The Annals of Statistics
- Ann. Statist.
- Volume 44, Number 3 (2016), 1131-1164.
Randomization-based models for multitiered experiments: I. A chain of randomizations
Full-text: Open access
Abstract
We derive randomization-based models for experiments with a chain of randomizations. Estimation theory for these models leads to formulae for the estimators of treatment effects, their standard errors and expected mean squares in the analysis of variance. We discuss the practicalities in fitting these models and outline the difficulties that can occur, many of which do not arise in two-tiered experiments.
Article information
Source
Ann. Statist., Volume 44, Number 3 (2016), 1131-1164.
Dates
Received: August 2013
Revised: May 2015
First available in Project Euclid: 11 April 2016
Permanent link to this document
https://projecteuclid.org/euclid.aos/1460381689
Digital Object Identifier
doi:10.1214/15-AOS1400
Mathematical Reviews number (MathSciNet)
MR3485956
Zentralblatt MATH identifier
1338.62028
Subjects
Primary: 62B15: Theory of statistical experiments 62J05: Linear regression
Secondary: 62J10: Analysis of variance and covariance
Keywords
Analysis of variance expected mean square mixed model multiphase experiments multitiered experiments randomization-based model REML structure tier
Citation
Bailey, R. A.; Brien, C. J. Randomization-based models for multitiered experiments: I. A chain of randomizations. Ann. Statist. 44 (2016), no. 3, 1131--1164. doi:10.1214/15-AOS1400. https://projecteuclid.org/euclid.aos/1460381689
References
- [1] Alejandro, P. P., Bailey, R. A. and Cameron, P. J. (2003). Association schemes and permutation groups. Discrete Math. 266 47–67.Mathematical Reviews (MathSciNet): MR1991706
Digital Object Identifier: doi:10.1016/S0012-365X(02)00798-7 - [2] Bailey, R. A. (1981). A unified approach to design of experiments. J. Roy. Statist. Soc. Ser. A 144 214–223.
- [3] Bailey, R. A. (1991). Strata for randomized experiments (with discussion). J. R. Stat. Soc. Ser. B Stat. Methodol. 53 27–78.Mathematical Reviews (MathSciNet): MR1094275
- [4] Bailey, R. A. (2008). Design of Comparative Experiments. Cambridge Univ. Press, Cambridge.Mathematical Reviews (MathSciNet): MR2422352
- [5] Bailey, R. A., Praeger, C. E., Rowley, C. A. and Speed, T. P. (1983). Generalized wreath products of permutation groups. Proc. Lond. Math. Soc. (3) 47 69–82.
- [6] Bardin, A. and Azaïs, J.-M. (1990). Une hypothèse minimale pour une théorie des plans d’expériences randomisés. Rev. Stat. Appl. 38 21–41.Mathematical Reviews (MathSciNet): MR1080504
- [7] Bates, D., Maechler, M. and Bolker, B. (2016). lme4: Linear mixed-effects models using S4 classes. Available at http://cran.at.r-project.org/web/packages/lme4/index.html (R package version 1.1-10, accessed January 8, 2016).
- [8] Brien, C. J. (1983). Analysis of variance tables based on experimental structure. Biometrics 39 51–59.
- [9] Brien, C. J. (1992). Factorial linear model analysis. Ph.D. thesis, Department of Plant Science, The University of Adelaide, Adelaide, South Australia. Available at http://hdl.handle.net/2440/37701 (accessed January 8, 2016).
- [10] Brien, C. J. (2015). dae: functions useful in the design and ANOVA of experiments. Available at http://cran.at.r-project.org/web/packages/dae/index.html (R package version 2.7-0, accessed April 26, 2015).
