Prediction of live weight in beef heifers using a body volume formula

The objective of this study was to develop and evaluate linear, quadratic, and allometric models to predict live weight (LW) using the body volume formula (BV) in crossbred heifers raised in southeastern Mexico. The LW (426.25±117.49kg) and BV (338.05±95.38 dm 3 ) were measured in 360 heifers aged between 3 and 30 months. Linear and non-linear regression were used to construct prediction models. The goodness-of-fit of the models was evaluated using the Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), coefficient of determination (R 2 ), mean squared error (MSE), and root MSE (RMSE). In addition, the developed models were evaluated through cross-validation ( k -folds). The ability of the fitted models to predict the observed values was evaluated based on the RMSEP, R 2, and mean absolute error (MAE). The quadratic model had the lowest values of AIC (2688.39) and BIC (2700.05). On the other hand, the linear model showed the lowest values of MSE (7954.74) and RMSE (89.19), and the highest values of AIC (2709.70) and BIC (2717.51). Despite this, all models presented the same R 2 value (0.87). The cross-validation (k-folds) evaluation of fit showed that the quadratic model had better values of MSEP (41.49), R 2 (0.85), and MAE (31.95). We recommend the quadratic model to predictive of the crossbred beef heifers' live weight using the body volume as the predictor.


INTRODUCTION
Live weight (LW) is one of the main information that assists in the decision-making process in cattle production systems, due to the direct relationship with the nutritional requirements of animals (Vieira et al., 2013;Nutrient…, 2021).In addition, monitoring the weight development of ruminants makes it possible to identify the phases in which the animal has a greater capacity to convert the food consumed into body tissue and the best time for its commercialization (Fernandes et al., 2012;Gurgel et al., 2021a).Regarding management, LW measurement is important in establishing nutritional and animal health programs (Sabbioni et al., 2020;Canul-Solís et al., 2022).
On the other hand, the high cost of acquiring and maintaining scales has been considered an obstacle to directly obtaining the live weight of animals (Salazar-Cuytun et al., 2022;Canul-Solís et al., 2022).In most cases, this leads to a subjective animal weight estimation, which leads to errors in live weight estimation and affects the profitability of production systems (Málková et al., 2021).In this sense, biometric measurements are a viable option to predict live weight due to the strong positive correlation between these characteristics and the live weight of animals (Chay-Canul et al., 2019;Gurgel et al., 2021b).Some studies were carried out to develop linear and multiple equations to estimate the live weight of ruminants through biometric measurements (Chay-Canul et al., 2019;Salazar-Cuytun et al., 2022;Canul-Solís et al., 2022).Also, several authors concluded that thoracic perimeter is the most important biometric measure for predicting the live weight of animals (Chico-Alcudia et al., 2022); however, the association of this parameter with other biometric measures can increase the accuracy and precision of predictive models (Málková et al., 2021;Gurgel et al., 2021b).
Another approach to the use of biometric measurements to predict the live weight of cattle is from the body volume (BV), obtained through the formula for calculating the volume of a cylinder, including body measurements of thoracic perimeter and body length (Paputungan et al., 2015).Paputungan et al. (2018) reported that 96% of the weight variation of native Indonesian cattle is explained by the BV measure, a value higher than any other biometric measure used alone.Thus, the authors recommended estimating the LW of cattle using a first-degree linear regression model through BV as the only predictor measure.It is noteworthy that the authors did not test and evaluate a quadratic or allometric equation to estimate LW through BV.
Sheep studies have shown that a second-degree linear model provides more accurate estimates of LW using the BV measure (Salazar-Cuytun et al., 2021, 2022).Thus, the tested hypothesis was that the LW of crossbred cattle presents a quadratic relationship with the measurement of BV and that a second-degree linear equation more accurately estimates the LW of these animals.Therefore, the objective of this study was to develop and evaluate linear, quadratic, and allometric mathematical models to predict LW using the BV formula in crossbred heifers raised in humid tropical conditions in Mexico.

