Edwin A. Iguodala¹, Eghwerido Joseph Thomas^2*, Austin Edokpayi¹, Obilade Titilola³

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¹ Nigerian Institute for Oil Palm Research, Benin City, Nigeria.
² Department of Mathematics and Computer Science, Federal University of Petroleum Resources, Effurun, Nigeria.
³ Department of Mathematics, Obafemi Awolowo University, Ile-Ife, Nigeria.
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Abstract:

Mixed model analysis procedure was used to analyze the effect of fertilizer application on the Fresh Fruit Bunch (FFB) yield of oil palm. This was with a view to achieve the most appropriate and a robust model for analyzing yield response for fertilizer application in oil palm. In this study, a mixed model analysis procedure was used to analyze yield data obtained from a fertilizer trial conducted between 1997 and 2005. In mixed effect model, replicates and years were used as block. In contrast the fixed effect ANOVA model usually lumped up replicates and years as a random error. In the model replicates were used as block with no block interaction, replicates as block with allowance for block-fertilizer interaction, years as block with allowance for block-fertilizer interaction, and years and replicates as block with allowance for year fertilizer and replicate-fertilizer interaction. Mixed model theory was also used to provide the explicit description of the design matrices in the models. Also, hypotheses relevant to each model were formulated and used to test for specific effects in the models such as, fixed part, random part and interacting parts using appropriate error terms as determined by the derived Expected Mean Squares (EMS). The results revealed that at 5% significant level (p < 0.05), the combination of Potassium (K) at 3.5 kg and magnesium (Mg) at 1.7 kg was sufficient for bunch yield of oil palm as the effect of fertilizer application was significant in the interactions of K and Mg due to treatment.

Keywords: Fixed Model, Random Model, Mixed Model, Oil Palm, Fertilizer Trial

Cite this paper: Iguodala, E. , Joseph Thomas, E. , Edokpayi, A. and Titilola, O. (2016) A Mixed Model Analysis of a Fertilizer Experiment on Oil Palm in Nigeria. Agricultural Sciences, 7, 521-530. doi: 10.4236/as.2016.78052.

References

[1]   [1]Airy, G.B. (1861) On the Algebraical and Numerical Theory of Errors of Observations and the Combinations of Observations. MacMillan, London.

[2]   Bailey, R.A. (1981) A Unified Approach to Design of Experiments. Journal of the Royal Statistical Society, Series A, 144, 214-223.
http://dx.doi.org/10.2307/2981920

[3]   Saville, B.R. and Herring, A.H. (2009) Testing Random Effects in the Linear Mixed Model Using Approximate Bayes factors. Biometrika, 65, 369-376.

[4]   Chi, E.M. and Reinsel, G.C. (1989) Models for Longitudinal Data with Random Effects and AR(1) Errors. Journal of the American Statistical Association, 84, 452-459.
http://dx.doi.org/10.1080/01621459.1989.10478790

[5]   Chiarandini, M. and Goegebeur, Y. (2010) Mixed Models for the Analysis of Optimization Algorithms. In: Bartz-Beielstein, T., Ed., Experimental Method for the Analysis of Optimization Algorithms, Springer, Berlin Heidelberg, 225-264.
http://dx.doi.org/10.1007/978-3-642-02538-9_10

[6]   Crump, S.L. (1947) The Estimation of Variance in Multiple Classification. PhD Thesis, Department of Statistics, Iowa State University, Ames.

[7]   Dempster, A.P., Laird, N.M. and Rubin, D.B. (1977) Maximum Likelihood from Incomplete Data via the EM Algorithm. Journal of the Royal Statistical Society, Series B, 39, 138.

[8]   Thisted, R.A. (1988) Elements of Statistical Computing. Chapman Hall, London.

[9]   Lindstrom, M.J. and Bates, D.M. (1988) Newton-Raphson and EM Algorithms for Linear Mixed-Effects Models for Repeated-Measures Data. Journal of the American Statistical Association, 83, 1014-1022.

[10]   Duchateau, L., Janssen, P. and Rowlands, J. (1998) Linear Mixed Models, an Introduction with Applications in Veterinary Research. International Livestock Research Institute, Kenya.

[11]   Fisher, R.A. (1925) Statistical Methods for Research Workers. Oliver and Boyd, London.

[12]   Fisher, R.A. (1935) Contribution to the Discussion of “Complex Experiments” by F. Yates. J. R. Statist. Soc, 2, 23-31

[13]   Foster, H.L. and Tarmizi, M.A. (1988) Variation in the Fertilizer Requirements of Oil Palm in Peninsular Malaysia. I. Within the Same Series. PORIM Bulletin, 16, 1-9.

