**Cuban Journal of Agriculltural Science 49(2): 117-125, 2015, ISSN:2079-3480**

**Applied Mathematics in researches from the Instituto de Ciencia Animal. Fifty years of experience**

**La Matemática Aplicada en las investigaciones del Instituto de Ciencia Animal, cincuenta años de experiencia**

**Verena Torres,**^{I}** R. Cobo,**^{I}

^{I}Instituto de Ciencia Animal, Apartado Postal 24, San José de las Lajas, Mayabeque, Cuba.

**ABSTRACT**

This study reviews advances in Applied Mathematics, developed between 1973 and 2014, at the Instituto de Ciencia Animal, and published in the Cuban Journal of Agricultural Science. These contributions are about estimation and hypothesis testing, experimental designs, informatics and automatic systems, sampling techniques and sample size, mathematical simulation and modeling, econometrics and multivariate methods. During the cited period, the percentages of articles per subject were 8.4, 11.6, 5.3, 16.8, 27.4, 13.7 and 16.8, respectively. Modeling and simulating, sampling techniques and sample size, and multivariate methods were the most published subjects.

**Key words:** Biometry, Economy and Informatics.

**RESUMEN**

Se reseñan estudios de Matemática Aplicada, desarrollados entre 1973 y 2014, en el Instituto de Ciencia Animal y publicados en la Revista Cubana de Ciencia Agrícola. Las temáticas de estas contribuciones versan sobre la estimación y prueba de hipótesis, diseños experimentales, informática y sistemas automatizados, técnicas de muestreo y tamaño de muestra, modelación y simulación matemática, econometría y métodos multivariados. En el período citado, el porcentaje de artículos por temática fue 8.4, 11.6, 5.3, 16.8, 27.4, 13.7 y 16.8, respectivamente. Las más publicadas fueron modelación y simulación, técnicas de muestreo y tamaño de muestra y métodos multivariados.

**Palabras
clave:**
Biometría, Economía e Informática.

**INTRODUCTION**

Since the foundation of the Instituto de Ciencia Animal in 1965, a department of Biometry was created with the main objective of giving statistical advice to the research developed in the institute. Since the first papers published in the Cuban Journal of Agricultural Science, the experts in statistics appear as coauthors. Studies of Willis *et al.* (1971), Gómez *et al.* (1971), Tomeu *et al.* (1973) and Willis *et al.* (1973), among others, can be cited. The company of the experts remains currently, which evidences the importance of expert advice in this field.

This study will refer to the papers of Statistics and Mathematics published in the Cuban Journal of Agricultural Science, divided into different areas of knowledge.

**ESTIMATION AND HYPOTHESIS TESTING**

The subject of Biometry is included on Cuban Journal of Agricultural Science in 1973, when the first papers on hypothesis tests appear. These studies began to deal with aspects related to basis hypothesis (Menchaca 1973). After 1980, researches developed in specific topics, like milk production and crops, are included and framed in the area of estimation. Menchaca (1981) introduced a method of correction per bias to the Wood method, and compared different estimation procedures of milk production. Later, Fernández *et al.* (2001) studied the performance of milk production of Siboney breed of Cuba during its first lactation, during rainy and dry seasons.

The study of pests in species and varieties of grasses was treated through statistical methods, due to the damage they provoke to these crops. Martínez Machin *et al.* (2002) studied the space and time distribution of *Heteropsylla cubana* (Crawford) in *Leucaena leucocephala* (Lam).

Torres (1980) showed the estimation of soil homogeneity index in agricultural experiments. Torres and Jordán (1989) estimated the dry matter of Coast cross 1 bermuda grass, regarding other yield components. These authors used the Ridge regression method.

Most of the statistical methods are based on the process of hypothesis tests, which are accompanied by four basic theoretical hypothesis, which include the experimental errors that should be distributed, homogeneous and independent, with a model that should be additive.

Classic literature on this topic proposed to apply Box transformations to the variables under study, when some of these assumptions are not fulfilled. Aguila *et al.* (1998) proposed a transformation based on the integration of a polygonal function, which relates treatment variances with their corresponding means.

Guerra *et al.* (2000) compared parametric and non-parametric procedures through the index of asymptotic relative efficiency (ARE). These authors concluded that the test of Kruskal-Wallis and the Fisher F showed similar efficiency indexes, but lower than that of Friedman. During the 40´s of last century, some techniques were developed, which do not perform numerous or deep suppositions of the population under study. Therefore, they were classified as methods of free distribution, which is widely used in the agricultural and livestock field.

The relationship between errors of type I and II are known after performing a hypothesis testing. However, generally, the inverse relation between these two probabilities is not considered and only the significance levels obtained in the analysis of variance are referred, without regarding that low values of α can obtain high values of β which endangers the decision-making. Torres and Segui (2001) discussed these topics and their relation with sample size and power function. These authors proposed a practical criterion for determining the *a posteriori* power function, which allowed to analyze the reliability of the results of researches and to design strategies for future studies

**EXPERIMENTAL DESIGN **

One of the most important topics of experimental design is the definition of the amount of observations per treatment to compare. Menchaca (1975) published procedures for determining sample size in classic designs (simple classification, random blocks and Latin squares design). Venéreo (1976) and Caballero (1979) studied the number of replications to use in balanced square designs.

The response surfaces, as a combination of regression analysis with experimental design, provide economical means to locate a group of conditions for an optimal response. Martínez Machín and Marrero (2000) made a comparison between the factorial design with factorial arrangement and the surface response design, with factorial arrangement of three levels, in order to determine the optimal response of the concentration of short chain fatty acids (SCFA).

