#  Modeling and Analysis of Discrete Response Data: Applications to Public
Opinion on Marijuana Legalization in the United States

Mohit Batham<EMAIL_ADDRESS>Soudeh Mirghasemi
<EMAIL_ADDRESS>Manini Ojha<EMAIL_ADDRESS>Mohammad Arshad
Rahman<EMAIL_ADDRESS><EMAIL_ADDRESS>Walmart, India.
Department of Economics, Hofstra University, Hempstead, USA  Jindal School of
Government and Public Policy, O.P. Jindal Global University, Sonipat, India.
Department of Economic Sciences, Indian Institute of Technology Kanpur, India.

###### Abstract

This chapter presents an overview of a specific form of limited dependent
variable models, namely discrete choice models, where the dependent (response
or outcome) variable takes values which are discrete, inherently ordered, and
characterized by an underlying continuous latent variable. Within this
setting, the dependent variable may take only two discrete values (such as 0
and 1) giving rise to binary models (e.g., probit and logit models) or more
than two values (say $j=1,2,\ldots,J$, where $J$ is some integer, typically
small) giving rise to ordinal models (e.g., ordinal probit and ordinal logit
models). In these models, the primary goal is to model the probability of
responses/outcomes conditional on the covariates. We connect the outcomes of a
discrete choice model to the random utility framework in economics, discuss
estimation techniques, present the calculation of covariate effects and
measures to assess model fitting. Some recent advances in discrete data
modeling are also discussed. Following the theoretical review, we utilize the
binary and ordinal models to analyze public opinion on marijuana legalization
and the extent of legalization – a socially relevant but controversial topic
in the United States. We obtain several interesting results including that
past use of marijuana, belief about legalization and political partisanship
are important factors that shape the public opinion.

###### keywords:

††journal: arXiv

##  1 Introduction

This chapter will discuss settings in which the dependent variable we seek to
model takes on a range of values that are restricted, broadly defined as
limited dependent variable models. Within the class of limited dependent
variables, a special case arises when the outcome is no longer a continuous
measure but a discrete variable. Such data often arise as individuals making a
choice from a set of potential discrete outcomes, thus earning the name
discrete choice models. The most common case of such models occurs when $y$ is
a binary response and takes on the values zero and one, indicating whether or
not the event has occurred, giving rise to binary models111Binary models are
special cases of both ordinal and multinomial models with more than two
categories.. Consider for example, participation in the labour force, whether
or not an individual will buy a vehicle, whether or not a country is part of
free trade agreement. In other cases, $y$ may take on multiple (more than two)
discrete values, with no natural ordering. Consider for example, choice of
brand of toothpaste or mode of transportation. These are referred to as
multinomial models. We refer the readers to Train (2009) for a detailed
discussion on multinomial models, their estimation and inference. Further,
there could be situations where $y$ takes on multiple (more than two) discrete
values that are inherently ordered or ranked. For example, scores attached to
opinion on surveys (oppose, neutral, support), classification of educational
attainment, or ratings on bonds. These give rise to ordinal models or ordered
choice models. Here, we discuss four discrete choice models – ordinal probit,
ordinal logit, binary probit and binary logit models.

Discrete choice models have their foundations in the theory of choice in
economics, which itself is inherently related with the random utility model
(Luce, 1959; Luce and Suppes, 1965; Marschak, 1960). The random utility
framework involves a utility maximizing rational individual whose objective is
to choose an alternative from a set of mutually exclusive and completely
exhaustive alternatives. The utilities attached with each alternative are
completely known to the decision maker and the agent chooses the same
alternative in replications of the experiment. However, to a researcher the
utilities are unknown, since s/he only observes a vector of characteristics
(such as age, gender, income etc.) of the decision maker, referred to as
representative utility. This forms the systematic component. The unobserved
factors form the stochastic part. The stochastic component is assigned a
distribution, typically continuous, to make probabilistic statements about the
observed choices conditional on the representative utility. The distributional
specification implies that there exists a continuous latent random variable
(or a continuous latent utility) that underlies the discrete outcomes.

When the set of alternatives or outcomes are inherently ordered or ranked,
individual choice of a particular alternative can be associated as the latent
variable crossing a particular threshold or cut-point. This latent variable
threshold-crossing formulation of the ordered choices elegantly connects
individual choice behavior and ordinal data models serve as a useful tool in
the estimation process. While the theoretical support relates to choice and
random utility theory, the econometric techniques are completely general and
applicable when the ordering conditions of the data are met.

To understand the application of discrete choice models, we consider the case
of legalization of marijuana in the United States. The debate around
legalization of marijuana has been an important yet controversial policy
issue. Marijuana has been proved to be effective in treatment of several
diseases and a wealth of new scientific understanding regarding its medicinal
benefits are documented in Berman et al. (2004); Wilsey et al. (2013); Abrams
et al. (2003, 2007); Ellis et al. (2009); Johnson et al. (2010); McAllister et
al. (2011); Guzmán (2003); Duran et al. (2010). 222The reader is directed to
the website www.procon.org for a list of 60 peer-reviewed articles
(http://medicalmarijuana.procon.org/view.resource.php?resourceID=000884) on
the effect of marijuana in treatment of the above mentioned diseases. However,
despite the medicinal benefits, smoking or consumption of marijuana is not
completely benign and may cause harmful effects, especially associated with
respiratory illnesses and cognitive development (Kalant, 2004; Polen et al.,
1993; Meier et al., 2012).333A list of peer reviewed articles on the public
health consequences of marijuana can be obtained from the Office of National
Drug Control Policy. Refer to https://www.whitehouse.gov/ondcp/marijuana. As a
result, several surveys have been conducted to assess public opinion on the
matter. For the purpose of this chapter, we specifically utilize poll data
collected by the Pew Research Center for the periods 2013 and 2014 to
demonstrate the application of binary models and ordinal models. While there
is an increasing trend in favor of legalizing marijuana based on public
opinion, it is noteworthy to study these specific time periods given that the
year 2013 marked the first time in more than four decades that majority of
Americans favored legalizing the use of marijuana in the United States (Dimock
et al., 2013).

##  2 Ordinal Models

In _ordinal_ regression models, the outcomes of a dependent variable are
categorical and follow a natural ordering. Each outcome or category is
assigned a score (value or number) with the characteristic that the scores
have an ordinal meaning but hold no cardinal interpretation. Therefore, the
difference between categories is not directly comparable. For example, the
study presented in Section 6.2 codifies the public response to marijuana
legalization as follows: 1 for ‘oppose legalization’, 2 for ‘legal only for
medicinal use’, and 3 for ‘legal for personal use’. Here, a score of 2 implies
more support for legalization as compared to 1, but we cannot interpret a
score of 2 as twice the support compared to a score of 1. Similarly, the
difference in support between 2 and 1 is not the same as that between 3 and 2.

While the ordinal regression model has a dependent variable that takes
discrete values, the model can be conveniently expressed in terms of a
continuous latent variable $z_{i}$444The continuous latent construct may
represent underlying latent utility, some kind of propensity, or strength of
preference (Greene and Hensher, 2010). as follows:

$z_{i}=x^{\prime}_{i}\beta+\epsilon_{i},\hskip
54.2025pt\forall\;i=1,\cdots,n,$ (1)

where $x_{i}$ is a $k\times 1$ vector of covariates, $\beta$ is a $k\times 1$
vector of unknown parameters and $n$ denotes the number of observations. Like
most applications, the stochastic term $\epsilon_{i}$ is assumed to be
_independently and identically distributed_ (_iid_) as a standard normal
distribution, i.e., $\epsilon_{i}\mathop{\sim}\limits^{iid}N(0,1)$ for
$i=1,\ldots,n$; which gives rise to an _ordinal probit model_ (also known as
_ordered probit model_). The latent variable $z_{i}$ is related to the
observed discrete response $y_{i}$ as follows:

$\gamma_{j-1}<z_{i}\leq\gamma_{j}\;\Rightarrow\;\emph{$y_{i}$ = j},\hskip
54.2025pt\forall\;i=1,\cdots,n;\;j=1,\cdots,J,$ (2)

where $-\infty=\gamma_{0}<\gamma_{1}\cdots<\gamma_{J-1}<\gamma_{J}=\infty$ are
the cut-points (or thresholds) and $y_{i}$ is assumed to have $J$ categories
or outcomes. A visual representation of the outcome probabilities (for the
case of marijuana legalization) and the cut-points are presented in Figure 1.
One may observe from Figure 1 that different combinations of $(\beta,\gamma)$
can produce the same outcome probabilities giving rise to parameter
identification problem. We therefore need to anchor the location and scale of
the distribution to identify the model parameters. The former is achieved by
setting $\gamma_{1}=0$ and the latter by assuming
$\textrm{var}(\epsilon_{i})=1$. Other identification schemes are possible and
the reader is referred to Jeliazkov et al. (2008) and Jeliazkov and Rahman
(2012) for details.

