We propose a multivariate regression model to handle multiple continuous bounded outcomes. We adopted the maximum likelihood approach for parameter estimation and inference. The model is specified by the product of univariate probability distributions and the correlation between the response variables is obtained through the correlation matrix of the random intercepts. For modeling continuous bounded variables on the interval we considered the beta and unit gamma distributions. The main advantage of the proposed model is that we can easily combine different marginal distributions for the response variable vector. The computational implementation is performed using Template Model Builder, which combines the Laplace approximation with automatic differentiation. Therefore, the proposed approach allows us to estimate the model parameters quickly and efficiently. We conducted a simulation study to evaluate the computational implementation and the properties of the maximum likelihood estimators under different scenarios. Moreover, we investigate the impact of distribution misspecification in the proposed model. Our model was motivated by a data set with multiple continuous bounded outcomes, which refer to the body fat percentage measured at five regions of the body. Simulation studies and data analysis showed that the proposed model provides a general and rich framework to deal with multiple continuous bounded outcomes.
RanasingheCGamagePKatulandaP, et al. Relationship between body mass index (BMI) and body fat percentage, estimated by bioelectrical impedance, in a group of Sri Lankan adults: A cross sectional study. BMC Public Health2013; 13: 797.
2.
AhimaRSLazarMA. The health risk of obesity-better metrics imperative. Science2013; 341: 856–858.
3.
De SchutterALavieCJMilaniRV. The impact of obesity on risk factors and prevalence and prognosis of coronary heart disease–the obesity paradox. Prog Cardiovasc Dis2014; 56: 401–408.
4.
JacksonAStanforthPGagnonJ, et al. The effect of sex, age and race on estimating percentage body fat from body mass index: The heritage family study. Int J Obes2002; 26: 789.
5.
OliveiraCCdRorizAKCMoreiraPdA, et al. Anthropometric indicators associated with hypertriglyceridemia in the prediction of visceral fat. Rev Bras de Cineantropometria e Desempenho Hum2014; 16: 485–493.
6.
PadwalRLeslieWDLixLM, et al. Relationship among body fat percentage, body mass index, and all-cause mortality: A cohort study. Ann Intern Med2016; 164: 532–541.
7.
BradburyKEGuoWCairnsBJ, et al. Association between physical activity and body fat percentage, with adjustment for BMI: A large cross-sectional analysis of UK Biobank. BMJ Open2017; 7: 1–9. https://bmjopen.bmj.com/content/7/3/e011843.abstract.
8.
SwiftDLJohannsenNMLavieCJ, et al. The role of exercise and physical activity in weight loss and maintenance. Prog Cardiovasc Dis2014; 56: 441–447.
9.
MyersAGibbonsCFinlaysonG, et al. Associations among sedentary and active behaviours, body fat and appetite dysregulation: Investigating the myth of physical inactivity and obesity. Br J Sports Med2017; 51: 1540–1544.
10.
GalvisDMBandyopadhyayDLachosVH. Augmented mixed beta regression models for periodontal proportion data. Stat Med2014; 33: 3759–3771.
11.
HungerMDöringAHolleR. Longitudinal beta regression models for analyzing health-related quality of life scores over time. BMC Med Res Methodol2012; 12: 144.
12.
CanchoVGBazánJLDeyDK. A new class of regression model for a bounded response with application in the study of the incidence rate of colorectal cancer. Stat Methods Med Res2020; 29: 2015–2033.
13.
FerrariSCribari-NetoF. Beta regression for modelling rates and proportions. J Appl Stat2004; 31: 799–815.
14.
EspinheiraPLFerrariSLCribari-NetoF. On beta regression residuals. J Appl Stat2008; 35: 407–419.
15.
SimasABBarreto-SouzaWRochaAV. Improved estimators for a general class of beta regression models. Comput Stat Data Anal2010; 54: 348–366.
16.
RochaASimasA. Influence diagnostics in a general class of beta regression models. TEST2011; 20: 95–119.
17.
Barndorff-NielsenOEJørgensenB. Some parametric models on the simplex. J Multivar Anal1991; 39: 106–116.
18.
KieschnickRMcCulloughBD. Regression analysis of variates observed on (0, 1): Percentages, proportions and fractions. Stat Model2003; 3: 193–213.
19.
BandyopadhyayDGalvisDMLachosVH. Augmented mixed models for clustered proportion data. Stat Methods Med Res2017; 26: 880–897.
20.
BonatWHLopesJEShimakuraSE, et al. Likelihood analysis for a class of simplex mixed models. Chil J Stat2018; 9: 3–17.
21.
MitnikPABaekS. The Kumaraswamy distribution: Median-dispersion re-parameterizations for regression modeling and simulation-based estimation. Stat Pap2013; 54: 177–192.
22.
MazucheliJMenezesAFernandesL, et al. The unit-Weibull distribution as an alternative to the Kumaraswamy distribution for the modeling of quantiles conditional on covariates. J Appl Stat2020; 47: 954–974.
