Flexible Mixture-Amount Models for Business and Industry Using Gaussian Processes

Many products and services can be described as mixtures of ingredients whose proportions sum to one. Specialized models have been developed for linking the mixture proportions to outcome variables, such as preference, quality and liking. In many scenarios, only the mixture proportions matter for the outcome variable. In such cases, mixture models suffice. In other scenarios, the total amount of the mixture matters as well. In these cases, one needs mixture- amount models. As an example, consider advertisers who have to decide on the advertising media mix (e.g. 30% of the expenditures on TV advertising, 10% on radio and 60% on online advertising) as well as on the total budget of the entire campaign. To model mixture-amount data, the current strategy is to express the response in terms of the mixture proportions and specify mixture parameters as parametric functions of the amount. However, specifying the functional form for these parameters may not be straightforward, and using a flexible functional form usually comes at the cost of a large number of parameters.In this paper, we present a new modeling approach which is flexible but parsimonious in the number of parameters. The model is based on so-called Gaussian processes and avoids the necessity to a-priori specify the shape of the dependence of the mixture parameters on the amount. We show that our model encompasses two commonly used model specifications as extreme cases. Finally, we demonstrate the model's added value when compared to standard models for mixture-amount data. We consider two applications. The first one deals with the reaction of mice to mixtures of hormones applied in different amounts. The second one concerns the recognition of advertising campaigns. The mixture here is the particular media mix (TV and magazine advertising) used for a campaign. As the total amount variable, we consider the total advertising campaign exposure