Background
Method
Source of data and study design
Variables of the study
Independent variable
Statistical method
Zero inflated regression model
Zero-inflated Poisson regression model
Zero-inflated negative binomial regression model
Hurdle model
Tests for ZIP and ZINB regression models
Bayesian ZINB and negative binomial–logit hurdle model
The convergence of the algorithm
Results
Test of overdispersion
Likelihood ratio Test for nested models | ||||
---|---|---|---|---|
Model | Hypothesis | p-value | Preferable model | |
ZIP = PLH vs ZINB = NBLH | H0:\(\varnothing \) =0 VS H1:\(\varnothing \) >0 | < 2.2e−16 | ZINB = NBLH |
Vuong test for non-nested models | ||||
---|---|---|---|---|
Model | Hypothesis | Vuong statistics | p-value | Comment |
ZIP vs PLH | Model1 = model2 vs Model1 > model2 | 1.216179 | 0.11196 | ZIP = PLH |
ZINB vs NBLH | Model1 = model2 vs Model1 > model2 | 1.234438 | 0.10852 | ZINB = NBLH |
ZIP | PLH | ZINB | NBLH | |
---|---|---|---|---|
AIC | 7591.194 | 7591.697 | 7590.496 | 7590.006 |
Log-Likelihood(df) | − 3764.597 (df = 30) | − 3764.849 (df = 30) | − 3765.248 (df = 31) | − 3765.003 (df = 31) |
Results of Bayesian ZINB and NBLH Model
Count model coefficients (NBLH) | Count model coefficients (ZINB) | ||||||
---|---|---|---|---|---|---|---|
Parameters | Stationarity Test | P-Value | Half-width test | Parameters | Stationarity Test | P-Value | Half-width test |
C | C | ||||||
\(\varnothing \) | Passed | 0.517 | Passed | \(\varnothing \) | Passed | 0.24 | Passed |
\({\beta }_{0}\) | Passed | 0.456 | Passed | \({\beta }_{0}\) | Passed | 0.574 | Passed |
\({\beta }_{1}\) | Passed | 0.322 | Passed | \({\beta }_{1}\) | Passed | 0.38 | Passed |
\({\beta }_{2}\) | Passed | 0.263 | Passed | \({\beta }_{2}\) | Passed | 0.425 | Passed |
\({\beta }_{3}\) | Passed | 0.188 | Passed | \({\beta }_{3}\) | Passed | 0.354 | Passed |
\({\beta }_{4}\) | Passed | 0.289 | Passed | \({\beta }_{4}\) | Passed | 0.371 | Passed |
\({\beta }_{5}\) | Passed | 0.515 | Passed | \({\beta }_{5}\) | Passed | 0.226 | Passed |
\({\beta }_{6}\) | Passed | 0.677 | Passed | \({\beta }_{6}\) | Passed | 0.353 | Passed |
\({\beta }_{7}\) | Passed | 0.462 | Passed | \({\beta }_{7}\) | Passed | 0.645 | Passed |
\({\beta }_{8}\) | Passed | 0.51 | Passed | \({\beta }_{8}\) | Passed | 0.71 | Passed |
\({\beta }_{9}\) | Passed | 0.39 | Passed | \({\beta }_{9}\) | Passed | 0.586 | Passed |
\({\beta }_{10}\) | Passed | 0.397 | Passed | \({\beta }_{10}\) | Passed | 0.286 | Passed |
\({\beta }_{11}\) | Passed | 0.635 | Passed | \({\beta }_{11}\) | Passed | 0.386 | Passed |
\({\beta }_{12}\) | Passed | 0.516 | Passed | \({\beta }_{12}\) | Passed | 0.897 | Passed |
\({\beta }_{13}\) | Passed | 0.371 | Passed | \({\beta }_{13}\) | Passed | 0.582 | Passed |
\({\beta }_{14}\) | Passed | 0.172 | Passed | \({\beta }_{14}\) | Passed | 0.375 | Passed |
Zero hurdle model coefficients (binomial with logit link) | Zero-inflation model coefficients | ||||||
---|---|---|---|---|---|---|---|