Figure 1: The two cut-points ($\gamma_{1},\gamma_{2}$) divide the area under
the curve into three parts, with each part representing the probability of a
response falling in the three response categories. The three probabilities
P$(y_{i}=1)$, P$(y_{i}=2)$ and P$(y_{i}=3)$ correspond to ‘oppose
legalization’, ‘legal only for medical use’ and ‘legal for personal use’,
respectively. Note that for each individual $i$ the mean $x^{\prime}_{i}\beta$
will be different and so will be the category probabilities.

Given a data vector $y$ = $(y_{1},\cdots,y_{n})^{\prime}$, the likelihood for
the ordinal probit model expressed as a function of unknown parameters
$(\beta,\gamma)$ is the following,

$\begin{split}\ell(\beta,\gamma;y)&=\prod_{i=1}^{n}\prod_{j=1}^{J}\Pr(y_{i}=j|\beta,\gamma)^{I(y_{i}=j)},\\\
&=\prod_{i=1}^{n}\prod_{j=1}^{J}\bigg{[}\Phi(\gamma_{j}-x^{\prime}_{i}\beta)-\Phi(\gamma_{j-1}-x^{\prime}_{i}\beta)\bigg{]}^{I(y_{i}=j)},\end{split}$
(3)

where $\Phi(\cdot)$ denotes the _cumulative distribution function (cdf)_ of a
standard normal distribution and $I(y_{i}=j)$ is an indicator function, which
equals 1 if the condition within parenthesis is true and 0 otherwise. The
parameter estimates for ($\beta,\gamma$) are obtained by maximizing the
logarithm of the likelihood (equation 3) using numerical techniques such as
the Newton-Raphson method or BHHH procedure (Train, 2009). The principle
behind maximizing the likelihood – known as maximum likelihood (ML) estimation
– is to obtain those parameter values that are most probable to have produced
the data under the assumed statistical model. Note that it is convenient to
work with the logarithm of the likelihood (log-likelihood) since logarithm
being a monotonic function, the maximum of the log-likelihood and the
likelihood occur at the same parameter values. Once the parameter estimates
are available, they may be used to calculate the covariate effects, make
predictions or assess model fitting. Interested readers may look into Greene
and Hensher (2010) or Johnson and Albert (2000, Chap. 4) for a detailed review
of ordinal data modeling.

Thus far, we have described the ordinal probit model but the framework can be
transformed into an _ordinal logit model_ (or _ordered logit model_) by simply
assuming that the error follows a logistic distribution (McKelvey and Zavoina,
1975; McCullagh, 1980). Therefore, for the model in equation (1), we now
assume that $\epsilon_{i}\sim L(0,1)$ for $i=1,\ldots,n$, where $L$ denotes a
logistic distribution with mean $0$ and variance $\pi^{2}/3$. Like the normal
distribution, the logistic distribution is symmetric but has heavier tails
relative to a normal distribution. The likelihood for the ordinal logit model
has the same structure as equation (3) with $\Phi(w)$ replaced by
$\Lambda(w)=\exp(w)/[1+\exp(w)]$, where $w$ is the argument inside the
parenthesis. Analogous to the ordinal probit model, the parameters are
estimated using the ML technique.

An interesting property of the ordinal logit model is that the ratio of odds
of not exceeding a certain category (say $j$) for any two individuals is
constant across response categories. This earns it the name _proportional odds
model_. To see this property in effect, let $\theta_{ij}=\Pr(y_{i}\leq j)$
denote the cumulative probability that individual $i$ chooses category $j$ or
below. For the ordinal logit model, this implies:
$\theta_{ij}=\exp{(\gamma_{j}-x^{\prime}_{i}\beta)}/\big{[}1+\exp{(\gamma_{j}-x^{\prime}_{i}\beta)}\big{]}$,
and
$\theta_{ij}/(1-\theta_{ij})=\exp\big{[}\gamma_{j}-x^{\prime}_{i}\beta\big{]}$,
where the latter represents the odds for the event $y_{i}\leq j$. Accordingly,
for any two individuals (say $1$ and $2$), the ratio of odds is,

$\frac{\theta_{1j}/(1-\theta_{1j})}{\theta_{2j}/(1-\theta_{2j})}=\exp\big{[}-(x_{1}-x_{2})^{\prime}\beta\big{]}.$
(4)

The odds ratio presented in equation (4) does not depend on the response
category $j$ and is proportional to $(x_{1}-x_{2})$ with $\beta$ being the
constant of proportionality. Thus, the name _proportional odds model_.

##  3 Binary Models

_Binary_ models are a simplification of ordinal models and are designed to
deal with situations where the outcome (response or dependent) variable is
dichotomous i.e., takes only two values, typically coded as 1 for ‘success’
and 0 for ‘failure’. For example, the application presented in Section 6.1
models the response as a ‘success’ if an opinion is in favor of legalization
and a ‘failure’ otherwise. The general set up of a binary probit model (or
simply probit model) is similar to an ordinal probit model and can be written
in terms of a continuous latent variable $z_{i}$555The continuous latent
variable can be interpreted as the difference between utilities from choice
$1$ and $0$ i.e., $z_{i}=U_{i1}-U_{i0}$, where $U$ denotes utility (Jeliazkov
and Rahman, 2012). as follows,

$\begin{split}z_{i}&=x^{\prime}_{i}\beta+\epsilon_{i},\hskip
54.2025pt\forall\;i=1,\cdots,n,\\\
y_{i}&=\left\\{\begin{array}[]{ll}1&\textrm{if}\;z_{i}>0,\\\
0&\textrm{otherwise}.\end{array}\right.\end{split}$ (5)

where $\epsilon_{i}\sim N(0,1)$ for $i=1,\ldots,n$. With only two responses,
there is a single cut-point which is typically fixed at 0 for the sake of
simplicity. A pictorial representation of the binary outcome probabilities for
marijuana legalization is shown in Figure 2. The figure also shows that the
cut-point $\gamma_{1}$ is fixed at 0 to anchor the location of the
distribution. Besides, the scale is fixed by assuming the variance of the
normal distribution is 1. Both the restrictions, as mentioned earlier, are
necessary to identify the model parameters.

Figure 2: The cut-point $\gamma_{1}$ divides the area under the curve into two
parts, the probability of failure and probability of success. In our study,
P$(y_{i}=0)$ and P$(y_{i}=1)$ correspond to probability of opposing and
supporting marijuana legalization, respectively. Note that for each individual
$i$ the mean $x^{\prime}_{i}\beta$ and hence the probabilities, P$(y_{i}=0)$
and P$(y_{i}=1)$, will be different.

The likelihood for the binary probit model can be expressed as,

$\begin{split}\ell(\beta;y)&=\prod_{i=1}^{n}\big{\\{}\Pr(y_{i}=0|x^{\prime}_{i}\beta)^{(1-y_{i})}\Pr(y_{i}=1|x^{\prime}_{i}\beta)^{y_{i}}\big{\\}},\\\
&=\prod_{i=1}^{n}\big{\\{}\Phi(-x^{\prime}_{i}\beta)^{(1-y_{i})}\Phi(x^{\prime}_{i}\beta)^{y_{i}}\big{\\}}.\end{split}$
(6)

Given the likelihood, the model parameters are estimated using the ML
technique i.e., by maximizing the log-likelihood (equation 6) with respect to
the parameter vector $\beta$. Once the parameter estimates are available, we
may calculate objects of interest, such as the covariate effects and predicted
probabilities. Measures for goodness of fit can also be calculated to assess
model fitting. Readers interested in further details about binary data
modeling may look into Johnson and Albert (2000, Chap. 3).

