Minimum Number of Replications for Tests in Four-Way ANOVA in Cross Classification and Split-Plot Design

Rob Verdooren; Dieter Rasch

doi:doi:10.11648/j.ajtas.20221101.17

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Minimum Number of Replications for Tests in Four-Way ANOVA in Cross Classification and Split-Plot Design

Rob Verdooren, Dieter Rasch

Published in American Journal of Theoretical and Applied Statistics (Volume 11, Issue 1)

Received: 17 January 2022 Accepted: 9 February 2022 Published: 25 February 2022

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Abstract

In statistical books for the analysis of designed experiments one can finds sometimes also the computation of the number of replications for balanced one-factor and two-factors designs. Later there were papers published concerning the computation of the number of replications of at most three-factors crossed or nested balanced designs. In 2011 the book “Optimal experimental design with R” was published; further a special R- program OPDOE was made to do the computation for these designs and the OPDOE program was used in this book. In this paper an extension of the determination of the minimum number of replications for balanced designs is given for four-factor crossed designs. The balanced cross classification of the four-way analysis of variance of the following models are investigated: Model 1 The factors A, B, C and D are all fixed; Model 2 D is random A, B and C are fixed; Model 3 C and D are random, A and B are fixed; Model 4 B, C and D are random, A is fixed. For these models small R-programs are given to compute the minimal number of the replications for testing the fixed effects using the non-centrality parameter λ of the non-central F- distribution F(df₁, df₂, λ). Further balanced Split-Plot design with one or two fixed factors in the main-plots are considered. The Blocks are denoted with B. The F statistics for testing the significance of the fixed factors are described and small R-programs for the determination of the minimal number of replications are given using the non-centrality parameter λ of the non-central F- distribution F(df₁, df₂, λ).

Published in	American Journal of Theoretical and Applied Statistics (Volume 11, Issue 1)
DOI	10.11648/j.ajtas.20221101.17
Page(s)	45-57
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Balanced Four-way ANOVA, Cross Classification, Split-plot Designs, Non-centrality Parameter λ of the Non-central, F-distribution F(df₁, df₂, λ), Minimal Number of Replications

References

[1]	Kuehl, R. O. (1994), Statistical Principles of Research Design and Analysis”, Duxburry Press, Belmont, CA.
[2]	Kutner, M. H., Nachtsheim. C. J., Neter, J. and Li, W. (2005), Applied Linear Statistical Models, 5^th edition, McGraw-Hill Irwin, New York.
[3]	Lindman, H. R. (1992), Analysis of Variance in Experimental Design, Springer-Verlag, New York.
[4]	Ott, R. L. and Longnecker, M. (2001), An Introduction to Statistcal Methods and Data Analyis, 5^thedition,Duxbury, Pacific Grove, CA.
[5]	Owen, D. B. (1962), Handbook of Statistical Tables, Addison-Wesley Publishing Company, Reading, MA.
[6]	Pearson, E. S. and Hartley, H. O. (1972), Biometrika Tables for Statisticians, Volume II, Cambridge, The University Press.
[7]	Rasch, D., Herrendörfer, G., Bock, J., Victor, N. and Guiard, V. (2008), Verfahrensbibliothek Versuchsplanung und ‑ auswertung, 2. verbesserte Auflage in einem Band mit CD, R. Oldenbourg Verlag, München, Wien.
[8]	Rasch, D. und Schott, D. (2016), Mathematische Statistik, Wiley – VCH, Weinheim.
[9]	Rasch, D. and Schott, D. (2018), Mathematical Statistics, Wiley, Oxford.
[10]	Rasch, D., Kubinger, K.D. and Yanagida, T. (2011), Statistics in Psychology using R and SPSS, Wiley, New York.
[11]	Rasch, D., Pilz, J., Verdooren, R. and Gebhardt, A. (2011), Optimal experimental design with R, Chapman and Hall, London.
[12]	Rasch, D., Spangl, B. and Wang, M. (2012), Minimal experimental size in the three way ANOVA cross classification model with approximate F-tests, Communications in Statistics- Simulation and Computation 41 (7): 1120–1130.
[13]	Rasch, D. and Verdooren, R. (2020), Determination of Minimum and Maximum Experimental Size in One-, Two- and Three-way ANOVA with Fixed and Mixed Models by R. Journal of Statistical Theory and Practice 14, 57 (2020). https://doi.org/10.1007/s42519-020-00088-6.
[14]	Rasch, D, Verdooren, R. and Pilz, R. (2019), Applied Statistics, Wiley, Oxford.
[15]	Spangl, B., Kaiblinger, N., Ruckdeschel, P. and Rasch, D. (2021), Minimum sample size in balanced ANOVA models of crossed, nested and mixed classifications, Communications in Statistics – Theory and Methods, https://doi.org/10.1080/03610926.2021.1938126.
[16]	Tang, P.C. (1938), The power function of the analysis of variance tests with tables and illustrations of their use, Statistical Research Memoirs, 2, 126-149.
[17]	Wang, M., Rasch, D. and Verdooren, R. (2005), Determination of the size of a balanced experiment in mixed ANOVA models using the modified approximate F-test. Journal of Statistical Planning and Inference 132 (1-2): 183-201.

