Hierarchical Bayes Framework Research Articles

Note: A new approach to the modeling of spatially dependent and heterogeneous geographical regions

We propose a new spatial modeling approach to calibrate the potential impact of spatial dependency and heterogeneity on the underlying drivers of customer service and/or satisfaction measurement. The newly proposed procedure derives regionally varying coefficients, provides more flexible fitting, improves calibration fit and predictive validation, and can potentially result in augmented managerial implications compared to existing procedures by utilizing a hierarchical Bayes framework with geographical boundary effects. Using synthetic datasets, we illustrate how the proposed model outperforms four relevant benchmark models including ordinary linear regression, a Spatially Dependent Segmentation model (Govind, Rabikar, and Mittal 2018), classic Geographically Weighted Regression, and Bayesian Geographically Weighted Regression. The improved performance is most prominent when there exist significant differences between geographic boundaries and/or irregular patterns of observation locations. In our automobile customer satisfaction application study, the proposed approach also demonstrates favorable performance compared to these benchmark models. We find a dramatically heterogeneous pattern regarding two covariates in the Mountain U.S. geographic division: dealership service is more important in urban areas (e.g., Phoenix, Salt Lake City and Denver) than in rural areas, but vice-versa concerning vehicle quality.

Fully moderated t-statistic in linear modeling of mixed effects for differential expression analysis

BackgroundGene expression profiling experiments with few replicates lead to great variability in the estimates of gene variances. Toward this end, several moderated t-test methods have been developed to reduce this variability and to increase power for testing differential expression. Most of these moderated methods are based on linear models with fixed effects where residual variances are smoothed under a hierarchical Bayes framework. However, they are inadequate for designs with complex correlation structures, therefore application of moderated methods to linear models with mixed effects are needed for differential expression analysis.ResultsWe demonstrated the implementation of the fully moderated t-statistic method for linear models with mixed effects, where both residual variances and variance estimates of random effects are smoothed under a hierarchical Bayes framework. We compared the proposed method with two current moderated methods and show that the proposed method can control the expected number of false positives at the nominal level, while the two current moderated methods fail.ConclusionsWe proposed an approach for testing differential expression under complex correlation structures while providing variance shrinkage. The proposed method is able to improve power by moderation and controls the expected number of false positives properly at the nominal level.

Estimating Local Intervention Impact with Individual Level National Data by Using GBD Studies to Guide Models

Background: Non-communicable diseases account for the majority of the global burden of disease. Interventions, such as exercise and healthy diet, have the potential to decrease the prevalence and burden of diseases. While global academic studies represent the domain knowledge and provide the marginal effect of risk factors, individual level observational data reflects the joint distribution of local behaviors and socioeconomics, and their local impact on disease prevalence. Methods: We developed a method to evaluate the potential impact of a public health intervention by guiding individual level disease prevalence models of local observational data with the published meta-analysis results of Global Burden of Disease (GBD) studies. We use a Hierarchical Bayes framework to ensure that the model parameters are in agreement with the qualitative patterns in the meta-analysis results. The method provides impact estimates with uncertainty intervals at the population or any targetable subpopulation level. The intervention impact accounts for the burden due to multiple diseases without double counting and controls for demographics, socioeconomics and behaviors. We evaluate the potential impact of behavioral interventions involving smoking, physical exercise and diet on prevalence and disability adjusted life years (DALY) due to Ischemic Heart Disease (IHD) and Diabetes in the Turkish adult population. Findings: As expected, eliminating smoking and secondhand smoking and encouraging exercise have the potential to reduce DALYs associated with IHD and Diabetes by 10% (8% to 12%) and 8% (7% to 10%), respectively in the overall population. Somewhat surprisingly, increasing fruit and vegetable consumption would increase the expected DALYs for about one in five individuals due to its adverse effect on Diabetes, while bringing about a 2% (1% to 3%) reduction at the population level DALYs. Interpretation: The findings are in line with the qualitative patterns of GBD meta-analysis results, but differ quantitatively for some risk factors and demographic groups. Beyond evaluating the potential impact at the national level, the method allows the decision maker to see the heterogeneity of the intervention impact due to joint distribution of behaviors and other socioeconomic/ lifestyle factors in addition to different relative risk estimates by age and gender. Funding Statement: The authors are employees of Koc University and there was no additional funding from other sources for this research. Declaration of Interests: The authors do not have any competing interest. Ethics Approval Statement: We received a waiver from the Koc University Ethics Committee. We also got approval from the Turkish Statistical Institute for the study to be able to use the micro-level data.

