This paper presents a Bayesian analysis of bivariate ordered probit regression model with excess of zeros. Specifically, in the context of joint modeling of two ordered outcomes, we develop zero-inflated bivariate ordered probit model and carry out estimation using Markov Chain Monte Carlo techniques. Using household tobacco survey data with substantial proportion of zeros, we analyze the socioeconomic determinants of individual problem of smoking and chewing tobacco. In our illustration, we find strong evidence that accounting for excess zeros provides good fit to the data. The example shows that the use of a model that ignores zero-inflation masks differential effects of covariates on nonusers and users.
This paper is concerned with joint modeling of two ordered data outcomes allowing for excess zeros. Economic, biological, and social science studies often yield data on two ordered categorical variables that are jointly dependent. Examples include the relationship between desired and excess fertility [
Many ordered discrete data sets are characterized by excess of zeros, both in terms of the proportion of nonusers and relative to the basic ordered probit or logit model. The zeros may be attributed to either corner solution to consumer optimization problem or errors in recording. In the case of individual smoking behavior, for example, the zeros may be recorded for individuals who never smoke cigarettes or for those who either used tobacco in the past or are potential smokers. In the context of individual patents applied for by scientists during a period of five years, zero patents may be recorded for scientists who either never made patent applications or for those who do but not during the reporting period [
The univariate as well as bivariate zero-inflated count data models are well established in the literature for example, Lambert [
This paper presents a Bayesian analysis of bivariate ordered probit model with excess of zeros. Specifically, we develop a zero-inflated ordered probit model and carry out the analysis using the Bayesian approach. The Bayesian analysis is carried out using Markov Chain Monte Carlo (MCMC) techniques to approximate the posterior distribution of the parameters. Bayesian analysis of the univariate zero-inflated ordered probit will be treated as a special case of the zero-inflated bivariate order probit model. The proposed models are illustrated by analyzing the socioeconomic determinants of individual choice problem of bivariate ordered outcomes on smoking and chewing tobacco. We use household tobacco prevalence survey data from Bangladesh. The observed proportion of zeros (those identifying themselves as nonusers of tobacco) is about 76% for smoking and 87% for chewing tobacco.
The proposed approach is useful for the analysis of ordinal data with natural zeros. The empirical analysis clearly shows the importance of accounting for excess zeros in ordinal qualitative response models. Accounting for excess zeros provides good fit to the data. In terms of both the signs and magnitudes of marginal effects, various covariates have differential impacts on the probabilities associated with the two types of zeros, nonparticipants and zero-consumption. The usual analysis that ignores excess of zeros masks these differential effects, by just focusing on observed zeros. The empirical results also show the importance of taking into account the uncertainty in the parameter estimates. Another advantage of the Bayesian approach to modeling excess zeros is the flexibility, particularly computational, of generalizing to multivariate ordered response models.
The rest of the paper is organized as follows. Section
We consider the basic Bayesian approach to a bivariate latent variable regression model with excess of zeros. To develop notation, let
We introduce inflation at the point
Let
Regarding identification of the parameters in the model defined by (
About 2/3 of the observations in our tobacco application below have a double-zero-state,
It is common to use marginal or partial effects to interpret covariate effects in nonlinear models; see, for example, Liu et al. [
From a practical point of view, we are less interested in the marginal effects of explanatory variables on the joint probabilities of choice from ZIBOP. Instead, we focus on the marginal effects associated with the marginal distributions of
Regressor
Continuing with the case of common covariate, the marginal effects of
Now consider case 2, where a generic independent variable
For case 3, where
As noted by a referee, it is important to understand the sources of covariate effects and the relationship between the marginal effects and the coefficient estimates. Since
Since the zero-inflated univariate ordered probit (ZIOP) model has not been analyzed previously in the Bayesian framework, we provide a brief sketch of the basic framework for ZIOP. The univariate ordered probit model with excess of zeros can be obtained as a special case of the ZIBOP model presented previously. To achieve this, let
Zero-inflation is now introduced at point
Assume that
Different choices of the specification of the joint distribution of
Assuming that
The Bayesian hierarchical model requires prior distributions for each parameter in the model. For this purpose, we can use noninformative conjugate priors. There are two reasons for adopting noninformative conjugate priors. First, we prefer to let the data dictate the inference about the parameters with little or no influence from prior distributions. Secondly, the noninformative priors facilitate resampling using Markov Chain Monte Carlo algorithm (MCMC) and have nice convergence properties. We assume noninformative (vague or diffuse) normal priors for regression coefficients
In choosing prior distributions for the threshold parameters,
The only unknown parameter associate with the distribution of
For carrying out a Bayesian inference, the joint posterior distribution of the parameters of the ZIBOP model in (
Full conditional posterior distributions are required to implement the MCMC algorithm [ fixed effects: zero state:
nonzero state:
thresholds:
bivariate correlation:
The MCMC algorithm simulates direct draws from the above full conditionals iteratively until convergence is achieved. A single long chain [
The Bayesian analysis of the univariate ZIOP follows as a special case of that of the ZIBOP presented above. In particular, the joint posterior distribution of the parameters of the ZIOP model in (
Apart from Bayesian estimation of the regression parameters, the posterior distributions of other quantities of interest can be obtained. These include posteriors for marginal effects and probabilities for nonparticipation, zero-consumption, and joint outcomes of interest. These will be considered in the application section. Next, we summarize model selection procedure.
