Isotonic regression analysis of Guzerá cattle growth curves

Submitted on November 30 , 2015 and accepted on November 10 , 2017. 1 Work funded by FAPEMIG (PPM-00563-16) and CNPq (445539/2014-3 and 309188/2014-8). 2 Universidade Federal de Lavras, Departamento de Ciências Exatas, Lavras, Minas Gerais, Brazil. adrianorodrigues155@yahoo.com.br; lucas@dex.ufla.br 3 Universidade Federal de Viçosa, Departamento de Zootecnia, Viçosa, Minas Gerais, Brazil. fabyanofonseca@ufv .br; darlene.duarte@ufv .br 4 Universidade Federal de Minas Gerais, Departamento de Zootecnia, Belo Horizonte, Minas Gerais, Brazil. idalmo.garcia@gmail.com 5 Associação Brasileira dos Criadores de Zebu (ABCZ), Uberaba, Minas Gerais, Brazil. henrique@abcz.org.br * Corresponding author: fabyanofonseca@ufv .br Isotonic regression analysis of Guzerá cattle growth curves 1


INTRODUCTION
In general, a characteristic growth curve of livestock animals has the sigmoid shape and therefore, the nonlinear models, such as Logistic, Richards, Gompertz, Brody, and Von Bertalanffy, have been widely used in the literature in this type of data in the literature (Silveira et al., 2011).Moreover, these models have parameters with biological significance such as adult weight and maturity rate, and their estimates can be used for the evaluation of the growth efficiency of the herd or as phenotypic observations in breeding programs.
The presence of disturbances in the weight-age data contributes to the non-convergence of the algorithms used to estimate the parameters of the growth models and lead to a low goodness of fit of these models (Silveira et al., 2011).These disturbances are characterized by decreases in weight over time due to environmental effects, including lack of nutrients, diseases, and improper management in general.Different studies regarding growth curves in Animal Science mention these decreases, among them are a study by Mendes et al. (2009) that worked with Hereford cattle raised in the south of the country, and a study by Silveira et al. (2011), that worked with data from crossbred lambs (Dorper x Rabo Largo) raised in Bahia.
One way to overcome this problem is to use a data transformation method that takes into account the expected trajectory of the growth curve.Ramsay (1998) and Hussian et al. (2004) suggested a data transformation based on isotonic regression as a technique feasible for precorrection of longitudinal observations with such disturbances.Gunn & Dunson (2005) also applied this transformation to progesterone concentration data and were able to remove oscillations that were preventing the fit of normal regression models.
From the foregoing, our objective was to apply a data transformation methodology by using isotonic regression in Guzerá cattle growth curves studies whose data presented disturbances characterized by decreased body weight in certain age groups, aiming to avoid possible problems of convergence and to improve the fit of the nonlinear models tested.

