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Jiani (Tiffany) Li Discipline of Business Analytics The University of Sydney |
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Assoc. Prof. Minh-Ngoc Tran Discipline of Business Analytics The University of Sydney |
Understanding attitudes and preferences is an important topic in many fields. One of the most commonly used analytical methods for measuring preferences is the discrete-choice experiment (DCE) [5]. In these experiments, respondents are given hypothetical scenarios involving several products with different combinations of attributes, such as brand name and price. In each trial, they are asked to choose the product they are most likely to buy. The most commonly used model to analyze these choice data and extract the information behind the choices is the multinomial logit (MNL) model [2]. This model is developed based on the expected utility theory, assuming that each option has a perceived utility and that individuals will choose the option with the highest expected utility [4]. Although the MNL model has been widely adopted in discrete response analyses, it provides only limited insight into understanding the cognitive processes of decision making [3]. This issue can be overcome by using the linear ballistic accumulator (LBA) model in cognitive science, which incorporates not only the choices made but also the time required to make those choices [1].
The LBA model assumes that decision making is an evidence accumulation process [1]. Evidence is accumulated separately and simultaneously for each option, and respondents choose the option whose evidence accumulator first reaches its corresponding evidence threshold. In addition to the choice itself, the LBA model allows for inferences about the latent cognitive aspects of the decision of choice, such as the rate at which evidence accumulates for each option and the amount of evidence each option requires to trigger a choice response.
The key elements of the LBA are the starting points (k), the drift rate (d), the response threshold (b), and the response time (t). The starting points (k) refer to the amount of evidence at the beginning of the decision-making process. Different accumulators have different starting points, which are the random values generated from a uniform distribution: U(0, A). The drift rates (d) refer to the speed of evidence accumulation, which are independently sampled from various normal distributions, with different means v1, v2, ..., vn, for different accumulators and a common variance s. The response threshold (b) is the amount of evidence required for each accumulator to trigger a response. The response time (t) consists of two parts. The first is a non-decision time (τ), which occurs before and after the evidence accumulation, such as stimulus encoding and motor-response production. The second is the time taken by the accumulator to reach its threshold. We always assume that the response time of non-decision processes to be constant across trials. The accumulation process is illustrated in Figure 1.
Figure 1: Graph demonstration of a binary-choice version of the LBA [1] |
Since both individual and group-wide preferences are crucial for marketing strategies, two levels of parameters are introduced, that is, individual-level parameters αj = (αj, 1, ..., αj, 5)T and group-level parameters μα, Σα of the LBA. A normal distribution is applied for the individual-level parameters to capture the between-subject variation, that is, αj ∼ (μα, Σα). The individual-level parameters capture the within-subject variation, while the group-level parameters capture the between-subject variation in the underlying cognitive processes of decision-making.
One barrier to applying the LBA to marketing is that it is not practical to obtain response time information, i.e., the time it takes for a customer to make the purchasing choice. However, the response time information is required to compute the likelihood. To resolve this problem, the likelihood function of the original LBA needs to be modified to fit the situation where only choice data is available. We do so by using importance sampling , which produces an unbiased estimate of the intractable likelihood that incorporates only the choice data. The likelihood for each individual p(REj|αj) can be estimated unbiasedly as in Equation below, where RTk are the samples from g(RT) and g(RT) is a proposal density which is easy to sample from.
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The model parameters include the group-level parameters μα, Σα and the individual-level parameters α1, ..., αm, where m is the number of individuals. In order to approximate the full joint posterior p(μα, Σα, α1, ..., αm| D), we use the manifold Gaussian variational Bayes (Manifold GVB), which is a stable algorithm and less sensitive to the initialisation [6]. The pseudo-code implementation is summarised in Algorithm 1.
| Algorithm 1: Manifold GVB with natural gradient [6] 
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The above analysis provides insight into the decision-making processes of past choices. In addition, we are also interested in the choices that individuals are likely to make in the future. In order to predict future choices based on past choices, the posterior predictive distribution is used. The posterior predictive distribution for new choices y* can be computed using the below Equation.
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Note that p(y*|D) could be estimated unbiasedly by sampling from the VB approximation of p(αj|D). Therefore, the pseudo-code implementation to estimate the posterior predictive distribution can be summarised as in Algorithm 2.
| Algorithm 2: Estimation of the posterior predictive distribution 
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We provide marketers with a tool to analyze how various factors affect consumer decision-making and the likelihood of each individual purchasing a specific product. This information can be used to help develop marketing strategies based on the goals marketers aim to achieve. For instance, using predictive probability information, marketers could identify the target groups most likely to purchase their products. By analyzing the characteristics of that specific group, such as gender or age, marketing strategies could be designed accordingly to address a larger group with similar characteristics. Another example is that because preference strength provides information about how sensitive individuals are towards each attribute (e.g., brand name or price), marketers could use this information and incorporate more highly preferred attributes into their marketing strategies.
References:
- Brown, S. and Heathcote, A. (2008). The simplest complete model of choice response time: Linear ballistic accumulation. Cognitive psychology, 57(3):153–178.
- Hausman, J. and McFadden, D. (1984). Specification tests for the multinomial logit model. Econometrica: Journal of the Econometric Society, pages 1219–1240.
- Hawkins, G. E., Marley, A., Heathcote, A., Flynn, T. N., Louviere, J. J., and Brown, S. D. (2014). Integrating cognitive process and descriptive models of attitudes and preferences. Cognitive science, 38(4):701–735.
- McFadden, D. and Train, K. (2000). Mixed mnl models for discrete response. Journal of applied Econometrics, 15(5):447–470.
- Mueller, S., Lockshin, L., and Louviere, J. J. (2010). What you see may not be what you get: Asking consumers what matters may not reflect what they choose. Marketing Letters, 21(4):335–350.
- Tran, M.-N., Nguyen, D. H., and Nguyen, D. (2019). Variational bayes on manifolds. arXiv preprint arXiv:1908.03097.
- Tran, M.-N., Nott, D. J., and Kohn, R. (2017). Variational bayes with intractable likelihood. Journal of Computational and Graphical Statistics, 26(4):873–882.