\({\gamma }_{0}\) | Passed | 0.398 | Passed | \({\gamma }_{0}\) | Passed | 0.447 | Passed |
\({\gamma }_{1}\) | Passed | 0.272 | Passed | \({\gamma }_{1}\) | Passed | 0.595 | Passed |
\({\gamma }_{2}\) | Passed | 0.281 | Passed | \({\gamma }_{2}\) | Passed | 0.692 | Passed |
\({\gamma }_{3}\) | Passed | 0.094 | Passed | \({\gamma }_{3}\) | Passed | 0.258 | Passed |
\({\gamma }_{4}\) | Passed | 0.151 | Passed | \({\gamma }_{4}\) | Passed | 0.292 | Passed |
\({\gamma }_{5}\) | Passed | 0.294 | Passed | \({\gamma }_{5}\) | Passed | 0.524 | Passed |
\({\gamma }_{6}\) | Passed | 0.3 | Passed | \({\gamma }_{6}\) | Passed | 0.494 | Passed |
\({\gamma }_{7}\) | Passed | 0.489 | Passed | \({\gamma }_{7}\) | Passed | 0.464 | Passed |
\({\gamma }_{8}\) | Passed | 0.365 | Passed | \({\gamma }_{8}\) | Passed | 0.427 | Passed |
\({\gamma }_{9}\) | Passed | 0.218 | Passed | \({\gamma }_{9}\) | Passed | 0.299 | Passed |
\({\gamma }_{10}\) | Passed | 0.377 | Passed | \({\gamma }_{10}\) | Passed | 0.329 | Passed |
\({\gamma }_{11}\) | Passed | 0.378 | Passed | \({\gamma }_{11}\) | Passed | 0.446 | Passed |
\({\gamma }_{12}\) | Passed | 0.546 | Passed | \({\gamma }_{12}\) | Passed | 0.797 | Passed |
\({\gamma }_{13}\) | Passed | 0.658 | Passed | \({\gamma }_{13}\) | Passed | 0.387 | Passed |
\({\gamma }_{14}\) | Passed | 0.659 | Passed | \({\gamma }_{14}\) | Passed | 0.398 | Passed |
Application of Bayesian negative binomial-logit hurdle model
Bayesian count model coefficients
Para | Bayesian negative binomial-logit hurdle (BNBLH) | ||||||||
---|---|---|---|---|---|---|---|---|---|
Count model coefficients | Para | Zero hurdle model coefficients | |||||||
Mean | SD | HPD of 95% CrI | OR | Mean | SD | HPD of 95% CrI | OR | ||
\(\varnothing \) | 0.029 | 0.016 | (0.003, 0.065) | 1.029 | |||||
\({\upbeta }_{0}\) | 0.869 | 0.163 | (0.558, 1.189) | 2.385 | \({\upgamma }_{0}\) | − 3.299 | 0.298 | (− 3.902, − 2.753) | 0.037 |
\({\upbeta }_{1}\) | 0.026 | 0.085 | (− 0.139, 0.193) | 1.026 | \({\upgamma }_{1}\) | 1.022 | 0.156 | (0.721, 1.334) | 2.779 |
\({\upbeta }_{2}\) | 0.090 | 0.087 | (− 0.078, 0.257) | 1.094 | \({\upgamma }_{2}\) | 1.434 | 0.164 | (1.123, 1.766) | 4.195 |
\({\upbeta }_{3}\) | − 0.161 | 0.073 | (− 0.304, − 0.021) | 0.851 | \({\upgamma }_{3}\) | − 0.709 | 0.141 | (− 0.992, − 0.433) | 0.492 |
\({\upbeta }_{4}\) | − 0.035 | 0.050 | (− 0.133, 0.064) | 0.966 | \({\upgamma }_{4}\) | 0.369 | 0.102 | (0.174, 0.573) | 1.446 |
\({\upbeta }_{5}\) | 0.133 | 0.089 | (− 0.041, 0.305) | 1.142 | \({\upgamma }_{5}\) | − 0.518 | 0.149 | (− 0.812, − 0.223) | 0.596 |
\({\upbeta }_{6}\) | 0.223 | 0.078 | (0.075, 0.376) | 1.25 | \({\upgamma }_{6}\) | − 0.099 | 0.138 | (− 0.366, 0.179) | 0.906 |
\({\upbeta }_{7}\) | 0.056 | 0.067 | (− 0.077, 0.188) | 1.058 | \({\upgamma }_{7}\) | − 0.250 | 0.134 | (− 0.515, 0.007) | 0.779 |
\({\upbeta }_{8}\) | − 0.068 | 0.070 | (− 0.206, 0.064) | 0.934 | \({\upgamma }_{8}\) | − 0.313 | 0.131 | (− 0.569, − 0.057) | 0.731 |