Torres and Chongo (1996) presented a mathematical model for consecutive measuring in the same experimental unit, with the purpose of avoiding wrong statistical inferences. Later, Torres *et al.* (2003) proposed the statistical methods of analysis of univariate and multivariate variance, as well as the analysis of indicator sum or average of areas under the curve, in the study of longitudinal data.

Gómez *et al.* (2012a) reviewed the most used statistical procedures in the analysis of designs of repeated measures in the agricultural and livestock field, and recommended the analysis of variance of fixed effects through the use of mix models, where the experimental units are considered as a random factor and time as a fixed factor. These last one included the correlations between repeated measures and the presence of heterogeneous variances. These authors pointed possible methods for estimating the parameters of these models. Nevertheless, they recommended the model of Restricted Maximum Likelihood (REML), and offered criteria of necessary information for selecting the best models.

Gómez *et al.* (2012b), in a study with mutant strains of *Trichoderma viride* cellulolitic fungi, compared the results of models of fixed and mixed effects in experiments with repeated measures.

**INFORMATICS**

Roche *et al.* (1999) developed a program to optimize resources in the nutrition of ruminants, with the maximum use of grass in the ration and ability of balanced ingestion. Ajete *et al.* (2000) showed a computer system for pig management and their population control. Sotolongo *et al.* (2004) developed a program that guarantees the individual technical control of cattle.

Informatics favored the creation of databases with different information, facilitating its promotion and spreading. Torres *et al.* (2001) created databases with scientific information published in Cuba about milk and meat production, based on grasses, forages and sugar cane. Grenón *et al.* (2008) developed an information system of extensive animal husbandry, with a support on a website, as a new tool that complements the extension process from the Instituto de Ciencia Animal (figure 1).

**SAMPLING TECHNIQUES AND SAMPLE SIZE **

The topic of population sampling is essential in statistics. For studying grasses and forages, this topic is very important in subjective sampling method, like visual ones, developed by Haydock and Shaw (1975), to determine availability and chemical composition of grasses, respectively. Torres and Jordán (1982) compared variants of visual sampling method to estimate availability of creeping grasses. Torres and Martínez (1986) and Torres (1987) performed studies of precision and determination of sample sizes. Torres *et al.* (1988) developed a subjective method, where the concept of grass volume was included, and designed equipment known as MEDIDEN (figure 2). Jordán *et al.* (1989) and Torres *et al.* (1998) applied visual methods to researches on grasses for studying different indicators.

Tables were developed for researching on non-ruminant animals, which allow to calculate sample size for experiments, according to completely randomized designs in pre-fattening and fattening pigs (Muñiz 1997).

Remote sensing, through aerial and satellite photos allowed the inventory of large extensions of agricultural areas. Ferrer *et al.* (1988) used these tools for identifying grasslands and Torres *et al.* (1991ab and 1992) studied the correlation degree between the optical density photometric indicator and the grassland indicators, together with the photographic interpretation of synthesized photographs, obtained with a multizonal aerial camera MKS-4. These authors stated productivity criteria of grasslands and spectral indexes of vegetation. Their studies took part of the experiment “Caribe Intercosmos 1988”, sponsored by the Cuban Academy of Sciences.

As a result of the researches, Torres *et al.* (1994) presented a methodology for obtaining a physical and geographical description and for performing the photographic interpretation of regions, in order to carry out inventories of different land uses and diagnosis of productivity (figure 3).

**MATHEMATICAL SIMULATION AND MODELING**

Mathematical modeling is a tool for estimating parameters of biological process. These techniques have been widely used in the field of animal and agricultural production, and allow the development of simulation and prognosis of productive results. The first modeling studies that appear as references in the journal are related to a multiplicative model and its application on the control of lactation curves effect (Menchaca and Jerez 1986), liveweight and intake of dairy cows (Menchaca and Ruiz 1987). López and Menchaca (1989) studied the modeling of growth of claves and heifers, and its variability within time.

Menchaca (1990) proposed stage models for animal growth, allowing the description of animal development curves according to life stages, with differences in feeding and handling that influence on growth rate. These models were created because the birth-age growth curve could not be properly represented by classic models. Later, as a continuation of this study, this author proposed the use of logarithmic transformation for finding estimators with optimal properties and stabilizing variances (Menchaca 1991ab). This same author studied multiplicative models for controlling perpendicular (season) and systematic effects, which affect animal growth. This author, in 1992, also formulated the union of stage model with the multiplicative model for studying growth in weight of calves in a feeding handling system, applied to growing animals under grazing conditions (Menchaca *et al.* 1993).

La O *et al.* (2013) retaken the modeling of liveweight curves in Cuban goats fed with natural shrubs and grasses from Granma province. This author introduces the concept of elasticity of Gompertz, logistic and non-linear models, in order to achieve a better interpretation of animal performance.

Menchaca and Ruiz (1990) presented a diagram of a simulation model that described the interphase between a model of grass intake (ingestions of dry matter, crude fiber and metabolizable energy in dairy cows) and the animal model, which describes the characteristics of this category (estimation of ME requirements, and estimation of CP ingestion ability).

In order to know the performance of ammonia release through dungs (kg of Nha-1) in the Voisin rational grazing, Torres *et al.* (1996) used non-linear models, regarding the days of dungs.

The use of non-conventional diets in non-ruminant species is an alternative for using by-products and cheapening feeding costs. In growing pigs, from 30 to 90 d, Larduet and Savón (1995) developed a model for simulating growth in this category. These authors considered the partition of ingested nitrogen during maintenance, growth and provision of energy.

