P 1. a = a vector of means of the distribution of individuals' p k = the probability of an individual choosing the kth X 1

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1 USING SAS IML TO PERFORM HIERARCIDCAL BAYES ESTIMATION FOR DISCRETE CHOICE MODELING David Steenhard, LexisNexis, Louisville, KY Nan-Ting Chou, University of Louisville, Louisville, KY ABSTRACT Most methods for analyzing choice-based conjoint data do so by combining data for all individuals. There could be extreme weakness in analyzing data this way, for it could obscure important individual aspects of the data. Hierarchical Bayes Estimation is one method of estimating individual part-worths. This method can reasonably estimate individual part-worths even with relatively little data from each respondent. In this paper we provide an introduction to Hierarchical Bayes Estimation and use this algorithm written in SAS IML to perform a choice-based conjoint aualysis. Sufficient details are provided to allow the readers to easily use Hierarchical Bayes Estimation as a tool for discrete choice modeling. INTRODUCTION Disaggregate or individual discrete choice modeling is fast becoming a favorite topic among market research professionals due to the technique's ability to answer a wide range of marketing questions. Discrete choice analysis consists of a series of questions, which ask respondents to choose between two or more hypothetical products. The products are described in each question by a list of attributes, which enable the respondent to easily compare between alternative products. Discrete choice modeling is typically done by using aggregate models (models that assume that all respondents have the same preferences). The resulting analysis provides a model of the choice behavior of a representative, or average respondent. However, respondents tend to differ across socio-demographic and/or attitudinal characteristics. The difficulty with an aggregate model is that by assuming all respondents have the same preferences, we assume away the important differences among respondents, which can potentially lead to problems with the model's accuracy and a loss of valuable marketing information. Important recent advances in statistics, however, have provided a way around the data deficiency of aggregate models. Hierarchical Bayes Estimation is one method of estimating individual partworths. Landmark articles by Allen by and Ginter ( 1995) and Link, DeSarbo, Green, and Young (1996) describe the estimation of individual partworths using Hierarchical Bayes (HB) models. This approach seemed extremely promising, since it could estimate reasonable individual part-worths even with relatively little amount of data from each respondent. However, this method is computationally intensive, and usually requires many thousands of iterations before convergence is met This paper illustrates how Hierarchical Bayes Estimation in SAS IML can be used to solve a discrete choice modeling problem and outlines the mechanics of Hierarchical Bayes Estimation in detail. IDERARCHICAL BAYES MODEL The HB model used here is called "hierarchical" because it has two levels. At the upper level, we assume that individuals' part-worths are described by a mnltivariate normal distribution with the following notation. where: p,- N(a,D) P 1 =a vector ofpart-worths for the ith individual. a = a vector of means of the distribution of individuals' part-worths. D = a matrix of variances and covariances of the distribution of part-worths across individuals. At the lower level we assume that, given an individual's part-worths, his/her probabilities of choosing particular alternatives are governed by a multinomiallogit model. The probability of the ith individual choosing the kth alternative in a particular task is: where: Pt = exp( x, 'P,)IL 1 exp(x/p,) (I) p k = the probability of an individual choosing the kth concept in a particular choice task. X 1 = a vector of values describing the jth alternative in that choice task. The parameters to be estimated are the vectors P 1 partworths for each individual, the vector a means ofthe distribution of part-worths, and the matrix D the variances and co variances of that distribution. ESTIMATION OF THE PARAMETERS The parameters P, a, and D are estimated by an itemtive process. This process is quite robust, and its results do not depend on starting values. However; to make the process converge as quickly as possible, one should start with estimates of the parameters that are reasonably close to fmal values. Initial estimates of the P 1 were set equal to the parameters of the aggregate multinomiallogit model. Initial estimates of a is the avemge of the P 1 Our initial estimates of D consists of variance and covariances of the aggregate multinomiallogit model. 146