MATERIAL AND METHODS
Isotonic regression refers to the fact that an increase in the independent variable will result in an increase in the response variable.Thus, assume that X = {t 1 , t 2 , .... ,t k } is a finite set of times in which t i < t j for i < j.A function f : X R is called isotonic or non-decreasing if f(t i ) < f(t j ) for i < j.If g is a function in X, a function g* is the isotonic regression of g with weights ù if, and only if, g* is a monotonic function that minimizes the following sum of squares: . Thus, it is said that g* is the non-decreasing function closest to the function g weighted by the weights ω.
A widely used Algorithm for isotonic regression is the Pool-Adjacent-Violators algorithm (PAVA) (Barlow et al., 1972).It starts with the function g(t), and if g(t) is isotonic, then g(t) = g*(t).Otherwise, there must be some index i such that g(t i-1 ) > g(t i ), being the value g(t i-1 ) called violator.These two values are replaced by their weighted averages, Av(i -1, i), given by: .After calculating this weighted average, the weights ω(t i-1 ) and ω(t i ) are replaced by the sum ω(t i-1 ) + ω(t i ).If this new set, with values k -1 is isotonic, then and g*(ti-1) = g*(ti) = (Av(i -1, i)) and g*(tj) = g(tj), with j i, i -1.Otherwise, there is a violator and the preceding process is repeated until a set with isotonic values is obtained.Note that the values of g* are the weighted averages of the blocks of values for which a violator existed.In this way, isotonic regression is obtained by a simple procedure, with statistical significance given by the use of local averages for the values that presented violators (disturbances in the expected behavior of the curve).
For the growth curves, consider the observations y i referring to the weights of an animal taken at the times t i , i = 1, 2,..., k, and assume y = f(t,θ) + ε is a non-linear regression model where: The concept for the use of such weights is that if y i * is distant from y i , that is, y i is a discrepant value in the data set, the weight ω(t i ) will be relatively small and will decrease its influence in obtaining the isotonized data.
To achieve a transformation even more robust to the influence of the outliers, the above described process can be successively applied to obtain an iterative process as follows.
Step 1: Transform the original data y = [y 1 , y 2 ,...,y k ] via isotonic regression with equal weights to give y ** = ]; Step m: Transform the original data y = [y 1 , y 2 ,...,y k ] via isotonic regression with different weights ω( ,...,y k *(m) ].Hence, the process is applied until there are no significant differences between the values isotonized in the (m-1) th step and the m th step.
The data used in the present study refer to the weightage observations of Guzerá cattle provided by the Brazilian Association of Zebu Breeders (ABCZ) located in Curvelo-MG.These data were collected from a weight gain test (WGT) performed at Meleiro Farm, which is located in the same municipality.The animal evaluations were performed according to the WGT methodology defined by ABCZ.A total of 45 newly weaned Guzerá males were evaluated, with mean initial weight of 219.9 kg and standard deviations of 38.05 kg and mean age of 325.8 days with standard deviation of 28.0 days.WGT was carried out with the animals fed on Brachiaria brizantha pastures and multiple supplementation.In addition to receiving the appropriate sanitary treatment, with the newly weaned animals kept under uniform conditions of management and feeding for 294 days.During the dry and wet seasons, the animals were fed protein salt and underwent usual vermifuge treatment.The animals were weighed at the beginning of WGT and, after an adaptation period of 70 days, they were weighed again to start the test itself.The evaluations were carried out every 56 days during the 224 testing days.Thus, each animal was weighed six times: one initial and five others for the weight gain test.
The growth models Gompertz (y i = A exp(-b exp(-kti)) + ei), Logistic (y i = A/(b exp(-kti)) + ei)), and Von Bertalanffy (y i = A(1 -b exp(-kti)) + ei)) were adjusted to the weightage data using the nls function of R (R Development Core Team, 2015), using the generalized ordinary least squares method for nonlinear regression with the iterative Gauss-Newton algorithm.The weighted and unweighted isotonic regressions were also implemented in R according to the functions developed by Rodrigues et al. (2010).

RESULTS AND DISCUSSION
The Table 1 shows the goodness of fit of the models tested, whose Mean Squared Errors (MSE) and R 2 values are represented by the means of values obtained from individual adjustments.
Although the data used in this study presented disturbances in the growth curve shape, no improvements were observed in the percentage of convergence using the isotonization methods when adjusting the Logistic and Gompertz models, once that 100% convergence (Table 1) was found for the three methodologies used (original data, corrected data by isotonic regression with equal weights and corrected data by isotonic regression with different weights).A small improvement in percentage of convergence was found for the Von Bertalanffy model when corrected via isotonic regression with different weights (84%) in relation to the original data (80%).
The Logistic model showed the lowest MSEs (Table 1) regardless of the methodology used.In addition, the isotonic transformations had the expected effect on the fit of this model, so the original data, data transformed by isotonic regression with equal weights, and data transformed by isotonic regression with different weights presented the values of 924.7541, 864.0190 and 630.1000, respectively.Although the differences between the models are small, the Logistic model also showed the best R 2 , especially when using the isotonic transformation with different weights.Thus, based on MSE and R 2 , it is possible to infer that the logistic model via isotonization of the data with different weights best described the weight of Guzerá animals in the conditions of this study.In addition, it also provided 100% convergence, indicating that besides the quality of fit, this model also has benefits in terms of ease of convergence.

,
and θ is the vector of parameters of the model.Since data y i are not necessarily in non-decreasing order due to environmental oscillations, data transformation to obtain increasing values can be done via isotonic regression, which is called isotonization.With this procedure, the original values y = [y 1 , y 2 ,...,y k ] are transformed into values y = [y * 1 are nondecreasing in relation to the evaluation dates.Therefore, the growth models will be adjusted to the data set y = [y * et al.(2010)  argued that the isotonic regression theory is sufficiently robust to the effects of outliers when the weights (ω(t i )) are given by the inverse of the difference between the value of the initial datum y i and the corresponding value after the isotonization y= y * 1 , ω(t i ) = 1/ y 1 -y i * |, if y i y * i ].With these weights, the original data can be again isotonized, obtaining new values y ** = [y ** 1

Table 1 :
Goodness of fit (Mean Squared Error, MSE; and Determination Coefficient, R 2 ) and percentage of convergence (% C) of Logistic, Von Bertalanffy, and Gompertz models fitted for growth curves of Guzerá cattle using different transformation methods (isotonization) for weight-age data

Table 2 :
Mean, standard deviations and coefficient of variation (as %) for individual estimates of parameters of the logistic model fitted for growth curves of Guzerá cattle using different transformation methods (isotonization) for weight-age data