García-Saavedra, Yuri M.

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    Modelos no lineales mixtos con variables de respuesta con distribución beta
    (2012-06) García-Saavedra, Yuri M.; Macchiavelli, Raúl E.; College of Arts and Sciences - Sciences; Santana Morant, Dámaris; Acuña Fernández, Edgar; Department of Mathematics; Canabal Torres, María Y.
    There are many situations in which random variables involving some kind of percentage, ratio, or fraction. Many studies have shown that linear regression models are not appropriate to model this type of data. However, the beta distribution is very useful for modeling data that are continuous and restricted to the interval (0,1), and also can be explained by other variables through a regression structure. Due to this, Ferrari and Cribari-Neto (2004) proposed a beta linear regression model in which the response variable is distributed as a beta using a different parameterization of its density function, thus obtaining a regression structure for the mean of the response with a constant precision parameter. In this study we extend this theory, proposing a beta nonlinear mixed regression model, where the conditional distribution of observations is assumed beta and the distribution of the random effects is assumed normal. We study the induced marginal distribution and the properties of the model proposed by means of simulation, which are compared with those obtained by a nonlinear mixed regression model assuming normal distribution. The estimates are obtained using the maximum likelihood, until quasi-Newton technique for the optimization and Gaussian quadrature for the integration. Standard errors of the model parameters were estimated using the Hessian matrix. Finally, we apply these results to studies of disease severity (relative amount of affected tissue in a given time) in plantain crops in Puerto Rico (2006-2007), which usually estimates the percentage of leaf area affected. Nonlinear models yield a better fit of the disease progress. Furthermore, since the parameters of the nonlinear curve vary, this variability is reflected with the inclusion of one or two random effects plant. This generates a correlation between observations from the same plant, so implicitly the correlation between repeated measures are incorporated into modeled.