2016, 13(1): 67-82. doi: 10.3934/mbe.2016.13.67

KL-optimal experimental design for discriminating between two growth models applied to a beef farm

1. 

Institute of Mathematics Applied to Science and Engineering, University of Castilla-La Mancha, 13071-Ciudad Real, Spain, Spain

Received  March 2015 Revised  June 2015 Published  October 2015

The body mass growth of organisms is usually represented in terms of what is known as ontogenetic growth models, which represent the relation of dependence between the mass of the body and time. The paper is concerned with a problem of finding an optimal experimental design for discriminating between two competing mass growth models applied to a beef farm. T-optimality was first introduced for discrimination between models but in this paper, KL-optimality based on the Kullback-Leibler distance is used to deal with correlated obsevations since, in this case, observations on a particular animal are not independent.
Citation: Santiago Campos-Barreiro, Jesús López-Fidalgo. KL-optimal experimental design for discriminating between two growth models applied to a beef farm. Mathematical Biosciences & Engineering, 2016, 13 (1) : 67-82. doi: 10.3934/mbe.2016.13.67
References:
[1]

M. Amo-Salas, J. López-Fidalgo and E. Porcu, Optimal designs for some stochastic processes whose covariance is a function of the mean,, Test, 22 (2013), 159. doi: 10.1007/s11749-012-0311-5. Google Scholar

[2]

M. Amo-Salas, J. López-Fidalgo and V. López-Ríos, Optimal Designs for Two Nested Pharmacokinetic Models with Correlated Observations,, Communications in Statistics - Simulation and Computation, 41 (2012), 944. doi: 10.1080/03610918.2012.625743. Google Scholar

[3]

S. Aparecida-Santos, G. Da Silva e Souza, M. Rubén de Oliveira and J. Robson-Sereno, Using nonlinear models to describe height growth curves in pantaneiro horses,, Pesquisa Agropecuária Brasileira, 7 (1999), 1133. Google Scholar

[4]

J. Arango and L. D. Van Vleck, Size of beef cows: early ideas, new developments,, Genetics and Molecular Research, 7 (2002), 51. Google Scholar

[5]

A. C. Atkinson and V. V. Fedorov, The designs of experiments for discriminating between two rival models,, Biometrika, 62 (1975), 57. doi: 10.1093/biomet/62.1.57. Google Scholar

[6]

J. J. Beltrán, W. T. Butts, T. A. Olson and M. Koger, Growth patterns of two lines of Angus cattle celected ucing predicted growth parameters,, Journal of Animal Science, 70 (1992), 734. Google Scholar

[7]

L. V. Bertalanffy, Untersuchungen über die Gesetzlichkeit des Wachstums I. Allgemeine Grundlagen der Theorie. Mathematisch-physiologische Gesetzlichkeiten des Wachstums bei Wassertieren,, Wilhelm Roux's archives of developmental biology, 131 (1934), 613. Google Scholar

[8]

A. Boukouvalas, D. Cornford and M. Stehlik, Optimal Design for Correlated processes with input-dependent noise,, Computational Statistics and Data Analysis, 71 (2014), 1088. doi: 10.1016/j.csda.2013.09.024. Google Scholar

[9]

U. N. Brimkulov, G. K. Krug and V. L. Savanov, Design of Experiments in Investigating Random fields and Processes,, Nauka, (1986). Google Scholar

[10]

S. Brody, Bioenergetics and Growth,, Reinhold, (1945). Google Scholar

[11]

J. E. Brown, H. A. Fitzhugh and T. C. Cartwright, A comparison of nonlinear models for describing weight-age relationships in cattle,, Journal of Animal Science, 42 (1976), 810. Google Scholar

[12]

H. Chernoff, Locally Optimal design for estimating parameters,, Annals of Mathematical Statistics, 24 (1953), 586. doi: 10.1214/aoms/1177728915. Google Scholar

[13]

N. Cressie, Statistics for Spatial Data,, John Wiley and Sons, (1993). doi: 10.1002/9781119115151. Google Scholar

[14]

R. I. Cue, D. Pietersma, D. Lefebvre, R. Lacroix, K. Wade, D. Pellerin, A. M. De Passillé and J. Rushen, Growth modeling of dairy heifers in Québec based on random regression,, Canadian Journal of Animal Science, 92 (2012), 143. doi: 10.4141/cjas2011-083. Google Scholar

[15]

E. Demidenko, Mixed Models: Theory and Applications,, Wiley Series in Probability and Statistics, (2004). doi: 10.1002/0471728438. Google Scholar

[16]