Similar to ordinal models, the framework for the binary probit model given by
equation (5) can be utilized to describe a binary logit model (or simply logit
model) with the modification that the errors follow a logistic distribution
(Hosmer et al., 2013). Both the location and scale restrictions still apply to
the logit model, but note that the variance is now fixed at $\pi^{2}/3$ as
compared to 1 in a probit model. To obtain the logit likelihood, the normal
_cdf_ $\Phi(x^{\prime}_{i}\beta)$ in equation (6) is replaced by the logistic
_cdf_ :
$\Lambda(x^{\prime}_{i}\beta)=\exp{(x^{\prime}_{i}\beta)}/\big{[}1+\exp{(x^{\prime}_{i}\beta)}\big{]}$.
We can maximize the resulting log-likelihood to obtain the parameter estimates
for the logit model.

The logit model is appealing to researchers in many fields (including
epidemiologists) because of the ease in interpreting its slope coefficient. To
see this, let $\theta_{i}=\Pr(y_{i}=1|x^{\prime}_{i}\beta)$ denote the
probability of success and $x_{.,l}$ be a continuous covariate (or independent
variable). Then the logarithm of the odds (log-odds) of success can be
expressed as,

$\log\bigg{(}\frac{\theta_{i}}{1-\theta_{i}}\bigg{)}=x^{\prime}_{i}\beta=x_{i,l}\,\beta_{l}+x^{\prime}_{i,-l}\,\beta_{-l}.$

where $x_{i}=(x_{i,l},\,x_{i,-l})$, $\beta=(\beta_{l},\,\beta_{-l})$, and $-l$
in the subscript denotes all covariates/parameters except the $l$-th
covariate/parameter. If we differentiate the log-odds with respect to the
$l$-th covariate, we obtain $\beta_{l}$. Therefore, the slope coefficient
$\beta_{l}$ represents the log-odds for a 1 unit change in the $l$-th
covariate.

Similarly, the coefficient of an indicator variable (dummy or dichotomous
variable) has an interesting interpretation. Let $x_{.,m}$ be an indicator
variable, $\theta_{i}^{1}$ be the probability of success when $x_{i,m}=1$,
$\theta_{i}^{0}$ be the probability of success when $x_{i,m}=0$. Our goal is
to find the expression for the odds-ratio, which measures the odds of success
among those with $x_{i,m}=1$ compared to those with $x_{i,m}=0$. Then the
logarithm of the odds-ratio is,

$\log\bigg{(}\frac{\theta_{i}^{1}/(1-\theta_{i}^{1})}{\theta_{i}^{0}/(1-\theta_{i}^{0})}\bigg{)}=\beta_{m}+x^{\prime}_{i,-m}\,\beta_{-m}-x^{\prime}_{i,-m}\,\beta_{-m}=\beta_{m}.$

The odds-ratio is better understood with the help of an example. Suppose $y$
denotes the presence or absence of a heart disease and $x_{.,m}$ denotes
whether the person is a smoker or non-smoker. Then, an odds-ratio $=2$ implies
that heart disease is twice as likely to occur among smokers as compared to
non-smokers for the population under study.

##  4 Covariate Effects and Model Fitting

In ordinal models, the coefficients do not give covariate effects because the
link function is non-linear and non-monotonic. Consequently, we need to
calculate the covariate effect for each outcome. Let $x_{.,l}$ be a continuous
covariate, then covariate effect for the $i$-th observation (or individual) in
an ordinal probit model is calculated as,

$\begin{split}\frac{\partial\Pr(y_{i}=j)}{\partial
x_{i,l}}&=-\beta_{l}\;\big{[}\phi(\gamma_{j}-x^{\prime}_{i}\beta)-\phi(\gamma_{j-1}-x^{\prime}_{i}\beta)\big{]}\\\
&\simeq-\hat{\beta}_{l}\;\big{[}\phi(\hat{\gamma}_{j}-x^{\prime}_{i}\hat{\beta})-\phi(\hat{\gamma}_{j-1}-x^{\prime}_{i}\hat{\beta})\big{]},\end{split}$
(7)

where $\phi(\cdot)$ denotes the probability density function (_pdf_) of a
standard normal distribution and $(\hat{\beta},\hat{\gamma})$ are the ML
estimates of the parameters $(\beta,\gamma)$. The average covariate effect is
computed by averaging the covariate effect in equation (7) across all
observations. If the covariate is an indicator variable (say $x_{.,m})$, then
the covariate effect for the $i$-th observation on outcome $j$ ($=1,\ldots,J$)
is calculated as,

$\begin{split}&\Pr(y_{i}=j|x_{i,-m},x_{i,m}=1)-\Pr(y_{i}=j|x_{i,-m},x_{i,m}=0)\\\
&=\big{[}\Phi(\gamma_{j}-{x^{\prime}}^{\dagger}_{i}\beta)-\Phi(\gamma_{j-1}-{x^{\prime}}^{\dagger}_{i}\beta)\big{]}-\big{[}\Phi(\gamma_{j-1}-{x^{\prime}}^{\ddagger}_{i}\beta)-\Phi(\gamma_{j-1}-{x^{\prime}}^{\ddagger}_{i}\beta)\big{]}\\\
&\simeq\big{[}\Phi(\hat{\gamma}_{j}-{x^{\prime}}^{\dagger}_{i}\hat{\beta})-\Phi(\hat{\gamma}_{j-1}-{x^{\prime}}^{\dagger}_{i}\hat{\beta})\big{]}-\big{[}\Phi(\hat{\gamma}_{j-1}-{x^{\prime}}^{\ddagger}_{i}\hat{\beta})-\Phi(\hat{\gamma}_{j-1}-{x^{\prime}}^{\ddagger}_{i}\hat{\beta})\big{]},\end{split}$
(8)

where ${x}^{\dagger}_{i}=(x_{i,-m},\,x_{i,m}=1)$ and
${x}^{\ddagger}_{i}=(x_{i,-m},\,x_{i,m}=0)$. The average covariate effect is
calculated by averaging the covariate effect given in equation (8) across all
observations. Note that for ordinal models, the sign of the regression
coefficient translates unambiguously into the sign of covariate effect only
for the lowest and highest categories of the response variable. Covariate
effect for the middle categories cannot be known _a priori_.

Moving on to binary probit model, the expressions for the covariate effects
simplify. For a continuous variable, the covariate effect is given by the
expression,

$\frac{\partial\Pr(y_{i}=1)}{\partial
x_{i,l}}=\beta_{l}\,\phi(x^{\prime}_{i}\beta)\simeq\hat{\beta}_{l}\;\phi(x^{\prime}_{i}\hat{\beta}),$
(9)

and the same for an indicator variable is given by the expression,

$\begin{split}&\Pr(y_{i}=1|x_{i,-m},x_{i,m}=1)-\Pr(y_{i}=1|x_{i,-m},x_{i,m}=0)\\\
&=\Phi({x^{\prime}}^{\dagger}_{i}\beta)-\Phi({x^{\prime}}^{\ddagger}_{i}\beta)\simeq\Phi({x^{\prime}}^{\dagger}_{i}\hat{\beta})-\Phi({x^{\prime}}^{\ddagger}_{i}{\beta}),\end{split}$
(10)

where all the notations have been explained in the previous paragraph. Once
again, the average covariate effect is computed by averaging across all
observations. While the discussion on covariate effects has considered ordinal
and binary probit models because of their implementation in the applications,
covariate effects for the ordinal and binary logit models can be calculated
analogously by replacing the normal _pdf_ ’s and _cdf_ ’s with logistic _pdf_
’s and _cdf_ ’s at appropriate places.

To assess the goodness of model fit, we calculate three measures: likelihood
ratio (LR) test statistic, McFadden’s R-square (McFadden, 1974) and hit-rate
(Johnson and Albert, 2000). For the null hypothesis
$H_{0}:\beta_{2}=\ldots=\beta_{k}=0$, the LR test statistic $\lambda_{LR}$ is
defined as follows:

$\lambda_{LR}=-2[\ln L_{0}-\ln
L_{\mathrm{fit}}]\quad\mathop{\sim}\limits^{H_{0}}\quad\chi^{2}_{k-1}$

where $\ln L_{\mathrm{fit}}$ is the log-likelihood of the fitted model and
$\ln L_{0}$ is the log-likelihood of the intercept-only model. Under the null
hypothesis, $\lambda_{LR}$ follows a chi-square distribution with degrees of
freedom equal to $k-1$, i.e., the number of restrictions under the null
hypothesis. So, we calculate the statistic $\lambda_{LR}$ and compare it with
$\chi^{2}_{k-1}$ for a given level of significance. If
$\lambda_{LR}>\chi^{2}_{k-1}$, then we reject the null hypothesis. Otherwise,
we do not reject the null hypothesis.

