Outline Introduction Analysis of the mixed logit choice model with Improving the efficiency of individualized designs for the covariates mixed logit choice model by including covariates Including covariates in experimental design for the mixed logit choice model Marjolein CRABBE Simulation Martina VANDEBROEK study Results Conclusions DEMA 2011 August 30 - September 2
Outline Outline Introduction Analysis of the mixed logit choice model with covariates • Introduction Including covariates in • Analysis of the mixed logit choice model with covariates experimental design for the • Including covariates in experimental design for the mixed mixed logit choice model logit choice model Simulation study • Simulation study Results • Conclusions Conclusions
Introduction Outline Introduction Analysis of the mixed logit • In choice-based conjoint studies, not only the attributes of choice model with the product profiles in the choice sets, but also covariates covariates may influence respondents’ choice behavior Including covariates in ◦ Demographics (age, gender, ...) experimental design for the ◦ Socio-economic data (income level, employment, ...) mixed logit ◦ Other individual-specific characteristics (brand or store choice model loyalty, ...) Simulation study • Taking choice related respondent characteristics into Results account in the setup and analysis of the discrete choice Conclusions experiment to increase the accuracy of the parameter estimates
Introduction Outline Introduction • The mixed logit or random-effects discrete choice model Analysis of the mixed logit to analyze choice data choice model with • Previous work: the inclusion of covariates in the covariates random-effects distribution to estimate the mixed logit Including covariates in choice model experimental design for the mixed logit • This research incorporates covariates in the construction of choice model efficient individualized designs for the mixed logit choice Simulation study model Results Conclusions ⇒ Can we improve the accuracy of the estimates for the individual-specific partworths in the mixed logit choice model by taking covariates into account in both design and estimation?
Analysis of the mixed logit choice Outline model with covariates Introduction Analysis of the mixed logit choice model with covariates • Hierarchical model with two levels Including covariates in ◦ Lower respondent level experimental design for the → Models individual choice behavior by the conditional logit mixed logit choice model (CL) model Simulation ◦ Upper population level study → Models preference heterogeneity in the population by Results assuming a random-effects distribution over the individual Conclusions partworths
Analysis of the mixed logit choice Outline model with covariates Introduction Lower respondent level Analysis of the mixed logit • Each person is assigned an individual-specific parameter choice model with vector β n , constant over all choice sets covariates Including • Conditional on β n , the probability that individual n covariates in experimental chooses alternative k in choice set s (CL model) design for the mixed logit choice model exp( x ′ ksn β n ) p ksn ( β n ) = Simulation � K study i =1 exp( x ′ isn β n ) Results • The likelihood of respondent n ’s series of choices y S Conclusions n for the S choice sets in the experimental design S K � � L ( β n | y S n , X S ( p ksn ( β n )) y ksn n ) = s =1 k =1
Analysis of the mixed logit choice Outline model with covariates Introduction Upper population level Analysis of the mixed logit • We assume the individual partworths depend on covariates choice model with covariates β n = Θz n + ξ n Including covariates in experimental design for the mixed logit ◦ q × 1 vector z n with covariates for respondent n choice model ◦ p × q matrix Θ with regression parameters Simulation study ◦ N ( ξ n | 0 , Σ ) a p -variate normal distribution Results • The individual-specific partworths follow a multivariate Conclusions normal distribution N ( β n | Θz n , Σ ) • The unconditional likelihood of respondent n ’s y S n � L ( Θ , Σ | y S n , X S L ( β n | y S n , X S n ) φ ( β n | Θz n , Σ ) d β n n , z n ) =
Analysis of the mixed logit choice Outline model with covariates Introduction Analysis of the mixed logit choice model with covariates Including covariates in Mixed logit choice model Mixed logit choice model experimental design for the without covariates with covariates mixed logit choice model Simulation U ksn = x ′ U ksn = x ′ ksn β n + ε ksn ksn β n + ε ksn study Results β n ∼ N ( µ , Σ ) β n ∼ N ( Θz n , Σ ) Conclusions