The commonly used criteria for model selection like BIC and AIC are not appropriate for the multilevel models (in the presence of random effects), which complicates the counting of the true number of free parameters. To overcome such a hurdle, Spiegelhalter et al. [
We consider an application to tobacco consumption behavior of individuals based on the 2001 household Tobacco Prevalence survey data from Bangladesh. The Survey was conducted in two administrative districts of paramount interest for tobacco production and consumption in the country. Data on daily consumption of smoking and chewing tobacco along with other socioeconomic and demographic characteristics and parental tobacco consumption habits were collected from respondents of 10 years of age and above. The data set has been used previously by Gurmu and Yunus [
The ordinal outcomes
Bivariate frequency distribution for intensity of tobacco use.
Smoke group | Chew group | Total ( | ||
0 | 1 | 2 | ||
0 | 3931 | 302 | 324 | 4557 |
1 | 265 | 12 | 6 | 283 |
2 | 526 | 35 | 37 | 598 |
3 | 498 | 29 | 35 | 562 |
Total ( | 5220 | 378 | 402 | 6000 |
Table
Definition and summary statistics for independent variables.
Name | Definition | Meanb | St. Dev. |
---|---|---|---|
Agea | Age in years | 30.35 | (14.9) |
Educationa | Number of years of formal schooling | 6.83 | (4.7) |
Income | Monthly family income in 1000s of Taka | 7.57 | (10.3) |
Male | = 1 if male | 54.6 | |
Married | = 1 if married | 57.2 | |
Muslim | = 1 if religion is Islam | 78.4 | |
Father use | = 1 if father uses tobacco | 54.0 | |
Mother use | = 1 if mother uses tobacco | 65.1 | |
Region | = 1 if Rangpur resident, | 49.7 | |
Urban | = 1 if urban resident | 38.0 | |
Agriservice | = 1 if agriculture labor or service occupation | 23.2 | |
Self-employed | = 1 if self-employed or household chores | 30.7 | |
Student | = 1 if student | 26.8 | |
Other | = 1 if business or other occupations (control) | 19.3 |
aIn implementation, we also include age squared and education squared.
bThe means for binary variables are in percentage.
Among the variables given in Table
We estimate the standard bivariate ordered probit (BOP) and zero-inflated bivariate ordered probit regression models for smoking and chewing tobacco and report estimation results for parameters, marginal effects, and choice probabilities, along measures of model selection. An earlier version of this paper reports results from the standard ordered probit model as well as the uncorrelated and correlated versions of the univariate zero-inflated ordered probit model for smoking tobacco. Convergence of the generated samples is assessed using standard tools (such as trace plots and ACF plots) within WinBUGS software. After initial 10,000 burn-in iterations, every 10th MCMC sample thereafter was retained from the next 100,000 iterations, obtaining 10,000 samples for subsequent posterior inference of the unknown parameters. The slowest convergence is observed for some parameters in the inflation submodel. By contrast, the autocorrelations functions for most of the marginal effects die out quickly relative to those for the associated parameters.
Table
Goodness-of-fit statistics via DIC.
Model | Dbar | Dhat | pD | DIC |
---|---|---|---|---|
Bivariate ordered probit (BOP) | 11417.1 | 11386.9 | 30.1 | 11447.2 |
Zero-inflated BOP | 11301.1 | 11270.3 | 29.8 | 11329.9 |
Dbar: Posterior mean of deviance, Dhat: Deviance evaluated at the posterior mean of the parameters, pD: Dbar-Dhat, the effective number of parameters, and DIC: Deviance information criterion.
Posterior mean, standard deviation, and 95% credible intervals of parameters from zibop for smoking and chewing tobacco.