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APA Style

Rob Verdooren, Dieter Rasch. (2022). Minimum Number of Replications for Tests in Four-Way ANOVA in Cross Classification and Split-Plot Design. American Journal of Theoretical and Applied Statistics, 11(1), 45-57. https://doi.org/10.11648/j.ajtas.20221101.17

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Rob Verdooren; Dieter Rasch. Minimum Number of Replications for Tests in Four-Way ANOVA in Cross Classification and Split-Plot Design. Am. J. Theor. Appl. Stat. 2022, 11(1), 45-57. doi: 10.11648/j.ajtas.20221101.17

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AMA Style

Rob Verdooren, Dieter Rasch. Minimum Number of Replications for Tests in Four-Way ANOVA in Cross Classification and Split-Plot Design. Am J Theor Appl Stat. 2022;11(1):45-57. doi: 10.11648/j.ajtas.20221101.17

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@article{10.11648/j.ajtas.20221101.17,
  author = {Rob Verdooren and Dieter Rasch},
  title = {Minimum Number of Replications for Tests in Four-Way ANOVA in Cross Classification and Split-Plot Design},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {11},
  number = {1},
  pages = {45-57},
  doi = {10.11648/j.ajtas.20221101.17},
  url = {https://doi.org/10.11648/j.ajtas.20221101.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20221101.17},
  abstract = {In statistical books for the analysis of designed experiments one can finds sometimes also the computation of the number of replications for balanced one-factor and two-factors designs. Later there were papers published concerning the computation of the number of replications of at most three-factors crossed or nested balanced designs. In 2011 the book “Optimal experimental design with R” was published; further a special R- program OPDOE was made to do the computation for these designs and the OPDOE program was used in this book. In this paper an extension of the determination of the minimum number of replications for balanced designs is given for four-factor crossed designs. The balanced cross classification of the four-way analysis of variance of the following models are investigated: Model 1 The factors A, B, C and D are all fixed; Model 2 D is random A, B and C are fixed; Model 3 C and D are random, A and B are fixed; Model 4 B, C and D are random, A is fixed. For these models small R-programs are given to compute the minimal number of the replications for testing the fixed effects using the non-centrality parameter λ of the non-central F- distribution F(df1, df2, λ). Further balanced Split-Plot design with one or two fixed factors in the main-plots are considered. The Blocks are denoted with B. The F statistics for testing the significance of the fixed factors are described and small R-programs for the determination of the minimal number of replications are given using the non-centrality parameter λ of the non-central F- distribution F(df1, df2, λ).},
 year = {2022}
}

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TY  - JOUR
T1  - Minimum Number of Replications for Tests in Four-Way ANOVA in Cross Classification and Split-Plot Design
AU  - Rob Verdooren
AU  - Dieter Rasch
Y1  - 2022/02/25
PY  - 2022
N1  - https://doi.org/10.11648/j.ajtas.20221101.17
DO  - 10.11648/j.ajtas.20221101.17
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 45
EP  - 57
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20221101.17
AB  - In statistical books for the analysis of designed experiments one can finds sometimes also the computation of the number of replications for balanced one-factor and two-factors designs. Later there were papers published concerning the computation of the number of replications of at most three-factors crossed or nested balanced designs. In 2011 the book “Optimal experimental design with R” was published; further a special R- program OPDOE was made to do the computation for these designs and the OPDOE program was used in this book. In this paper an extension of the determination of the minimum number of replications for balanced designs is given for four-factor crossed designs. The balanced cross classification of the four-way analysis of variance of the following models are investigated: Model 1 The factors A, B, C and D are all fixed; Model 2 D is random A, B and C are fixed; Model 3 C and D are random, A and B are fixed; Model 4 B, C and D are random, A is fixed. For these models small R-programs are given to compute the minimal number of the replications for testing the fixed effects using the non-centrality parameter λ of the non-central F- distribution F(df1, df2, λ). Further balanced Split-Plot design with one or two fixed factors in the main-plots are considered. The Blocks are denoted with B. The F statistics for testing the significance of the fixed factors are described and small R-programs for the determination of the minimal number of replications are given using the non-centrality parameter λ of the non-central F- distribution F(df1, df2, λ).
VL  - 11
IS  - 1
ER  -

