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**Gaussian copula joint models to analysis mixed correlated longitudinal count and continuous responses.**
*(English)*
Zbl 07532211

Summary: Gaussian Copula joint models for analyzing mixed correlated longitudinal continuous and count responses with random effects are presented where the count response is defined by a latent variable approach, and its distribution is a member of the power series family of distributions. A copula-based joint model is proposed that accounts for associations between count and continuous responses. A full likelihood-based inference method for estimation is used by which maximum likelihood estimation of parameters is obtained. To illustrate the utility of the models, some simulation studies are performed. Finally, the proposed models are motivated by analyzing a medical data set where the correlated responses are the number of joint damaged obtained from the severity of osteoporosis (count response) and Body Mass Index (continuous response). Effects of some covariates on responses are investigated simultaneously. Besides, identifiability of parameters, detection of outlines by examining residuals and sensitivity analyses are fully investigated.

### MSC:

62-XX | Statistics |

### Keywords:

Gaussian Copula; mixed correlated responses; latent variables; longitudinal responses; random effect
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\textit{F. Rezaee} et al., Commun. Stat., Theory Methods 50, No. 23, 5499--5516 (2021; Zbl 07532211)

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### References:

[1] | Bahrami Samani, E.; Ganjali, M.; Khodaddadi, A., A latent variable model for mixed continuous and ordinal responses, Journal of Statistical Theory and Applications, 7, 3, 337-49 (2008) |

[2] | Catalano, P. J.; Ryan, L. M., Bivariate latent variable models for clustered discrete and continuous outcomes, Journal of the American Statistical Association, 87, 419, 651-58 (1992) |

[3] | Cook, R. D., Assessment of local influence, Journal of the Royal Statistical Society: Series B (Methodological), 48, 2, 133-69 (1986) · Zbl 0608.62041 |

[4] | Cox, D. R.; Wermuth, N., Response models for mixed binary and quantitative variables, Biometrika, 79, 3, 441-61 (1992) · Zbl 0766.62042 |

[5] | de Leon, A. R.; Carriègre, K., General mixed-data model: Extension of general location and grouped continuous models, Canadian Journal of Statistics, 35, 4, 533-48 (2007) · Zbl 1143.62323 |

[6] | de Leon, A. R.; Wu, B., Copula-based regression models for a bivariate mixed discrete and continuous outcome, Statistics in Medicine, 30, 2, 175-85 (2011) |

[7] | Embrechts, P.; Lindskog, F.; McNeil, A. (2001) |

[8] | Fletcher, R., Practical methods of optimization (1987), New York: John Wiley & Sons, New York · Zbl 0905.65002 |

[9] | Frey, R.; McNeil, A. J.; Nyfeler, M., Copulas and credit models, Risk, 10, 1, 111-14 (2001) |

[10] | Gueorguieva, R.; Sanacora, G., Joint analysis of repeatedly observed continuous and ordinal measures of disease severity, Statistics in Medicine, 25, 8, 1307-22 (2006) |

[11] | Gueorguieva, R. V.; Agresti, A., A correlated probit model for joint modeling of clustered binary and continuous responses, Journal of the American Statistical Association, 96, 455, 1102-12 (2001) · Zbl 1072.62612 |

[12] | Gunawan, D.; Khaled, M. A.; Kohn, R., Mixed marginal copula modeling, Journal of Business & Economic Statistics, 137-147 (2020) |

[13] | He, J.; Li, H.; Edmondson, A. C.; Rader, D. J.; Li, M., A gaussian copula approach for the analysis of secondary phenotypes in case-control genetic association studies, Biostatistics, 13, 3, 497-508 (2012) · Zbl 1244.62156 |

[14] | Heckman, J. J.1977. Dummy endogenous variables in a simultaneous equation system. Econometrica46 (4):931-60. · Zbl 0382.62095 |

[15] | Jafari, N.; Tabrizi, E.; Samani, E. B., Gaussian copula mixed models with non-ignorable missing outcomes, Applications & Applied Mathematics, 10, 1, 39-56 (2015) · Zbl 1326.62121 |

[16] | Jiryaie, F.; Withanage, N.; Wu, B.; De Leon, A., Gaussian copula distributions for mixed data, with application in discrimination, Journal of Statistical Computation and Simulation, 86, 9, 1643-59 (2016) · Zbl 07184692 |