Torres *et al.* (1999) described growth dynamics of Cynodon nlemfuensis (star grass), through the fixing of linear and non-linear models, with the use of different statistic criteria and the derivative of biomass dry weight according to time.

Regarding the different statistic criteria to use, when regression models are selected, Guerra *et al.* (2003) presented 14 criteria for reaching adequate theoretical and practical applications. Torres *et al.* (2012) stated other criteria for comparing and selecting non-linear models.

Fernández *et al.* (2004) continued the modeling studies with lactation curve characterization for the genotype of Siboney from Cuba (5/8 H and 3/8 Z). In 2005, these authors determined the factors affecting monthly weighings and estimated the model for lactation curve, corrected for these factors. Torres *et al.* (2009) developed a stochastic approximation of the logistic model, in order to estimate the productive perfomance of water buffaloes in Cuba during growth-fattening stage.

Torres and Ortiz (2005) carried out a summary with the applications of modeling and simulation on the process of production and feeding of farm animals. These authors proposed that every country or region should design and develop their own models for fitting their conditions, in a way that these models can become an useful tool for decision-making, always using statistical criteria that guarantees reliability of the proposed models.

Regarding growth of grass species, Rodríguez *et al.* (2011 and 2013) modelled the growth of *Pennisetum purpureum* cv. Cuba CT-169 and of *Pennisetum purpureum* cv. king-grass, respectively. These authors used linear and non-linear models during rainy and dry periods in the occidental part of Cuba. Also Ruiz *et al.* (2012abc), in three studies, presented the results of growth modeling of *Thitonia diversipholia* plant material during rainy and dry seasons. These authors characterized the best performances in yield components of this variety.

Modeling has also been frequently used for characterizing the dynamics of ruminal degradation. Although many authors have presented several models in *in vitro* gas production, the main objective of these researches is the comparison of feeding systems or of different materials composing these systems. Jay *et al.* (2012ab) carried out four homogeneity tests of non-linear regression models, and presented an evaluation of fixed range tests (lowest significant distance LSD, Tukey´s honest significant distance HSD, Sheffé´s significant distance SSD, and Bonferroni-corrected significant distance BSD) for the multiple comparison of treatment groups for curves, starting from the square mean distance.

**MULTIVARIATE METHODS **

Most of the scientific researches need the analysis of simultaneous relations among three or more variables. The statistical analysis of these variables will probably suggest, at the beginning, the modification of stated hypothesis. In this process, variables are continuously added and removed, which have a multivariate nature, so they correspond to the measuring on the same individuals.

Torres *et al.* (1993ab) presented the first two results on the application of multivariate techniques, with examples of analysis of principal components and that of multivariate variance, on the selection of variables capable of expressing phenotypic differences found among 16 king grass clones, obtained by tissue culture, and on the study of repeated measuring (years) in the comparison of grass species, respectively.

Varela and Torres (2005) performed a generalization of the principal components analysis, called PCA, for the analysis of several data matrixes, where the three modes were identified according to the interest or characteristics of the studied performed.

For the analysis of variables that explain better the chemical composition and anti-nutritional factors of grains from 14 temporal legumes, Camelo *et al.* (2007 and 2008) used the PCA and classified varieties into four groups. This allowed their characterization for their further use in feeding non-ruminant animals.

Torres *et al.* (2008 and 2013) introduced the Statistical Model of Impact Measuring (SMIM), which is a coherent and harmonic combination of multivariate methods, in order to achieve the double purpose of identifying variables and indicators, as well as typifying the performance of productive units. The application of this model allows making the correct decisions for establishing models of efficient management, which are appropriate for the characteristics of the ecosystems where cattle rearing systems are located. In addition, it evaluates the impact of processes of technological innovation that are introduced on the productive
chains.

After these papers, some studies on the application of this model are also performed. Authors like Benitez *et al.* (2008) used it for establishing the factors that determined productive efficiency of cattle farms in mountain areas of Granma province. Febles *et al.* (2011ab) applied this model to determine the importance of edaphoclimatic factors and to analyze impact indexes of seed production of tropical grasses. Martínez *et al.* (2012) also used this model to evaluate the effect of technologies of *Pennisetum purpureum* (Cuba CT-115) biomass banks on milk production in Villa Clara province. figures 4 and 5 show the performance of the effect of biomass bank in milk producing units of “Desembarco del Granma” enterprise.

The SMIM has also been applied in different countries like Mexico. Ruiz *et al.* (2012) has used it in this region, in order to characterize beef production systems in Mixquiahuala de Juárez, Hidalgo, through impact indexes reached with the application of different technologies. Vargas *et al.* (2011) reported the use of this model in the typification of cattle farms in Cotopaxi and Los Ríos provinces, Ecuador. Chivangulula *et al.* (2013) applied it on the analysis of sustainability of a familiar pig production system in Kaala, Angola, which allowed to meet and identify the main problems that limit pig production in this province.

Cobo and Borroto (2013) used the SMIM for analyzing the bio-economical efficiency of milk production through the Data Envelopment Analysis (DEA).

Recently, Segura and Torres (2014ab) published two papers with the addition of two new procedures to the SMIM for the treatment of lost and atypical values of databases and the validation of the classification and typification of double purpose farms from the Ecuadorian Amazon.

The table 1 sums up of quantitative form the frequency of described works and the years in which they have been published.

**ECONOMY**

Crespo (1976) insisted on the use of fertilization, and recommended, since that moment, the need of evaluating the costs due to the prices of inputs. This fact led to studies on economical efficiency of fertilization in grasses and forage. Cino *et al.* (1985) performed an economical study on the response of N fertilization in *Digitaria decumbens* (pangola grass). These authors determined production functions through square regressions, fitted by the minimum squares, in order to obtain higher gains regarding N doses.