2 Given these initial values, each iteration consists of the following three steps. Using the estimates of the fl 1 and D, generate a new estimate of a. assuming that a is distributed nonnally with mean equal to the average. Using the estimates of the a and fl 1, generate a new estimate of D, from the inverse Wishart distribution. Using the estimates of the a and D, generate a new estimate of fl 1, from a procedure known as "Metropolis Hasting Algorithm" Which will be discuss in detail in the next section. For each of these three steps we re-estimate one set of the parameters (a, fl 1 or D) based on current values of the other two sets. This technique is know as "Gibbs sampling", and converges to the correct distribution for each of the three sets of parameters. Another name for this procedure is a "Monte Carlo Markov Chain", because the fact that the estimates in each iteration are determined from those of the previous iteration by a constant set of transition rules. This process is carried out for a large number of iterations. The first few thousand are used to achieve convergence, with successive iterations fitting the data better and better. These iterations are called "burn-in" or "transitory" iterations. After the transitory iterations are completed we start to save the estimates of the fl 1,a, and D for each iteration. To get a point estimates of the part-worths for each respondent, we take the average of the fl 1 from these iterations. METROPOLIS HASTINGS ALGORITHM The Metropolis Hasting Algorithm is used to draw each new set of betas for each individual. We use the symbol flold to indicate the previous estimate of fl 1 We then generate a trial value for the new estimate of fl 1, which we call fl NEW, and then test whether it represents an improvement. If so we accept it as our next estimate, if not we accept or reject it with probability depending on how much worse it is than the previous estimate. To get flnew we draw a random vector d of"differences" from a distribution with mean of zero and covariance matrix proportioual to D, and let flnew = flow + d We then calculate the probability of the data given each set ofpartworths, fl OLD and fl NEW, using the formula for the multinomial logit model (1). That is done by calculating the probability of each choice that individual made, using the multinomiallogit formula for Pk and then multiplying all these probabilities together and call these resulting values p OLD and p NEW, respectively. Next we calculate the relative density of the distribution of the betas corresponding to fl OLD and fl NEW, given current estimates of parameters a and D (these serve as priors in the Bayesian updating). Call these values dow and d NEW. The relative density of the distribution at the location of a point fl is given by the following fonnula: Relative Density=exrt-Yz (fl -a)'d- 1 (fl -a)j Finally calculate the ratio: r = PNEWdNEW I Powdow From Bayesian updating the posterior probabilities are proportional to the product of the likelihood times the priors. The probabilities p NEW and pow are the likelihood's of the data given the parameter estimates flow and fl NEW. The densities dow and d NEW are proportional to the probabilities of drawing those values of flow and flnew, respectively, from the distribution of part-worths, and play the role of priors. Therefore, r is the ratio of posterior probabilities of flow and flnew. If r is greater than unity, the new estimate has a higher posterior probability than the previous one, and we accept fl NEW. If r is less than unity we accept fl NEW with probability equal tor. Two influences are at work in deciding whether to accept the new estimate ofbeta. First, if a respondent's choices fit well, their estimated fl 1 depends mostly on his own data and is influenced less by the population distribution (relative density). But if their choices fit poorly then their estimated fl 1 depends more on the population distribution and is influenced less by their data. In this way HB makes use of every respondent's data in producing estimates for each individual. This sharing of information is what gives HB the ability to produce reasonable estimates for each respondent even when there may be inadequate information for each individual. The following SAS IML code performs the Metropolis Hasting Algorithm. start beta(nind,subj,set,x,beta,alpha,d, jd,umean,arate); /* Help find the vector delb by finding the factor that is proportional to the covariance matrix */ ucov=jd*d; accept=o; decline=o; invd=inv(d); seed=int(ranuni(o)*loooo); * Break all the information by individual respondents; do i=l to nind; xi=x[loc(subj=i),]; seti=set[loc(subj=i),]; /* To get the new estimate for beta draw a random vector delb from a multivariate normal distribution with mean of zero and covariance matrix proportional to D, 147