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Author Information

Rob Verdooren

Danone Nutricia Research, Division Data Sciences, Utrecht, the Netherlands
Dieter Rasch

Institute of Statistics, University of Natural Resources and Life Sciences, Vienna, Austria

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APA Style

Rob Verdooren, Dieter Rasch. (2022). Minimum Number of Replications for Tests in Four-Way ANOVA in Cross Classification and Split-Plot Design. American Journal of Theoretical and Applied Statistics, 11(1), 45-57. https://doi.org/10.11648/j.ajtas.20221101.17

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ACS Style

Rob Verdooren; Dieter Rasch. Minimum Number of Replications for Tests in Four-Way ANOVA in Cross Classification and Split-Plot Design. Am. J. Theor. Appl. Stat. 2022, 11(1), 45-57. doi: 10.11648/j.ajtas.20221101.17

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AMA Style

Rob Verdooren, Dieter Rasch. Minimum Number of Replications for Tests in Four-Way ANOVA in Cross Classification and Split-Plot Design. Am J Theor Appl Stat. 2022;11(1):45-57. doi: 10.11648/j.ajtas.20221101.17

Copy | Download

@article{10.11648/j.ajtas.20221101.17,
  author = {Rob Verdooren and Dieter Rasch},
  title = {Minimum Number of Replications for Tests in Four-Way ANOVA in Cross Classification and Split-Plot Design},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {11},
  number = {1},
  pages = {45-57},
  doi = {10.11648/j.ajtas.20221101.17},
  url = {https://doi.org/10.11648/j.ajtas.20221101.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20221101.17},
  abstract = {In statistical books for the analysis of designed experiments one can finds sometimes also the computation of the number of replications for balanced one-factor and two-factors designs. Later there were papers published concerning the computation of the number of replications of at most three-factors crossed or nested balanced designs. In 2011 the book “Optimal experimental design with R” was published; further a special R- program OPDOE was made to do the computation for these designs and the OPDOE program was used in this book. In this paper an extension of the determination of the minimum number of replications for balanced designs is given for four-factor crossed designs. The balanced cross classification of the four-way analysis of variance of the following models are investigated: Model 1 The factors A, B, C and D are all fixed; Model 2 D is random A, B and C are fixed; Model 3 C and D are random, A and B are fixed; Model 4 B, C and D are random, A is fixed. For these models small R-programs are given to compute the minimal number of the replications for testing the fixed effects using the non-centrality parameter λ of the non-central F- distribution F(df1, df2, λ). Further balanced Split-Plot design with one or two fixed factors in the main-plots are considered. The Blocks are denoted with B. The F statistics for testing the significance of the fixed factors are described and small R-programs for the determination of the minimal number of replications are given using the non-centrality parameter λ of the non-central F- distribution F(df1, df2, λ).},
 year = {2022}
}

Copy | Download

TY  - JOUR
T1  - Minimum Number of Replications for Tests in Four-Way ANOVA in Cross Classification and Split-Plot Design
AU  - Rob Verdooren
AU  - Dieter Rasch
Y1  - 2022/02/25
PY  - 2022
N1  - https://doi.org/10.11648/j.ajtas.20221101.17
DO  - 10.11648/j.ajtas.20221101.17
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 45
EP  - 57
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20221101.17
AB  - In statistical books for the analysis of designed experiments one can finds sometimes also the computation of the number of replications for balanced one-factor and two-factors designs. Later there were papers published concerning the computation of the number of replications of at most three-factors crossed or nested balanced designs. In 2011 the book “Optimal experimental design with R” was published; further a special R- program OPDOE was made to do the computation for these designs and the OPDOE program was used in this book. In this paper an extension of the determination of the minimum number of replications for balanced designs is given for four-factor crossed designs. The balanced cross classification of the four-way analysis of variance of the following models are investigated: Model 1 The factors A, B, C and D are all fixed; Model 2 D is random A, B and C are fixed; Model 3 C and D are random, A and B are fixed; Model 4 B, C and D are random, A is fixed. For these models small R-programs are given to compute the minimal number of the replications for testing the fixed effects using the non-centrality parameter λ of the non-central F- distribution F(df1, df2, λ). Further balanced Split-Plot design with one or two fixed factors in the main-plots are considered. The Blocks are denoted with B. The F statistics for testing the significance of the fixed factors are described and small R-programs for the determination of the minimal number of replications are given using the non-centrality parameter λ of the non-central F- distribution F(df1, df2, λ).
VL  - 11
IS  - 1
ER  -

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