R. S. K. DeNise and J. S. Brinks, Genetic and environmental aspects of the growth curve parameters in beef cows,, Journal of Animal Science, 61 (1985), 1431. Google Scholar

[17]

H. Dette and A. Pepelyshev, Efficient experimental designs for sigmoidal growth models,, Journal of statistical planning and inference, 138 (2008), 2. doi: 10.1016/j.jspi.2007.05.027. Google Scholar

[18]

H. Dette, A. Pepelyshev and A. Zhigljavsky, Optimal Design for linear models with correlated observations,, Annals of Statistics, 41 (2013), 143. doi: 10.1214/12-AOS1079. Google Scholar

[19]

H. Dette and S. Titoff, Optimal discrimination designs,, Annals of Statistics, 37 (2009), 2056. doi: 10.1214/08-AOS635. Google Scholar

[20]

H. A. Fitzhugh Jr., Analysis of growth curves and strategies for altering their shape,, Journal of Animal Science, 42 (1976), 1036. Google Scholar

[21]

B. Gompertz, On the nature of the function expressive of the law of human mortality,, Mathematical Demography, 6 (1977), 279. doi: 10.1007/978-3-642-81046-6_30. Google Scholar

[22]

L. A. Goonewardene, R. T. Berg and R. T. Hardin, A growth study of beef cattle,, Canadian Journal of Animal Science, 61 (1981), 1041. doi: 10.4141/cjas81-128. Google Scholar

[23]

P. Goos and B. Jones, Optimal Design of Experiments: A Case-Study Approach,, Wiley, (2011). doi: 10.1002/9781119974017. Google Scholar

[24]

H. Hirooka and Y. Yamada, A general simulation model for cattle growth and beef production,, Asian-Australasian Journal of Animal Sciences, 3 (1990), 205. doi: 10.5713/ajas.1990.205. Google Scholar

[25]

Z. B. Johson, C. J. Brown and A. H. Brown Jr., Evaluation of growth pattern of beef cattle,, Agricultural Experiment Station, (1990). Google Scholar

[26]

M. Kaps, W. O. Herring and W. R. Lamberson, Genetic and environmental parameters for mature weight in Angus cattle,, Journal of Animal Science, 77 (1999), 569. Google Scholar

[27]

S. Kullback and R. A. Leibler, On information and sufficiency,, The Annals of Mathematical Statistics, 22 (1951), 79. doi: 10.1214/aoms/1177729694. Google Scholar

[28]

G. López de la Torre, J. J. Candotti, A. Reverter, M. M. Bellido, P. Vasco, L. J. García and J. S. Grinks, Effectc of growth parameters on cow efficiency,, Journal of Animal Science, 70 (1992), 2668. Google Scholar

[29]

J. López-Fidalgo, I. M. Ortíz-Rodríguez and W. K. Wong, Design issues for population growth models,, Journal of Applied Statistics, 38 (2011), 501. doi: 10.1080/02664760903521419. Google Scholar

[30]

J. López-Fidalgo, C. Tommasi and P. C. Trandafir, An optimal experimental design criterion for discriminating between non-normal models,, Journal of the Royal Statistical Society: Series B, 69 (2007), 231. doi: 10.1111/j.1467-9868.2007.00586.x. Google Scholar

[31]

C. H. McCulloch and S. H. Searle, Generalized, Linear and Mixed Models,, Wiley Series in Probability and Statistics, (2001). Google Scholar

[32]

G. Matheron, Traite de Geostatistique Appliquee,, Editions Technip., (1962). Google Scholar

[33]

C. May and C. Tommasi, Model selection and parameter estimation in non-linear nested models,, Statistica Sinica, 24 (2014), 63. Google Scholar

[34]

W. G. Müller, Collecting Spatial Data: Optimum Design of Experiments for Random Fields,, Physica-Verlag, (2001). Google Scholar

[35]

J. A. Nelder, The fitting of a generalization of the logistic curve,, Biometrics, 17 (1961), 89. doi: 10.2307/2527498. Google Scholar

[36]

T. C. Nelsen, C. R. Long and T. C. Cartwright, Postinflection growth in straight red and crossbred cattle,, Journal of Animal Science, 55 (1982), 2280. Google Scholar

[37]

D. G. Nicholls and S. J. Ferguson, Bioenergetics,, Academic Press Inc., (2004). Google Scholar

[38]

A. Pázman, Foundations of Optimum Experimental Design,, D. Reidel publishing company, (1986). Google Scholar

[39]

A. Pepelyshev, The Role of the Nugget Term in the Gaussian Process Method,, mODa 9. Advances in Model-Oriented Design and Analysis. Contributions to Statistics, 1 (2010), 149. doi: 10.1007/978-3-7908-2410-0_20. Google Scholar