Herding in the consumption and purchase of digital goods and moderators of the herding bias

Digital goods are increasingly produced by average individuals in a serialized fashion. However, it is unclear whether users engage in herding in the consumption and purchase of such digital goods and what the moderators of the herding effect are. Thus, we propose a simultaneous equations model based on herding theory to theoretically examine users’ potential herding behavior through two competing effects: the private signal effect and the sequential actions effect, which refer to the impact of the private signals and observed sequential actions of others on user quality inference and herding, respectively. The model is implemented in a hierarchical Bayes framework, and it is estimated using data from the top Chinese literature site. The empirical results suggest that users engage in rational herding in both digital book consumption and purchase on the focal site and that the herding bias is surprisingly stronger for purchasing. Product features significantly mitigate the herding bias, while the reputation of the producer and competition exacerbate the herding effect. The impact of this rational herding is also quantified. This study offers new insights and important theoretical and managerial implications for marketing researchers, amateur producers, marketing managers, and publishers.

Modeling the effects of dynamic group influence on shopper zone choice, purchase conversion, and spending

In many retail contexts, social interaction plays an important role in the shopping process. We propose a three-stage dynamic linear model that captures the influence of group discussion on shopper behavior within a hierarchical Bayes framework. The model is tested using a video tracking and transaction dataset from a specialty apparel store. The research reveals that group conversations have a significant impact on the shopper’s department or “zone” choice, purchase likelihood, and spending over time. This group influence is magnified by the size of the group (particularly for zone penetration and purchase conversion), and is also moderated by group composition and cohesiveness. The conversations of mixed-age groups and groups who stay together while shopping have a significant influence on shopper behavior across all three stages, while discussions by adult groups exhibit a marginal carryover effect for purchase conversion. When shoppers have repeated discussions in a specific department, they are more likely to return to and buy from this department, while the cumulative number of discussions in the store drives higher spending levels. We also observe that group shoppers visit more departments than their solo counterparts; and mixed-age groups and solo shoppers are more likely to buy than adults-only or teen groups. This study has important implications for how retailers manage shopper engagement and group interaction in their stores.

Bayesian estimation on semiparametric models with shape restricted B-splines

ABSTRACTIn this paper, we develop a Bayesian estimation procedure for semiparametric models under shape constrains. The approach uses a hierarchical Bayes framework and characterizations of shape-constrained B-splines. We employ Markov chain Monte Carlo methods for model fitting, using a truncated normal distribution as the prior for the coefficients of basis functions to ensure the desired shape constraints. The small sample properties of the function estimators are provided via simulation and compared with existing methods. A real data analysis is conducted to illustrate the application of the proposed method.

Nonparametric Bayesian modeling for monotonicity in catch ratio

ABSTRACTThis article proposes nonparametric Bayesian approaches to monotone function estimation. This approach uses a hierarchical Bayes framework and a characterization of stick-breaking process that allows unconstrained estimation of the monotone function. In order to avoid the limitation of parametric modeling, a general class of prior distributions, called stick-breaking priors, is considered. It accommodates much more flexible forms and can easily deal with skewness, multimodality, etc., of the dependent variable response. The proposed approach is incorporated to model the catch ratio based on automatic weather station (AWS) data.

Space-time modeling for post-fire vegetation recovery

Recently, there has been increased interest in the behavior of wildfires. Behavior includes explaining: incidence of wildfires; recurrence times for wildfires; sizes, scars, and directions of wildfires; and recovery of burned regions after a wildfire. We study this last problem. In particular, we use the annual normalized difference vegetation index (NDVI) to provide a picture of vegetative levels. Employed post-wildfire, it provides a picture of vegetative recovery. The contribution here is to model post-fire vegetation recovery from a different perspective. What exists in the literature specifies a parametric monotone form for the recovery function and then fits it to the available data. However, recovery need not be monotone; NDVI levels may increase or decrease annually according to climate variables. Furthermore, when there is recovery, it need not follow a simple parametric form. Instead, we view recovery in a relative way. We model what NDVI would look like over the fire scar in the absence of a wildfire. Then, we can examine NDVI recovery locally, employing the observed NDVI recovery at a location relative to the predictive distribution of NDVI at that location. We work with wildfire data from the Communidad Autonomia of Aragon in Spain. We develop our approach in two stages. First, we validate the predictability of NDVI in the absence of wildfire. Then, we study annual recovery and evolution of recovery for an illustrative wildfire region. We work within a hierarchical Bayes framework, adopting suitable dynamic spatial models, attaching full uncertainty to our inference on recovery.

