Main Article Content
Abstract
In this paper, we use a regression model for modeling bounded outcome scores (BOS), where the outcome is Kumaraswamy distributed. Similar to the Beta distribution, this distribution can take a variety of shapes while being computationally easier to use. Thus, it is deemed as a suitable alternative distribution to the Beta in modeling bounded random processes. In the proposed model, the median of a bounded response is modeled by the linear predictors which is defined through regression parameters and explanatory variables. We obtained the maximum likelihood estimates (MLE) of the parameters, provided closed-form expressions for the score functions and Fisher information matrix, and presented some diagnostic measures. We conducted Monte Carlo simulations to investigate the finite-sample performance of the MLEs of the parameters. Finally, two practical applications of this model to the real data sets are presented and discussed.
Keywords
Article Details

This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
References
-
Arostegui, I., Núñez‐Antón, V., & Quintana, J. M. (2007). Analysis of the short form‐36 (SF‐36): the beta‐binomial distribution approach. Statistics in medicine, 26(6), 1318-1342.
Atkinson, A. C., & Atkinson, A. C. (1985). Plots, transformations and regression; an introduction to graphical methods of diagnostic regression analysis. Retrieved from
Bottai, M., Cai, B., & McKeown, R. E. (2010). Logistic quantile regression for bounded outcomes. Statistics in medicine, 29(2), 309-317.
Cordeiro, G. M., Nadarajah, S., & Ortega, E. M. (2012). The Kumaraswamy Gumbel distribution. Statistical Methods & Applications, 21(2), 139-168.
Cordeiro, G. M., Ortega, E. M., & Nadarajah, S. (2010). The Kumaraswamy Weibull distribution with application to failure data. Journal of the Franklin Institute, 347(8), 1399-1429.
Cramer, J. A., Perrine, K., Devinsky, O., Bryant‐Comstock, L., Meador, K., & Hermann, B. (1998). Development and cross‐cultural translations of a 31‐item quality of life in epilepsy inventory. Epilepsia, 39(1), 81-88.
de Pascoa, M. A., Ortega, E. M., & Cordeiro, G. M. (2011). The Kumaraswamy generalized gamma distribution with application in survival analysis. Statistical Methodology, 8(5), 411-433.
Ferrari, S., & Cribari-Neto, F. (2004). Beta regression for modelling rates and proportions. Journal of Applied Statistics, 31(7), 799-815.
Griffiths, W., Carter Hill, R., & Judge, G. G. (1993). Learning and practicing econometrics.
Hu, C., Yeilding, N., Davis, H. M., & Zhou, H. (2011). Bounded outcome score modeling: application to treating psoriasis with ustekinumab. Journal of pharmacokinetics and pharmacodynamics, 38(4), 497-517.
Hunger, M., Baumert, J., & Holle, R. (2011). Analysis of SF-6D index data: is beta regression appropriate? Value in Health, 14(5), 759-767.
Hunger, M., Döring, A., & Holle, R. (2012). Longitudinal beta regression models for analyzing health-related quality of life scores over time. BMC medical research methodology, 12(1), 144.
Hutton, J. L., & Stanghellini, E. (2011). Modelling bounded health scores with censored skew‐normal distributions. Statistics in medicine, 30(4), 368-376.
Jones, M. (2009). Kumaraswamy’s distribution: A beta-type distribution with some tractability advantages. Statistical Methodology, 6(1), 70-81.
Kieschnick, R., & McCullough, B. D. (2003). Regression analysis of variates observed on (0, 1): percentages, proportions and fractions. Statistical modelling, 3(3), 193-213.
Kızılaslan, F., & Nadar, M. (2016). Estimation and prediction of the Kumaraswamy distribution based on record values and inter-record times. Journal of Statistical Computation and Simulation, 86(12), 2471-2493.
Kumaraswamy, P. (1980). A generalized probability density function for double-bounded random processes. Journal of Hydrology, 46(1-2), 79-88.
Lesaffre, E., Rizopoulos, D., & Tsonaka, R. (2006). The logistic transform for bounded outcome scores. Biostatistics, 8(1), 72-85.
McCullagh, P., & Nelder, J. A. (1989). Generalized Linear Models, no. 37 in Monograph on Statistics and Applied Probability: Chapman & Hall.
Mitnik, P. A., & Baek, S. (2013). The Kumaraswamy distribution: median-dispersion re-parameterizations for regression modeling and simulation-based estimation. Statistical Papers, 1-16.
Nocedal, J., & Wright, S. (1999). Numerical Optimization Springer-Verlag. New York.
Press, W., Teukolsky, S., Vetterling, W., & Flannery, B. (2007). Section 10.9: Quasi-newton or variable metric methods in multidimensions. Numerical recipes The art of scientific computing.
Smithson, M., & Verkuilen, J. (2006). A better lemon squeezer? Maximum-likelihood regression with beta-distributed dependent variables. Psychological methods, 11(1), 54.
Verkuilen, J., & Smithson, M. (2012). Mixed and mixture regression models for continuous bounded responses using the beta distribution. Journal of Educational and Behavioral Statistics, 37(1), 82-113.
Wewers, M. E., & Lowe, N. K. (1990). A critical review of visual analogue scales in the measurement of clinical phenomena. Research in nursing & health, 13(4), 227-236.
Xu, X. S., Samtani, M., Yuan, M., & Nandy, P. (2014). Modeling of bounded outcome scores with data on the boundaries: Application to disability assessment for dementia scores in Alzheimer’s disease. The AAPS journal, 16(6), 1271-1281.
Xu, X. S., Samtani, M. N., Dunne, A., Nandy, P., Vermeulen, A., De Ridder, F., & Initiative, A. s. D. N. (2013). Mixed-effects beta regression for modeling continuous bounded outcome scores using NONMEM when data are not on the boundaries. Journal of pharmacokinetics and pharmacodynamics, 40(4), 537-544.