[40]

A. C. Ponce de León and A. C. Atkinson, Optimum experimental design for discrimination between tow rival models in the presence of prior information,, Biometrika, 78 (1991), 601. doi: 10.1093/biomet/78.3.601. Google Scholar

[41]

F. J. Richards, A flexible growth function for empirical use,, Journal of Experimental Botany., 10 (1959), 290. doi: 10.1093/jxb/10.2.290. Google Scholar

[42]

B. D. Ripley, Spatial Statistics,, John Wiley and Sons. Wiley Series in Probability and Mathematical Statistics, (1981). Google Scholar

[43]

J. Sacks and D. Ylvisaker, Designs for regression problems with correlated errors,, The Annals of Mathematical Statistics, 41 (1970), 2057. doi: 10.1214/aoms/1177696705. Google Scholar

[44]

T. S. Stewart and T. G. Martin, Optimal mature size of Angus cows for maximum cow productivity,, Journal of Animal Production, 37 (1983), 179. doi: 10.1017/S0003356100001707. Google Scholar

[45]

M. Stehlík, J. M. Rodríguez-Díaz, W. G. Muller and J. López-Fidalgo, Optimal allocation of bioassays in the case of parametrized covariance functions: an application to lung's retention of radioactive particles,, Test, 17 (2008), 56. doi: 10.1007/s11749-006-0022-x. Google Scholar

[46]

C. Tommasi, J. M. Rodríguez-Díaz and M. T. Santos-Martín, Integral approximations for computing optimum designs in random effects logistic regression models,, Computational Statistics and Data Analysis 71 (2014), 71 (2014), 1208. doi: 10.1016/j.csda.2012.05.024. Google Scholar

[47]

D. Uciński and A. C. Atkinson, Experimental design for time-dependent models with correlated observations,, Studies in Nonlinear Dynamics and Econometrics, 8 (2004), 1. Google Scholar

[48]

D. Uciński and B. Bogacka, T-optimum designs for discrimination between two multiresponse dynamic models,, Journal of the Royal Statistical Society. Series B, 67 (2005), 3. doi: 10.1111/j.1467-9868.2005.00485.x. Google Scholar

[49]

O. D. Vergara, J. M. Flórez-Murillo, M. J. Hernández-Pérez, E. M. Arboleda-Zapata and A. Calderón-Rangel, Description of crossbred cattle growth using Brody model,, Livestock Research for Rural Development, 25 (2013). Google Scholar

[50]

A. Zhigljavsky, H. Dette and A. Pepelyshev, A new approach to optimal design for linear models with correlated observations,, Journal of the American Statistical Association, 105 (2010), 1093. doi: 10.1198/jasa.2010.tm09467. Google Scholar

show all references

References:
[1]

M. Amo-Salas, J. López-Fidalgo and E. Porcu, Optimal designs for some stochastic processes whose covariance is a function of the mean,, Test, 22 (2013), 159. doi: 10.1007/s11749-012-0311-5. Google Scholar

[2]

M. Amo-Salas, J. López-Fidalgo and V. López-Ríos, Optimal Designs for Two Nested Pharmacokinetic Models with Correlated Observations,, Communications in Statistics - Simulation and Computation, 41 (2012), 944. doi: 10.1080/03610918.2012.625743. Google Scholar

[3]

S. Aparecida-Santos, G. Da Silva e Souza, M. Rubén de Oliveira and J. Robson-Sereno, Using nonlinear models to describe height growth curves in pantaneiro horses,, Pesquisa Agropecuária Brasileira, 7 (1999), 1133. Google Scholar

[4]

J. Arango and L. D. Van Vleck, Size of beef cows: early ideas, new developments,, Genetics and Molecular Research, 7 (2002), 51. Google Scholar

[5]

A. C. Atkinson and V. V. Fedorov, The designs of experiments for discriminating between two rival models,, Biometrika, 62 (1975), 57. doi: 10.1093/biomet/62.1.57. Google Scholar

[6]

J. J. Beltrán, W. T. Butts, T. A. Olson and M. Koger, Growth patterns of two lines of Angus cattle celected ucing predicted growth parameters,, Journal of Animal Science, 70 (1992), 734. Google Scholar

[7]

L. V. Bertalanffy, Untersuchungen über die Gesetzlichkeit des Wachstums I. Allgemeine Grundlagen der Theorie. Mathematisch-physiologische Gesetzlichkeiten des Wachstums bei Wassertieren,, Wilhelm Roux's archives of developmental biology, 131 (1934), 613. Google Scholar

