### HB-Reg 貝斯階層回歸軟體

In the analysis of marketing research data, there are many occasions when the researcher has a sample of respondents, stores, or other experimental units, and wishes to estimate separate regression coefficients for each unit.

Consider three examples,

- In full-profile conjoint analysis, respondents give preference ratings for hypothetical product concepts. Regression analysis is often used, where the independent variables are columns of a "design matrix" describing the concepts, and the dependent variable consists of preference ratings.
- Respondents in a customer satisfaction study provide ratings of several companies. Some ratings are on "explanatory" variables, such as customer service, product durability, convenience of use, etc. Other ratings are more general, such as overall satisfaction with the companies' products. One goal of the study is to infer the relative importance of each explanatory factor in determining overall satisfaction.
- During a pricing experiment in grocery stores, the prices of several products are varied systematically in different time periods, and sales of each product are measured with scanner data. The independent variables are product prices and other factors such as the presence of displays, coupons, and newspaper features. The dependent variables are product sales.

In each situation, respondents or stores may have different regression functions. In the past, researchers have often tried to handle this problem by ignoring heterogeneity among individuals, pooling all the data, and estimating a single set of regression coefficients that describe the "average" individual. However, an alternative solution has recently become available to marketing researchers with the introduction of "hierarchical Bayes" (HB) methods.

Aggregate regression confounds heterogeneity (true differences between respondents/stores) with noise. Because HB-Reg can distinguish heterogeneity from noise, it results in more stable individual- AND aggregate-level estimates of betas. HB-Reg also is more robust in the case of multicolinearity among the independent variables than aggregate regression.

Several recent articles have shown that hierarchical Bayes estimation can do a creditable job of estimating individual parameters even when there are more parameters than observations per individual. This is done by considering each individual to be a sample from a population of similar individuals, and "borrowing" information from other individuals in the estimation for each one.

HB-Reg is a generalized analytical tool. The user provides the data as a matrix of independent variables and a dependent variable column in a text-only file. HB-Reg offers parameter constraints, meaning the ability to constrain certain parameters to be larger (smaller) than others, or to be greater than or less than zero. Advanced users can also control the prior variance and covariances, and degrees of freedom for the prior covariance matrix. These features will permit more reasonable estimation of parameters, even when relatively sparse information is available within the unit of analysis.

When using the full-size system, up to 1000 parameters per individual can be estimated. HB-Reg requires the Microsoft .NET framework.