23.
MazucheliJBapatSRMenezesAFB. A new one-parameter unit-lindley distribution. Chil J Stat2020; 11: 53–67.
24.
MousaAMEl-SheikhAAAbdel-FattahMA. A gamma regression for bounded continuous variables. Adv Appl Stat2016; 49: 305.
25.
LemonteAJBazánJL. New class of Johnson distributions and its associated regression model for rates and proportions. Biom J2016; 58: 727–746.
26.
ManuguerraMHellerGZMaJ. Continuous ordinal regression for analysis of visual analogue scales: The R package ordinalcont. J Stat Softw2020; 96: 1–25.
27.
LohseTRohrmannSFaehD, et al. Continuous outcome logistic regression for analyzing body mass index distributions. F1000Research2017; 6: 1933.
28.
LemonteAJMoreno-ArenasG. On a multivariate regression model for rates and proportions. J Appl Stat2019; 46: 1084–1106.
29.
Cepeda-CuervoEAchcarJALoperaLG. Bivariate beta regression models: Joint modeling of the mean, dispersion and association parameters. J Appl Stat2014; 41: 677–687.
30.
SouzaDFMouraFA. Multivariate beta regression with application in small area estimation. J Off Stat2016; 32: 747.
31.
PetterleRRBonatWHScarpinCT, et al. Multivariate quasi-beta regression models for continuous bounded data. Int J Biostat2021; 17: 39–53.
32.
GebregziabherMZhaoYDismukeC, et al. Joint modeling of multiple longitudinal cost outcomes using multivariate generalized linear mixed models. Health Serv Outcomes Res Methodol2013; 13: 39–57.
33.
JaffaMAGebregziabherMJaffaAA. Analysis of multivariate longitudinal kidney function outcomes using generalized linear mixed models. J Transl Med2015; 13: 1–12.
34.
JaffaMAGebregziabherMLuttrellDK, et al. Multivariate generalized linear mixed models with random intercepts to analyze cardiovascular risk markers in type-1 diabetic patients. J Appl Stat2016; 43: 1447–1464.
35.
GebregziabherMEckertMAMatthewsLJ, et al. Joint modeling of multivariate hearing thresholds measured longitudinally at multiple frequencies. Commun Stat-Theory Methods2018; 47: 5418–5434.
36.
R Core TeamR: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria, 2020. https://www.R-project.org/.
37.
KristensenKNielsenABergCW, et al. TMB: Automatic differentiation and Laplace approximation. J Stat Softw2016; 70: 1–21.
38.
TierneyLKadaneJB. Accurate approximations for posterior moments and marginal densities. J Am Stat Assoc1986; 81: 82–86.
39.
LiuQPierceDA. A note on Gauss–Hermite quadrature. Biometrika1994; 81: 624–629.
40.
BreslowNELinX. Bias correction in generalised linear mixed models with a single component of dispersion. Biometrika1995; 82: 81–91.
NikuJWartonDIHuiFK, et al. Generalized linear latent variable models for multivariate count and biomass data in ecology. J Agric Biol Environ Stat2017; 22: 498–522.
44.
NikuJBrooksWHerliansyahR, et al. Efficient estimation of generalized linear latent variable models. PLoS ONE2019; 14: e0216129.
45.
ThorsonJTIanelliJNLarsenEA, et al. Joint dynamic species distribution models: A tool for community ordination and spatio-temporal monitoring. Glob Ecol Biogeogr2016; 25: 1144–1158.
46.
JonsenIMcMahonCPattersonT, et al. Movement responses to environment: Fast inference of variation among southern elephant seals with a mixed effects model. Ecology2019; 100: e02566.
FournierDASkaugHJAnchetaJ, et al. AD model builder: Using automatic differentiation for statistical inference of highly parameterized complex nonlinear models. Optim Methods Softw2012; 27: 233–249.
PinheiroJCChaoEC. Efficient Laplacian and adaptive Gaussian quadrature algorithms for multilevel generalized linear mixed models. J Comput Graph Stat2006; 15: 58–81.
54.
ClaassenEA. A reduced bias method of estimating variance components on generalized linear mixed models. PhD Dissertation, University of Nebraska, Lincoln, NE, 2014.
55.
HandayaniDNotodiputroKASadikK, et al. A comparative study of approximation methods for maximum likelihood estimation in generalized linear mixed models (GLMM). In: AIP conference proceedings, vol. 1827, 2014, p. 020033. AIP Publishing LLC https://doi.org/10.1063/1.4979449
56.
SmithHGnanadesikanRHughesJB. Multivariate analysis of variance (MANOVA). Biometrics1962; 18: 22–41.
57.
BonatWHPetterleRRBalbinotP, et al. Modelling multiple outcomes in repeated measures studies: Comparing aesthetic eyelid surgery techniques. Stat Model2020; 0. (Online). https://doi.org/10.1177/1471082X20943312
Supplementary Material
Please find the following supplemental material available below.
For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.
For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.
