Individual-level Parameters Research Articles

Interactions among Size-Structured Populations in a Whole-Lake Experiment: Size- and Scale-Dependent Processes

Interactions in size-structured populations are characterized by a mixture of predatory and competitive interactions dependent on the size of the individual organism. We analyzed this mixture of size-structured interactions for a species constellation consisting of perch (Perca fluviatilis) and roach (Rutilus rutilus) in a replicated whole lake experiment over 4 yr. Roach are preyed upon by large perch, but at the same time compete with small perch for zooplankton. Predictions regarding the effects of roach on perch performance and resource dynamics in the whole-lake experiment were based on results from previous pond and enclosure experiments carried out over short (months) time periods. Variables measured in the whole-lake experiment included both individual level parameters (diet and growth of perch) as well as population level parameters (mortality, population numbers and size structures of perch, abundance and biomass of zooplankton and benthic macroinvertebrates). The population size of perch ≥ 2 yr old decreased to less than 10% of the pre-treatment levels in treatment lakes after the introduction of roach and remained low over the whole study period. The mortality in perch one-year-old and older increased with size. Larger perch had a higher growth decrease than smaller perch, but also the individual growth rate of young-of-the-year (YOY) perch was lower in treatment lakes than in control lakes. YOY perch were almost excluded during the winter and spring following the roach introduction. The few perch of this year class that survived had the following years a higher growth than the corresponding year class in control lakes. This could be related to an increased availability of predator-sensitive macroinvertebrates in treatment lakes. Perch in treatment lakes fed to a larger extent on macroinvertebrates and less on pelagic zooplankton than perch in control lakes. As expected, zooplankton abundances decreased in treatment lakes the year following the roach introduction, but thereafter zooplankton abundances were higher in treatment lakes due to the absence of perch recruitment in these lakes. No incidence of perch predation on roach was recorded during the study period. We attribute this to the spatial scale dependent escape ability and habitat use of vulnerable size classes of roach. YOY perch had a stronger impact on between-year variation in zooplankton than YOY roach and this can be related to the higher foraging (handling) capacity of the former and the restricted habitat use of the latter. Individual level parameters (diet, growth) had a higher statistical power than population level parameters (abundance, biomass, mortality), and small-scale experiments successfully predicted a higher proportion (81%) of the individual level parameters. However, the lower success in predicting population level responses (38%) was directly connected to a failure to predict individual level parameters (large perch diet and growth). Our results suggest that the small spatial scale was the major limitation on the predictive ability of the small-scale experiments.

The Impact of Heterogeneity in Purchase Timing and Price Responsiveness on Estimates of Sticker Shock Effects

The notion that individuals have an internal reference price against which they compare observed prices is well supported by several psychological theories. Empirically, several papers in the marketing literature, employing scanner panel data, have modeled the impact of reference prices on brand choice via the sticker shock formulation, in which consumers evaluate choice alternatives using differences between shelf prices and reference prices. Most of the studies reported thus far have not accounted for heterogeneity in price response among consumers and have typically imputed reference prices from the shelf prices of brands that a consumer is supposed to have “observed” on previous purchases in the category. Since category marketing activity can differentially affect the purchase timing of households, we argue that this measure of reference price may follow certain systematic patterns across consumers and, when combined with unaccounted for price response heterogeneity, may result in a spurious sticker shock effect. Specifically, we show that estimates of sticker shock are biased upward if households that are price-sensitive in the brand choice decision are also more responsive to category promotion activity in their purchase timing decision. We discuss some general conditions under which the bias occurs and conduct a simulation experiment to confirm our specific hypotheses. Our simulation results show that changes in purchase timing are a critical determinant of the bias in the sticker shock effect. We also show that unaccounted for price response heterogeneity can in itself result in a biased sticker shock parameter; however, this requires very large differences in price sensitivities across consumers, far greater than what is normally observed. We develop a hierarchical Bayes version of the nested logit model, which models heterogeneity via individual-level parameters in a continuous random effects framework. We estimate the model on scanner panel data from the yogurt and ketchup categories. We find, in both categories, that the 95% probability interval of the posterior distribution of the mean sticker shock coefficient contains the value zero. Therefore, at least for the data used in this study, there is no evidence for the sticker shock effect at the aggregate level. In contrast, the corresponding coefficient from a standard model (which ignores this heterogeneity) is highly significant and supports the existence of a (possibly spurious) sticker shock effect. Consistent with our explanation of the underlying cause of the bias, households that are more price-sensitive in the choice decision are also found to be more responsive to category promotion activity in their decision to purchase in the category. The results highlight the measurement problems associated with imputing reference prices from past prices. Since the frequency, duration, and price level of a retailer's promotional program depend on its size and prevalence, accurate estimates of the sticker shock effect are essential for formulating optimal promotion strategies. An adequate accounting of consumer heterogeneity is critical to this effort.