Aguilar *et al.* (1994) applied linear programming to the ration formulation for poultry, taking into account the requirements of amino acids. From an economical point of view, Cino *et al.* (1999) studied some species of grain legumes, and considered everything from meal production process up to their inclusion on poultry
diets.

Continuing the mathematical and economical applications, Cino *et al.* (1994) evaluated intercropping of different forage species at the sowing moment of a brachiaria grassland with. Later, Cino and Valdés (1995) made an economical comparison between the Voisin rational grazing system and the traditional fattening system in grazing. There was also an economical evaluation of dairy systems with protein banks of *Neonotonia wightii* (Glycine) and *Leucaena leucocephala* (Cino *et al.* 1996 and Cino and Castillo 1999).

The economical evaluation of fattening technologies was developed by Cino *et al.* (2001), who demonstrated that the cost per animal and per kilogram of liveweight was inferior in systems of low inputs, and the benefit/cost relation showed the best indexes for the technologies of high inputs, due to the best animal performance and the lowest duration of the fattening period. Similar results were found by Rey and Reyes (2003) after studying the economical effect of two methods of rotational grazing. These authors stated that using low inputs, there is no bio-economical stability because of the productivity of animals. Later, Cino *et al.* (2011) found that silvopastoral systems with *Leucaena leucocephala* could be feasible economical option to increase biomass production in units of milk and meat production, which are destined for cattle pre-fattening and fattening.

Regarding the application of econometric methods on cost analysis of milk production, Cobo *et al.* (2011) proposed the regression methods, and used the minimum squares for estimating total costs, regarding other cost elements, so they could be reduced and the volume of incomes could be increased.

**REFERENCES**

Aguiar I., Larduet R. & Fraga L. M. 1994. ‘‘A note on the economic effect of the utilization of amino acids in broiler rations’’. *Cuban Journal of Agricultural Science*, 28 (1), pp. 15–20, ISSN: 2079-3480.

Aguila B., Torres V., Brito M., Sarduy L. & Noda A. 1998. ‘‘A transformation to homogenize the variances’’. *Cuban Journal of Agricultural Science*, 32, p. 11, ISSN: 2079-3480.

Ajete A., Díaz C. P. & Moyongo J. C. 2000. ‘‘SCPorcino. Computer programme for the control of the swine herd’’. *Cuban Journal of Agricultural Science*, 34, p. 101.

Benitez D., Fernandez J. L., Ray J., Ramirez A., Torres V., Tandron I., Diaz M. & Guerra J. 2008. ‘‘Determinant factors in the biomass production of three pasture species in rational grazing systems in the Cauto Valley, Cuba’’. *Cuban Journal of Agricultural Science*, 41 (3), pp. 221–224, ISSN: 2079-3480.

Caballero A. 1979. ‘‘Sample size for completely randomized and randomized block designs where the experimental unit is a group of animals’’. *Cuban Journal of Agricultural Science*, 13 (3), pp. 227–238, ISSN: 2079-3480.

Camelo S., Torres V. & Diaz M. F. 2007. ‘‘Multivariate analysis of the chemical composition of seasonal legume grains’’. *Cuban Journal of Agricultural Science*, 41 (2), pp. 107–111, ISSN: 2079-3480.

Camelo S., Torres V. & Diaz M. F. 2008. ‘‘Multivariate analysis of the anti-nutritional factors of the seasonal legumes grains’’. *Cuban Journal of Agricultural Science*, 42 (4), pp. 329–331, ISSN: 2079-3480.

Chivangulula M., Torres V., Morais J., Mário J. N. & Gabriel R. 2013. ‘‘Multivariate evaluation of the family pig production system in Caála, Angola’’. *Cuban Journal of Agricultural Science*, 47 (3), p. 279, ISSN: 2079-3480.

Cino D., Larduet R. & Jordan H. 1996. ‘‘Economical results in a dairy system with a protein bank of Glycine (*Neonotonia wightii*)’’. *Cuban Journal of Agricultural Science*, 30 (3), pp. 241–246, ISSN: 2079-3480.

Cino D. M. & Castillo E. 1999. ‘‘A note on costs and benefits of rotational cattle fattening systems with leucaena (*Leucaena leucocephala*) under non-irrigation conditions’’. *Cuban Journal of Agricultural Science*, 33 (4), pp. 339–342, ISSN: 2079-3480.

Cino D. M., Crespo G. & Sardiñas O. 1985. ‘‘A study on economical efficiency of N fertilizationto pangola grass (*Digitaria decumbens*) during the rainy season’’. *Cuban Journal of Agricultural Science*, 19 (3), pp. 323–329, ISSN: 2079-3480.

Cino D. M., Díaz A., Castillo E. & Hernández J. L. 2011. ‘‘Cattle fattening in Leucaena leucocephala grazing: some economic and financial indicators for making decisions’’. *Cuban Journal of Agricultural Science*, 45 (1), p. 7, ISSN: 2079-3480.

Cino D. M., Diaz M. F., Lon-Wo E. & Gonzalez A. 1999. ‘‘Economical evaluation of raw legume grain meals and their potential use in poultry feeding’’. *Cuban Journal of Agricultural Science*, 33 (2), pp. 127–133, ISSN: 2079-3480.

Cino D. M., Sierra D., Martín P. C. & Valdés G. 2001. ‘‘An economic evaluation of beef production alternatives’’. *Cuban Journal of Agricultural Science*, 35 (2), pp. 123–127, ISSN: 2079-3480.