Variable | Mean | St. dev. | 2.50% | Median | 97.50% |
---|---|---|---|---|---|
Main ( | |||||
Age/10 | 0.672 | 0.119 | 0.444 | 0.685 | |
Age square/100 | −0.070 | 0.012 | −0.093 | −0.071 | |
Education | −0.071 | 0.014 | −0.097 | −0.071 | |
Education square | 0.001 | 0.001 | −0.002 | 0.001 | |
Income | 0.000 | 0.002 | −0.005 | 0.000 | |
Male | 2.092 | 0.086 | 1.925 | 2.091 | |
Married | 0.213 | 0.070 | 0.074 | 0.213 | |
Muslim | −0.053 | 0.052 | −0.157 | −0.053 | |
Region | −0.007 | 0.048 | −0.102 | −0.007 | |
Urban | −0.096 | 0.051 | −0.198 | −0.097 | |
Agriservice | −0.234 | 0.056 | −0.345 | −0.233 | |
Self-employed | −0.246 | 0.087 | −0.414 | −0.247 | |
student | −0.476 | 0.137 | −0.742 | −0.478 | |
0.284 | 0.017 | 0.252 | 0.283 | ||
0.987 | 0.030 | 0.928 | 0.987 | ||
Main ( | |||||
Age/10 | 0.649 | 0.133 | 0.382 | 0.658 | |
Age square/100 | −0.046 | 0.013 | −0.071 | −0.046 | |
Education | −0.020 | 0.016 | −0.052 | −0.020 | |
Education square | −0.002 | 0.001 | −0.005 | −0.002 | |
Income | 0.001 | 0.003 | −0.004 | 0.002 | |
Male | −0.479 | 0.081 | −0.641 | −0.479 | |
Married | −0.025 | 0.075 | −0.171 | −0.025 | |
Muslim | −0.072 | 0.056 | −0.181 | −0.072 | |
Region | 0.417 | 0.051 | 0.317 | 0.418 | |
Urban | −0.080 | 0.058 | −0.194 | −0.079 | |
Agriservice | 0.052 | 0.074 | −0.096 | 0.052 | |
Self-employed | 0.127 | 0.092 | −0.058 | 0.126 | |
Student | −0.450 | 0.221 | −0.887 | −0.448 | |
0.484 | 0.023 | 0.439 | 0.484 | ||
Inflation ( | |||||
Age/10 | −0.012 | 2.044 | −4.755 | 0.253 | |
Age square/100 | 0.509 | 0.552 | −0.197 | 0.398 | |
Education | −0.218 | 0.115 | −0.476 | −0.204 | |
Education square | 0.028 | 0.011 | 0.010 | 0.026 | |
Income | 0.006 | 0.022 | −0.027 | 0.003 | |
Male | 0.239 | 0.827 | −1.582 | 0.417 | |
Married | 2.306 | 4.478 | −0.416 | 0.500 | |
Muslim | −0.528 | 0.356 | −1.331 | −0.494 | |
Mother | −0.170 | 0.267 | −0.716 | −0.164 | |
Father | −0.119 | 0.330 | −0.664 | −0.160 | |
Region | 0.630 | 0.291 | 0.061 | 0.625 | |
Urban | 0.040 | 0.357 | −0.737 | 0.071 | |
Agrservice | 5.312 | 5.416 | 1.017 | 2.674 | |
Self-employed | 3.783 | 5.025 | 0.124 | 1.275 | |
Sstudent | −0.344 | 0.411 | −1.154 | −0.339 | |
−0.185 | 0.033 | −0.249 | −0.186 | ||
Select probabilities: | |||||
0.760 | 0.004 | 0.752 | 0.760 | ||
0.871 | 0.004 | 0.864 | 0.871 | ||
0.662 | 0.005 | 0.652 | 0.662 | ||
0.242 | 0.048 | 0.151 | 0.243 |
Results for the constant terms in the main and inflation parts have been suppressed for brevity.
To facilitate interpretation of results, we report in Tables
Posterior mean, standard deviation, and 95% credible intervals of marginal effects of covariates on probability of smoking and chewing tobacco (ZIBOP model).
Variable | Probability | Mean | St. dev. | 2.50% | Median | 97.50% |
---|---|---|---|---|---|---|
Age | Nonparticipation | −0.0259 | 0.0129 | −0.0556 | −0.0236 | −0.0078 |
Zero-consumption, | 0.0463 | 0.0102 | 0.0294 | 0.0453 | 0.0687 | |
All zeros, | 0.0204 | 0.0059 | 0.0078 | 0.0213 | 0.0304 | |
0.0058 | 0.0035 | 0.0009 | 0.0053 | 0.0138 | ||
−0.0014 | 0.0029 | −0.0057 | −0.0019 | 0.0055 | ||
