This chapter utilises the RPANA model to simultaneously infer ANA, and handle preference heterogeneity.^{[1]} Consider a choice task wherein the choice alternatives are described by K attributes, from which the analyst models non-attendance to K* attributes. Using a latent class/probabilistic decision process approach, the unconditional probability of respondent n choosing an alternative (or sequence of alternatives across multiple choice tasks) can be decomposed into the probability of that respondent exhibiting a certain pattern of attendance and non-attendance across attributes, and the probability of choosing the alternative or sequence of alternatives, conditional on belonging to a specific class of ANA behaviour.

The class assignment component of the model can instead be considered as the ANA assignment component, as it controls the inferred non-attendance rates for each of the K* attributes. It will be shown how the ANA assignment can be broken down into further components; however, these can be combined to form the final ANA assignment component. In this, each class m represents a unique combination of the taste coefficients being constrained to zero, to reflect the specific combination of non-attendance across attributes that the class represents. When not constrained to zero, these coefficients are either constrained to be equal across classes (Scarpa et ah, 2009), or unique coefficients are estimated for each class (Hensher & Greene, 2010). The former approach is the most common in the literature and is used herein. The unconditional probabilities can be obtained by multiplying the final ANA assignment probabilities by the choice probabilities that are conditioned on the ANA assignment, and integrating over all ANA assignment classes.

The most common structural form for the ANA assignment component utilises a conventional latent class approach, wherein a single multinomial logit (MNL) model is estimated, in which each alternative represents a specific combination of attendance and non-attendance across the K* attributes. However, as K* increases, the parametric cost of this approach may become prohibitive. An alternative approach was implemented by Hole (2011), in which a binary logit model is estimated to infer non-attendance for each of the K* attributes. The final ANA assignment probability is then the product of the relevant binary logit probabilities, for any given pattern of attendance. This approach has the benefit of parsimony, as non-attendance can be modelled with as few as K* parameters, however it relies on the assumption that non-attendance is independent across attributes. Collins et al. (2013) present a generalised model, in which the number of ANA assignment models can lie between 1 (as with the conventional latent class approach) and K* (as with Hole, 2011). Non-attendance is assumed to be independent between but not within subsets of the attributes. This allows parsimony to be maximised, whilst allowing correlation in non-attendance to be introduced as required. This chapter will rely on Hole's approach, whilst introducing random parameters as in Collins et al. (2013).

We will now formalise the RPANA model, starting with the ANA assignment component. Define A as the number of ANA assignment models. Each ANA assignment model a controls the non-attendance associated with attributes, and is specified with classes. Since we use the approach of Hole (2011), and for all a.^{[2]} Define as the probability of respondent n belonging to class c in ANA assignment model a. The probability is calculated with an MNL model, such that

A parameter serves as a constant term, capturing the assignment to class c that is not explained by socio-demographic and other influences, . To ensure identification, is constrained to zero for one class, typically that which represents full attendance. In the final ANA assignment model, each class m will represent a unique pattern of ANA over all K* attributes for which ANA is modelled. This pattern of ANA will be represented by a unique set of ANA assignment model classes, . The probability of respondent n belonging to class m is

We now consider the choice probabilities conditional on assignment to a class in the final ANA assignment model. These will first be presented with the MNL model, with random coefficients being introduced subsequently. Consider first the total utility of alternative i for respondent n, , which is composed of the representative utility , and the unobserved component of utility, . The representative component is associated with a vector of observed variables, . The utility associated with these variables is estimated with a vector of taste coefficients , such that the representative utility is. For the MNL model, the probability that alternative i will be chosen is

The variables that enter into the representative utility contain the K attributes that describe the choice alternatives. The taste coefficients in the vector represent the sensitivities to the associated variables. For any choice model that is conditioned on a combination of ANA over K* attributes, some elements of may be constrained to zero to represent ANA to one or more attributes.

Next, we partition the full set of taste coefficients into one or more subsets. First, is composed of the taste coefficients for the K – K* attributes for which ANA is not modelled. Then introduce A subsets, each denoted , which are composed of the taste coefficients associated with the attributes for which ANA is controlled by ANA assignment model a. Each a controls assignment to classes, each representing a unique combination of ANA over attributes. Each combination will represent a unique pattern of censoring of . For each a, introduce sets, each denoted . The elements of are either zero, representing ANA, or the taste coefficients drawn from the same position in , representing attendance to the attribute. That is, the taste coefficients that are not censored are constrained to be equal across the sets. The variables to enter into the representative utility, , are similarly partitioned into either set , for attributes for which ANA is not modelled, or one of A subsets , for attributes for which ANA is modelled. Conditional on assignment to classes in each of the A ANA assignment models, the representative utility of alternative j for respondent n now becomes

This serves to censor the taste coefficients associated with the attributes that are ignored in the class of the final ANA assignment model upon which the representative utility is conditioned.

For panel data, we can specify the choice probabilities with respect to a sequence of choices of alternatives over T time periods, . Assuming that the unobserved component of utility is independently and identically extreme value type 1 distributed over alternatives, respondents and time, the probability of a sequence of choices of alternatives, conditional on assignment to classes , is

The unconditional probability of a sequence of choices for respondent n is obtained by taking the product of two probabilities: the probability of a combination

of ANA, and the probability of the sequence of choices, conditional on assignment to that combination of ANA; then integrating over all analyst specified combinations of ANA. This can be expressed as

To capture preference heterogeneity amongst decision-makers that attend to the attributes, we can introduce random parameters, such that the taste coefficients β vary over decision-makers with density A distribution is specified for each taste coefficient, and the moments of these distributions are estimated with structural parameters. This combination of random parameters and ANA results in the RPANA model, in which the unconditional probability of a sequence of choices for respondent n is

Under full attendance for all attributes, the RPANA model becomes the regular RPL model, where this property facilitates comparisons of the two using likelihood ratio tests.

[1] The reader is referred to Collins et al. (2013) for a more detailed exposition of the model.

[2] The model notation could be simplified somewhat given the exclusive use of binary logit models in the ANA assignment component; however the benefit of the current approach is that it remains consistent with Collins et al. (2013).

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