Cino D. M., Sistachs M. & Melendez J. F. 1994. ‘‘Economical evaluation of the use of intercropped cultures for the feeding of dairy cows in milk production systems’’. *Cuban Journal of Agricultural Science*, 28 (2), pp. 153–159, ISSN: 2079-3480.

Cino D. M. & Valdes G. 1995. ‘‘Simulation of the economical feasibility of the utilization of the Voisin grazing system in beef fattening’’. *Cuban Journal of Agricultural Science*, 29 (2), pp. 145–151, ISSN: 2079-3480.

Cobo R. & Borroto O. 2013. ‘‘Determination of the bio-economical efficiency of milk production throughout data envelopment analysis models’’. *Cuban Journal of Agricultural Science*, 47 (3), p. 233, ISSN: 2079-3480.

Cobo R., Torres V., Machado Y. & Fraga M. 2011. ‘‘Econometric methods in the analysis of duary total production costs’’. *Cuban Journal of Agricultural Science*, 45 (3), p. 227, ISSN: 2079-3480.

Crespo G. 1976. ‘‘Differed nitrogen fertilization and annual production of pangola (*Digitaria decumbens* Stent) and guinea (*Panicum maximum* Jacq.) grass pastures’’. *Cuban Journal of Agricultural Science*, 10, p. 223, ISSN: 2079-3480.

Febles G., Torres V., Baños R., Ruiz T. E., Yáñez S. & Echeverría J. 2011a. ‘‘Multivariate analysis application to determine the preponderance of edaphoclimatic factors in the production of seeds from tropical prairie grasses’’. *Cuban Journal of Agricultural Science*, 45 (1), p. 45, ISSN: 2079-3480.

Febles G., Torres V., Baños R., Ruiz T. E., Yañez S. & Echeverría J. 2011b. ‘‘Utilization of the impact index to interpret the relative influence of edaphoclimatic factors on the production of tropical pasture seeds’’. *Cuban Journal of Agricultural Science*, 45 (1), p. 53, ISSN: 2079-3480.

Fernández L., Buxadera A. M. & Guerra C. W. 2004. ‘‘Comparative study of different functions for the analysis of the lactation curve in the genotype Siboney de Cuba’’. *Cuban Journal of Agricultural Science*, 38 (4), pp. 343–351, ISSN: 2079-3480.

Fernandez L., Buxadera A. M. & Guerra C. W. 2005. ‘‘Factors affecting milk yield in the Siboney de Cuba genotype. Linear models with controlled lactation curve effect’’. *Cuban Journal of Agricultural Science*, 39 (3), pp. 255–262, ISSN: 2079-3480.

Fernandez L., Menéndez A., Guerra W. & Suárez M. 2001. ‘‘Estimation of the standard lactation curves of the Siboney de Cuba breed for their use in lactation extensions’’. *Cuban Journal of Agricultural Science*, 35 (2), pp. 93–97, ISSN: 2079-3480.

Ferrer E., Torres V. & San Martín E. 1988. ‘‘Preliminary study on the application of remote sensing in grassland identification.’’. *Cuban Journal of Agricultural Science*, 22 (3), pp. 235–241, ISSN: 2079-3480.

Gómez J., López D., Menchaca M. & Rico C. 1971. ‘‘Estimation of optimum slaughter age in broilers. 1. Feed and protein efficiency related to live weight and carcass traits’’. *Cuban Journal of Agricultural Science*, 5, p. 313.

Gómez S., Torres V., García Y., Fraga L. M., Sarduy L. & Savón L. L. 2012a. ‘‘Comparison of models of fixed and mixed effects on the analysis of an experiment with mutant strains of cellulotic fungus Trichoderma viride’’. *Cuban Journal of Agricultural Science*, 46 (2), p. 127, ISSN: 2079-3480.

Gómez S., Torres V., García Y. & Navarro J. A. 2012b. ‘‘Statistical procedures most used in the analysis of measures repeated in time in the agricultural sector’’. *Cuban Journal of Agricultural Science*, 46 (1), ISSN: 2079-3480, Available: <http://www.ciencia-animal.org/cuban-journal-of-agricultural-science/articles/V46-N1-Y2012-P001-Sarai-Gomez.pdf>, [Accessed: March 16, 2016].

Grenon D. A., Lizazo D. & Torres V. 2008. ‘‘Proposal of the System of Information and Extension in Extensive Cattle Rearing (SIEGE)’’. *Cuban Journal of Agricultural Science*, 42 (4), pp. 339–342, ISSN: 2079-3480.

Guerra C. W., Cabrera A. & Fernández L. 2003. ‘‘Criteria for the selection of statistical models in scientific research’’. *Cuban Journal of Agricultural Science*, 37 (1), pp. 3–9, ISSN: 2079-3480.

Guerra C. W., Calzadilla J. de & Torres V. 2000. ‘‘Efficiency indicators regarding non-parametric methods in statistics’’. *Cuban Journal of Agricultural Science*, 34 (1), pp. 1–4, ISSN: 2079-3480.

Haydock K. & Shaw N. 1975. ‘‘The comparative yield method for estimating dry matter yield of pasture’’. *Australian Journal of Experimental Agriculture*, 15 (76), pp. 663–670.

Jay O., Torres V., Marrero Y. & Torres J. P. 2012a. ‘‘Sensibility analysis of homogeneity tests of in vitro gas production curves by Monte Carlo simulation’’. *Cuban Journal of Agricultural Science*, 46 (1), p. 15, ISSN: 2079-3480.