A Bayesian approach to the spatial representation of market structure from consumer choice data

This paper is concerned with the spatial representation of market structure calibrated on actual or intended choice data. Previous models developed for that purpose accommodate consumer heterogeneity by estimating parameters for each consumer, typically using the method of maximum likelihood. This approach to heterogeneity avoids assuming any particular distribution of the individual level parameters across the sample, but leads to problems related to the consistency of the estimates, sufficient degrees of freedom, and the validity of asymptotic standard errors and test statistics. Of greater concern is the assumption of independence of the choice observations within the same individual. This assumption is necessary in a maximum likelihood (MLE) framework to make the estimation computationally feasible. However, the marketing and psychology literature (cf. Manrai, 1995; Tversky and Simonson, 1993; Kim et al., forth coming) demonstrates that dependencies among choice alternatives may exist, and negligence to take such covariance into account may lead to inconsistent estimates, reduced predictive validity, and incorrect managerial action. We develop a new multidimensional scaling (MDS) model that estimates spatial market structures from pick-any/ J choice data, provides for individual level parameters, and allows for correlations among the choice alternatives across individuals. We provide a Bayesian estimation method that overcomes the traditional problems associated with estimating models with such correlated alternatives. We provide an application to pick-any/ J data for various brands of portable telephones. In a comparative analysis, we show that the proposed model outperforms one in which the utilities are assumed to be uncorrelated across choice alternatives.

A Hierarchical Bayes Model of Primary and Secondary Demand

Product design, pricing policies, and promotional activities influence the primary and secondary demand for goods and services. Brand managers need to develop an understanding of the relationships between marketing mix decisions and consumer decisions of whether to purchase in the product category, which brand to buy, and how much to consume. Knowledge about factors most effective in influencing primary and secondary demand of a product allows firms to grow by enhancing their market share as well their market size. The purpose of this paper is to develop an individual level model that allows an investigation of both the primary and secondary aspects of consumer demand. Unlike models of only primary demand or only secondary demand, this more comprehensive model offers the opportunity to identify changes in product features that will result in the greatest increase in demand. It also offers the opportunity to differentially target consumer segments depending upon whether consumers are most likely to enter the market, increase their consumption level, or switch brands. In the proposed hierarchical Bayes model, an integrative framework that jointly models the discrete choice and continuous quantity components of consumer decision is employed instead of treating the two as independent. The model includes parameters that capture individual specific reservation value, attribute preference, and expenditure sensitivity. The model development is based upon the microeconomic theory of utility maximization. Heterogeneity in model parameters across the sample is captured by using a random effects specification guided by the underlying microeconomic model. This requires that some of the effects are strictly positive. This is accommodated through the use of a gamma distribution of heterogeneity for some of the parameters. A normal distribution of heterogeneity is used for the remaining parameters. Gibbs sampling is used to estimate the model. The key methodological contribution of this paper is that we show how to specify a hierarchical Bayes continuous random effects model that integrates consumer choice and quantity decisions such that individual-level parameters can be estimated. Individual level estimates are desirable because insights into primary demand involve nonlinear functions of model parameters. For example, consumers not in the market are those whose utilities for the choice alternatives fall below some reservation value. The proposed methodology yields individual specific estimates of reservation values and expenditure sensitivity, which allow assessment of the origins of demand other than the switching behavior of consumers. The methodology can also be used to help identify changes in product features most likely to bring new customers into a market. Our work differs from previous research in this area as we lay the framework needed to obtain individual-level parameter estimates in a continuous random effects model that integrates choice and quantity. The methodology is demonstrated with survey data collected about consumer preferences and consumption for a food item. For the data available, a large response heterogeneity was observed across all model parameters. In spite of limited data available at the individual level, a majority of the individual level estimates were found to be significant. Predictive tests demonstrated the superiority of the proposed model over existing latent class and aggregate models. Particularly, significant gains in predictive accuracy were observed for the “no-buy” behavior of the respondents. These gains demonstrate that by structurally linking the choice and quantity models results in a more accurate characterization of the market than existing finite mixture approaches that model choice and quantity independently. We show that our joint model makes more efficient use of the available data and results in better parameter estimates than those that assume independence. Finally, the individual level demand analysis is illustrated through a simple example involving a $1.00 price cut. We demonstrate practical usefulness of the model for targeting by developing the demographic, attitudinal, and behavioral profiles of consumer groups most likely to increase consumption, enter the market, or switch brands because of a price cut decision.