[8]

A. Boukouvalas, D. Cornford and M. Stehlik, Optimal Design for Correlated processes with input-dependent noise,, Computational Statistics and Data Analysis, 71 (2014), 1088. doi: 10.1016/j.csda.2013.09.024. Google Scholar

[9]

U. N. Brimkulov, G. K. Krug and V. L. Savanov, Design of Experiments in Investigating Random fields and Processes,, Nauka, (1986). Google Scholar

[10]

S. Brody, Bioenergetics and Growth,, Reinhold, (1945). Google Scholar

[11]

J. E. Brown, H. A. Fitzhugh and T. C. Cartwright, A comparison of nonlinear models for describing weight-age relationships in cattle,, Journal of Animal Science, 42 (1976), 810. Google Scholar

[12]

H. Chernoff, Locally Optimal design for estimating parameters,, Annals of Mathematical Statistics, 24 (1953), 586. doi: 10.1214/aoms/1177728915. Google Scholar

[13]

N. Cressie, Statistics for Spatial Data,, John Wiley and Sons, (1993). doi: 10.1002/9781119115151. Google Scholar

[14]

R. I. Cue, D. Pietersma, D. Lefebvre, R. Lacroix, K. Wade, D. Pellerin, A. M. De Passillé and J. Rushen, Growth modeling of dairy heifers in Québec based on random regression,, Canadian Journal of Animal Science, 92 (2012), 143. doi: 10.4141/cjas2011-083. Google Scholar

[15]

E. Demidenko, Mixed Models: Theory and Applications,, Wiley Series in Probability and Statistics, (2004). doi: 10.1002/0471728438. Google Scholar

[16]

R. S. K. DeNise and J. S. Brinks, Genetic and environmental aspects of the growth curve parameters in beef cows,, Journal of Animal Science, 61 (1985), 1431. Google Scholar

[17]

H. Dette and A. Pepelyshev, Efficient experimental designs for sigmoidal growth models,, Journal of statistical planning and inference, 138 (2008), 2. doi: 10.1016/j.jspi.2007.05.027. Google Scholar

[18]

H. Dette, A. Pepelyshev and A. Zhigljavsky, Optimal Design for linear models with correlated observations,, Annals of Statistics, 41 (2013), 143. doi: 10.1214/12-AOS1079. Google Scholar

[19]

H. Dette and S. Titoff, Optimal discrimination designs,, Annals of Statistics, 37 (2009), 2056. doi: 10.1214/08-AOS635. Google Scholar

[20]

H. A. Fitzhugh Jr., Analysis of growth curves and strategies for altering their shape,, Journal of Animal Science, 42 (1976), 1036. Google Scholar

[21]

B. Gompertz, On the nature of the function expressive of the law of human mortality,, Mathematical Demography, 6 (1977), 279. doi: 10.1007/978-3-642-81046-6_30. Google Scholar

[22]

L. A. Goonewardene, R. T. Berg and R. T. Hardin, A growth study of beef cattle,, Canadian Journal of Animal Science, 61 (1981), 1041. doi: 10.4141/cjas81-128. Google Scholar

[23]

P. Goos and B. Jones, Optimal Design of Experiments: A Case-Study Approach,, Wiley, (2011). doi: 10.1002/9781119974017. Google Scholar

[24]

H. Hirooka and Y. Yamada, A general simulation model for cattle growth and beef production,, Asian-Australasian Journal of Animal Sciences, 3 (1990), 205. doi: 10.5713/ajas.1990.205. Google Scholar

[25]

Z. B. Johson, C. J. Brown and A. H. Brown Jr., Evaluation of growth pattern of beef cattle,, Agricultural Experiment Station, (1990). Google Scholar

[26]

M. Kaps, W. O. Herring and W. R. Lamberson, Genetic and environmental parameters for mature weight in Angus cattle,, Journal of Animal Science, 77 (1999), 569. Google Scholar

[27]

S. Kullback and R. A. Leibler, On information and sufficiency,, The Annals of Mathematical Statistics, 22 (1951), 79. doi: 10.1214/aoms/1177729694. Google Scholar

[28]

G. López de la Torre, J. J. Candotti, A. Reverter, M. M. Bellido, P. Vasco, L. J. García and J. S. Grinks, Effectc of growth parameters on cow efficiency,, Journal of Animal Science, 70 (1992), 2668. Google Scholar

[29]

J. López-Fidalgo, I. M. Ortíz-Rodríguez and W. K. Wong, Design issues for population growth models,, Journal of Applied Statistics, 38 (2011), 501. doi: 10.1080/02664760903521419. Google Scholar