Jay O., Torres V., Marrero Y., Torres P. & others. 2012b. ‘‘Tests assessment for multiple comparisons of in vitro gas curves, from the root of the mean square distance’’. *Cuban J. Agric. Sci*, 46, p. 133.

Jordan H., Torres V. & Perez I. 1989. ‘‘Some considerations on visual samplings with 5 reference frames. 1. Relationship between reference frames and various indicators of Coast cross bermuda grass No. 1’’. *Cuban Journal of Agricultural Science*, 23 (1), pp. 83–88, ISSN: 2079-3480.

La O A. M. A., Guevara F., Fonseca N., Rodríguez L., Pinto R., Gómez H., Medina F. J. & Hernández A. 2013. ‘‘Application of the logistical and Gompertz models to the analysis of live weight curves in Cuban Creole kids’’. *Cuban Journal of Agricultural Science*, 47 (1), ISSN: 2079-3480, Available: <https://www.researchgate.net/profile/R_Pinto-Ruiz/publication/262451533_Aplicacin_de_los_modelos_logstico_y_Gompertz_al_anlisis_de_curvas_de_peso_vivo_en_cabritos_criollos_cubanos/links/0f3175380098309b81000000.pdf>, [Accessed: March 16, 2016].

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López V. & Menchaca M. A. 1989. ‘‘Statistical characterization of calf and heifer growth throughout liveweight performance’’. *Cuban Journal of Agricultural Science*, 23 (2), pp. 131–135, ISSN: 2079-3480.

Machin L. M., Valenciaga N., Ruiz T. E., Mora C. & Noda A. 2002. ‘‘Localization in time and space of *Heteropsylla cubana* (Crawford) in a *Leucaena leucocephala* (Lam) de Wit grassland’’. *Cuban Journal of Agricultural Science*, 36 (1), pp. 3–6, ISSN: 2079-3480.

Martinez Machin L. & Marrero Y. 2000. ‘‘Application of the response surface methodology on the study of the in vitro ruminal fermentation.’’. *Cuban Journal of Agricultural Science*, 34 (4), pp. 279–284, ISSN: 2079-3480, CABDirect2.

Martínez O., Torres V. & Aguilar P. I. 2012. ‘‘Impact of biomass banks with Pennisetum purpureum (Cuba CT-115) on milk production’’. *Cuban Journal of Agricultural Science*, 46 (3), ISSN: 2079-3480, Available: <http://cjascience.com/index.php/CJAS/article/view/40>, [Accessed: March 16, 2016].

Menchaca M. 1973. ‘‘A short method for analysis of statistical transformations’’. *Cuban Journal of Agricultural Science*, 7, p. 141.

Menchaca M. 1981. ‘‘Comparison of estimation methods and sampling intervals in the prediction of milk production’’. *Cuban Journal of Agricultural Science*, 15, p. 1.

Menchaca M. A. 1975. ‘‘Determination of sample size in Latin square designs’’. *Cuban Journal of Agricultural Science*, 9, p. 1.

Menchaca M. A. 1980. ‘‘Bias correction in Wood’s method for the estimation of milk yield in total lactation’’. *Cuban Journal of Agricultural Science*, 14 (2), pp. 105–109, ISSN: 2079-3480.

Menchaca M. A. 1990. ‘‘The use of stage models for describing animal growth curves’’. *Cuban Journal of Agricultural Science*, 24 (1), pp. 31–36, ISSN: 2079-3480, CABDirect2.

Menchaca M. A. 1991a. ‘‘Modelling of the bovine weight growth. 2. Multiplicative model controlling the growth curve and other effects’’. *Cuban Journal of Agricultural Science*, 25 (3), pp. 231–236, ISSN: 2079-3480.

Menchaca M. A. (Instituto de C. A. 1991b. ‘‘Modelling of the bovine weight growth. 1. An intrinsically linear model for growth representation’’. *Cuban Journal of Agricultural Science*, 25 (2), pp. 125–128, ISSN: 2079-3480.

Menchaca M. A. & Jerez I. 1986. ‘‘Evaluation of three tropical grasses. 1. Statistical treatment’’. *Cuban Journal of Agricultural Science*, 20 (3), pp. 225–231, ISSN: 2079-3480.

Menchaca M. A. & Ruiz R. 1987. ‘‘A note on algebraic representation of live weight and consumption during lactation of dairy cows’’. *Cuban Journal of Agricultural Science*, 21 (1), pp. 1–4, ISSN: 2079-3480.

Menchaca M. A. & Ruiz R. 1990. ‘‘Simulation of dry matter, crude fibre and metabolizable energy consumptions in an experiment with dairy cows grazing Coast Cross 1 bermuda grass (*Cynodon dactylon* Pers)’’. *Cuban Journal of Agricultural Science*, 24 (3), pp. 251–259, ISSN: 2079-3480, CABDirect2.

Menchaca M. A., Valdes G. & Brito M. 1993. ‘‘A study on the performance of grazing animals through the use of a growth multiplicative model’’. *Cuban Journal of Agricultural Science*, 27 (1), pp. 11–16, ISSN: 2079-3480.

Muniz M. 1997. ‘‘Tables used to estimate sample size in performance experiments with pigs in the pre-fattening and fattening stages’’. *Cuban Journal of Agricultural Science*, 31 (1), pp. 1–9, ISSN: 2079-3480.

Rey S. & Reyes J. J. 2003. ‘‘Economical effect of two rotational grazing methods with dairy cows and two grazing intensities’’. *Cuban Journal of Agricultural Science*, 37 (2), pp. 107–113, ISSN: 2079-3480.