3 and let the new betan = beta + delb, where beta is the previous estimate of betan *I call vnormal(delb,umean,ucov,l,seed); delb=delb'; seed=seed+i; betao=beta [, i I ; betan=betao+delb; * Find the exponential of the utilities for the new and old estimates; eutilo=exp(xi*betao); eutiln=exp(xi*betan); I* Find the probability of each choice and for each choice task that the individual made then multiply all the probabilities together *I maxseti=max(seti); po=l; pn=l; do j=l to maxseti; tutilo=eutilo[loc(seti=j),]; tutiln=eutiln[loc(seti=j),); I* Find the sum of all of the exponential utilities for each choice task *I sutilo=sum(tutilo); sutiln=sum(tutiln); I* Find the probability of each choice that the individual made then calculate the product of the probabilities for each individual *I ptempn= tutilnlsutiln; ptempo= tutilolsutilo; do m=l to nrow(ptempn); po=po*ptempo[m,); pn=pn*ptempn[m,); I* Calculate the relative density of the distribution of the betas corresponding to betao and betan given current estimates of parameters alpha and D. Call these values etmpo and etmpn. Finally calculate the ratio (pn*etmpn)/po*etmpo) *I diffo=betao-alpha; diffn=betan-alpha; tmpo=diffo'*invd*diffo; tmpn=diffn'*invd*diffn; etmpo=exp(-o.s*tmpo); etmpn=exp(-o.s*tmpn); I* Select either betao or betan for the new estimate of beta based on the ratio. If this ratio is greater than or equal to unity accept betan as the new estimate for beta for that individual. If the ratio less than unity, then use a random process to decide whether to accept betan or retain betao. Accept betan with probability equal to the ratio. *I ratio=(pn*etmpn)l(po*etmpo); minr=min(ratio,l); rand=uniform(o); I* Determine if you want to save the new estimate of beta or not *I if rand <= minr then do; beta[,i)=betan; accept=accept+l; else do; beta[,i)=betao; decline=decline + 1; arate=accept/(accept+decline); free xi seti delb eutilo eutiln maxseti po pn tutilo tutiln sutilo sutiln ptempn ptempo invd diffo diffn tmpo tmpn etmpo etmpn ratio minr rand accept decline; finish; DISCRETE CHOICE MODEL EXAMPLE A discrete choice model was developed to find the perceptions of 530 customers that conduct company research. These customers have either used a fee-based product within the past six months, or would find such a product of value. In order to qualify for this study, respondents had to be at least somewhat involved in the decision to purchase fee-based research product for their company. The survey was developed to present respondents with different trade-offs of attnbutes, such as: features and functions, content, price plans, and brands. Survey respondents were asked to evaluate descriptions of a set of attributes and indicate which one they would most likely buy. The feature levels and prices for each alternative are systematically varied. Each respondent evaluate sixteen different ''buying situations" or choice sets. Subsequent analysis of the choices made by each individual allows us to estimate the importance of the various levels of each attribute of the product or service offering. A key deliverable from discrete choice modeling is a market simulator that allows managers to continue to test "what if' scenarios by varying the attributes such as price, features and bundling services. In this study, ten different attributes were included in the trade-off design (see appendix A for a description of the attributes). Each attribute has at least two different levels. Combinations of attribute levels that were not feasible were omitted. Two scenarios were simulated for this analysis. The first simulation was with "Basic" Brand A and its three major competitors: Brands B, C, and D. This simulation was designed to represent the product offering available in the current marketplace. The second simulation was with "Deluxe Brand A" and its three major competitors. "Deluxe Brand A" includes: upgrades of screening capabilities and downloading capabilities; increased coverage of geographic area; and raising the subscription prices. The competition's product offerings remained the same for both simulations. 148