−0.0690 | 0.0235 | −0.1223 | −0.0658 | −0.0344 | ||
Zero-consumption, | 0.0403 | 0.0116 | 0.0195 | 0.0386 | 0.0675 | |
All zeros, | 0.0145 | 0.0064 | 0.0018 | 0.0149 | 0.0264 | |
−0.0034 | 0.0021 | −0.0071 | −0.0035 | 0.0008 | ||
−0.0019 | 0.0014 | −0.0043 | −0.0020 | 0.0011 | ||
Education | Nonparticipation | −0.2823 | 0.0768 | −0.4260 | −0.2837 | −0.1252 |
Zero-consumption, | 0.2447 | 0.0749 | 0.0917 | 0.2459 | 0.3851 | |
All zeros, | −0.0377 | 0.0241 | −0.0853 | −0.0374 | 0.0094 | |
0.0498 | 0.0141 | 0.0231 | 0.0494 | 0.0789 | ||
0.0241 | 0.0102 | 0.0045 | 0.0239 | 0.0444 | ||
−0.5557 | 0.1536 | −0.8415 | −0.5588 | −0.2417 | ||
Zero-consumption, | 0.3136 | 0.0772 | 0.1561 | 0.3159 | 0.4546 | |
All zeros, | 0.0313 | 0.0161 | −0.0009 | 0.0315 | 0.0618 | |
−0.0134 | 0.0080 | −0.0288 | −0.0135 | 0.0027 | ||
−0.0222 | 0.0119 | −0.0455 | −0.0221 | 0.0009 | ||
Income | Nonparticipation | −0.0004 | 0.0015 | −0.0038 | −0.0002 | 0.0022 |
Zero-consumption, | 0.0003 | 0.0014 | −0.0022 | 0.0002 | 0.0035 | |
All zeros, | −0.0001 | 0.0004 | −0.0009 | −0.0001 | 0.0008 | |
0.0001 | 0.0003 | −0.0005 | 0.0000 | 0.0008 | ||
0.0000 | 0.0002 | −0.0003 | 0.0000 | 0.0004 | ||
−0.0007 | 0.0030 | −0.0075 | −0.0004 | 0.0044 | ||
Zero-consumption, | 0.0001 | 0.0016 | −0.0025 | 0.0000 | 0.0036 | |
All zeros, | −0.0002 | 0.0005 | −0.0011 | −0.0002 | 0.0007 | |
0.0001 | 0.0002 | −0.0003 | 0.0001 | 0.0004 | ||
0.0001 | 0.0001 | −0.0002 | 0.0001 | 0.0003 | ||
Male | Nonparticipation | −0.0254 | 0.0599 | −0.1268 | −0.0305 | 0.1012 |
Zero-consumption, | −0.3595 | 0.0611 | −0.4900 | −0.3540 | −0.2565 | |
All zeros, | −0.3849 | 0.0116 | −0.4078 | −0.3849 | −0.3618 | |
0.0630 | 0.0040 | 0.0555 | 0.0630 | 0.0711 | ||
0.1560 | 0.0065 | 0.1435 | 0.1559 | 0.1689 | ||
0.1659 | 0.0083 | 0.1503 | 0.1657 | 0.1829 | ||
Zero-consumption, | 0.1012 | 0.0623 | −0.0309 | 0.1064 | 0.2075 | |
All zeros, | 0.0758 | 0.0126 | 0.0511 | 0.0759 | 0.1004 | |
0.0501 | 0.0033 | 0.0438 | 0.0500 | 0.0567 | ||
−0.1258 | 0.0112 | −0.1478 | −0.1258 | −0.1040 | ||
Married | Nonparticipation | −0.0680 | 0.0777 | −0.2274 | −0.0433 | 0.0346 |
Zero-consumption, | 0.0200 | 0.0705 | −0.0778 | −0.0001 | 0.1692 | |
All zeros, | −0.0480 | 0.0149 | −0.0796 | −0.0472 | −0.0207 | |
0.0056 | 0.0035 | 0.0006 | 0.0047 | 0.0132 | ||
0.0161 | 0.0061 | 0.0060 | 0.0154 | 0.0296 | ||
0.0263 | 0.0073 | 0.0116 | 0.0264 | 0.0406 | ||
Zero-consumption, | 0.0709 | 0.0791 | −0.0371 | 0.0474 | 0.2349 | |
All zeros, | 0.0028 | 0.0119 | −0.0200 | 0.0026 | 0.0269 | |
0.0628 | 0.0032 | 0.0566 | 0.0627 | 0.0693 | ||
−0.0656 | 0.0115 | −0.0888 | −0.0654 | −0.0434 | ||
Muslim | Nonparticipation | 0.0393 | 0.0243 | −0.0050 | 0.0384 | 0.0900 |
Zero-consumption, | −0.0239 | 0.0247 | −0.0752 | −0.0231 | 0.0216 | |
All zeros, | 0.0154 | 0.0090 | −0.0016 | 0.0153 | 0.0334 | |
−0.0023 | 0.0011 | −0.0044 | −0.0022 | −0.0002 | ||
−0.0053 | 0.0027 | −0.0106 | −0.0053 | −0.0001 | ||
−0.0078 | 0.0060 | −0.0200 | −0.0077 | 0.0036 | ||
Zero-consumption, | −0.0260 | 0.0258 | −0.0797 | −0.0253 | 0.0222 | |