Another popular goodness of fit measure for discrete choice models is
McFadden’s R-square ($R^{2}_{M}$), due to McFadden (1974). The McFadden’s
R-square, also referred to as pseudo R-square or likelihood ratio index, is
defined as follows,

$R^{2}_{M}=1-\frac{\ln L_{\mathrm{fit}}}{\ln L_{0}}$

The $R^{2}_{M}$ is intuitively appealing because it is bounded between 0 and
1, similar to the coefficient of determination $(R^{2})$ in linear regression
models. When all slope coefficients are zero, the $R^{2}_{M}$ equals zero; but
in discrete choice models $R^{2}_{M}$ can never equal 1, although it can come
close to 1. While higher values of $R^{2}_{M}$ implies better fit, the value
as such has no natural interpretation in sharp contrast to $R^{2}$ which
denotes the proportion of variation in the dependent variable explained by the
covariates.

While both LR test statistic and McFadden’s R-square are commonly used in
applied studies, the hit-rate is relatively uncommon. The hit-rate (HR) is
defined as the percentage of correct predictions i.e., percentage of
observations for which the model correctly assigns the highest probability to
the observed response category. Mathematically, the HR can be defined as
follows,

$HR=\frac{1}{n}\sum_{i=1}^{n}I\bigg{(}\Big{(}\max_{j}\;\\{\hat{p}_{ij}\\}_{j=1}^{J}\Big{)}=y_{i}\bigg{)},$

where $\hat{p}_{ij}$ is the predicted probability that individual $i$ selects
outcome $j$, and $I(\cdot)$ is the indicator function as defined earlier.

##  5 Some Advances in Discrete Choice Modeling

Till now, we have looked at ordinal and binary models and their Classical (or
Frequentist) approach to estimation via the maximum likelihood technique,
which only involves the likelihood function (say $f(y|\theta)$, where $\theta$
is the parameter vector). The Classical approach assumes that model parameters
are unknown but have fixed values and hence the parameters cannot be treated
as random variables. In contrast, Bayesian approach to estimation utilizes the
Bayes’ theorem,

$\pi(\theta|y)=\frac{f(y|\theta)\pi(\theta)}{\int
f(y|\theta)\pi(\theta)\;d\theta}\;,$

to update the belief/information about $\theta$ (considered a random variable)
by combining information from the observed sample (via the likelihood
function) and non-sample or prior beliefs (arising from previous studies,
theoretical consideration, researcher’s belief, etc.) represented by the prior
distribution $\pi(\theta)$. Inference is based on the posterior distribution
$\pi(\theta|y)$. Bayesian approach provides several advantages including
finite sample inference, working with likelihoods which are difficult to
evaluate, and advantages in computation. Interested readers may look into
Greenberg (2012) for details on the Bayesian approach and Poirier (1995) for a
comparison of Classical and Bayesian estimation methods.

The posterior density for ordinal and binary models do not have a tractable
density and so the parameters cannot be sampled directly. While the
Metropolis-Hastings (MH) algorithm (Metropolis et al., 1953; Hastings, 1970)
can be employed to sample the parameters, the standard and more convenient
approach is to consider _data augmentation_ (Tanner and Wong, 1987). In this
approach, the joint posterior density is augmented by a latent variable $z$
and the augmented joint posterior $\pi(\theta,z|y)$ is written as,

$\pi(\theta,z|y)\propto\pi(\theta)f(z,y|\theta)=\pi(\theta)f(z|\theta)f(y|z,\theta).$

For an ordinal probit model, $\theta=(\beta,\gamma)$ and
$f(y|z,\beta,\gamma)\equiv f(y|z,\gamma)$; whereas for a binary probit model
$\theta=\beta$ and $f(y|z,\beta)\equiv f(y|z)$. The two equivalencies arise
because, given a latent observation $z_{i}$, $y_{i}$ is known with certainty
regardless of $\beta$ for $i=1,\ldots,n$. This de-linking of the likelihood
function from $\beta$, made possible through data augmentation, simplifies the
estimation procedure and allows sampling of $\beta$ through a Gibbs process
(Geman and Geman, 1984) – a well known Markov chain Monte Carlo (MCMC)
technique. The latent variable $z$ is sampled element-wise from a truncated
normal distribution. For ordinal models, a monotone transformation of the cut-
points, $\gamma$, is sampled using an MH algorithm. The MCMC algorithms for
estimating ordinal and binary probit models outlined here were introduced in
Albert and Chib (1993). Other notable references that describe the Bayesian
modeling and estimation of ordinal and binary responses in great detail
include Johnson and Albert (2000), Greenberg (2012), and Jeliazkov and Rahman
(2012). Bayesian estimation of logit model is based on the same principle and
presented in Holmes and Held (2006) and Jeliazkov and Rahman (2012).

The ordinal and binary models considered in this chapter, whether estimated
using the Classical or the Bayesian techniques, provide information on the
average probability of outcomes conditional on the covariates. However,
interests in the quantiles of the response variable as a robust alternative to
mean regression have grown enormously since the introduction of quantile
regression in Koenker and Bassett (1978). Quantile modeling gained further
momentum with the development of Bayesian quantile regression by Yu and Moyeed
(2001), where the authors create a working likelihood by assuming that the
errors follow an asymmetric Laplace (AL) distribution (Yu and Zhang, 2005).
Binary quantile regression was proposed by Kordas (2006) and its Bayesian
formulation was presented by Benoit and Poel (2010). Rahman (2016) introduced
Bayesian quantile regression with ordinal responses and estimated the model
using MCMC techniques. The corresponding R package _bqror_ for estimating the
quantile ordinal model is described in Maheshwari and Rahman (2023) along with
the computation of marginal likelihood for comparing alternative quantile
models. A flexible form of Bayesian ordinal quantile regression was proposed
in Rahman and Karnawat (2019). Some recent research on ordinal and binary
quantile regression in the panel/longitudinal set up include Rahman and
Vossmeyer (2019), and Bresson et al. (2021). Two applied studies employing
ordinal and binary quantile framework are Omata et al. (2017) and Ojha and
Rahman (2021), respectively. Interested readers may explore the above
mentioned papers and references therein to develop a thorough understanding on
ordinal and binary quantile modeling and their applications.

##  6 Application: Public Opinion on Legalization of Marijuana in the United
States

In the United States (US), marijuana is illegal under the federal law as per
the Controlled Substances Act of 1970. The Act classifies marijuana as a
schedule I drug i.e., a drug with no accepted medical value, high potential
for abuse and not safe to use even under medical supervision (Drug Enforcement
Administration, 2011). However, state laws pertaining to marijuana have
evolved overtime. Up until 2016, 29 states had either legalized, allowed
access for medical reasons or decriminalized its use (See Figure 3). More
legalization efforts are appearing in the remaining states of US. Such
revisions in state laws represent a change in public attitude that is aptly
reflected in survey data collected by independent polling agencies such as Pew
Research Center, General Social Survey and Gallup. Figure 4 shows an
increasing trend in favor of legalizing marijuana based on public opinion.
Besides, political standing on marijuana has also popularized the debate on
its legalization and may have affected public opinion regarding it. The
gradual growth in support of marijuana perhaps echoes a better public insight
on the medicinal value of marijuana (Earleywine, 2005) and the social cost of
prohibition that includes illegal trade, racially skewed arrests of African
Americans and huge enforcement cost (Shepard and Blackley, 2007).

Figure 3: Marijuana state laws in the US as of 6th January, 2016.

Figure 4: Public opinion on marijuana legalization for the period 1969-2015.
The black dashed line is the 50 percent benchmark. Data source: Pew Research
Center, General Social Survey and Gallup. We have averaged the percentages for
years with multiple surveys. The combined percentage of the two opinions is
below 100 since on average 4 percent of respondents answered “don’t know” or
“refused to answer”.

