Welcome to RprobitB! This vignette introduces the R package and defines the underlying model.
With RprobitB1 you can
analyze choices made by deciders among a discrete set of alternatives,
estimate (latent class) (mixed) (multinomial) probit models in a Bayesian framework,
model heterogeneity by approximating any underlying mixing distributions through a mixture of normal distributions,
identify latent classes of decision makers.
Run install.packages("RprobitB") in your R console to install the latest version of RprobitB.
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.
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 \(\tilde{\Sigma}\).
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
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.
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 \(\sum_c s_c=1\).
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
The package name RprobitB is a portmanteau, combining R (the programming language), probit (the model class) and B (for Bayes, the estimation method).↩︎
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.↩︎
RprobitB provides an alternative to fixing an error term variance in order to normalize with respect to scale by fixing an element of \(\alpha\).↩︎
We use the abbreviation \((\beta_n)_n\) as a shortcut to \((\beta_n)_{n =1,...,N}\) the collection of vectors \(\beta_n,n=1,...,N\).↩︎
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 \(J=2\).↩︎