All zeros, | 0.0133 | 0.0092 | −0.0046 | 0.0133 | 0.0315 | |
0.0613 | 0.0030 | 0.0554 | 0.0613 | 0.0674 | ||
−0.0746 | 0.0091 | −0.0926 | −0.0746 | −0.0569 | ||
Father use | Nonparticipation | 0.0122 | 0.0187 | −0.0251 | 0.0124 | 0.0487 |
Zero-consumption, | −0.0102 | 0.0158 | −0.0411 | −0.0104 | 0.0214 | |
All zeros, | 0.0020 | 0.0030 | −0.0040 | 0.0019 | 0.0082 | |
−0.0005 | 0.0008 | −0.0022 | −0.0005 | 0.0011 | ||
−0.0009 | 0.0014 | −0.0037 | −0.0009 | 0.0018 | ||
−0.0005 | 0.0008 | −0.0023 | −0.0005 | 0.0011 | ||
Zero-consumption, | −0.0116 | 0.0179 | −0.0464 | −0.0118 | 0.0240 | |
All zeros, | 0.0006 | 0.0011 | −0.0012 | 0.0003 | 0.0033 | |
−0.0003 | 0.0006 | −0.0019 | −0.0002 | 0.0007 | ||
−0.0002 | 0.0005 | −0.0014 | −0.0001 | 0.0005 | ||
Mother use | Nonparticipation | 0.0129 | 0.0257 | −0.0343 | 0.0123 | 0.0634 |
Zero-consumption, | −0.0106 | 0.0215 | −0.0527 | −0.0103 | 0.0298 | |
All zeros, | 0.0024 | 0.0043 | −0.0047 | 0.0020 | 0.0115 | |
−0.0006 | 0.0012 | −0.0031 | −0.0006 | 0.0014 | ||
−0.0011 | 0.0019 | −0.0051 | −0.0009 | 0.0022 | ||
−0.0007 | 0.0012 | −0.0033 | −0.0005 | 0.0012 | ||
Zero-consumption, | −0.0119 | 0.0242 | −0.0587 | −0.0118 | 0.0338 | |
All zeros, | 0.0010 | 0.0016 | −0.0007 | 0.0004 | 0.0053 | |
−0.0006 | 0.0009 | −0.0030 | −0.0002 | 0.0005 | ||
−0.0004 | 0.0007 | −0.0023 | −0.0001 | 0.0002 | ||
Region | Nonparticipation | −0.0480 | 0.0240 | −0.0963 | −0.0470 | −0.0040 |
Zero-consumption, | 0.0412 | 0.0237 | −0.0039 | 0.0406 | 0.0889 | |
All zeros, | −0.0068 | 0.0079 | −0.0222 | −0.0068 | 0.0086 | |
0.0021 | 0.0011 | 0.0001 | 0.0021 | 0.0046 | ||
0.0033 | 0.0025 | −0.0016 | 0.0033 | 0.0083 | ||
0.0013 | 0.0052 | −0.0087 | 0.0014 | 0.0114 | ||
Zero-consumption, | −0.0206 | 0.0252 | −0.0672 | −0.0217 | 0.0301 | |
All zeros, | −0.0686 | 0.0078 | −0.0840 | −0.0686 | −0.0533 | |
0.0756 | 0.0038 | 0.0682 | 0.0755 | 0.0832 | ||
−0.0070 | 0.0070 | −0.0207 | −0.0070 | 0.0072 | ||
Urban | Nonparticipation | −0.0062 | 0.0261 | −0.0595 | −0.0054 | 0.0428 |
Zero-consumption, | 0.0217 | 0.0258 | −0.0271 | 0.0211 | 0.0733 | |
All zeros, | 0.0155 | 0.0088 | −0.0018 | 0.0155 | 0.0324 | |
−0.0007 | 0.0012 | −0.0029 | −0.0008 | 0.0017 | ||
−0.0042 | 0.0028 | −0.0096 | −0.0042 | 0.0014 | ||
−0.0106 | 0.0056 | −0.0215 | −0.0106 | 0.0006 | ||
Zero-consumption, | 0.0181 | 0.0275 | −0.0337 | 0.0178 | 0.0739 | |
All zeros, | 0.0119 | 0.0090 | −0.0062 | 0.0120 | 0.0295 | |
0.0597 | 0.0036 | 0.0528 | 0.0597 | 0.0668 | ||
−0.0716 | 0.0075 | −0.0864 | −0.0717 | −0.0566 | ||
Agriservice | Nonparticipation | −0.1989 | 0.0521 | −0.3092 | −0.1960 | −0.1062 |
Zero-consumption, | 0.2102 | 0.0506 | 0.1202 | 0.2075 | 0.3161 | |
All zeros, | 0.0113 | 0.0098 | −0.0084 | 0.0115 | 0.0297 | |
0.0058 | 0.0018 | 0.0026 | 0.0057 | 0.0097 | ||
0.0023 | 0.0033 | −0.0039 | 0.0021 | 0.0092 | ||
−0.0194 | 0.0060 | −0.0311 | −0.0194 | −0.0077 | ||
Zero-consumption, | 0.1838 | 0.0530 | 0.0871 | 0.1811 | 0.2940 | |
All zeros, | −0.0151 | 0.0126 | −0.0400 | −0.0150 | 0.0091 | |
0.0680 | 0.0049 | 0.0588 | 0.0678 | 0.0782 | ||