While policies on marijuana use are in the early stages of formulation as
states evaluate its costs and benefits (Winterbourne, 2012), more states
decriminalizing its use may cause a major policy shift at the federal level
(Ferner, 2015). In this regard, some scholars argue that public policies ought
to be guided by public opinion such that mass opinion and democracy is upheld
(Monroe, 1998; Paletz et al., 2015). Besides, Shapiro (2011) cites a large
number of studies to argue that public opinion influences government policy
making in the US. Therefore, it is imperative to study and identify the
factors that significantly impact US public opinion towards marijuana
legalization.

In the next section, we employ a binary probit model to analyze public opinion
on marijuana legalization and thereafter implement the ordinal probit model to
analyze public opinion on the extent of marijuana legalization. The choice of
probit models over their logit counterparts is driven by practical
considerations – the probit models are tractable in univariate cases and can
be generalized to multivariate and hierarchical settings (Jeliazkov et al.,
2008). In contrast, logistic model based on logistic distribution cannot model
correlations in multivariate settings.

###  6.1 Binary Probit Model

####  6.1.1 Data

We utilize the March 2013 Political Survey data from the Pew Research Center
to analyze public opinion on marijuana and identify the factors that
significantly impact the probability of supporting its legalization. The
survey was conducted during the period March 13-17, 2013, by Abt SRBI
(Schulman, Ronca & Buculvas, Inc) for the Pew Research Center for the People
and the Press. The survey selected and interviewed a representative sample of
1,501 adults living in the US. Of the 1501 adults, 750 individuals were
interviewed over land line and the remaining 751 individuals over cell phone.
The available sample had several respondents with missing values (“don’t know”
or “refused to answer”) on the variables of interest, along with 49
respondents who were unsure about marijuana legalization. After removing data
on these respondents, we have a sample of 1182 observations available for the
study.

Table 1: Descriptive summary of the variables (March 2013 Political Survey). variable |  | mean | std
---|---|---|---
log age |  | 3.86 | 0.40
log income |  | 10.64 | 0.98
household size |  | 2.72 | 1.44
| category | counts | percentage
past use |  | 554 | 46.87
male |  | 570 | 48.22
| bachelors & above | 426 | 36.04
education | below bachelors | 360 | 30.46
| high school & below | 396 | 33.50
tolerant states |  | 374 | 31.64
| white | 938 | 79.36
race | african american | 142 | 12.01
| other races | 102 | 8.63
| republican | 353 | 29.86
party affiliation | democrat | 404 | 34.18
| independent & others | 425 | 35.96
| protestant | 494 | 41.79
| roman catholic | 258 | 21.83
religion | christian | 138 | 11.68
| conservative | 72 | 6.09
| liberal | 220 | 18.61
public opinion | favor legalization | 622 | 52.62
| oppose legalization | 560 | 47.38

In this application, the dependent variable is response to the question: “Do
you think the use of marijuana should be made legal, or not?”. The responses
were recorded as ‘yes, legal’ (i.e. favor legalization), ‘no, illegal’ (i.e.,
oppose legalization) or ‘don’t know or refused’. We remove the last category
as it constitutes missing responses. This makes the response variable binary
and hence a binary probit model is utilized to analyze the response based on
the following set of covariates: age, income, household size, past use of
marijuana, gender, education, state of residence, race, party affiliation and
religion.

Age was recorded in years. Income (measured in US Dollars) was reported as
belonging to one of 9 groups (0-10k, 10k-20k, $\cdots$, 40k-50k, 50k-75k,
75k-100k, 100k-150k, 150k and above, where ‘k’ denotes thousand). We convert
income to a continuous variable by taking the mid-point of the first 8 income
groups and impute 150k for the last group. Household size represents the
number of members in the family. Past use of marijuana and gender are
indicator variables in the model. Educational attainment of the respondents is
classified into three categories and the category ‘high school and below’
forms the base or reference category in the regressions. The variable
‘tolerant states’ indicates if a respondent lives in one of the 20 states,
where recreational usage is legal, possession is decriminalized and/or allowed
for medical use only.666The list of tolerant states before the date of the
survey include Alaska, Arizona, California, Colorado, Connecticut, Delaware,
Hawaii, Maine, Maryland, Massachusetts, Michigan, Montana, Nevada, New Jersey,
New Mexico, Oregon, Rhode Island, Vermont, Washington, Washington DC. Source:
https://www.whitehouse.gov/ondcp/state-laws-related-to-marijuana. Race is
classified into three categories and ‘White’ race is used as the base category
in the regressions. The category ‘Other Races’ comprises of Asian, Hispanic,
native American, Pacific Islanders and remaining races. Party affiliation is
also classified into three categories and Republican Party is used as the
reference category in the models. Religion is classified into five categories
and ‘Protestant’ is used as the base category. Here, the category
‘Conservative’ comprises of respondents belonging to one of the following
religions: Buddhist, Hindu, Islam, Jew, Mormon and orthodox church. The
category ‘Liberal’ comprises of respondents who claim to be Agnostic, Atheist,
Universalist or nothing in particular. The descriptive statistics for all the
variables are presented in Table 1.

Let us now look at the socio-demographic characteristics of a typical
respondent in the sample. An average respondent is about 51 years old and s/he
belongs to a household of size 3 with an annual income of 60,527 US Dollars.
The sample is almost evenly split between males and females and 46.87 percent
of respondents have a history of marijuana use. In the sample, the largest
proportion of respondents (36.04 percent) have ‘bachelors and above’ degree,
followed by ‘below bachelors’ degree (30.46 percent). A significant fraction
of the respondents (31.64 percent) reside in states that have some favorable
laws towards marijuana. The sample is predominantly White (79.36 percent) with
a good representation (12.01 percent) of the African American population.
Amongst the respondents, almost 30 percent consider themselves as Republican,
about 34 percent declare themselves as Democrats and the remaining are
Independent or belong to other parties. With respect to religious
codification, the largest proportion (41.79 percent) is Protestants, followed
by Roman Catholics (21.83 percent). A good proportion, 11.68 percent, declare
themselves to be simply Christian. The Liberal category forms about 18.61
percent and the Conservatives have the lowest fraction at 6.09 percent.

####  6.1.2 Estimation

We estimate four different binary probit models and present the estimated
coefficients and standard errors for each model in Table 2. To begin with,
Model 1 considers a basic set of covariates that includes log age, income,
past use of marijuana, gender, education categories, household size and state
of residence of the respondents. Subsequent models generalize Model 1 by
adding more variables to the basic set of regressors. Specifically, Model 2
adds the race variable, Model 3 adds party affiliation to the list of
variables in Model 2, and Model 4 incorporates religious denomination to the
regressors in Model 3. All the four models have high LR statistic as shown in
Table 2 and hence we reject the null hypothesis that the coefficients are
jointly zero in each model. The two other goodness of fit statistics,
McFadden’s $R^{2}$ and hit-rate, also show that all models provide a good fit,
with each subsequent model providing a better fit than the previous model.

####  6.1.3 Results

We focus on the results from Model 4, since it provides the best model fit.
Table 2 shows that log age has a negative coefficient and is statistically
significant at 5 percent level.777The default significance level is 5 percent
and henceforth reference to significance level will be omitted. This implies
that young people are more supportive of marijuana legalization. This is not
surprising since risk taking or deviant behavior is high amongst the younger
population and is well documented in the literature (Brown et al., 1974).
Moreover, Saieva (2008) reports a negative association between age and
probability of supporting legalization, while Alfonso and Dunn (2007) and
Delforterie et al. (2015) note that marijuana prevalence rate is higher among
the younger population.