An Examination of Social Influence on Shopper Behavior Using Video Tracking Data

This research investigates how the social elements of a retail store visit affect shoppers’ product interaction and purchase likelihood. The research uses a bivariate model of the shopping process, implemented in a hierarchical Bayes framework, which models the customer and contextual factors driving product touch and purchase simultaneously. A unique video tracking database captures each shopper's path and activities during the store visit. The findings reveal that interactive social influences (e.g., salesperson contact, shopper conversations) tend to slow the shopper down, encourage a longer store visit, and increase product interaction and purchase. When shoppers are part of a larger group, they are influenced more by discussions with companions and less by third parties. Stores with customers present encourage product interaction up to a point, beyond which the density of shoppers interferes with the shopping process. The effects of social influence vary by the salesperson's demographic similarity to the shopper and the type of product category being shopped. Several behavioral cues signal when shoppers are in a potentially high need state and may be good sales prospects.

Climate Sensitivity Distributions Dependence on the Possibility that Models Share Biases

Abstract Uncertainty about biases common across models and about unknown and unmodeled feedbacks is important for the tails of temperature change distributions and thus for climate risk assessments. This paper develops a hierarchical Bayes framework that explicitly represents these and other sources of uncertainty. It then uses models’ estimates of albedo, carbon cycle, cloud, and water vapor–lapse rate feedbacks to generate posterior probability distributions for feedback strength and equilibrium temperature change. The posterior distributions are especially sensitive to prior beliefs about models’ shared structural biases: nonzero probability of shared bias moves some probability mass toward lower values for climate sensitivity even as it thickens the distribution’s positive tail. Obtaining additional models of these feedbacks would not constrain the posterior distributions as much as narrowing prior beliefs about shared biases or, potentially, obtaining feedback estimates having biases uncorrelated with those impacting climate models. Carbon dioxide concentrations may need to fall below current levels to maintain only a 10% chance of exceeding official 2°C limits on global average temperature change.

Analyzing spatial point patterns subject to measurement error

We address the issue of inference for a noisy point pattern. The unobserved true point process is modelled as a nonhomogeneous Poisson process. For modeling the underlying intensity surface we use a scaled Gaussian mixture distribution. The noise that creeps in during the measurement procedure causes random displacement of the true locations. We consider two settings. With a bounded region of interest, (i) this displacement may cause a true location within the boundary to be associated with an ‘observed’ location outside of the region and thus missed and (ii) we have the possibility in (i) but also vice versa; the displacement may bring in an observed location whose true location lies outside the region. Under (i), we can only lose points and, depending on the variability in the measurement error as well as the number of true locations close to boundary, this can cause a significant number of locations to be lost from our recorded set of data. Estimation of the intensity surface from the observed data can be misleading especially near the boundary of our domain of interest. Under (ii), the modeling problem is more difficult; points can be both lost and gained and it is challenging to characterize how we may gain points with no data on the underlying intensity outside the domain of interest. In both cases, we work within a hierarchical Bayes framework, modeling the latent point pattern using a Cox process and, given the process realization, introducing a suitable measurement error model. Hence, the specification includes the true number of points as an unknown. We discuss choice of measurement error model as well as identifiability problems which arise. Models are fitted using an markov chain Monte Carlo implementation. After validating our method against several synthetic datasets we illustrate its application for two ecological datasets.

A Bayesian Approach to Non-Parametric Monotone Function Estimation

SummaryThe paper proposes two Bayesian approaches to non-parametric monotone function estimation. The first approach uses a hierarchical Bayes framework and a characterization of smooth monotone functions given by Ramsay that allows unconstrained estimation. The second approach uses a Bayesian regression spline model of Smith and Kohn with a mixture distribution of constrained normal distributions as the prior for the regression coefficients to ensure the monotonicity of the resulting function estimate. The small sample properties of the two function estimators across a range of functions are provided via simulation and compared with existing methods. Asymptotic results are also given that show that Bayesian methods provide consistent function estimators for a large class of smooth functions. An example is provided involving economic demand functions that illustrates the application of the constrained regression spline estimator in the context of a multiple-regression model where two functions are constrained to be monotone.