−0.0529 | 0.0096 | −0.0716 | −0.0530 | −0.0338 | ||
Self-employed | Nonparticipation | −0.1287 | 0.0693 | −0.2542 | −0.1191 | −0.0122 |
Zero-consumption, | 0.1590 | 0.0686 | 0.0431 | 0.1508 | 0.2845 | |
All zeros, | 0.0303 | 0.0166 | −0.0034 | 0.0305 | 0.0627 | |
0.0005 | 0.0025 | −0.0042 | 0.0005 | 0.0058 | ||
−0.0075 | 0.0060 | −0.0192 | −0.0075 | 0.0043 | ||
−0.0233 | 0.0089 | −0.0398 | −0.0237 | −0.0046 | ||
Zero-consumption, | 0.1034 | 0.0704 | −0.0179 | 0.0941 | 0.2327 | |
All zeros, y2=0 | −0.0254 | 0.0147 | −0.0546 | −0.0251 | 0.0035 | |
0.0684 | 0.0047 | 0.0594 | 0.0681 | 0.0781 | ||
−0.0430 | 0.0118 | −0.0660 | −0.0431 | −0.0195 | ||
Student | Nonparticipation | 0.0305 | 0.0357 | −0.0312 | 0.0270 | 0.1076 |
Zero-consumption, | 0.0548 | 0.0434 | −0.0353 | 0.0564 | 0.1354 | |
All zeros, | 0.0852 | 0.0206 | 0.0437 | 0.0855 | 0.1247 | |
−0.0090 | 0.0027 | −0.0149 | −0.0089 | −0.0041 | ||
−0.0295 | 0.0079 | −0.0455 | −0.0294 | −0.0143 | ||
−0.0468 | 0.0106 | −0.0657 | −0.0475 | −0.0244 | ||
Zero-consumption, | 0.0284 | 0.0448 | −0.0686 | 0.0313 | 0.1073 | |
All zeros, | 0.0588 | 0.0239 | 0.0065 | 0.0610 | 0.0995 | |
0.0390 | 0.0102 | 0.0207 | 0.0383 | 0.0604 | ||
−0.0979 | 0.0142 | −0.1211 | −0.0994 | −0.0659 |
Posterior mean, standard deviation and 95% credible intervals of parameters from BOP for smoking and chewing tobacco.
Variable | Mean | St. Dev. | 2.50% | Median | 97.50% |
---|---|---|---|---|---|
Smoking ( | |||||
Age/10 | 1.029 | 0.095 | 0.828 | 1.030 | |
Age square/100 | −0.104 | 0.010 | −0.123 | −0.105 | |
Education | −0.078 | 0.014 | −0.105 | −0.078 | |
Education square | 0.002 | 0.001 | 0.000 | 0.002 | |
Income | 0.000 | 0.002 | −0.004 | 0.000 | |
Male | 2.066 | 0.091 | 1.888 | 2.067 | |
Married | 0.221 | 0.064 | 0.093 | 0.220 | |
Muslim | −0.083 | 0.049 | −0.177 | −0.083 | |
Region | 0.041 | 0.043 | −0.044 | 0.041 | |
Urban | −0.091 | 0.048 | −0.186 | −0.091 | |
Agriservice | −0.121 | 0.050 | −0.219 | −0.122 | |
Self-employed | −0.149 | 0.087 | −0.318 | −0.150 | |
Sstudent | −0.720 | 0.093 | −0.905 | −0.719 | |
0.270 | 0.015 | 0.241 | 0.270 | ||
0.956 | 0.028 | 0.901 | 0.956 | ||
Chewing ( | |||||
Age/10 | 0.797 | 0.091 | 0.609 | 0.801 | |
Age square/100 | −0.059 | 0.010 | −0.079 | −0.059 | |
Education | −0.023 | 0.016 | −0.055 | −0.023 | |
Education square | −0.002 | 0.001 | −0.005 | −0.002 | |
Income | 0.002 | 0.003 | −0.004 | 0.002 | |
Male | −0.441 | 0.074 | −0.586 | −0.441 | |
Married | −0.010 | 0.073 | −0.153 | −0.011 | |
Muslim | −0.077 | 0.056 | −0.187 | −0.077 | |
Region | 0.430 | 0.049 | 0.334 | 0.430 | |
Urban | −0.082 | 0.056 | −0.193 | −0.081 | |
Agriservice | 0.078 | 0.073 | −0.067 | 0.078 | |
Self employed | 0.177 | 0.087 | 0.010 | 0.176 | |
Student | −0.715 | 0.177 | −1.070 | −0.710 | |
0.480 | 0.023 | 0.436 | 0.480 | ||
−0.178 | 0.034 | −0.244 | −0.179 |
Each equation includes father use and mother use variables as well as a constant term.
Posterior mean, standard deviation, and 95% credible intervals of marginal effects of covariates on probability of smoking and chewing tobacco (BOP model).
