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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[9]

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

[10]

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

[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.

[12]

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

[13]

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

[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.

[15]

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

[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.

[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.

[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.

[19]

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

[20]

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

[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.

[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.

[23]

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

[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.

[25]

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

[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.

[27]

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

[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.

[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.

[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.

[31]

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

[32]

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

[33]

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

[34]

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

[35]

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

[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.

[37]

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

[38]

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

[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.

[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.

[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.

[42]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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).

[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.

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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[9]

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

[10]

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

[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.

[12]

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

[13]

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

[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.

[15]

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

[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.

[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.

[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.

[19]

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

[20]

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

[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.

[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.

[23]

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

[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.

[25]

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

[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.

[27]

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

[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.

[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.

[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.

[31]

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

[32]

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

[33]

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

[34]

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

[35]

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

[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.

[37]

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

[38]

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

[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.

[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.

[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.

[42]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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).

[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.

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