IDEAS home Printed from https://ideas.repec.org/a/spr/jagbes/v24y2019i2d10.1007_s13253-019-00360-8.html
   My bibliography  Save this article

Quasi-beta Longitudinal Regression Model Applied to Water Quality Index Data

Author

Listed:
  • Ricardo Rasmussen Petterle

    (Paraná Federal University)

  • Wagner Hugo Bonat

    (Paraná Federal University)

  • Cassius Tadeu Scarpin

    (Paraná Federal University (UFPR))

Abstract

We propose a new class of regression models to deal with longitudinal continuous bounded data. The model is specified using second-moment assumptions, and we employ an estimating function approach for parameter estimation and inference. The main advantage of the proposed approach is that it does not need to assume a multivariate probability distribution for the response vector. The fitting procedure is easily implemented using a simple and efficient Newton scoring algorithm. Thus, the quasi-beta longitudinal regression model can easily handle data in the unit interval, including exact zeros and ones. The covariance structure is defined in terms of a matrix linear predictor composed by known matrices. A simulation study was conducted to check the properties of the estimating function estimators of the regression and dispersion parameter estimators. The NORTA algorithm (NORmal To Anything) was used to simulate correlated beta random variables. The results of this simulation study showed that the estimators are consistent and unbiased for large samples. The model is motivated by a data set concerning the water quality index, whose goal is to investigate the effect of dams on the water quality index measured on power plant reservoirs. Furthermore, diagnostic techniques were adapted to the proposed models, such as DFFITS, DFBETAS, Cook’s distance and half-normal plots with simulated envelope. The R code and data set are available in the supplementary material.