Variable | Probability | Mean | St. dev. | 2.50% | Median | 97.50% |
---|---|---|---|---|---|---|
Age | All zeros, | 0.0368 | 0.0038 | 0.0288 | 0.0369 | 0.0438 |
−0.0004 | 0.0003 | −0.0009 | −0.0004 | 0.0002 | ||
−0.0073 | 0.0010 | −0.0093 | −0.0073 | −0.0054 | ||
−0.0292 | 0.0029 | −0.0345 | −0.0292 | −0.0230 | ||
All zeros, | 0.0213 | 0.0043 | 0.0125 | 0.0215 | 0.0298 | |
−0.0056 | 0.0013 | −0.0082 | −0.0057 | −0.0031 | ||
−0.0030 | 0.0008 | −0.0047 | −0.0030 | −0.0014 | ||
Education | All zeros, | −0.0342 | 0.0236 | −0.0803 | −0.0340 | 0.0126 |
0.0038 | 0.0025 | −0.0011 | 0.0039 | 0.0086 | ||
0.0130 | 0.0084 | −0.0039 | 0.0129 | 0.0293 | ||
0.0174 | 0.0128 | −0.0076 | 0.0172 | 0.0428 | ||
All zeros, | 0.0322 | 0.0156 | 0.0002 | 0.0326 | 0.0616 | |
−0.0150 | 0.0077 | −0.0296 | −0.0151 | 0.0009 | ||
−0.0201 | 0.0113 | −0.0418 | −0.0203 | 0.0024 | ||
Income | All zeros, | −0.0001 | 0.0004 | −0.0009 | −0.0001 | 0.0007 |
0.0000 | 0.0000 | −0.0001 | 0.0000 | 0.0001 | ||
0.0000 | 0.0001 | −0.0002 | 0.0000 | 0.0002 | ||
0.0000 | 0.0002 | −0.0004 | 0.0000 | 0.0005 | ||
All zeros, | −0.0003 | 0.0005 | −0.0012 | −0.0003 | 0.0007 | |
0.0001 | 0.0002 | −0.0002 | 0.0001 | 0.0004 | ||
0.0001 | 0.0002 | −0.0002 | 0.0001 | 0.0004 | ||
Male | All zeros, | −0.3824 | 0.0121 | −0.4064 | −0.3826 | −0.3586 |
0.0641 | 0.0040 | 0.0567 | 0.0640 | 0.0722 | ||
0.1540 | 0.0065 | 0.1416 | 0.1540 | 0.1667 | ||
0.1643 | 0.0083 | 0.1487 | 0.1641 | 0.1807 | ||
All zeros, | 0.0721 | 0.0123 | 0.0481 | 0.0721 | 0.0962 | |
0.0500 | 0.0032 | 0.0438 | 0.0500 | 0.0565 | ||
−0.1222 | 0.0108 | −0.1430 | −0.1220 | −0.1014 | ||
Married | All zeros, | −0.0416 | 0.0124 | −0.0666 | −0.0415 | −0.0174 |
0.0039 | 0.0013 | 0.0015 | 0.0038 | 0.0067 | ||
0.0131 | 0.0042 | 0.0053 | 0.0130 | 0.0218 | ||
0.0246 | 0.0070 | 0.0106 | 0.0247 | 0.0385 | ||
All zeros, | 0.0018 | 0.0118 | −0.0207 | 0.0018 | 0.0254 | |
0.0622 | 0.0031 | 0.0563 | 0.0622 | 0.0685 | ||
−0.0640 | 0.0114 | −0.0873 | −0.0640 | −0.0420 | ||
Muslim | All zeros, | 0.0154 | 0.0092 | −0.0029 | 0.0154 | 0.0331 |
−0.0013 | 0.0008 | −0.0028 | −0.0013 | 0.0002 | ||
−0.0044 | 0.0026 | −0.0093 | −0.0044 | 0.0008 | ||
−0.0097 | 0.0058 | −0.0211 | −0.0097 | 0.0018 | ||
All zeros, | 0.0126 | 0.0093 | −0.0051 | 0.0125 | 0.0313 | |
0.0613 | 0.0031 | 0.0555 | 0.0613 | 0.0675 | ||
−0.0739 | 0.0091 | −0.0922 | −0.0739 | −0.0562 | ||
Father use | All zeros, | 0.7604 | 0.0042 | 0.7521 | 0.7604 | 0.7684 |
0.0477 | 0.0027 | 0.0426 | 0.0477 | 0.0531 | ||
0.0982 | 0.0035 | 0.0915 | 0.0982 | 0.1051 | ||
0.0937 | 0.0032 | 0.0874 | 0.0936 | 0.1000 | ||
All zeros, | 0.8713 | 0.0039 | 0.8635 | 0.8713 | 0.8789 | |
0.0623 | 0.0030 | 0.0566 | 0.0623 | 0.0684 | ||
0.0664 | 0.0030 | 0.0607 | 0.0664 | 0.0724 | ||
Mother use | All zeros, | 0.7604 | 0.0042 | 0.7521 | 0.7604 | 0.7684 |
0.0477 | 0.0027 | 0.0426 | 0.0477 | 0.0531 | ||
0.0982 | 0.0035 | 0.0915 | 0.0982 | 0.1051 | ||
0.0937 | 0.0032 | 0.0874 | 0.0936 | 0.1000 | ||
All zeros, | 0.8713 | 0.0039 | 0.8635 | 0.8713 | 0.8789 | |
0.0623 | 0.0030 | 0.0566 | 0.0623 | 0.0684 | ||
0.0664 | 0.0030 | 0.0607 | 0.0664 | 0.0724 | ||
Region | All zeros, | −0.0075 | 0.0079 | −0.0229 | −0.0075 | 0.0080 |
0.0006 | 0.0007 | −0.0007 | 0.0006 | 0.0020 | ||
