---
title: "Model definition"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Model definition}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
bibliography: ../inst/REFERENCES.bib
link-citations: yes
---

```{r, label = setup, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  fig.path = "img/",
  fig.align = "center",
  fig.dim = c(8, 6),
  out.width = "75%"
)
library("RprobitB")
options("RprobitB_progress" = FALSE)
```

## The probit model

The probit model^[The name *probit* is a portmanteau of *prob*ability and un*it*. [@Bliss1934]] is a regression-type model where the dependent variable only takes a finite number of values and the error term is normally distributed [@Agresti2015]. Its purpose is to estimate the probability that the dependent variable takes a certain, discrete value. The most common application are discrete choice scenarios. The dependent variable here is one of finitely many and mutually exclusive alternatives, and explanatory variables typically are characteristics of the deciders or the alternatives.

To be concrete, assume that we possess data of $N$ decision makers which choose between $J \geq 2$ alternatives^[To be precise, the model name gets the prefix *multinomial* in the case $J > 2$.] at each of $T$ choice occasions^[For notational simplicity, the number of choice occasions $T$ is assumed to be the same for each decision maker here. However, we are not restricted to this case: `{RprobitB}` allows for unbalanced panels, i.e. varying $T$. Of course, the cross-sectional case $T = 1$ is possible.]. Specific to each decision maker, alternative and choice occasion, we furthermore observe $P$ choice attributes that we use to explain the choices. The continuous choice attributes cannot be linked directly to the discrete choices but must take a detour over a latent variable. In the discrete choice setting, this variable can be interpreted as the decider's utility of a certain alternative. Decider $n$'s utility $U_{ntj}$ for alternative $j$ at choice occasion $t$ is modeled as

\begin{equation}
  U_{ntj} = X_{ntj}'\beta + \epsilon_{ntj}
\end{equation}

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

- $X_{ntj}$ is a (column) vector of $P$ characteristics of $j$ as faced by $n$ at $t$,

- $\beta \in {\mathbb R}^{P}$ is a vector of coefficients,

- and $(\epsilon_{nt:}) = (\epsilon_{nt1},\dots,\epsilon_{ntJ})' \sim \text{MVN}_{J} (0,\Sigma)$ is the model's 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 $\Sigma$.

Now 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^[This in fact is a critical assumption because many studies show that humans do not decide in this rational sense in general, see for example @Hewig2011], the decisions are linked to the utilities via

\begin{equation}
y_{nt} = {\arg \max}_{j = 1,\dots,J} U_{ntj}.
\end{equation}

In the ordered probit case, the concept of decider's having separate utilities for each alternative is no longer natural [@Train2009]. Instead, we model only a single utility value
\begin{align*}
  U_{nt} = X_{nt}'\beta_n + \epsilon_{nt}
\end{align*}
per decider $n$ and choice occasion $t$, which we interpret as the "level of association" that $n$ has with the choice question. The utility value falls into discrete categories, which in turn are linked to the ordered alternatives $j=1,\dots,J$. Formally,
\begin{align*}
   y_{nt} = \sum_{j = 1,\dots,J} j \cdot I(\gamma_{j-1} < U_{nt} \leq \gamma_{j}),
\end{align*}
with end points $\gamma_0 = -\infty$ and $\gamma_J = +\infty$, and thresholds $(\gamma_j)_{j=1,\dots,J-1}$. To ensure monotonicity of the thresholds, we rather estimate logarithmic threshold increments $d_j$ with $\gamma_j = \sum_{i=1,\dots,j} \exp{d_i}$, $j=1,\dots,J-1$.

## Choice behavior heterogeneity

Note that the coefficient vector $\beta$ is constant across decision makers. This assumption is too restrictive for many applications.^[For example, consider the case of modeling the choice of a means of transportation to work: It is easily imaginable that business people and pensioners do not share the same sensitivities towards cost and time.] Heterogeneity in choice behavior can be modeled by imposing a distribution on $\beta$ such that each decider can have their own preferences.

Formally, we define $\beta = (\alpha, \beta_n)$, where $\alpha$ are $P_f$ coefficients that are constant across deciders and $\beta_n$ are $P_r$ decider-specific coefficients. Consequently, $P = P_f + P_r$. Now if $P_r>0$, $\beta_n$ is distributed according to some $P_r$-variate distribution, the so-called mixing distribution.

Choosing an appropriate mixing distribution is a notoriously difficult task of the model specification. In many applications, different types of standard parametric distributions (including the normal, log-normal, uniform and tent distribution) are tried in conjunction with a likelihood value-based model selection, cf., @Train2009, Chapter 6. Instead, `{RprobitB}` implements the approach of [@Oelschlaeger:2020] to approximate any underlying mixing distribution by a mixture of (multivariate) Gaussian densities. More precisely, the underlying mixing distribution $g_{P_r}$ for the random coefficients $(\beta_n)_{n}$ is approximated 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}

Here, $(s_c)_{c}$ 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 \land \beta_n \mid z,b,\Omega \sim \phi_{P_r}(\cdot \mid b_{z_n},\Omega_{z_n}).
\end{equation}

We call the resulting model the *latent class mixed multinomial probit model*. 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 *binary probit model* if additionally $J=2$.

## Model normalization

As is well known, any utility model needs to be normalized with respect to level and scale in order to be identified [@Train2009]. Therefore, we consider the transformed model

\begin{equation}
\tilde{U}_{ntj} = \tilde{X}_{ntj}' \beta + \tilde{\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) $\tilde{U}_{ntj} = U_{ntj} - U_{ntJ}$, $\tilde{X}_{ntj} = X_{ntj} - X_{ntJ}$, and $\tilde{\epsilon}_{ntj} = \epsilon_{ntj} - \epsilon_{ntJ}$, where $(\tilde{\epsilon}_{nt:}) = (\tilde{\epsilon}_{nt1},...,\tilde{\epsilon}_{nt(J-1)})'  \sim \text{MVN}_{J-1} (0,\tilde{\Sigma})$ and $\tilde{\Sigma}$ denotes a covariance matrix with the top-left element restricted to one.^[`{RprobitB}` provides an alternative to fixing an error term variance in order to normalize with respect to scale by fixing an element of $\alpha$.]

## Parameter labels

In `{RprobitB}`, the probit model parameters are saved as an `RprobitB_parameter` object. Their labels are consistent with their definition in this vignette. For example:

```{r, RprobitB-parameter-example}
RprobitB:::RprobitB_parameter(
  P_f = 1,
  P_r = 2,
  J = 3,
  N = 10,
  C = 2, # the number of latent classes
  alpha = c(1), # the fixed coefficient vector of length 'P_f'
  s = c(0.6, 0.4), # the vector of class weights of length 'C'
  b = matrix(c(-1, 1, 1, 2), nrow = 2, ncol = 2),
  # the matrix of class means as columns of dimension 'P_r' x 'C'
  Omega = matrix(c(diag(2), 0.1 * diag(2)), nrow = 4, ncol = 2),
  # the matrix of class covariance matrices as columns of dimension 'P_r^2' x 'C'
  Sigma = diag(2), # the differenced error term covariance matrix of dimension '(J-1)' x '(J-1)'
  # the undifferenced error term covariance matrix is labeled 'Sigma_full'
  z = rep(1:2, 5) # the vector of the allocation variables of length 'N'
)
```

Mind that the matrix `Sigma_full` is not unique and can be any matrix that results into `Sigma` after the differencing, see the non-exported function `RprobitB:::undiff_Sigma()`.

## References