\({\upbeta }_{9}\) | − 0.080 | 0.069 | (− 0.217, 0.054) | 0.923 | \({\upgamma }_{9}\) | − 0.887 | 0.137 | (− 1.162, − 0.624) | 0.412 |
\({\upbeta }_{10}\) | 0.027 | 0.094 | (− 0.156, 0.209) | 1.027 | \({\upgamma }_{10}\) | − 0.371 | 0.181 | (− 0.731, − 0.018) | 0.69 |
\({\upbeta }_{11}\) | 0.105 | 0.082 | (− 0.050, 0.271) | 1.111 | \({\upgamma }_{11}\) | − 0.249 | 0.151 | (− 0.547,0.045) | 0.780 |
\({\upbeta }_{12}\) | 0.221 | 0.065 | (0.096, 0.349) | 1.247 | \({\upgamma }_{12}\) | 2.121 | 0.112 | (1.908,2.345) | 8.339 |
\({\upbeta }_{13}\) | 0.149 | 0.065 | (0.025, 0.279) | 1.161 | \({\upgamma }_{13}\) | − 0.383 | 0.128 | (− 0.629, − 0.120) | 0.682 |
\({\upbeta }_{14}\) | 0.005 | 0.047 | (− 0.088, 0.098) | 1.005 | \({\upgamma }_{14}\) | − 0.227 | 0.095 | (− 0.410, − 0.041) | 0.797 |
Para | Bayesian zero-inflated negative binomial (BZINB) | ||||||||
---|---|---|---|---|---|---|---|---|---|
Posterior count model coefficients | Para | Posterior zero-inflation model coefficients | |||||||
Mean | SD | HPD of 95% CrI | OR | Mean | SD | HPD of 95% CrI | OR | ||
\(\varnothing \) | 0.029 | 0.016 | (0.003, 0.065) | 1.029 | \({\upgamma }_{0}\) | 3.210 | 0.302 | (2.570, 3.799) | 24.78 |
\({\upbeta }_{0}\) | 0.901 | 0.163 | (0.588, 1.215) | 2.462 | \({\upgamma }_{1}\) | − 1.021 | 0.156 | (− 1.339, − 0.725) | 0.360 |
\({\upbeta }_{1}\) | 0.019 | 0.087 | (− 0.154, 0.185) | 1.019 | \({\upgamma }_{2}\) | − 1.426 | 0.165 | (− 1.767, − 1.110) | 0.240 |
\({\upbeta }_{2}\) | 0.083 | 0.088 | (− 0.092, 0.251) | 1.087 | \({\upgamma }_{3}\) | 0.697 | 0.136 | (0.431, 0.965) | 2.008 |
\({\upbeta }_{3}\) | − 0.170 | 0.075 | (− 0.321, − 0.029) | 0.844 | \({\upgamma }_{4}\) | − 0.378 | 0.103 | (− 0.577, − 0.173) | 0.685 |
\({\upbeta }_{4}\) | − 0.036 | 0.051 | (− 0.135, 0.066) | 0.965 | \({\upgamma }_{5}\) | 0.529 | 0.152 | (0.238, 0.828) | 1.697 |
\({\upbeta }_{5}\) | 0.122 | 0.092 | (− 0.053, 0.306) | 1.13 | \({\upgamma }_{6}\) | 0.121 | 0.143 | (− 0.148, 0.408) | 1.129 |
\({\upbeta }_{6}\) | 0.213 | 0.082 | (0.059, 0.376) | 1.237 | \({\upgamma }_{7}\) | 0.254 | 0.132 | (− 0.007, 0.514) | 1.289 |
\({\upbeta }_{7}\) | 0.053 | 0.068 | (− 0.080, 0.186) | 1.054 | \({\upgamma }_{8}\) | 0.306 | 0.130 | (0.053, 0.563) | 1.358 |
\({\upbeta }_{8}\) | − 0.075 | 0.069 | (− 0.212, 0.059) | 0.928 | \({\upgamma }_{9}\) | 0.889 | 0.133 | (0.625, 1.152) | 2.433 |
\({\upbeta }_{9}\) | − 0.088 | 0.072 | (− 0.233, 0.052) | 0.916 | \({\upgamma }_{10}\) | 0.360 | 0.191 | (− 0.003, 0.748) | 1.433 |
\({\upbeta }_{10}\) | 0.011 | 0.096 | (− 0.175, 0.200) | 1.011 | \({\upgamma }_{11}\) | 0.251 | 0.156 | (− 0.050, 0.560) | 1.285 |
\({\upbeta }_{11}\) | 0.098 | 0.082 | (− 0.063, 0.261) | 1.103 | \({\upgamma }_{12}\) | − 2.096 | 0.113 | (− 2.315, − 1.875) | 0.123 |
\({\upbeta }_{12}\) | 0.221 | 0.065 | (0.098, 0.353) | 1.247 | \({\upgamma }_{13}\) | 0.406 | 0.127 | (0.151, 0.651) | 1.501 |
\({\upbeta }_{13}\) | 0.152 | 0.071 | (0.016, 0.293) | 1.164 | \({\upgamma }_{14}\) | 0.231 | 0.094 | (0.050, 0.413) | 1.26 |
\({\upbeta }_{14}\) | 0.004 | 0.047 | (-0.088, 0.098) | 1.004 |