Table 2: Estimation results for the binary probit model. |  | model 1 |  | model 2 |  | model 3 |  | model 4
---|---|---|---|---|---|---|---|---
|  | coef | se |  | coef | se |  | coef | se |  | coef | se
intercept |  | $1.74^{**}$ | $0.61$ |  | $1.70^{**}$ | $0.64$ |  | $0.11$ | $0.66$ |  | $0.51$ | $0.69$
log age |  | $-0.41^{**}$ | $0.11$ |  | $-0.41^{**}$ | $0.11$ |  | $-0.39^{**}$ | $0.11$ |  | $-0.29^{**}$ | $0.12$
log income |  | $-0.06$ | $0.05$ |  | $-0.06$ | $0.05$ |  | $-0.02$ | $0.05$ |  | $-0.03$ | $0.05$
past use |  | $0.81^{**}$ | $0.08$ |  | $0.82^{**}$ | $0.08$ |  | $0.82^{**}$ | $0.08$ |  | $0.81^{**}$ | $0.08$
male |  | $0.18^{**}$ | $0.08$ |  | $0.17^{**}$ | $0.08$ |  | $0.21^{**}$ | $0.08$ |  | $0.17^{**}$ | $0.08$
bachelors & above |  | $0.23^{**}$ | $0.10$ |  | $0.23^{**}$ | $0.10$ |  | $0.23^{**}$ | $0.11$ |  | $0.21^{*}$ | $0.11$
below bachelors |  | $0.19^{*}$ | $0.10$ |  | $0.19^{*}$ | $0.10$ |  | $0.24^{**}$ | $0.10$ |  | $0.26^{**}$ | $0.10$
household size |  | $-0.04$ | $0.03$ |  | $-0.05$ | $0.03$ |  | $-0.04$ | $0.03$ |  | $-0.03$ | $0.03$
tolerant states |  | $0.13$ | $0.08$ |  | $0.12$ | $0.08$ |  | $0.12$ | $0.08$ |  | $0.08$ | $0.09$
african american |  | $..$ | $..$ |  | $-0.05$ | $0.12$ |  | $-0.26^{**}$ | $0.13$ |  | $-0.14$ | $0.13$
other races |  | $..$ | $..$ |  | $0.12$ | $0.14$ |  | $0.03$ | $0.14$ |  | $0.02$ | $0.15$
democrat |  | $..$ | $..$ |  | $..$ | $..$ |  | $0.68^{**}$ | $0.10$ |  | $0.57^{**}$ | $0.11$
other parties |  | $..$ | $..$ |  | $..$ | $..$ |  | $0.48^{**}$ | $0.10$ |  | $0.40^{**}$ | $0.10$
roman catholic |  | $..$ | $..$ |  | $..$ | $..$ |  | $..$ | $..$ |  | $0.18^{*}$ | $0.10$
christian |  | $..$ | $..$ |  | $..$ | $..$ |  | $..$ | $..$ |  | $-0.06$ | $0.13$
conservative |  | $..$ | $..$ |  | $..$ | $..$ |  |  | $..$ |  | $0.44^{**}$ | $0.18$
liberal |  | $..$ | $..$ |  | $..$ | $..$ |  | $..$ | $..$ |  | $0.66^{**}$ | $0.12$
LR ($\chi^{2}$) statistic |  |  | $164.37$ |  |  | $165.38$ |  |  | $211.44$ |  |  | $248.55$
mcfadden’s $R^{2}$ |  |  | $0.10$ |  |  | $0.10$ |  |  | $0.13$ |  |  | $0.15$
hit-rate |  |  | $66.67$ |  |  | $66.58$ |  |  | $67.51$ |  |  | $69.03$
$\ast\ast$ p $<$ 0.05, $\ast$ p $<$ 0.10 |  |  |  |  |  |  |  |

The coefficient for income is negative, but statistically insignificant as
also documented in Nielsen (2007) and Saieva (2008). As such, we do not note a
significant relationship between income and probability of supporting
marijuana legalization. This is in disagreement with the hypothesis that the
economically weaker section will support legalization since marijuana is
popular within the lower income group. Past use of marijuana may drive support
for legalization, but this was not controlled either in Nielsen (2007) or
Saieva (2008). Controlling for this variable in our models, we find that the
coefficient for past use is largest amongst all variables and highly
significant. This finding provides support to the hypothesis that individuals
who have used marijuana in the past strongly favor its legalization and the
large coefficient value implies that past use is an important factor in
favoring legalization. Males are more supportive of legalizing marijuana
compared to females and the coefficient is significant, a result also
documented in Nielsen (2007) and Delforterie et al. (2015). Similarly,
Rodríguez (2015) finds that boys are more likely to use marijuana during their
adolescent years. Since past use is an important determinant for supporting
legalization and marijuana use is more prevalent among males, it is not
surprising to find that males are more supportive of legalization.

The indicator variables for higher education i.e., ‘bachelors & above’ and
‘below bachelors’ have positive coefficients and are statistically significant
(either at 10 or 5 percent significance level) relative to the base category,
‘high school & below’. Thus, higher education leads to increased support for
legalization possibly because a more educated individual can better understand
the costs and medicinal benefits of marijuana. However, some studies have
found that early use of marijuana leads to lower educational attainment and
poor performance in school (Lynskey and Hall, 2000; Van Ours and Williams,
2009; Horwood et al., 2010). Household size has a negative effect, but the
coefficient is not statistically significant. Similarly, the coefficient for
‘tolerant states’ is positive, but insignificant. This implies that residing
in one of 20 states that offers some relaxation on marijuana use/offence does
not statistically increase the probability of supporting legalization.

The coefficient for African American and ‘Other Races’, are not statistically
different from the base category, White, with the exception of Model 3. The
negative coefficient for African American, although insignificant, is rather
surprising because one would expect African Americans to support legalization
in order to curtail the large number of marijuana related arrests from the
African American community. In line with this, Chen and Killeya-Jones (2006)
also examine the extent of marijuana use across race and find that marijuana
use is higher among suburban White students compared to their African American
counterparts. Moreover, Nasim et al. (2007) document the cultural orientation
for African American young women and find that traditional religious beliefs
and practices could be the reason behind less marijuana usage among African
American.

Political affiliation often represents ideological differences towards any
public policy and several poll studies conducted by Pew and Gallup have found
that Republicans (Democrats) are more likely to oppose (favor) legalization of
marijuana. We also arrive at a similar conclusion, with the coefficients for
Democrat and ‘Other Parties’ being positive and significant. This suggests
that individuals having either political orientations are more supportive of
legalization compared to Republicans and the result is consistent with Nielsen
(2007). Lastly, we look at the effect of religious affiliations since
religious beliefs sometime acts as a protective factor against alcohol usage
and smoking. The results show that the coefficient for Roman Catholic is
positive and statistically significant at 10 percent, while coefficient for
Christian is negative but statistically insignificant. In contrast, the
coefficients for Conservative and Liberal are both positive and statistically
significant and hence both groups are more supportive of legalization compared
to Protestants. However, individual opinions often do not strictly adhere to
religious codes and conducts, so these results may vary significantly across
samples.

Table 3: Average covariate effects from Model 4. covariate |  | $\Delta$P(favor legalization)
---|---|---
age, 10 years |  | $-0.019$
past use |  | $0.285$
male |  | $0.057$
bachelors & above |  | $0.068$
below bachelors |  | $0.085$
democrat |  | $0.188$
other parties |  | $0.134$
roman catholic |  | $0.058$
conservative |  | $0.143$
liberal |  | $0.220$

The above discussion suggests that the signs of the coefficients, except the
race variables, are consistent with what one would typically expect. However,
the coefficients by themselves do not give the covariate effects (see Section
2 and 3). Table 3 presents the average covariate effects for all significant
variables, either at 5 or 10 percent level. Results show that an increase in
age by 10 years decreases the probability of support by 1.9 percent. The
highest positive impact comes from past use, which shows that an individual
who has used marijuana is 28.5 percent more likely to support legalization
relative to someone who has never used it. Males are 5.7 percent more likely
to support legalization relative to females. Higher education increases the
probability of support and an individual with bachelors or higher degree
(below bachelors) is 6.8 (8.5) percent more likely to support legalization
relative to an individual with a high school degree or below. Political
affiliation to the Democratic Party increases the probability of support by
18.8 percent. Similarly, an individual who identifies themselves with
Independent and other parties is 13.4 percent more likely to support
legalization compared to a Republican. Finally, an individual who is
Conservative (Liberal) is 14.3 (22) percent more likely to support
legalization relative to a Protestant.

###  6.2 Ordinal Probit Model - The Extent of Marijuana Legalization

The year 2013 was the first year in four decades that majority of Americans
favored legalization of marijuana. While its support has grown overtime as
shown in Figure 4, it is important to distinguish between different levels of
support. This distinction is crucial because support for personal use of
marijuana is stronger than support for its medical use and has different
policy implications. Individuals may opine to support marijuana for medicinal
benefits, but not for personal use. The February 2014 Political Survey
recorded individual response as a three level categorical variable, which
permits use of an ordinal probit model to study the effect of covariates on
public opinion about the extent of legalization.