Roche A., Larduet R., Torres V. & Ajete A. 1999. ‘‘CalRac: a microcomputer programme for the estimation of ruminant rations’’. *Cuban Journal of Agricultural Science*, 33 (1), pp. 13–19, ISSN: 2079-3480.

Rodríguez L., Larduet R., Martínez R. O., Torres V., Herrera M., Medina Y. & Noda A. 2013. ‘‘Modeling of the biomass accumulation dynamics in *Pennisetum purpureum* cv. king grass in the Western region of Cuba’’. *Cuban Journal of Agricultural Science*, 47, p. 119, ISSN: 2079-3480.

Rodríguez L., Torres V., Martínez O., Jay O., Noda A. C. & Herrera M. 2011. ‘‘Models to estimate the growth dynamics of Pennisetum purpureum cv. Cuba CT-169’’. *Cuban Journal of Agricultural Science*, 45 (4), p. 349, ISSN: 2079-3480.

Ruiz M., Ruiz J., Torres V. & Cach J. 2012a. ‘‘Study of beef meat production systems in a municipality of Hidalgo State, Mexico’’. *Cuban Journal of Agricultural Science*, 46 (3), p. 261, ISSN: 2079-3480.

Ruiz T. E., Torres V., Febles G., Díaz H. & González J. 2012b. ‘‘Use of modeling for studying the growth of Tithinia diversifolia collection 17’’. *Cuban Journal of Agricultural Science*, 46, p. 243.

Ruiz T. E., Torres V., Febles G., Díaz H. & González J. 2012c. ‘‘Use of modeling for studying the growth of Tithonia diversifolia collection 10’’. *Cuban Journal of Agricultural Science*, 46 (3), p. 237, ISSN: 2079-3480.

Ruiz T. E., Torres V., Febles G., Díaz H. & González J. 2012d. ‘‘Use of modeling to study the growth of the plant material 23 of Tithonia diversifolia’’. *Cuban Journal of Agricultural Science*, 46 (1), p. 23, ISSN: 2079-3480.

Segura E. O. & Torres V. 2013a. ‘‘Comparison criteria strengthened in classification and type representation, according to the Statistical Model of Impact Measuring, in a case study in Pastaza, Ecuador’’. *Cuban Journal of Agricultural Science*, 48 (4), ISSN: 2079-3480, Available: <http://ediciones.ica.edu.cu/index.php/CJAS/article/view/505>, [Accessed: March 16, 2016].

Segura E. O. & Torres V. 2013b. ‘‘Treatment of missing and atypical values in the application of the Statistical Model of Impact Measuring in a study of 90 dairy farms in Pastaza province, Ecuador.’’. *Cuban Journal of Agricultural Science*, 48 (4), ISSN: 2079-3480, Available: <https://ediciones.ica.edu.cu/index.php/CJAS/article/view/506>, [Accessed: March 16, 2016].

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Tomeu A., Pena J. A. & Menchaca M. 1972. ‘‘Diallel cross among the F3 of six sorghum crosses’’. *Revista Cubana de Ciencia Agricola*, 6 (2), pp. 267–278, ISSN: 2079-3480.

Torres V. 1980. ‘‘Estimation of the homogeneity index of the soil through random block experiments’’. *Cuban Journal of Agricultural Science*, 14, p. 213, ISSN: 2079-3480.

Torres V. 1987. ‘‘Visual method for estimating pasture availability. II Determination of sample size’’. *Cuban Journal of Agricultural Science*, 21, p. 113.

Torres V., Barbosa I., Meyer R., Noda A. & Sarduy L. 2012. ‘‘Criteria of goodness of fit test in the selection of non-linear models for the description of biological performances’’. *Cuban Journal of Agricultural Science*, 46, p. 345, ISSN: 2079-3480.

Torres V., Barsch H., Ytzerott S. & Weichett H. 1991a. ‘‘Remote sensing in grassland studies. 2. Productivity criterium’’. *Cuban Journal of Agricultural Science*, 25 (2), pp. 129–134, ISSN: 2079-3480, CABDirect2.

Torres V. & Chongo B. 1996. ‘‘A mathematical model for experiments with repeated measurements in the same experiment unit’’. *Cuban Journal of Agricultural Science*, 30 (1), pp. 13–17, ISSN: 2079-3480.

Torres V., Cino D., Ramos N., Ferrer E. & San Martin E. 1994. ‘‘Remote sensing for the inventory and evaluation of grasslands’’. *Cuban Journal of Agricultural Science*, 28 (3), pp. 259–264, ISSN: 2079-3480.

Torres V., Cobo R., Sanchez L. & Raez N. 2013. ‘‘Statistical tool for measuring the impact of milk production on the local development of a province in Cuba’’. *Livestock Research for Rural Development*, 25 (9), Available: <http://www.lrrd.cipav.org.co/lrrd25/9/torr25159.htm>, [Accessed: March 16, 2016].

Torres V., Crespo G. & Cuesta A. 1996. ‘‘A note the modelling of the ammonia losses of cow dung under Voisin´s rational grazing system’’. *Cuban Journal of Agricultural Science*, 30, p. 131.

Torres V., Crespo G., Martinez O., Martin P. C., Roche A., Vega Y., Sarduy L., Perez M. & Brito M. 2001. ‘‘A database for Cuban publications on beef and milk production using sugarcane, pastures and forages’’. *Cuban Journal of Agricultural Science*, 35 (1), pp. 9–12, ISSN: 2079-3480.