Suggested Citation

  • Ricardo Rasmussen Petterle & Wagner Hugo Bonat & Cassius Tadeu Scarpin, 2019. "Quasi-beta Longitudinal Regression Model Applied to Water Quality Index Data," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 24(2), pages 346-368, June.
    • Handle: RePEc:spr:jagbes:v:24:y:2019:i:2:d:10.1007_s13253-019-00360-8
    DOI: 10.1007/s13253-019-00360-8
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s13253-019-00360-8
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s13253-019-00360-8?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Pablo Mitnik & Sunyoung Baek, 2013. "The Kumaraswamy distribution: median-dispersion re-parameterizations for regression modeling and simulation-based estimation," Statistical Papers, Springer, vol. 54(1), pages 177-192, February.
    2. Zhenguo Qiu & Peter X.‐K. Song & Ming Tan, 2008. "Simplex Mixed‐Effects Models for Longitudinal Proportional Data," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 35(4), pages 577-596, December.
    3. Peter Xue-Kun Song & Ming Tan, 2000. "Marginal Models for Longitudinal Continuous Proportional Data," Biometrics, The International Biometric Society, vol. 56(2), pages 496-502, June.
    4. Wagner Hugo Bonat & Bent Jørgensen, 2016. "Multivariate covariance generalized linear models," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 65(5), pages 649-675, November.
    5. da-Silva, C.Q. & Migon, H.S. & Correia, L.T., 2011. "Dynamic Bayesian beta models," Computational Statistics & Data Analysis, Elsevier, vol. 55(6), pages 2074-2089, June.
    6. Barndorff-Nielsen, O. E. & Jørgensen, B., 1991. "Some parametric models on the simplex," Journal of Multivariate Analysis, Elsevier, vol. 39(1), pages 106-116, October.
    7. Siti Haslinda Mohd Din & Marek Molas & Jolanda Luime & Emmanuel Lesaffre, 2014. "Longitudinal profiles of bounded outcome scores as predictors for disease activity in rheumatoid arthritis patients: a joint modeling approach," Journal of Applied Statistics, Taylor & Francis Journals, vol. 41(8), pages 1627-1644, August.
    8. Bent Jørgensen & Sven Jesper Knudsen, 2004. "Parameter Orthogonality and Bias Adjustment for Estimating Functions," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 31(1), pages 93-114, March.
    9. Silvia Ferrari & Francisco Cribari-Neto, 2004. "Beta Regression for Modelling Rates and Proportions," Journal of Applied Statistics, Taylor & Francis Journals, vol. 31(7), pages 799-815.
    10. Figueroa-Zúñiga, Jorge I. & Arellano-Valle, Reinaldo B. & Ferrari, Silvia L.P., 2013. "Mixed beta regression: A Bayesian perspective," Computational Statistics & Data Analysis, Elsevier, vol. 61(C), pages 137-147.
    11. Ed McKenzie, 1985. "An Autoregressive Process for Beta Random Variables," Management Science, INFORMS, vol. 31(8), pages 988-997, August.
    12. W. H. Bonat & J. Olivero & M. Grande-Vega & M. A. Farfán & J. E. Fa, 2017. "Modelling the Covariance Structure in Marginal Multivariate Count Models: Hunting in Bioko Island," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 22(4), pages 446-464, December.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Josmar Mazucheli & Bruna Alves & Mustafa Ç. Korkmaz & Víctor Leiva, 2022. "Vasicek Quantile and Mean Regression Models for Bounded Data: New Formulation, Mathematical Derivations, and Numerical Applications," Mathematics, MDPI, vol. 10(9), pages 1-23, April.
    2. Patrícia L. Espinheira & Alisson Oliveira Silva, 2020. "Residual and influence analysis to a general class of simplex regression," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 29(2), pages 523-552, June.
    3. Aknouche, Abdelhakim & Dimitrakopoulos, Stefanos, 2021. "Autoregressive conditional proportion: A multiplicative-error model for (0,1)-valued time series," MPRA Paper 110954, University Library of Munich, Germany, revised 06 Dec 2021.
    4. Wagner Hugo Bonat & Paulo Justiniano Ribeiro & Walmes Marques Zeviani, 2015. "Likelihood analysis for a class of beta mixed models," Journal of Applied Statistics, Taylor & Francis Journals, vol. 42(2), pages 252-266, February.
    5. Phillip Li, 2018. "Efficient MCMC estimation of inflated beta regression models," Computational Statistics, Springer, vol. 33(1), pages 127-158, March.
    6. Hui Song & Yingwei Peng & Dongsheng Tu, 2017. "Jointly modeling longitudinal proportional data and survival times with an application to the quality of life data in a breast cancer trial," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 23(2), pages 183-206, April.
    7. Guillermo Ferreira & Jorge Figueroa-Zúñiga & Mário Castro, 2015. "Partially linear beta regression model with autoregressive errors," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(4), pages 752-775, December.
    8. Ceren Eda Can & Gul Ergun & Refik Soyer, 2022. "Bayesian Analysis of Proportions via a Hidden Markov Model," Methodology and Computing in Applied Probability, Springer, vol. 24(4), pages 3121-3139, December.
    9. Jay Verkuilen & Michael Smithson, 2012. "Mixed and Mixture Regression Models for Continuous Bounded Responses Using the Beta Distribution," Journal of Educational and Behavioral Statistics, , vol. 37(1), pages 82-113, February.
    10. Francesca Condino & Filippo Domma, 2023. "Unit Distributions: A General Framework, Some Special Cases, and the Regression Unit-Dagum Models," Mathematics, MDPI, vol. 11(13), pages 1-25, June.
    11. Wenting Liu & Huiqiong Li & Anmin Tang & Zixin Cui, 2023. "Bayesian Joint Modeling Analysis of Longitudinal Proportional and Survival Data," Mathematics, MDPI, vol. 11(16), pages 1-17, August.
    12. Abdelhakim Aknouche & Stefanos Dimitrakopoulos, 2023. "Autoregressive conditional proportion: A multiplicative‐error model for (0,1)‐valued time series," Journal of Time Series Analysis, Wiley Blackwell, vol. 44(4), pages 393-417, July.
    13. Rashad A. R. Bantan & Christophe Chesneau & Farrukh Jamal & Mohammed Elgarhy & Muhammad H. Tahir & Aqib Ali & Muhammad Zubair & Sania Anam, 2020. "Some New Facts about the Unit-Rayleigh Distribution with Applications," Mathematics, MDPI, vol. 8(11), pages 1-23, November.
    14. Maria Victoria Ibañez & Marina Martínez-Garcia & Amelia Simó, 2021. "A Review of Spatiotemporal Models for Count Data in R Packages. A Case Study of COVID-19 Data," Mathematics, MDPI, vol. 9(13), pages 1-23, July.
    15. Carolina Costa Mota Paraíba & Natalia Bochkina & Carlos Alberto Ribeiro Diniz, 2018. "Bayesian truncated beta nonlinear mixed-effects models," Journal of Applied Statistics, Taylor & Francis Journals, vol. 45(2), pages 320-346, January.
    16. Fabrizi, Enrico & Trivisano, Carlo, 2016. "Small area estimation of the Gini concentration coefficient," Computational Statistics & Data Analysis, Elsevier, vol. 99(C), pages 223-234.
    17. Zhou, Haiming & Huang, Xianzheng, 2022. "Bayesian beta regression for bounded responses with unknown supports," Computational Statistics & Data Analysis, Elsevier, vol. 167(C).
    18. Acocella, Angela & Caplice, Chris & Sheffi, Yossi, 2020. "Elephants or goldfish?: An empirical analysis of carrier reciprocity in dynamic freight markets," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 142(C).
    19. Zhenguo Qiu & Peter X.‐K. Song & Ming Tan, 2008. "Simplex Mixed‐Effects Models for Longitudinal Proportional Data," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 35(4), pages 577-596, December.
    20. Jorge I. Figueroa-Zúñiga & Cristian L. Bayes & Víctor Leiva & Shuangzhe Liu, 2022. "Robust beta regression modeling with errors-in-variables: a Bayesian approach and numerical applications," Statistical Papers, Springer, vol. 63(3), pages 919-942, June.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jagbes:v:24:y:2019:i:2:d:10.1007_s13253-019-00360-8. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.