The Innovation Diffusion Process in a Heterogeneous Population: A Micromodeling Approach

A model of the innovation diffusion process is developed using a micromodeling approach that explicitly considers the determinants of adoption at the individual level in a decision analytic framework, and incorporates heterogeneity in the population with respect to initial perceptions, preference characteristics, and responsiveness to information. The micromodelling approach provides a behavioral basis for explaining adoption at the disaggregate level and the consequent pattern of diffusion at the aggregate level. The analytical implications of the model are compared and contrasted with the traditional, aggregate-level, diffusion models. An advantage of our approach is its micro-theory driven flexibility in accommodating various patterns of diffusion. Examples are provided of conditions under which the model yields diffusion patterns identical to those of some well-known aggregate models. A pilot study is reported, outlining procedures for data collection and estimation of the individual-level parameters, and providing a preliminary test of the predictive performance of the model. Measurement of the individual parameters prior to product launch enables potential applications of the model for segmentation of the target population in terms of their expected adoption times.

The Multilevel Analysis of Trends with Repeated Cross-Sectional Data

The purpose of this paper is to develop a multilevel model for the analysis of trends within repeated cross-sectional samples. The model proposed by the authors uses a time series of macro-level variables to explain trends in the individual-level parameters estimated within each sample. This model falls within the general category of varying parameter models but in the present context additional methodological complications arise because of the potential for serial correlations among the errors in the macro-level equations and because the effects of the macro-level variables can be expected to differ across sectors of the population. The latter consideration leads the authors to propose a 3-level model permitting the effects of macro-level variables to be a function of a simple buffering variable that measures the sensitivity of different population groups to change. The authors develop a mixed estimator that allows for both types of methodological complications and they use it to describe and explain recent trends in the process of stratification. The results yield new insights into the structure of short-term trends in the mobility regime. (authors)

Old instruments and devices

This paper presents a framework for estimating and updating user preferences in the context of app-based recommender systems. We specifically consider recommender systems which provide personalized menus of options to users. A Hierarchical Bayes procedure is applied in order to account for inter- and intra-consumer heterogeneity, representing random taste variations among individuals and among choice situations (menus) for a given individual, respectively. Three levels of preference parameters are estimated: population-level, individual-level and menu-specific. In the context of a recommender system, the estimation of these parameters is repeated periodically in an offline process in order to account for trends, such as changing market conditions. Furthermore, the individual-level parameters are updated in real-time as users make choices in order to incorporate the latest information from the users. This online update is computationally efficient which makes it feasible to embed it in a real-time recommender system. The estimated individual-level preferences are stored for each user and retrieved as inputs to a menu optimization model in order to provide recommendations. The proposed methodology is applied to both Monte-Carlo and real data. It is observed that the online update of the parameters is successful in improving the parameter estimates in real-time. This framework is relevant to various recommender systems that generate personalized recommendations ranging from transportation to e-commerce and online marketing, but is particularly useful when the attributes of the alternatives vary over time.