Including covariates in Outline experimental design for the mixed Introduction logit choice model Analysis of the mixed logit choice model with covariates • Individually adapted sequential Bayesian conjoint choice Including designs (Yu et al. 2011) covariates in experimental • Superior to aggregate designs due to preference design for the mixed logit heterogeneity in the population choice model Simulation • Based on two-level structure of the mixed logit choice study model Results → Individual choice behavior modeled by the conditional logit Conclusions model • Two stages ◦ Initial static stage ◦ Adaptive sequential stage
Including covariates in Outline experimental design for the mixed Introduction logit choice model Analysis of the mixed logit choice model with covariates Initial static stage Including covariates in experimental • Construction of an individual initial Bayesian D -efficient design for the design X S 1 mixed logit n with S 1 choice sets for each respondent choice model • Minimizing the expectation of the D -error over a prior Simulation study distribution of the model parameters Results • Multivariate normal prior N ( β n | Θ 0 z n , Σ 0 ) Conclusions ◦ Covariate values for individual n in vector z n ◦ Prior values for hyperparameters Θ 0 and Σ 0 obtained from a pilot study, previous experiments or expert knowledge
Including covariates in Outline experimental design for the mixed Introduction logit choice model Analysis of the mixed logit choice model with covariates Adaptive sequential stage Including covariates in experimental • Initial experiment X S 1 n for individual n and corresponding design for the choices y S 1 mixed logit n choice model • Bayesian update of prior information Simulation study q ( β n | y S 1 n , X S 1 n , z n , Θ 0 , Σ 0 ) Results Conclusions L ( β n | y S 1 n , X S 1 n ) φ ( β n | Θ 0 z n , Σ 0 ) = L ( β n | y S 1 n , X S 1 � n ) φ ( β n | Θ 0 z n , Σ 0 ) d β n
Including covariates in Outline experimental design for the mixed Introduction logit choice model Analysis of the mixed logit choice model with covariates Including Adaptive sequential stage covariates in experimental • Consider all possible candidate sets for x S 1 +1 design for the n mixed logit • The additional choice set is obtained by minimizing the choice model expected D -error of the combined design ( X S 1 n , x S 1 +1 Simulation ) over n study the updated prior distribution q ( β n | y S 1 n , X S 1 n , z n , Θ 0 , Σ 0 ) Results • Recurring process of updating an individual’s prior Conclusions information by means of its observed choices and sequentially adding efficient choice sets
Simulation study Outline Aim Introduction • Comparing the performance of different design and Analysis of the mixed logit estimation strategies in obtaining accurate individual-level choice model with parameter estimates and predictions and verifying whether covariates the incorporation of covariates in design and/or estimation Including covariates in is valuable experimental design for the • Four different design and estimation combinations mixed logit choice model Simulation study Design Estimation Results 1 C-C IASB with covariates covariates Conclusions 2 NC-C IASB without covariates covariates 3 NC-NC(I) IASB without covariates no covariates 4 NC-NC(O) single nearly orthogonal no covariates • Designs of type 3 3 / 3 / 16 • For the IASB designs, five ( S 1 ) choice sets in initial designs
Simulation study Outline Setup Introduction • Pilot study Analysis of the 1. The 250 respondents from the main study are also used in mixed logit the pilot study choice model with 2. 100 additional respondents are used in the pilot study covariates (different from the 250 in the main study) Including covariates in • True choice behavior experimental A. Influenced by the covariate(s) design for the mixed logit B. Not influenced by the covariate(s) choice model Simulation study Pilot study Choice behavior Results I 1. 250 main resp A. influenced by covariate(s) Conclusions II B. not influenced by covariate(s) III 2. 100 additional resp A. influenced by covariate(s) IV B. not influenced by covariate(s) • Discussion of the results for one binary covariate • Similar results for the two-covariate case
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