# Introduction to RprobitB and model formulation

#### Lennart Oelschläger

#### 2021-11-12

Welcome to RprobitB! This vignette introduces the R package and defines the underlying model.

## Introdution

With RprobitB 1 you can

1. analyze choices made by deciders among a discrete set of alternatives,

2. estimate (latent class) (mixed) (multinomial) probit models in a Bayesian framework,

3. model heterogeneity by approximating any underlying mixing distributions through a mixture of normal distributions,

4. identify latent classes of decision makers.

Run `install.packages("RprobitB")` in your R console to install the latest version of RprobitB.

## Levels of model complexity

Why the notation (latent class) (mixed) (multinomial) probit model? Because RprobitB can fit probit models of increasing complexity:

- Most basic, modelling the choice between two alternatives (the probit model).

- Considering more than two alternatives leads to the multinomial probit model.

- If we incorporate random effects, the model gets the prefix mixed.

- The most general model is the latent class mixed multinomial probit model, which approximates the mixing distribution through a mixiture of normal distributions.

## Outline of the vignettes

1. How is the model defined?

2. How to prepare or simulate choice data?

3. How to fit a model?

## The model

Assume that we observe the choices of \(N\) decision makers which decide between \(J\) alternatives at each of \(T\) choice occasions.2 Specific to each decision maker, alternative and choice occasion, we furthermore observe \(P_f+P_r\) choice attributes that we use to explain the choices. The first \(P_f\) attributes are connected to fixed coefficients, the other \(P_r\) attributes to random coefficients following a joint distribution mixed across decision makers.

Person \(n\)’s utility \(\tilde{U}_{ntj}\) for alternative \(j\) at choice occasion \(t\) is modeled as \[\begin{equation} \tilde{U}_{ntj} = \tilde{W}_{ntj}'\alpha + \tilde{X}_{ntj}'\beta_n + \tilde{\epsilon}_{ntj} \end{equation}\]

for \(n=1,\dots,N\), \(t=1,\dots,T\) and \(j=1,\dots,J\), where

- \(\tilde{W}_{ntj}\) is a vector of \(P_f\) characteristics of \(j\) as faced by \(n\) at \(t\) corresponding to the fixed coefficient vector \(\alpha \in {\mathbb R}^{P_f}\),

- \(\tilde{X}_{ntj}\) is a vector of \(P_r\) characteristics of \(j\) as faced by \(n\) at \(t\) corresponding to the random, decision maker-specific coefficient vector \(\beta_n \in {\mathbb R}^{P_r}\), where \(\beta_n\) is distributed according to some \(P_r\)-variate distribution \(g_{P_r}\),

- and \((\tilde{\epsilon}_{nt:}) = (\tilde{\epsilon}_{nt1},\dots,\tilde{\epsilon}_{ntJ})' \sim \text{MVN}_{J} (0,\tilde{\Sigma})\) is the models’ error term vector for \(n\) at \(t\), which in the probit model is assumed to be multivariate normally distributed with zero mean and covariance matrix \(\tilde{\Sigma}\).

### Normalization

As is well known, any utility model needs to be normalized with respect to level and scale in order to be identified. Therefore, we consider the transformed model

\[\begin{equation} U_{ntj} = W_{ntj}'\alpha + X_{ntj}'\beta_n + \epsilon_{ntj}, \end{equation}\]

\(n=1,\dots,N\), \(t=1,\dots,T\) and \(j=1,\dots,J-1\), where (choosing \(J\) as the reference alternative) \(U_{ntj}=\tilde{U}_{ntj} - \tilde{U}_{ntJ}\), \(W_{ntj}=\tilde{W}_{ntj}-\tilde{W}_{ntJ}\), \(X_{ntj}=\tilde{X}_{ntj}-\tilde{X}_{ntJ}\) and \(\epsilon_{ntj}=\tilde{\epsilon}_{ntj}-\tilde{\epsilon}_{ntJ}\), where \((\epsilon_{nt:}) = (\epsilon_{nt1},...,\epsilon_{nt(J-1)})' \sim \text{MVN}_{J-1} (0,\Sigma)\) and \(\Sigma\) denotes a covariance matrix with the top-left element restricted to one.3

### Linking the latent utility to the choices

Let \(y_{nt}=j\) denote the event that decision maker \(n\) chooses alternative \(j\) at choice occasion \(t\). Assuming utility maximizing behavior of the decision makers, the decisions are linked to the utilities via \[\begin{equation} y_{nt} = \sum_{j=1}^{J-1} j\cdot 1 \left (U_{ntj}=\max_i U_{nti}>0 \right) + J \cdot 1\left (U_{ntj}<0 ~\text{for all}~j\right), \end{equation}\] where \(1(A)\) equals \(1\) if condition \(A\) is true and \(0\) else.

### Approximating the mixing distribution

We approximate the mixing distribution \(g_{P_r}\) for the random coefficients4 \(\beta=(\beta_n)_{n}\) by a mixture of \(P_r\)-variate normal densities \(\phi_{P_r}\) with mean vectors \(b=(b_c)_{c}\) and covariance matrices \(\Omega=(\Omega_c)_{c}\) using \(C\) components, i.e. \[\begin{equation} \beta_n\mid b,\Omega \sim \sum_{c=1}^{C} s_c \phi_{P_r} (\cdot \mid b_c,\Omega_c), \end{equation}\] where \((s_c)_{c}\) are weights satisfying \(0 < s_c\leq 1\) for \(c=1,\dots,C\) and \(\sum_c s_c=1\).

### Interpreting the latent classes

One interpretation of the latent class model is obtained by introducing variables \(z=(z_n)_n\) allocating each decision maker \(n\) to class \(c\) with probability \(s_c\), i.e. \[\begin{equation} \text{Prob}(z_n=c)=s_c \quad \text{and} \quad \beta_n \mid z,b,\Omega \sim \phi_{P_r}(\cdot \mid b_{z_n},\Omega_{z_n}). \end{equation}\]

We call this model the latent class mixed multinomial probit model.5

1. The package name RprobitB is a portmanteau, combining R (the programming language), probit (the model class) and B (for Bayes, the estimation method).↩︎

2. For notational simplicity, the number of choice occasions \(T\) is assumed to be the same for each decision maker here. However, RprobitB allows for a different number of choice occasions for each decision maker.↩︎

3. RprobitB provides an alternative to fixing an error term variance in order to normalize with respect to scale by fixing an element of \(\alpha\).↩︎

4. We use the abbreviation \((\beta_n)_n\) as a shortcut to \((\beta_n)_{n =1,...,N}\) the collection of vectors \(\beta_n,n=1,...,N\).↩︎

5. Note that the model collapses to the (normally) mixed multinomial probit model if \(P_r>0\) and \(C=1\), to the multinomial probit model if \(P_r=0\) and to the basic probit model if additionally \(J=2\).↩︎