0.0022 | 0.0023 | −0.0023 | 0.0022 | 0.0067 | ||
0.0047 | 0.0049 | −0.0050 | 0.0047 | 0.0144 | ||
All zeros, | −0.0691 | 0.0078 | −0.0846 | −0.0691 | −0.0539 | |
0.0756 | 0.0038 | 0.0684 | 0.0755 | 0.0832 | ||
−0.0065 | 0.0070 | −0.0200 | −0.0065 | 0.0072 | ||
Urban | All zeros, | 0.0167 | 0.0087 | −0.0003 | 0.0167 | 0.0339 |
−0.0014 | 0.0008 | −0.0030 | −0.0014 | 0.0000 | ||
−0.0049 | 0.0026 | −0.0100 | −0.0049 | 0.0001 | ||
−0.0104 | 0.0054 | −0.0210 | −0.0104 | 0.0002 | ||
All zeros, | 0.0130 | 0.0088 | −0.0041 | 0.0129 | 0.0303 | |
0.0592 | 0.0036 | 0.0524 | 0.0591 | 0.0664 | ||
−0.0721 | 0.0074 | −0.0866 | −0.0721 | −0.0576 | ||
Agriservice | All zeros, | 0.0218 | 0.0088 | 0.0043 | 0.0219 | 0.0390 |
−0.0018 | 0.0007 | −0.0032 | −0.0018 | −0.0004 | ||
−0.0062 | 0.0025 | −0.0110 | −0.0062 | −0.0013 | ||
−0.0138 | 0.0057 | −0.0250 | −0.0139 | −0.0027 | ||
All zeros, | −0.0127 | 0.0119 | −0.0366 | −0.0127 | 0.0106 | |
0.0656 | 0.0043 | 0.0572 | 0.0655 | 0.0742 | ||
−0.0528 | 0.0094 | −0.0711 | −0.0529 | −0.0338 | ||
Self employed | All zeros, | 0.0277 | 0.0162 | −0.0039 | 0.0277 | 0.0592 |
−0.0028 | 0.0018 | −0.0065 | −0.0027 | 0.0003 | ||
−0.0087 | 0.0053 | −0.0194 | −0.0086 | 0.0012 | ||
−0.0163 | 0.0093 | −0.0335 | −0.0165 | 0.0025 | ||
All zeros, | −0.0290 | 0.0144 | −0.0578 | −0.0286 | −0.0017 | |
0.0686 | 0.0046 | 0.0600 | 0.0685 | 0.0779 | ||
−0.0396 | 0.0116 | −0.0617 | −0.0398 | −0.0162 | ||
Student | All zeros, | 0.1287 | 0.0155 | 0.0980 | 0.1286 | 0.1588 |
−0.0173 | 0.0030 | −0.0235 | −0.0171 | −0.0118 | ||
−0.0475 | 0.0069 | −0.0614 | −0.0475 | −0.0343 | ||
−0.0639 | 0.0063 | −0.0758 | −0.0640 | −0.0510 | ||
All zeros, | 0.0855 | 0.0151 | 0.0531 | 0.0866 | 0.1117 | |
0.0278 | 0.0071 | 0.0154 | 0.0274 | 0.0428 | ||
−0.1133 | 0.0088 | −0.1288 | −0.1140 | −0.0944 |
Using (
Income has opposite effects on probability of nonparticipation and zero-consumption, predicting on average that tobacco is an inferior good for nonparticipants and a normal good for participants. However, the 95% credible interval contains zero, suggesting that the effect of income is weak. Generally, the opposing effects on probabilities of nonparticipation and zeroconsumption would have repercussions on both the magnitude and the statistical significance of the full effect of observing zero-consumption. Similar considerations apply to positive levels of consumption since the marginal effect on probability of observing consumption level
In this paper we analyze the zero-inflated bivariate ordered probit model in a Bayesian framework. The underlying model arises as a mixture of a point mass distribution at
The proposed zero-inflated bivariate model is particularly useful when most of the bivariate ordered outcomes are zero
For more details see Tables
WinBUGS Code for Fitting the Proposed Models (see Algorithm
The authors thank Alfonso Flores-Lagunes, the editor, two anonymous referees and seminar participants at the Conference on Bayesian Inference in Econometrics and Statistics, the Joint Statistical Meetings, the Southern Economics Association Conference, and Syracuse University for useful comments. Mohammad Yunus graciously provided the data used in this paper.