- [11] Brien, C. J. and Bailey, R. A. (2006). Multiple randomizations (with discussion). J. R. Stat. Soc. Ser. B Stat. Methodol. 68 571–609.Mathematical Reviews (MathSciNet): MR2301010
Digital Object Identifier: doi:10.1111/j.1467-9868.2006.00557.x - [12] Brien, C. J. and Bailey, R. A. (2009). Decomposition tables for experiments. I. A chain of randomizations. Ann. Statist. 37 4184–4213.Mathematical Reviews (MathSciNet): MR2572457
Digital Object Identifier: doi:10.1214/09-AOS717
Project Euclid: euclid.aos/1256303541 - [13] Brien, C. J. and Bailey, R. A. (2010). Decomposition tables for experiments. II. Two–one randomizations. Ann. Statist. 38 3164–3190.Mathematical Reviews (MathSciNet): MR2722467
Digital Object Identifier: doi:10.1214/09-AOS785
Project Euclid: euclid.aos/1284391761 - [14] Brien, C. J. and Demétrio, C. G. B. (2009). Formulating mixed models for experiments, including longitudinal experiments. J. Agric. Biol. Environ. Stat. 14 253–280.Mathematical Reviews (MathSciNet): MR2750840
Digital Object Identifier: doi:10.1198/jabes.2009.08001 - [15] Brien, C. J., Harch, B. D., Correll, R. L. and Bailey, R. A. (2011). Multiphase experiments with at least one later laboratory phase. I. Orthogonal designs. J. Agric. Biol. Environ. Stat. 16 422–450.Mathematical Reviews (MathSciNet): MR2843135
Digital Object Identifier: doi:10.1007/s13253-011-0060-z - [16] Brien, C. J. and Payne, R. W. (1999). Tiers, structure formulae and the analysis of complicated experiments. The Statistician 48 41–52.
- [17] Brien, C. J. and Payne, R. W. (2012). AMTIER Procedure. In GenStat Reference Manual Release 15, Part 3 Procedures 106–108. VSN International, Hemel Hempstead, UK. Available at http://www.vsni.co.uk/resources/documentation/genstat-reference-procedure-library/ (accessed January 8, 2016).
- [18] Butler, D., Cullis, B. R., Gilmour, A. R. and Gogel, B. J. (2009). Mixed Models for S Language Environments: ASReml-R Reference Manual. Version 3. DPI Publications, Brisbane. Available at http://www.vsni.co.uk/resources/documentation/ (accessed January 8, 2016).
- [19] Cox, D. R. (1958). The interpretation of the effects of non-additivity in the Latin square. Biometrika 45 69–73.
- [20] Curnow, R. N. (1959). The analysis of a two phase experiment. Biometrics 15 60–73.
- [21] Gilmour, S. G. and Goos, P. (2009). Analysis of data from non-orthogonal multistratum designs in industrial experiments. J. R. Stat. Soc. Ser. C Appl. Stat. 58 467–484.Mathematical Reviews (MathSciNet): MR2750088
Digital Object Identifier: doi:10.1111/j.1467-9876.2009.00662.x - [22] Grundy, P. M. and Healy, M. J. R. (1950). Restricted randomization and quasi-Latin squares. J. R. Stat. Soc. Ser. B 12 286–291.
- [23] Houtman, A. M. and Speed, T. P. (1983). Balance in designed experiments with orthogonal block structure. Ann. Statist. 11 1069–1085.
- [24] James, A. T. (1957). The relationship algebra of an experimental design. Ann. Math. Stat. 28 993–1002.Mathematical Reviews (MathSciNet): MR97142
Digital Object Identifier: doi:10.1214/aoms/1177706800
Project Euclid: euclid.aoms/1177706800 - [25] Littell, R. C., Milliken, G. A., Stroup, W. W., Wolfinger, R. D. and Schabenberger, O. (2006). SAS for Mixed Models, 2nd ed. SAS Press, Cary, NC.
- [26] McIntyre, G. A. (1955). Design and analysis of two phase experiments. Biometrics 11 324–334.
- [27] Monod, H. and Bailey, R. A. (1992). Pseudofactors: Normal use to improve design and facilitate analysis. J. R. Stat. Soc. Ser. C Appl. Stat. 41 317–336.
- [28] Nelder, J. A. (1954). The interpretation of negative components of variance. Biometrika 41 544–548.
- [29] Nelder, J. A. (1965). The analysis of randomized experiments with orthogonal block structure. I. Block structure and the null analysis of variance. Proc. Roy. Soc. Ser. A 283 147–162.