4 RESULTS Using the Hierarchical Bayes Estimation in SAS IML to analyze the response data from the 530 customers, we are able to derive the following results: Estimates of the preference shares for the different brands in the current market. This provides information about customers' preferences of the existing products. Estimates of the preference shares for the "Deluxe Brand A" product. It allows us to predict the change of customers' preference towards Brand A if the deluxe version is offered to the market. Run different "What If' scenarios by varying the different attributes and measuring the preference share it has on the different brands. Estimates of individual and overall average (across all individuals) utility values for each level of every attribute. This allows us to predict the potential improvement of preference share for each attribute of the existing product. Estimates of average (across all individuals) importance of each attribute. This provides information about which attributes are more important in the produces composition. The following table displays importance of an attribute. The importance is defined as its weight, or maximum influence it can have on a product choice, given the range of attribute levels defined in the study. Attribute importances are only accurate when individual or segment level utility data are available. Attribute Average Importance Price Plan 12.6% Brand 13.1% Screening Capabilities 4.6% Supplemental Content 9.0"/o Download/Editing Capabilities 9.7% Company Coverage 14.6% Updates oftime-sensitive lnfonnation 4.1% Linking to Full-Text Source Documents 5.0"/o Content Selection 3.1% Subscription Price 9.7% Transaction Price 5.8% Report Price 8.7% 100.0"/o From the importances table attributes price plan, brand, and company coverage are the top 3 attributes. CONCLUSIONS Discrete choice modeling has several advantages that make it appropriate for use in market research. It offers a more realistic shopping simulation-respondents are asked to choose between carefully chosen sets of complete products. They show the trade-offs the respondents make between attributes/features. This is much more realistic than the most common forms of ratings-based or preference based conjoint analysis exercises. Choice-based conjoint analysis allow participants to indicate to indicate that they would not purchase any of the alternatives. It is possible in choice-based conjoint analysis to have "product or alternative specific" attribute levels. For example in this paper we use different price plan, but the attributes describing the price plans are different among the different pricing plans. Hierarchical Bayes also provide individual-level estimates from choice data which have been shown to out-perform aggregate logit for predicting shares for holdout choices, even when there was very little heterogeneity in the data. Having individual-level utilities lends to segmented marketing strategies. REFERENCES Sawtooth Software (2000). The CBC/HB Module, For Hierarchical Bayes Estimation, Version 1.5, Sequim, W A:. Sawtooth Software. Sawtooth Software (1999). The Client Conjoint Simulator, version 1.0, Sequim, W A: Sawtooth Software. Allenby, G.M. and Ginter, J.L. (1995). ''Using Extremes to Design Products and Segment Markets," Journal of Marketing Research, 32, (November) Lenk, P.J., DeSarbo, W.S., Green P.E. and Young, M.R. (1996) "Hierarchical Bayes Conjoint Analysis: Recovery of Part-Worth Heterogeneity fro Reduced Experimental Designs," Marketing Science, 15, Chib, S. and Greenberg, E. (1995) ''Understanding the Metropolis-Hasting Algorithm," American Statistician, 49, (November) SAS Institute Inc. (1999). SASIIML User's Guide, Version 8, Cary NC: SAS Institute Inc. CONTACT INFORMATION Your comments and questions are valued and encouraged. Contact the authors at: David Steenhard Hidden Pond Ct. Louisville, KY david steenhard@lexis-nexis.com Nan-Ting Chou University of Louisville Economics Department University of Louisville Louisville, KY ntchouo SAS is a registered trademark ofsas Institute Inc. in the USA and other countries. indicates USA registration. 149

5 DISCRETE CHOICE DESIGN ATTRIBUTES (APPENDIX A) Price Plan: Flat rate by # of users Transactional per report Flat rate by# of committed reports Brand: Brand A Brand B Brand C Brand D Screening Capabilities: Sophisticated Limited Supplemental Content: Limited Moderate Extensive Extensive plus legal, patent & trademark Download/Editing Capabilities: Print only, no download Download, no edit Download and edit Company Coverage: U.S. public companies only All U.S. (public & private companies) All U.S. plus U.K. and Europe All U.S. plus U.K., Europe, Asia, & Latin America Subscription Price: Subscription Price Plan 1 Subscription Price Plan 2 Subscription Price Plan 3 Subscription Price Plan 4 Transaction Price: Price 1 Price 2 Price 3 Reports Price: Report Price Plan 1 Report Price Plan 2 Report Price Plan 3 Report Price Plan 4 Updates of Time-Sensitive Information: Real time Recent time Linking to Full-Text Source Documents: Cannot link Can link Content Selection: One part at a time Parts/entire report, in one step 150