[17] | Lee, H. E., Bayesian model comparison, Copula analysis of correlated counts, 325-48 (2014), UK: Emerald Group Publishing Limited |

[18] | McCulloch, C., Joint modelling of mixed outcome types using latent variables, Statistical Methods in Medical Research, 17, 1, 53-73 (2008) · Zbl 1154.62339 |

[19] | Nelsen, R. B., An introduction to copulas (2007), New York: Springer Science & Business Media, New York |

[20] | Olkin, I.; Tate, R. F., Multivariate correlation models with mixed discrete and continuous variables, The Annals of Mathematical Statistics, 32, 2, 448-65 (1961) · Zbl 0113.35101 |

[21] | Poon, W.-Y.; Lee, S.-Y., Maximum likelihood estimation of multivariate polyserial and polychoric correlation coefficients, Psychometrika, 52, 3, 409-30 (1987) · Zbl 0627.62060 |

[22] | Samani, E.; Tahmasebinejad, Z., Joint modelling of mixed correlated nominal, ordinal and continuous responses, Journal of Statistical Research, 45, 1, 37-47 (2011) |

[23] | Sammel, M. D.; Ryan, L. M.; Legler, J. M., Latent variable models for mixed discrete and continuous outcomes, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 59, 3, 667-78 (1997) · Zbl 0889.62043 |

[24] | Shemyakin, A.; Youn, H., Copula models of joint last survivor analysis, Applied Stochastic Models in Business and Industry, 22, 2, 211-24 (2006) · Zbl 1127.62091 |

[25] | Sklar, M., Fonctions de repartition an dimensions et leurs marges, Publications de l’Institut de Statistique de L’Université de Paris, 8, 229-31 (1959) · Zbl 0100.14202 |

[26] | Song, P. X.-K.; Li, M.; Yuan, Y., Joint regression analysis of correlated data using gaussian copulas, Biometrics, 65, 1, 60-68 (2009) · Zbl 1159.62049 |

[27] | Tang, X.-S.; Li, D.-Q.; Rong, G.; Phoon, K.-K.; Zhou, C.-B., Impact of copula selection on geotechnical reliability under incomplete probability information, Computers and Geotechnics, 49, 264-78 (2013) |

[28] | Tang, X.-S.; Li, D.-Q.; Zhou, C.-B.; Phoon, K.-K., Copula-based approaches for evaluating slope reliability under incomplete probability information, Structural Safety, 52, 90-99 (2015) |

[29] | Tang, X.-S.; Li, D.-Q.; Zhou, C.-B.; Phoon, K.-K.; Zhang, L.-M., Impact of copulas for modeling bivariate distributions on system reliability, Structural Safety, 44, 80-90 (2013) |

[30] | Tate, R. F., Correlation between a discrete and a continuous variable. point-biserial correlation, The Annals of Mathematical Statistics, 25, 3, 603-7 (1954) · Zbl 0056.36702 |

[31] | Tate, R. F., Applications of correlation models for biserial data, Journal of the American Statistical Association, 50, 272, 1078-95 (1955) · Zbl 0066.13103 |

[32] | Tate, R. F., The theory of correlation between two continuous variables when one is dichotomized, Biometrika, 42, 1-2, 205-16 (1955) · Zbl 0065.12901 |

[33] | Teixeira-Pinto, A.; Normand, S.-L T., Correlated bivariate continuous and binary outcomes: Issues and applications, Statistics in Medicine, 28, 13, 1753-73 (2009) |

[34] | Tutz, G., Modelling of repeated ordered measurements by isotonic sequential regression, Statistical Modelling: An International Journal, 5, 4, 269-87 (2005) · Zbl 1092.62131 |

[35] | Verbeke, G., Linear mixed models in practice, Linear mixed models for longitudinal data, 63-153 (1997), New York: Springer, New York |

[36] | Wu, B.; de Leon, A. R., Gaussian copula mixed models for clustered mixed outcomes, with application in developmental toxicology, Journal of Agricultural, Biological, and Environmental Statistics, 19, 1, 39-56 (2014) · Zbl 1303.62099 |

[37] | Yang, Y.; Kang, J.; Mao, K.; Zhang, J., Regression models for mixed poisson and continuous longitudinal data, Statistics in Medicine, 26, 20, 3782-800 (2007) |

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