[30]

J. López-Fidalgo, C. Tommasi and P. C. Trandafir, An optimal experimental design criterion for discriminating between non-normal models,, Journal of the Royal Statistical Society: Series B, 69 (2007), 231. doi: 10.1111/j.1467-9868.2007.00586.x. Google Scholar

[31]

C. H. McCulloch and S. H. Searle, Generalized, Linear and Mixed Models,, Wiley Series in Probability and Statistics, (2001). Google Scholar

[32]

G. Matheron, Traite de Geostatistique Appliquee,, Editions Technip., (1962). Google Scholar

[33]

C. May and C. Tommasi, Model selection and parameter estimation in non-linear nested models,, Statistica Sinica, 24 (2014), 63. Google Scholar

[34]

W. G. Müller, Collecting Spatial Data: Optimum Design of Experiments for Random Fields,, Physica-Verlag, (2001). Google Scholar

[35]

J. A. Nelder, The fitting of a generalization of the logistic curve,, Biometrics, 17 (1961), 89. doi: 10.2307/2527498. Google Scholar

[36]

T. C. Nelsen, C. R. Long and T. C. Cartwright, Postinflection growth in straight red and crossbred cattle,, Journal of Animal Science, 55 (1982), 2280. Google Scholar

[37]

D. G. Nicholls and S. J. Ferguson, Bioenergetics,, Academic Press Inc., (2004). Google Scholar

[38]

A. Pázman, Foundations of Optimum Experimental Design,, D. Reidel publishing company, (1986). Google Scholar

[39]

A. Pepelyshev, The Role of the Nugget Term in the Gaussian Process Method,, mODa 9. Advances in Model-Oriented Design and Analysis. Contributions to Statistics, 1 (2010), 149. doi: 10.1007/978-3-7908-2410-0_20. Google Scholar

[40]

A. C. Ponce de León and A. C. Atkinson, Optimum experimental design for discrimination between tow rival models in the presence of prior information,, Biometrika, 78 (1991), 601. doi: 10.1093/biomet/78.3.601. Google Scholar

[41]

F. J. Richards, A flexible growth function for empirical use,, Journal of Experimental Botany., 10 (1959), 290. doi: 10.1093/jxb/10.2.290. Google Scholar

[42]

B. D. Ripley, Spatial Statistics,, John Wiley and Sons. Wiley Series in Probability and Mathematical Statistics, (1981). Google Scholar

[43]

J. Sacks and D. Ylvisaker, Designs for regression problems with correlated errors,, The Annals of Mathematical Statistics, 41 (1970), 2057. doi: 10.1214/aoms/1177696705. Google Scholar

[44]

T. S. Stewart and T. G. Martin, Optimal mature size of Angus cows for maximum cow productivity,, Journal of Animal Production, 37 (1983), 179. doi: 10.1017/S0003356100001707. Google Scholar

[45]

M. Stehlík, J. M. Rodríguez-Díaz, W. G. Muller and J. López-Fidalgo, Optimal allocation of bioassays in the case of parametrized covariance functions: an application to lung's retention of radioactive particles,, Test, 17 (2008), 56. doi: 10.1007/s11749-006-0022-x. Google Scholar

[46]

C. Tommasi, J. M. Rodríguez-Díaz and M. T. Santos-Martín, Integral approximations for computing optimum designs in random effects logistic regression models,, Computational Statistics and Data Analysis 71 (2014), 71 (2014), 1208. doi: 10.1016/j.csda.2012.05.024. Google Scholar

[47]

D. Uciński and A. C. Atkinson, Experimental design for time-dependent models with correlated observations,, Studies in Nonlinear Dynamics and Econometrics, 8 (2004), 1. Google Scholar

[48]

D. Uciński and B. Bogacka, T-optimum designs for discrimination between two multiresponse dynamic models,, Journal of the Royal Statistical Society. Series B, 67 (2005), 3. doi: 10.1111/j.1467-9868.2005.00485.x. Google Scholar

[49]

O. D. Vergara, J. M. Flórez-Murillo, M. J. Hernández-Pérez, E. M. Arboleda-Zapata and A. Calderón-Rangel, Description of crossbred cattle growth using Brody model,, Livestock Research for Rural Development, 25 (2013). Google Scholar

[50]

A. Zhigljavsky, H. Dette and A. Pepelyshev, A new approach to optimal design for linear models with correlated observations,, Journal of the American Statistical Association, 105 (2010), 1093. doi: 10.1198/jasa.2010.tm09467. Google Scholar

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