####  6.2.1 Data

The February 2014 Political Survey was conducted during February 14-23, 2014
by the Princeton Survey Research Associates and sponsored by the Pew Research
Center for the People and the Press. In the survey, a representative sample of
1,821 adults living in the US were interviewed over telephone with 481 (1,340)
individuals interviewed over land line (cell phone, including 786 individuals
without a land line phone). The sampled data contain several missing
observations and many respondents were unsure about their opinion on
legalization. As before, we remove data on these respondents and are left with
a sample of 1,492 observations for the study.

Table 4: Descriptive summary of the variables (February 2014 Political Survey). variable |  | mean | std
---|---|---|---
log age |  | 3.72 | 0.44
log income |  | 10.63 | 0.98
household size |  | 2.74 | 1.42
| category | counts | percentage
past use |  | 719 | 48.19
male |  | 792 | 53.02
| bachelors & above | 551 | 36.93
education | below bachelors | 434 | 29.09
| high school & below | 507 | 33.98
tolerant states |  | 556 | 37.27
eventually legal |  | 1,154 | 77.35
| white | 1149 | 77.01
race | african american | 202 | 13.54
| other races | 141 | 9.45
| republican | 333 | 22.32
party affiliation | democrat | 511 | 34.25
| independent & others | 648 | 43.43
| protestant | 550 | 36.86
| roman catholic | 290 | 19.44
religion | christian | 182 | 12.20
| conservative | 122 | 8.18
| liberal | 348 | 23.32
| oppose legalization | 218 | 14.61
public opinion | legal only for medicinal use | 640 | 42.90
| legal for personal use | 634 | 42.49

The dependent variable in the model is the respondents’ answer to the
question, “Which comes closer to your view about the use of marijuana by
adults?”. The options provided were, ‘It should not be legal,’ ‘It should be
legal only for medicinal use,’ or ‘It should be legal for personal use’. The
fourth category labeled, ‘Don’t know/Refused’ is removed from the study.
Similar to the March 2013 Survey, the February 2014 Survey also collected
information on the age, income, household size, past use, gender, education,
race, party affiliation and religion. We use these variables as independent
variables in the models. All the definitions and categories for the variables
remain the same as in Section 6.1.1. We also include the indicator variable
‘tolerant states’, with the definition modified to include Illinois and New
Hampshire to the previous list of 20 states.888Note that marijuana related
laws were passed in Illinois and New Hampshire after the March 2013 Political
Survey, but before the February 2014 Political Survey. Finally, we include an
additional variable, labeled ‘eventually legal’, for which data was collected
only in the February 2014 Survey. This variable indicates whether respondents
expect marijuana to be legal irrespective of their individual opinion. We
present the descriptive statistics for all the variables in Table 4.

Upon exploration of the socio-demographic characteristics of the current
sample, we note that an average respondent is about 45.5 years old and s/he
belongs to a household of size 3 with an annual income of 60,647 US Dollars.
Thus, the typical respondent is about 6 years younger compared to the March
2013 data and approximately has the same household size and income. The
percentage of males is higher in the current sample by 5 percent, but still
close to a fair split between males and females. Similarly, the sample is
almost evenly split between respondents who have used marijuana and those who
have not. The largest proportion of respondents (36.93 percent) have
‘bachelors and above’ degree, followed by ‘high school & below’ degree (33.98
percent). A significant fraction of the respondents (37.27 percent) reside in
states that have some favorable law on marijuana. Looking at the additional
variable ‘eventually legal’, note that 77.35 percent of the surveyed people
expect marijuana to be legal irrespective of their opinion. Similar to the
earlier data, the sample is predominantly White (77.01 percent) with a good
representation (13.54 percent) of the African American population. Party
affiliation shows that 34.25 percent of the sample is comprised of Democrats,
22.32 percent Republicans and the remaining fraction are ‘independent &
others’. With respect to religious classifications, the largest proportion of
respondents are Protestant (36.86 percent), followed by Liberal (23.32
percent) and Roman Catholics (19.44 percent).

####  6.2.2 Estimation

The ordinal probit model results are presented in Table 5, which show the
coefficient estimates and standard errors of four different models. The
estimation of models follows a similar sequence as in Table 2. Model 5 is the
base model and contains log age, log income, past use of marijuana, male,
education categories, household size, tolerant states and eventually legal.
Model 6 adds the race categories to Model 5 and Model 7 adds party affiliation
to the list of variables in Model 6. Finally, Model 8 contains all the
variables in Model 7 and religious categories. The goodness of fit statistics
are presented in the last three rows of Table 5. The LR statistics are large
and each model fits better than the respective intercept model. The other two
measures, McFadden’s $R^{2}$ and hit-rate, show that Model 7 and Model 8
outperform the remaining two models. While Model 8 provides a better fit
compared to Model 7 as per McFadden’s $R^{2}$ (0.1187 compared to 0.1254),
Model 7 has a better hit-rate.

Table 5: Estimation results for the ordinal probit model. |  | model 5 |  | model 6 |  | model 7 |  | model 8
---|---|---|---|---|---|---|---|---
|  | coef | se |  | coef | se |  | coef | se |  | coef | se
intercept |  | $1.26^{**}$ | $0.44$ |  | $1.27^{**}$ | $0.45$ |  | $0.83^{*}$ | $0.46$ |  | $0.34$ | $0.48$
log age |  | $-0.44^{**}$ | $0.07$ |  | $-0.45^{**}$ | $0.07$ |  | $-0.45^{**}$ | $0.08$ |  | $-0.35^{**}$ | $0.08$
log income |  | $0.07^{*}$ | $0.03$ |  | $0.07^{*}$ | $0.03$ |  | $0.08^{**}$ | $0.03$ |  | $0.09^{**}$ | $0.04$
past use |  | $0.74^{**}$ | $0.06$ |  | $0.73^{**}$ | $0.06$ |  | $0.71^{**}$ | $0.06$ |  | $0.69^{**}$ | $0.06$
male |  | $0.06$ | $0.06$ |  | $0.06$ | $0.06$ |  | $0.07$ | $0.06$ |  | $0.06$ | $0.06$
bachelors & above |  | $0.26$ | $0.08$ |  | $0.26^{**}$ | $0.08$ |  | $0.25^{**}$ | $0.08$ |  | $0.24^{**}$ | $0.08$
below bachelors |  | $0.06$ | $0.08$ |  | $0.05^{*}$ | $0.08$ |  | $0.05$ | $0.08$ |  | $0.05$ | $0.08$
household size |  | $-0.04^{*}$ | $0.02$ |  | $-0.04$ | $0.02$ |  | $-0.03$ | $0.02$ |  | $-0.02$ | $0.02$
tolerant states |  | $0.11^{*}$ | $0.06$ |  | $0.13^{**}$ | $0.06$ |  | $0.09$ | $0.06$ |  | $0.07$ | $0.07$
eventually legal |  | $0.58^{**}$ | $0.07$ |  | $0.58^{**}$ | $0.07$ |  | $0.56^{**}$ | $0.07$ |  | $0.57^{**}$ | $0.07$
african american |  | $..$ | $..$ |  | $0.11$ | $0.09$ |  | $-0.01$ | $0.10$ |  | $0.03$ | $0.10$
other races |  | $..$ | $..$ |  | $-0.18^{*}$ | $0.11$ |  | $-0.26^{**}$ | $0.11$ |  | $-0.27^{**}$ | $0.11$
democrat |  | $..$ | $..$ |  | $..$ | $..$ |  | $0.48^{**}$ | $0.09$ |  | $0.44^{**}$ | $0.09$
other parties |  | $..$ | $..$ |  | $..$ | $..$ |  | $0.40^{**}$ | $0.08$ |  | $0.36^{**}$ | $0.08$
roman catholic |  | $..$ | $..$ |  | $..$ | $..$ |  | $..$ | $..$ |  | $0.10$ | $0.09$
christian |  | $..$ | $..$ |  | $..$ | $..$ |  | $..$ | $..$ |  | $0.16$ | $0.10$
conservative |  | $..$ | $..$ |  | $..$ | $..$ |  |  | $..$ |  | $0.09$ | $0.12$
liberal |  | $..$ | $..$ |  | $..$ | $..$ |  | $..$ | $..$ |  | $0.39^{**}$ | $0.09$
cut-point |  | $1.43^{**}$ | $0.05$ |  | $1.43^{**}$ | $0.05$ |  | $1.45^{**}$ | $0.05$ |  | $1.46^{**}$ | $0.05$
LR ($\chi^{2}$) statistic |  |  | $316.27$ |  |  | $321.47$ |  |  | $356.99$ |  |  | $377.02$
mcfadden’s $R^{2}$ |  |  | $0.10$ |  |  | $0.11$ |  |  | $0.12$ |  |  | $0.12$
hit-rate |  |  | $57.77$ |  |  | $57.44$ |  |  | $59.11$ |  |  | $58.91$
$\ast\ast$ p $<$ 0.05, $\ast$ p $<$ 0.10 |  |  |  |  |  |  |  |

####  6.2.3 Results

We focus on the results from Model 8 because it is the most general model,
provides the best fit according to McFadden’s $R^{2}$ and no variables change
sign. The results indicate that log age has a negative effect on the support
for personal use (third category) and is statistically significant at 5
percent level. Alternatively, log age has a positive effect on opposing
legalization (first category) and is statistically significant. However, the
effect of age on medicinal use only (second category) cannot be determined _a
priori_. Henceforth, we shall only discuss the effect on personal use and the
impact on opposing legalization will be opposite to that of personal use. As
before, the default level of significance used is 5 percent and further
discussion will omit reference to significance level.