[14]   Foster, H. (2003) Assessment of Oil Palm Fertilizer Requirements. In: Fairhurst, T.H. and Hardter, R., Eds, Oil Palm: Management for Large and Sustainable Yields, Potash and Phosphate Institute (PPI), Potash and Phosphate Institute of Canada (PPIC) and International Potash Institute, 231-257

[15]   Gelfand, A.E., Hills, S.E., Racine-Poon, A. and Smith, A.F.M. (1990) Illustration of Bayesian Inference in Normal Data Models Using Gibbs Sampling. Journal of the American Statistical Association, 85, 972-985.
http://dx.doi.org/10.1080/01621459.1990.10474968

[16]   Gelman, A. (2006) Analysis of Variance. Columbia University, New York.
http://dx.doi.org/10.1017/cbo9780511790942.028

[17]   Hartley, H.O. and Rao, J.N.K. (1967) Maximum-Likelihood Estimation for the Mixed Analysis of Variance Model. Biometrika, 54, 93-108.
http://dx.doi.org/10.1093/biomet/54.1-2.93

[18]   Harville, D.A. (1977) Maximum Likelihood Approaches to Variance Components Estimation and to Related Problems. Journal of the American Statistical Association, 72, 320-338.
http://dx.doi.org/10.1080/01621459.1977.10480998

[19]   Harville, D.A. and Mee, R.W. (1984) A Mixed-Model Procedure for Analyzing Ordered Categorical Data. Biometrics, 40, 393-408.
http://dx.doi.org/10.2307/2531393

[20]   Henderson, C.R. (1953) Estimation of Variance and Covariance Components. Biometrics, 9, 226-252.
http://dx.doi.org/10.2307/3001853

[21]   Henderson, C.R., Kempthorne, O., Searle, S.R. and von Krosigk, C.M. (1959) The Estimation of Environmental and Genetic Trends from Records Subject to Culling. Biometrics, 15, 192-218.
http://dx.doi.org/10.2307/2527669

[22]   Henderson, C.R. and Quaas, R.L. (1976) Multiple Trait Evaluation Using Relative’s Records. Journal of Animal Science, 43, 11-88.

[23]   Henderson, C.R. (1990) Statistical Method in Animal Improvement: Historical Overview. In: Gianola, D. and Hammond, K., Eds., Advances in Statistical Methods for Genetic Improvement of Livestock, Springer-Verlag, Berlin, 2-14.
http://dx.doi.org/10.1007/978-3-642-74487-7_1

[24]   Jennrich, R.I. and Schluchter, M.D. (1986) Unbalanced Repeated Measures Models with Structural Covariance Matrices. Biometrics, 42, 805-820.
http://dx.doi.org/10.2307/2530695

[25]   Laird, N.M. and Ware, J.H. (1982) Random-Effects Models for Longitudinal Data. Biometrics, 38, 963-974.
http://dx.doi.org/10.2307/2529876

[26]   Liang, K.-Y. and Zeger, S.L. (1986) Longitudinal Data Analysis Using Generalized Linear Models. Biometrika, 73, 13- 22.
http://dx.doi.org/10.1093/biomet/73.1.13

[27]   Makinde, E.A., Oluwa, O.K., Oke, A.O. and Duyile, P.O. (2010) Effects of Organic, Organomineral and NPK Fertilizer Treatments on Fresh and Dry Matter Yield of Amaranthus cruentus L. on Soil Types in Lagos, Nigeria. New York Science Journal, 3, 12-17.

[28]   Melaniel, L.B. and Grunwald, G.K. (2004) Mixed Model for the Analysis of Replicated Spatial Point Patterns. Bio-statistics, 5, 633-648.

[29]   Miller, J.J. (1977) Asymptotic Properties of Maximum Likelihood Estimates in the Mixed Model of the Analysis of Variance. The Annalds of Statistics, 5, 746-762.
http://dx.doi.org/10.1214/aos/1176343897

[30]   Patterson, H.D. and Thompson, R. (1971) Recovery of Inter-Block Information when Block Sizes Are Unequal. Biometrika, 78, 609-619.

[31]   Piepho, H.P., Buchse, A. and Emrich, K. (2003) A Hitchhiker’s Guide to Mixed Models for Randomized Experiments. Journal of Agronomy and Crop Science, 189, 310-322.
http://dx.doi.org/10.1046/j.1439-037X.2003.00049.x

[32]   Piepho, H.P., Buchse, A. and Richter, C. (2004) A Mixed Modelling Approach for Randomized Experiments with Repeated Measures. Journal of Agronomy and Crop Science, 190, 230-247.
http://dx.doi.org/10.1111/j.1439-037X.2004.00097.x

[33]   Smith, A.B., Cullis, B.R. and Thompson, R. (2005) The Analysis of Crop Cultivar Breeding and and Evaluation Trials: An Overview of Current Mixed Model Approaches. Journal of Agricultural Science, 143, 449-462.
http://dx.doi.org/10.1017/S0021859605005587

[34]   Wakefield, J.C., Smith, A.F.M., Racine-Poon, A. and Gelfand, A.E. (1994) Bayesian Analysis of Linear and Nonlinear Population Models Using the Gibbs Sampler, Applied Statistics. 143-152.

[35]   Witkovsky, V. (2001) MATLAB Algorithm Mixed.M for Solving Henderson’s Mixed Model Equations. Technical Report, Institute of Measurement Science, Slovak Academy of Sciences, Bratislava.

[36]   Yeboah, S., Akromah, R. and Quansah, C. (2012) Organic and Inorganic Fertilizers Application on the Growth and Yield of Artemisia annua L. in the Humid Tropics of Ghana. African Journal of Agricultural Research, 7, 177-182.
http://dx.doi.org/10.5897/AJAR11.679

[37]   Jun, Z. (1993) Mixed Model Approaches for Estimating Genetic Covariances between Two Traits with Unequal Design Matrices. Journal of Biomathematics, 8, 24-30.