MATERIALS AND METHODS
The animals were managed following the guidelines and regulations for animal experimentation set forth by the Academic Division of Agricultural Sciences at Universidad Juárez Autónoma de Tabasco (UJAT).All methods were performed according to in vivo animal research guidelines: ARRIVE 2.0 (Sert et al., 2020).
The animals included in the present study belonged to four production units located in the state of Chiapas, southern Mexico.The predominant climate in this region is hot and humid, with abundant rains in summer.
For the development of the equations, a total of 360 crossbred (Bos taurus × Bos indicus) replacement heifers with different breed compositions, aged between 3 to 30 months were used.The heifers were grazed on African bermudagrass (Cynodon nlemfuensis) and Brachiaria humidicola pastures, without concentrate supplementation.Live weight (LW, kg) was recorded in each heifer using a digital livestock scale (Revuelta ® , Nuevo León, Mexico), whereas thoracic perimeter (TP, cm), body length (BL, cm) was recorded using a flexible fiberglass tape measure (Truper ® , SA de CV, San Lorenzo, Mexico) considering the anatomical references described by Bautista-Diaz et al. (2020) and Salazar-Cuytun et al. (2022).
Body volume (BV) was estimated using the formula to calculate the volume of a cylinder, by including the measurements of TP and BL in its composition.The volume (dm 3 ) was thus calculated as follows: Radius (cm) = TP/ 2π Volume (dm 3 ) = (π × r 2 × BL)/1000, Where r = circumference radius (cm); π = 3.1416; TP = thoracic perimeter (cm); and BL = body length (cm).
Descriptive statistics were obtained using the description function of the "Psych" package in R software.The relationship between LW and BV was evaluated by linear and multiple regressions, using the LM function in R software.In addition, the following allometric equation was fitted: Y = a + bX, where Y represents LW, X represents BV, and a and b are parameters.The parameters of the allometric model were estimated by nonlinear regressions using the NLS function of R software.Residual analysis was included to identify outliers, which were detected by plotting the studentized residuals against the values predicted by the equation.Outliers were removed if the value of the studentized residuals was outside the range of −2.5 to 2.5.The models and their residuals were plotted in the ggplot2 package of R software.The quality-of-fit of the regression models was evaluated using the Akaike Information Criterion (AIC); Bayesian Information Criterion (BIC); mean square error (MSE); root MSE (RMSE).
The predictive capacity of the three models for LW was evaluated by cross-validating k-folds (k = 10) according to Steyerberg andHarrel (2016), andCanul-Solis et al. (2020).The ability of the fitted model to predict the actual observed values was evaluated using MSE, R 2 , and the mean absolute error (MAE).The k-folds crossvalidation was performed using the scikit-learn package (Pedregosa et al., 2011).
The quadratic model had the lowest values of AIC (2688.39)and BIC (2700.05).On the other hand, the linear model showed the lowest values of MSE (7954.74)and RMSE (89.19), and the highest values of AIC (2709.70)and BIC (2717.51).Despite this, all models presented the same value for the coefficient of determination (R2 =0.87) (Table 2), as demonstrated in Figure 1, which shows that all LW prediction equations using BV in crossbred heifers present the same variation.
The quality-of-fit using the k-folds technique (cross-validation) allowed us to identify that the three proposed models showed an adequate fit considering the internal validation (Table 3).Of these, the quadratic model had lower values of mean square error of prediction (MSEP = 41.49) and mean absolute error (MAE = 31.95),also, a higher coefficient of determination (R 2 = 0.85).

DISCUSSION
The present study proposed to develop and evaluate mathematical models based on BV to Arq.Bras.Med. Vet. Zootec., v.74, n.6, p.1127-1133, 2022 predict LW in crossbred beef heifers.The correlation coefficient between BV and LW (r = 0.93) is similar to that found in various previous studies.Paputungan et al. (2015) reported a correlation between LW and BV in crossbred Indian cows, above 0.97.Also, Paputungan et al.
(2018) determine a correlation between LW and BV in Indonesian Local-Bali grade cattle, above 0.96 regardless of the age of the animals.Several studies on other animal species have identified a high correlation between LW and BV (Takaendengan et al., 2012;Salazar-Cuytun et al., 2021, 2022).These findings reveal that the BV as a predictor variable can be a consistent parameter for predicting the LW of production animals.The practical implications are that the BV may represent better the body mass of the animal, which is directly related to nutrient requirements of maintenance (Chay-Canul et al., 2019).
All models tested were able to predict the weight of animals.As these proved to be equally accurate (Table 2), either of them can be used to predict the LW of crossbred heifers using BV measurement.However, the predictive capacity of the three models for LW was evaluated by cross-validation of k-folds (k = 10).This approach was performed by randomly dividing the set of observation values into non-overlapping k-folds of approximately the same size (Steyerberg and Harrel, 2016;Canul-Solis et al., 2020).The first fold is treated as a validation set and the model fits the remaining k-1 folds (training data).This procedure allowed the estimation of higher values of R 2 and lower values of MSPE and MAE for the quadratic model (Table 3).Therefore, the quadratic model was the best-performing mathematical model according to the adequacy assessment to predict the LW of beef heifers using BV calculated from TP and BL data (Fig. 1).Salazar-Cuytun et al. (2021, 2022) and reported that the LW of lambs was better adjusted by a linear equation of the second degree as a function of the measurement of BV.Likewise, Gurgel et al. (2021b) observed that the association of several biometric measures promotes better estimates of the live weight of Santa Inês sheep.These authors also reported that the square of the TP measurement provides better predictions of the LW of the lambs.In contrast, Paputungan et al. (2015Paputungan et al. ( , 2018) ) recommended the use of a firstdegree linear regression model to predict LW from body volume (LW = a + b × BV) in native Indonesian cattle.It is noteworthy that these authors did not test other equations to estimate the LW through the BV.The results confirm that the second-degree linear equation (quadratic) can be safely used to estimate the LW of crossbred heifers by means of the BV measurement.However, it should be considered that this equation was developed from data from heifers kept in tropical pastures.Therefore, its application is limited to animals raised under such conditions (Tedeschi, 2006).
For this, if we want to use the equation in other types of animals, breeds, or production systems, it would be necessary to evaluate its functionality under these specific conditions.Because body conformation and body fat deposition may differ between animals of different sexes and breedsaspects that may interfere with the correlation between BV and LW in ruminant animals (Paputungan et al., 2018;Salazar-Cuytun et al., 2021;Chico-Alcudia et al., 2022).For this reason, models should be developed for animals of different physiological conditions and sexes, in different management scenarios, to improve decision-making and the economic benefits provided by determining and monitoring the LW of domestic animals (Sherwin et al., 2021;Málková et al., 2021;Chico-Alcudia et al., 2022).

CONCLUSION
The BV can be used as a predictor of the LW of the crossbred beef heifers kept in tropical pastures.We recommend the quadratic model to predictive of the crossbred beef heifers' live weight using the body volume as the predictor.

Figure 1 .
Figure 1.Live weight (LW) prediction equations using the body volume formula (BV) in crossbred heifers raised in tropical humid conditions (n = 360).

Table 3 .
Internal k-folds cross-validation of the proposed models using body volume in crossbred heifers raised in tropical humid conditions