LONG-TERM CO2ENRICHMENT OF A FOREST ECOSYSTEM: IMPLICATIONS FOR FOREST REGENERATION AND SUCCESSION

The composition and successional status of a forest affect carbon storage and net ecosystem productivity, yet it remains unclear whether elevated atmospheric carbon dioxide (CO2) will impact rates and trajectories of forest succession. We examined how CO2 enrichment (+200 microL CO2/L air differential) affects forest succession through growth and survivorship of tree seedlings, as part of the Duke Forest free-air CO2 enrichment (FACE) experiment in North Carolina, USA. We planted 2352 seedlings of 14 species in the low light forest understory and determined effects of elevated CO2 on individual plant growth, survival, and total sample biomass accumulation, an integrator of plant growth and survivorship over time, for six years. We used a hierarchical Bayes framework to accommodate the uncertainty associated with the availability of light and the variability in growth among individual plants. We found that most species did not exhibit strong responses to CO2. Ulmus alata (+21%), Quercus alba (+9.5%), and nitrogen-fixing Robinia pseudoacacia (+230%) exhibited greater mean annual relative growth rates under elevated CO2 than under ambient conditions. The effects of CO2 were small relative to variability within populations; however, some species grew better under low light conditions when exposed to elevated CO2 than they did under ambient conditions. These species include shade-intolerant Liriodendron tulipifera and Liquidambar styraciflua, intermediate-tolerant Quercus velutina, and shade-tolerant Acer barbatum, A. rubrum, Prunus serotina, Ulmus alata, and Cercis canadensis. Contrary to our expectation, shade-intolerant trees did not survive better with CO2 enrichment, and population-scale responses to CO2 were influenced by survival probabilities in low light. CO2 enrichment did not increase rates of sample biomass accumulation for most species, but it did stimulate biomass growth of shade-tolerant taxa, particularly Acer barbatum and Ulmus alata. Our data suggest a small CO2 fertilization effect on tree productivity, and the possibility of reduced carbon accumulation rates relative to today's forests due to changes in species composition.

Modeling Movie Life Cycles and Market Share

We examine box-office sales in the context of a market share model. This is accomplished by developing a combination of a sliding-window logit model and a gamma diffusion pattern in a hierarchical Bayes framework. We show that accounting for the full choice set available every week not only increases the fit of weekly movie sales but also leads to parameter estimates that depict a richer picture of the movie industry. We show that movie studios appear to have a good understanding of the products they produce, knowing when to support them and when not to. We also show that the effect of the number of opening week screens is overestimated in traditional models. Our research indicates that actors have a direct and directors an indirect effect on consumers’ movie choice. Releasing a movie contemporaneously with other movies of the same genre adversely affects box-office performance all around. Releasing a movie against movies of the same Motion Picture Association of America (MPAA) rating hurts its sales in the beginning, but there is a displacement effect, which leads to a less severe sales loss in the long run.

Estimating the long-term trend in the extreme values of tropospheric ozone using a multivariate approach.

This paper deals with the identification and estimation of the long-term trend in the extreme values of tropospheric ozone after allowing for the confounding effects of meteorological conditions. The U. S. Environmental Protection Agency's (EPA's) National Ambient Air Quality Standard for ozone is stated in terms of exceedances of a specified threshold level. Therefore, the EPA is concerned with the long-term trend in the probability of an exceedance. A multivariate nonparametric probit regression model estimated within the framework of a hierarchical Bayes model is used to model the probability of an exceedance after allowing for the effects of changing meteorological conditions. There are three advantages to using this model. First, the trends estimated at each site in a region can be separated into a city-wide component that is common to all sites and a site-specific component that is unique to the individual site. Second, the hierarchical Bayes framework allows for combining information across monitoring sites to increase the information available regarding the trend at each individual site. Third, the nonparametric model does not require the a priori specification of the functional forms relating the probability of an exceedance to the meteorological variables. Ozone data from four Houston, Texas monitoring sites for the period 1981-1997 are analyzed. We find that there is a downward trend in the probability of an exceedance in the 1980s followed by a relatively flat trend in the 1990s.