- [30] Nelder, J. A. (1965). The analysis of randomized experiments with orthogonal block structure. II. Treatment structure and the general analysis of variance. Proc. Roy. Soc. Ser. A 283 163–178.
- [31] Nelder, J. A. (1968). The combination of information in generally balanced designs. J. R. Stat. Soc. Ser. B Stat. Methodol. 30 303–311.
- [32] Nelder, J. A. (1977). A reformulation of linear models. J. Roy. Statist. Soc. Ser. A 140 48–76.
- [33] Ojima, Y. (1998). General formulae for expectations, variances and covariances of the mean squares for staggered nested designs. J. Appl. Stat. 25 785–799.
- [34] Patterson, H. D. and Thompson, R. (1971). Recovery of inter-block information when block sizes are unequal. Biometrika 58 545–554.
- [35] Patterson, H. D. and Thompson, R. (1975). Maximum likelihood estimation of components of variance. In Proceedings of the Eighth International Biometric Conference (L. C. A. Corsten and T. Postelnicu, eds.) 197–207. Editura Academiei Republicii Socialiste România, Bucureşti.Mathematical Reviews (MathSciNet): MR468083
- [36] Payne, R. W. and Wilkinson, G. N. (1977). A general algorithm for analysis of variance. J. R. Stat. Soc. Ser. C Appl. Stat. 26 251–260.
- [37] Pinheiro, J. C. and Bates, D. M. (2000). Mixed Effects Models in S and S-PLUS. Springer, New York.
- [38] R Development Core Team (2016). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. Available at http://www.r-project.org/ (accessed January 8, 2016).
- [39] SAS Institute Inc. (2010). SAS/STAT® 9.22 User’s Guide. SAS Institute Inc., Cary, NC.
- [40] Searle, S. R., Casella, G. and McCulloch, C. E. (1992). Variance Components. Wiley, New York.Mathematical Reviews (MathSciNet): MR1190470
- [41] Smith, A., Cullis, B. and Gilmour, A. (2001). The analysis of crop variety evaluation data in Australia. Aust. N. Z. J. Stat. 43 129–145.Mathematical Reviews (MathSciNet): MR1855705
- [42] Speed, T. P. (1987). What is an analysis of variance? (with discussion). Ann. Statist. 15 885–941.Mathematical Reviews (MathSciNet): MR902237
Digital Object Identifier: doi:10.1214/aos/1176350472
Project Euclid: euclid.aos/1176350472 - [43] Speed, T. P. and Bailey, R. A. (1982). On a class of association schemes derived from lattices of equivalence relations. In Algebraic Structures and Applications (University of Western Australia, 1980) (P. Schultz, C. E. Praeger and R. P. Sullivan, eds.). Lecture Notes in Pure and Applied Mathematics 74 55–74. Dekker, New York.Mathematical Reviews (MathSciNet): MR647160
- [44] Speed, T. P. and Bailey, R. A. (1987). Factorial dispersion models. Int. Stat. Rev. 55 261–277.
- [45] Tjur, T. (1984). Analysis of variance models in orthogonal designs (with discusion). Int. Stat. Rev. 52 33–81.
- [46] VSN International (2015). GenStat for Windows, 18th ed. VSN International, Hemel Hempstead, UK. Available at GenStat.co.uk.
- [47] Wilkinson, G. N. (1970). A general recursive procedure for analysis of variance. Biometrika 57 19–46.
- [48] Wolde-Tsadik, G. and Afifi, A. A. (1980). A comparison of the “sometimes pool”, “sometimes switch” and “never pool” procedures in the two-way ANOVA random effects model. Technometrics 22 367–373.Mathematical Reviews (MathSciNet): MR585634
- [49] Wood, J. T., Williams, E. R. and Speed, T. P. (1988). Non-orthogonal block structure in two-phase designs. Aust. J. Stat. 30A 225–237.
- [50] Yates, F. (1936). A new method of arranging variety trials involving a large number of varieties. J. Agric. Sci. 26 424–455. Reprinted with additional author’s note in Yates, F. (1970) Experimental Design: Selected Papers, pages 147–180. Griffin, London.
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