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Modeling and Analysis of Discrete Response Data: Applications to Public Opinion on Marijuana Legalization in the United States
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 models\footnote{Binary 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-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-etal-2004, Wilsey-etal-2013, Abrams-etal-2003, Abrams-etal-2007, Ellis-etal-2009, Johnson-etal-2010, McAllister-etal-2011, Guzman-2003,Duran-etal-2010. \footnote{The reader is directed to the website www.procon.org for a list of 60 peer-reviewed articles (\url{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-etal-1993, Meier-etal-2012.\footnote{A 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-etal-2013.
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 (ref) 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 $z_{i}$\footnote{The continuous latent construct may represent underlying latent utility, some kind of propensity, or strength of preference Greene-Hensher-2010.} as follows:
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:
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 (ref). One may observe from Figure (ref) 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-etal-2008 and Jeliazkov-Rahman-2012 for details.
Given a data vector $y$ = $(y_{1}, \cdots, y_{n})'$, the likelihood for the ordinal probit model expressed as a function of unknown parameters $(\beta, \gamma)$ is the following,
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 (ref)) 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-Hensher-2010 or Johnson-Albert-2000 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-Zavoina-1975,McCullagh-1980. Therefore, for the model in equation (ref), 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 (ref) 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 $\theta_{ij} = \Pr(y_{i} \le j)$ denote the cumulative probability that individual $i$ chooses category $j$ or below. For the ordinal logit model, this implies: $\theta_{ij} = \exp{(\gamma_{j} - x'_{i}\beta)}/ \big[1 + \exp{(\gamma_{j} - x'_{i}\beta )} \big]$, and $ \theta_{ij}/(1-\theta_{ij}) = \exp \big[\gamma_{j} - x'_{i}\beta \big]$, where the latter represents the odds for the event $y_{i} \le j$. Accordingly, for any two individuals (say $1$ and $2$), the ratio of odds is,
The odds ratio presented in equation (ref) 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.
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 (ref) 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 $z_{i}$\footnote{The 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-Rahman-2012.} as follows,
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 (ref). 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.
The likelihood for the binary probit model can be expressed as,
Given the likelihood, the model parameters are estimated using the ML technique i.e., by maximizing the log-likelihood (equation (ref)) 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-Albert-2000.
Similar to ordinal models, the framework for the binary probit model given by equation (ref) can be utilized to describe a binary logit model (or simply logit model) with the modification that the errors follow a logistic distribution Hosmer-etal-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 $\Phi(x'_{i}\beta)$ in equation (ref) is replaced by the logistic cdf: $\Lambda (x'_{i}\beta) = \exp{(x'_{i}\beta)}/ \big[1 + \exp{(x'_{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 $\theta_{i} = \Pr(y_{i}=1|x'_{i}\beta)$ 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,
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,
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.
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,
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 (ref) 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,
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 (ref) 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,
and the same for an indicator variable is given by the expression,
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-Albert-2000. For the null hypothesis $H_{0}: \beta_{2} = \ldots = \beta_{k}=0$, the LR test statistic $\lambda_{LR}$ is defined as follows:
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,
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,
where $\hat{p}_{ij}$ is the predicted probability that individual $i$ selects outcome $j$, and $I(\cdot)$ is the indicator function as defined earlier.
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,
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 PoirierBook-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-etal-1953, Hastings-1970 can be employed to sample the parameters, the standard and more convenient approach is to consider data augmentation Tanner-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,
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-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-Chib-1993. Other notable references that describe the Bayesian modeling and estimation of ordinal and binary responses in great detail include Johnson-Albert-2000, Greenberg-2012, and Jeliazkov-Rahman-2012. Bayesian estimation of logit model is based on the same principle and presented in Holmes-Held-2006 and Jeliazkov-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-Basset-1978. Quantile modeling gained further momentum with the development of Bayesian quantile regression by Yu-Moyeed-2001, where the authors create a working likelihood by assuming that the errors follow an asymmetric Laplace (AL) distribution Yu-Zhang-2005. Binary quantile regression was proposed by Kordas-2006 and its Bayesian formulation was presented by Benoit-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-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-Karnawat-2019. Some recent research on ordinal and binary quantile regression in the panel/longitudinal set up include Rahman-Vossmeyer-2019, and Bresson-etal-2021. Two applied studies employing ordinal and binary quantile framework are Omata-etal-2017 and Ojha-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.
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 DEA-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 (ref)). 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 (ref) 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 EarleywineBook-2005 and the social cost of prohibition that includes illegal trade, racially skewed arrests of African Americans and huge enforcement cost Shepard-Blackley-2007.
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 FernerNews-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-etal-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-etal-2008. In contrast, logistic model based on logistic distribution cannot model correlations in multivariate settings.
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.
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.\footnote{The 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 (ref).
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.
We estimate four different binary probit models and present the estimated coefficients and standard errors for each model in Table (ref). 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 (ref) 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.
We focus on the results from Model 4, since it provides the best model fit. Table (ref) shows that log age has a negative coefficient and is statistically significant at 5 percent level.\footnote{The 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-etal-1974. Moreover, Saieva-2008 reports a negative association between age and probability of supporting legalization, while Alfonso-Dunn-2007 and Delforterie-etal-2015 note that marijuana prevalence rate is higher among the younger population.
The coefficient for income is negative, but statistically insignificant as also documented in Nielsen-2010 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-2010 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-2010 and Delforterie-etal-2015. Similarly, Rodriguez-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-Hall-2000, VanOurs-Williams-2009, Horwood-etal-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-KilleyaJones-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-etal-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-2010. 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.
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 (ref) and (ref)). Table (ref) 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.
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 (ref), 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.
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.
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 (ref). 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.\footnote{Note 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 (ref).
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).
The ordinal probit model results are presented in Table (ref), which show the coefficient estimates and standard errors of four different models. The estimation of models follows a similar sequence as in Table (ref). 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 (ref). 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.
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 (ref), 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 (ref). 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.
We mentioned in Section (ref) 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 (ref). 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.
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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