International Choice Modelling Conference, International Choice Modelling Conference 2015

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Flexible mixture-amount and mixture-process variable models for choice data
Aiste Ruseckaite, Dennis Fok, Peter Goos

Last modified: 11 May 2015


Many products and services that are considered in every day choices can be described as mixtures of ingredients. Examples are the mixture of different fruits composing a fruit salad (e.g. 50% of apples, 30% of wild berries and 20% of grapes) and the mixture of different transportation modes used by an individual on a particular trip (e.g. 70% of travel time by metro and 30% by bike). In some scenarios, the total amount of the mixture may also vary across alternatives. In such cases, the choice between different mixtures depends not only on the proportions but also on the total amount. This type of data is called mixture-amount data. For instance, advertisers have to decide on the advertising media mix in marketing (e.g. 30% of the expenditures on TV advertising, 10% on radio, and 60% on internet) as well as on the total budget of the entire campaign. The mix of transportation modes chosen by a traveler may also depend on the total travel time. In other scenarios, the choice of a mixture does not depend on the total amount of the mixture, but on some other quantitative variable. For instance, for a consumer, the choice of a fruit salad might depend not only on the fruits composing the salad, but also on its price. A consumer’s preference for bread may depend on its formulation as well as on the baking time and/or the baking temperature. In the literature on mixture experiments, these additional quantitative variables are referred to as process variables.

If the amount/process variable of a mixture affects the response, the utility parameters of the mixture ingredients in a model tend to vary with this variable. To capture this, the strategy has, until now, been to express the model's parameters as a parametric function of the amount/process variable. The effect of this variable on the response is then captured through its effect on the mixture parameters. By writing the parameters of traditional mixture models as functions of the total amount/process variable, new models that explain the effects of the amount/process variable on the blending properties of the components are obtained. In the literature, such models for mixture-amount data have only been proposed for a continuous dependent variable. However, as discussed above, in many choice settings, mixture-amount data are also very relevant. Our first goal in this paper is to discuss how to specify a choice model for this type of data.

Next to introducing mixture-amount models in the choice context, we also present an important extension to this model class. The models described above require a specification of the functional form of the function relating mixture parameters to the amount/process variable. Correctly specifying such a function may not be straightforward. Furthermore, for some functional forms, the number of parameters may explode.

The approach we present as an alternative is very flexible. It is based on so-called Gaussian processes and avoids the necessity to specify the shape of the functional form describing the relationship between the amount/process variable and the mixture parameters. In this specification, we only use a smoothness assumption for the relationship between the mixture parameters and the amount/process variable. The strength of this smoothness assumption is described by a parameter that can be estimated.  Another way to interpret our model is that we treat the parameters of the model as functions of the amount/process variable and consider a distribution directly over these functions.

Technically, the model specifies a separate parameter vector for every unique level of the amount/process variable. One such parameter vector specifies the impact of the mixture components on the choice behavior at a specific value of the amount/process variable. These parameter vectors are, however, not independent. The model incorporates the idea that for amount levels that are close to each other, the model parameters are expected to be rather similar. The Gaussian process formalizes this by specifying the correlation structure between all the parameter vectors, where the correlation depends on the difference between the amount/process variables. If two amount levels are very different, the correlation between the associated parameter vectors is very small. On the other hand, if the amount levels are almost the same, the correlation approaches one. The dependence between the parameters and the amount variables is specified by a function that is locally smooth with high probability. The correlation structure itself is governed by the so-called Gaussian kernel, which is parameterized by a single parameter. This parameter specifies the dependence of the correlations and the amount differences. This parameter, therefore, controls for the smoothness of the mixture parameters as a function of the amount/process variable.  Hence, only one parameter determines the shape of the relationship between the parameter values and the amount/process variable in contrast to earlier approaches where for some models the number of parameters may get very large. If one sets this parameter to 0 or to a large positive value, one can obtain existing models as special cases, namely, when the parameter equals 0 one obtains different and independent mixture parameters for each unique amount level. When the parameter approaches infinity, one obtains a single vector for mixture parameters so that the amount/process variable does not play a role.

Finally, apart from the correlation due to similar amount levels, the parameters of the model at a given amount might also exhibit some correlation structure. For instance, if one prefers apples in a fruit salad, s/he might like pears in the salad as well. We allow for this type of correlation, too. As a result, the overall covariance structure between the model’s parameters is defined via two terms. We demonstrate how this approach naturally leads to the model specification where the parameters of a model follow a matrix normal distribution. We estimate the model parameters by employing Bayesian techniques. Although we present the model in the choice context, it can be easily adapted for modeling a continuous response variable.

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