemporaneous influencing factors like current own and competitive prices or other marketing activities, but also on past prices representing customer response to price dynamics. On the other hand, non- or semiparametric regression models have been proposed in order to accommodate potential nonlinearities in price response, and related empirical findings for frequently purchased consumer goods indicate that price effects may show complex nonlinearities, which are difficult to capture with parametric models. In this contribution, we combine nonparametric price response modeling and behavioral pricing theory. In particular, we propose a semiparametric approach to flexibly estimating price-change or reference price effects based on store-level sales data. We compare different representations for capturing symmetric vs. asymmetric and proportional vs. disproportionate price-change effects following adaptation-level and prospect theory, and further compare our flexible autoregressive model specifications to parametric benchmark models. Functional flexibility is accommodated via P-splines, and all models are estimated within a fully Bayesian framework. In an empirical study, we demonstrate that our semiparametric dynamic models provide more accurate sales forecasts for most brands considered compared to competing benchmark models that either ignore price dynamics or just include them in a parametric way

Conjoint-Analyse und Marktsegmentierung

Die Marktsegmentierung zählt neben der Neuproduktplanung und Preisgestaltung zu den wesentlichen Einsatzgebieten der Conjoint-Analyse. Neben traditionell eingesetzten zweistufigen Vorgehensweisen, bei denen Conjoint-Analyse und Segmentierung in zwei getrennten Schritten erfolgen, stehen heute mit Methoden wie der Clusterwise Regression oder Mixture-Modellen neuere Entwicklungen, die eine simultane Segmentierung und Präferenzschätzung ermöglichen, zur Verfügung. Der Beitrag gibt einen Überblick über die vorliegenden methodischen Ansätze zur Verknüpfung von Conjoint-Analyse und Marktsegmentierung und zeigt die Vorzüge simultaner Conjointsegmentierungsmethoden gegenüber den in der Unternehmenspraxis noch immer weit verbreiteten zweistufigen Verfahren auf. Along with new product/concept identification and pricing, market segmentation ranks among the primary purposes in commercial conjoint applications. Traditionally, this conjoint segmentation has been accomplished by a two-step procedure, (1) either by first segmenting markets and subsequently estimating conjoint models at the segment level or (2) by first conducting conjoint analysis at the individual level and then clustering individual level part-worths. However, in recent years, some powerful techniques for simultaneously performing market segmentation and calibrating segment-level part-worths such as clusterwise regression procedures and mixture models have been proposed. This article provides an overview of existing conjoint segmentation methods and particularly focuses on the newer simultaneous approaches which offer substantial advantages compared to the traditional two-step procedures.Marktsegmentierung; Conjoint-Analyse ; Simultanverfahren;