The “Shopping Basket”: A Model for Multicategory Purchase Incidence Decisions

Consumers make multicategory decisions in a variety of contexts such as choice of multiple categories during a shopping trip or mail-order purchasing. The choice of one category may affect the selection of another category due to the complementary nature (e.g., cake mix and cake frosting) of the two categories. Alternatively, two categories may co-occur in a shopping basket not because they are complementary but because of similar purchase cycles (e.g., beer and diapers) or because of a host of other unobserved factors. While complementarity gives managers some control over consumers' buying behavior (e.g., a change in the price of cake mix could change the purchase probability of cake frosting), co-occurrence or co-incidence is less controllable. Other factors that may affect multi-category choice may be (unobserved) household preferences or (observed) household demographics. We also argue that not accounting for these three factors simultaneously could lead to erroneous inferences. We then develop a conceptual framework that incorporates complementarity, co-incidence and heterogeneity (both observed and unobserved) as the factors that could lead to multi-category choice. We then translate this framework into a model of multi-category choice. Our model is based on random utility theory and allows for simultaneous, interdependent choice of many items. This model, the multi variate probit model, is implemented in a Hierarchical Bayes framework. The hierarchy consists of three levels. The first level captures the choice of items for the shopping basket during a shopping trip. The second level captures differences across households and the third level specifies the priors for the unknown parameters. We generalize some recent advances in Markov chain Monte Carlo methods in order to estimate the model. Specifically, we use a substitution sampler which incorporates techniques such as the Metropolis Hit-and-Run algorithm and the Gibbs Sampler. The model is estimated on four categories (cake mix, cake frosting, fabric detergent and fabric softener) using multicategory panel data. The results disentangle the complementarity and co-incidence effects. The complementarity results show that pricing and promotional changes in one category affect purchase incidence in related product categories. In general, the cross-price and cross-promotion effects are smaller than the own-price and own-promotions effects. The cross-effects are also asymmetric across pairs of categories, i.e., related category pairs may be characterized as having a “primary” and a “secondary” category. Thus these results provide a more complete description of the effects of promotional changes by examining them both within and across categories. The co-incidence results show the extent of the relationship between categories that arises from uncontrollable and unobserved factors. These results are useful since they provide insights into a general structure of dependence relationships across categories. The heterogeneity results show that observed demographic factors such as family size influence the intrinsic category preference of households. Larger family sizes also tend to make households more price sensitive for both the primary and secondary categories. We find that price sensitivities across categories are not highly correlated at the household level. We also find some evidence that intrinsic preferences for cake mix and cake frosting are more closely related than preferences for fabric detergent and fabric softener. We compare our model with a series of null models using both estimation and holdout samples. We show that both complementarity and co-incidence play a significant role in predicting multicategory choice. We also show how many single-category models used in conjunction may not be good predictors of joint choice. Our results are likely to be of interest to retailers and manufacturers trying to optimize pricing and promotion strategies across many categories as well as in designing micromarketing strategies. We illustrate some of these benefits by carrying out an analysis which shows that the “true” impact of complementarity and co-incidence on profitability is significant in a retail setting. Our model can also be applied to other domains. The combination of item interdependence and individual household level estimates may be of particular interest to database marketers in building customized “cross-selling” strategies in the direct mail and financial service industries.

Binning Clones by Hybridization with Complex Probes: Statistical Refinement of an Inner Product Mapping Method

Molecular methods that use long-range information to solve genomics problems (i.e., top-down strategies) efficiently have become increasingly prominent in the genomics literature. One such method, an implementation of inner product mapping (IPM), uses noisy, long-range radiation hybrid (RH)/YAC overlap data and relatively noise-free RH/STS overlap data to localize clones to specific chromosomal regions. Because the molecular data are rarely noise-free, statistical models tailored to the top-down molecular methods make the methods far more effective. We develop two statistical models for IPM (or any other top-down strategy of similar form), a parametric logit model and a nonparametric order-restricted model, and show how these models can be implemented within a hierarchical Bayes framework. Using these models, we refine the chromosome 11 map reported in M. Perlinet al.(1995,Genomics28: 315–327). Our analyses improve the IPM map, both in terms of successful localization of clones and in terms of the confidence with which they are localized.