We note that the coefficient for log income is positive and statistically
significant. This implies that individuals with higher income are more likely
to support legalization for personal use. Past use of marijuana has a
statistically significant positive effect on the probability of supporting
personal use of marijuana. Moreover, the coefficient for past use is largest
among all the variables, a result which is similar to that obtained in the
binary probit model. Contrary to the finding in Section 6.1.3, the coefficient
for male is positive, but is not statistically significant. Thus, the current
data do not confirm any role of gender on public opinion towards marijuana.
Higher education is positively associated with support for personal use of
marijuana. However, only the coefficient for ‘bachelors degree & above’ is
statistically significant. The coefficients for household size and ‘tolerant
states’ are not significant and conform to the results from the binary probit
model. We find that ‘eventually legal’ has a significant positive effect,
indicating that if an individual expects marijuana to be legal irrespective of
his or her opinion, then s/he is more likely to support legalization for
personal use.

The race variables suggest that opinions of African Americans on personal use
of marijuana are not significantly different compared to the Whites. Such a
similarity of opinions across race was also observed in the binary probit
model. In contrast, ‘Other Races’ has a significant negative coefficient and
is more opposed to legalization as compared to Whites. The coefficients for
political party affiliations are in consonance with the results from Section
6.1.3. Affiliation to Democratic Party or ‘Other Parties’ increases the
support for personal use and the coefficients are significant. Lastly,
religious affiliations do not show a strong effect. Here, only the Liberals
are more supportive of personal use of marijuana, while the opinions of the
remaining religious categories are not significantly different from the base
category, Protestant.

Table 6: Average covariate effects from Model 8. covariate |  | $\Delta$P(not legal) | $\Delta$P(medicinal use) | $\Delta$P(personal use)
---|---|---|---|---
age, 10 years |  | $0.015$ | $0.012$ | $-0.028$
income, $10,000 |  | $-0.005$ | $-0.003$ | $0.008$
past use |  | $-0.129$ | $-0.113$ | $0.243$
bachelors & above |  | $-0.045$ | $-0.035$ | $0.080$
eventually legal |  | $-0.126$ | $-0.060$ | $0.186$
other races |  | $0.059$ | $0.031$ | $-0.089$
democrat |  | $-0.080$ | $-0.066$ | $0.147$
other parties |  | $-0.070$ | $-0.051$ | $0.121$
liberal |  | $-0.068$ | $-0.066$ | $0.134$

We mentioned in Section 2 that the coefficients of the ordinal probit model
only give the direction of impact for the first and last categories, but not
the remaining categories. The actual covariate effects need to be calculated
for all the categories. We compute the average covariate effects for all
significant variables and present them in Table 6. From the table, note that
past use, eventually legal, and identifying oneself as a Democrat are three
variables with the highest impact on public opinion. Past use of marijuana
increases support for personal use by 24.3 percent and decreases the support
for medicinal use and oppose legalization by 11.3 and 12.9 percents,
respectively. Similarly, a respondent who expects marijuana to be legal is
18.6 percent more likely to support marijuana for personal use. This increase
comes from a decrease in probability for medicinal use and oppose
legalization, which are 6.0 and 12.6 percents, respectively. In the same way,
a respondent who is a Democrat is 14.7 percent more likely to favor personal
use, and 6.6 and 8.0 percents less likely to favor medicinal use and oppose
legalization, respectively. The covariate effects for the remaining variables
can be interpreted similarly.

##  7 Conclusion

This chapter presents an overview of two popular ordinal models (ordinal
probit and ordinal logit models) as well as two widely used binary models
(probit and logit models). These models fall within the class of discrete
choice models and are extremely popular in several disciplines including
economics, epidemiology, finance, and sociology. The models are described
using the latent variable threshold-crossing framework since it elegantly
connects individual choice behavior with the random utility model in
economics. While we focus on the ordinal probit and binary probit models as
prototypes to derive the likelihood and outline the estimation procedure, the
approach is completely applicable to its logit counterparts. Some interesting
aspects about interpreting the coefficients of logit models are emphasized.
Since the models considered here are non-linear models, the coefficients
cannot be interpreted as covariate effects. We explain how to compute the
covariate effects when the covariates are continuous and when they are binary
(indicator variable). Measures to assess model fitting, namely, likelihood
ratio statistic, McFadden’s R-square, and hit-rate are also described. We also
include specific applications of discrete choice models, wherein we utilize
the ordinal probit and binary probit models to analyze public opinion on
marijuana legalization and extent of legalization in the United States – a
notable and contentious topic with credible arguments from proponents as well
as critics of the policy.

The first application utilizes a binary probit model to analyze the response
to legalization (‘oppose legalization’ or ‘favor legalization’) based on
individual demographic variables, educational background, racial
characteristics and political affiliation of the respondents. The data is
taken from the March 2013 Political Survey collected by the Pew Research
Center. The results suggest that while log age has a negative effect, past use
of marijuana, male, higher education, affiliation to the Democratic or
‘Independent & Other’ parties, as well as Roman Catholic, Conservative and
Liberal religious beliefs have a positive effect on the probability of
supporting legalization. Not surprisingly, past use of marijuana has the
highest positive effect and increases the probability of support by 28.5
percent. The proposed model performs well and correctly classifies 69 percent
of the responses.

The second study employs an ordinal probit model to analyze the ordered
response to legalization (‘oppose legalization’, ‘legal for medicinal use’ or
‘legal for personal use’) based on a similar set of covariates as in the first
study. Data for this study is taken from the February 2014 Political Survey
collected by the Pew Research Center. The results show that log age and
belonging to ‘Other Races’ (non-White and non-African American) negatively
(positively) affects the probability of supporting personal use (oppose
legalization) and the latter has the highest negative effect at $8.9$ percent.
The variables that have a positive (negative) effect on personal use (oppose
legalization) include income and indicators for past use of marijuana,
bachelors or higher education, individual expectation on eventual
legalization, Democratic Party, ‘Other Parties’ and Liberal religious beliefs.
Amongst these, past use of marijuana has the highest positive effect on
personal use at 24.3 percent. The proposed ordinal model performs well and
correctly classifies approximately 59 percent of the responses.

The insights from these studies are interesting and may assist policymakers to
better assess public preference regarding marijuana legalization, the extent
of legalization (particularly, medical marijuana) and the factors associated
with such preferences. For instance, the finding that educational attainment
has a positive impact on public support for legalization implies that
providing information on the medicinal findings on marijuana and emphasizing
college and university education is likely to increase support for it. If
people believe that the benefits outweigh the costs, then public opinion will
move further towards legalization and vice-versa. These findings may also be
helpful to various advocacy groups and opposition lobbies engaged in promoting
or opposing unrestricted legalization of marijuana respectively. A clear
understanding of the underlying factors that drive an individual’s opinion
will help these groups better plan their campaigns. For example, since support
for legalization is negatively related with age, groups opposing legalization
may consider not campaigning amongst the youth as it is unlikely to yield
support. Similarly, lobbies engaged in opposing legalization may optimize
their time use on better possibilities than convince an individual or group
with a history of marijuana use to oppose legalization.

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