Torres V., Ferrer E. & MARTIN E. 1992. ‘‘Remote sensing in grassland studies. 3. Plant indices from spectrometric measurements’’. *Cuban Journal of Agricultural Science*, 26 (1), pp. 1–4, ISSN: 2079-3480.

Torres V., Ferrer E. & San Martín E. 1991b. ‘‘Remote sensing in grassland studies. 1. Photographic interpretation and optical processing’’. *Cuban Journal of Agricultural Science*, 25 (1), pp. 1–6, ISSN: 2079-3480.

Torres V., Jerez I. & Valle R. 1988. ‘‘Method of subjective sampling for estimating creeping pasture availability’’. *Cuban Journal of Agricultural Science*, 22 (1), pp. 1–7, ISSN: 2079-3480.

Torres V. & Jordan H. 1982. ‘‘A comparison of variants of the visual sampling method in the estimation of the availability of Coast Cross 1 bermuda grass (*Cynodon dactylon* cv. Coast Cross 1)’’. *Cuban Journal of Agricultural Science*, 16 (3), pp. 233–236, ISSN: 2079-3480.

Torres V. & Jordan H. 1989. ‘‘Estimation of dry mater in Coast cross 1 bermuda grass as a function of other yield components: Ridge regression and least squares’’. *Cuban Journal of Agricultural Science*, 23 (1), pp. 1–6, ISSN: 2079-3480.

Torres V., Lazo J., Ruiz T. E. & Noda A. 1999. ‘‘Mathematical modelling to estimate C. nlemfuensis pasture availability’’. *Cuban Journal of Agricultural Science*, 33 (4), pp. 343–351, ISSN: 2079-3480.

Torres V., Lazo J., Ruiz T. & Noda A. 1998. ‘‘The extension of the sampling method of Haydock and Shaw to estimate the morphological components of a *Cynodon nlemfuensis* cv. Jamaican sward’’. *Cuban Journal of Agricultural Science*, 32 (3), pp. 251–254, ISSN: 2079-3480.

Torres V., Lopez V. & Noda A. 1993a. ‘‘Example for the application of multivariate techniques in different stages of the evaluation and screening of pasture species. 2. Multivariate analysis of variance’’. *Cuban Journal of Agricultural Science*, 27 (3), pp. 251–254, ISSN: 2079-3480.

Torres V. & Martínez J. 1986. ‘‘Visual method for estimating pasture availability. 1. Precision studies’’. *Cuban Journal of Agricultural Science*, 20 (1), pp. 1–7, ISSN: 2079-3480.

Torres V., Martínez R. O. & Noda A. 1993b. ‘‘Example for application of multivariate techniques in different stages of the evaluation and screening of pasture species. 1. Principal components’’. *Cuban Journal of Agricultural Science*, 27, p. 125.

Torres V., Navarro J. R. & Pérez T. 2003. ‘‘Statistical models for processing experiments with repeated measurements in the same experimental unit’’. *Cuban Journal of Agricultural Science*, 37 (3), pp. 225–230, ISSN: 2079-3480.

Torres V. & Ortiz J. 2005. ‘‘Application of modelling and simulation to the production and feeding of faro animals’’. *Cuban Journal of Agricultural Science*, 39, pp. 385–393, ISSN: 2079-3480.

Torres V., Ramos N., Lizazo D., Monteagudo F. & Noda A. 2008. ‘‘Statistical model for measuring the impact of innovation or technology transfer in agriculture’’. *Cuban Journal of Agricultural Science*, 42 (2), pp. 131–137, ISSN: 2079-3480.

Torres V., Sampaio I. & Fundora O. 2009. ‘‘Stochastic model of the productive performance in the growing stage of water buffaloes in Cuba’’. *Cuban Journal of Agricultural Science*, 43 (2), pp. 111–114, ISSN: 2079-3480.

Torres V. & Segui Y. 2001. ‘‘Practical procedure for the determination of the a posteriori power function’’. *Cuban Journal of Agricultural Science*, 35 (4), pp. 297–300, ISSN: 2079-3480.

Varela M. & Torres V. 2005. ‘‘Application of three-mode principal components analysis in the multivariate characterization of king grass somaclones’’. *Cuban Journal of Agricultural Science*, 39 (4), pp. 527–534, ISSN: 2079-3480.

Vargas J., Benítez D., Torres V., Velázquez F. & Erazo O. 2011. ‘‘Typification of the cattle farms in the mountain feet of Los Ríos and Cotopaxi provinces of the Republic of Ecuador’’. *Cuban Journal of Agricultural Science*, 45 (4), p. 381, ISSN: 2079-3480.

Venereo A. 1976. ‘‘Number of replications in Latin square change-over designs for the estimation of residual effects’’. *Cuban Journal of Agricultural Science*, 10, p. 231, ISSN: 2079-3480.

Willis M. B., Menchaca M. & Preston T. R. 1973. ‘‘The use of Brahman, Brown Swiss, Charolais, Criollo and Holstein bulls on Zebu cows: post weaning performance and carcass characteristics’’. *Cuban Journal of Agricultural Science*, 7, p. 1.

Willis M. B., Preston T. R. & Menchaca M. 1971. ‘‘The use of Brahman, Brown Swiss, Charolais, Criollo and Holstein bulls on Zebu cows: performance to weaning’’. *Cuban Journal of Agricultural Science*, 5, p. 247, ISSN: 2079-3480.

Received: January 1, 2015

Accepted: February 1, 2015

* Verena Torres,* Instituto de Ciencia Animal, San José de las Lajas, Mayabeque, Cuba.
Email: vtorres@ica.co.cu

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