# On the General Gauss-Markov Model for Experiments in Block Designs

Biometrical Letters (2012)

- Volume: 49, Issue: 1, page 1-36
- ISSN: 1896-3811

## Access Full Article

top## Abstract

top## How to cite

topTadeusz Caliński. "On the General Gauss-Markov Model for Experiments in Block Designs." Biometrical Letters 49.1 (2012): 1-36. <http://eudml.org/doc/268753>.

@article{TadeuszCaliński2012,

abstract = {The main estimation and hypothesis testing results related to the Gauss- Markov model, in its general form, are recalled and the application of these results to the analysis of experiments in block designs is considered. Special attention is given to the randomization-derived model for a general block design, and for a proper block design in particular. The question whether the randomization-derived model can be considered as a particular general Gauss-Markov model is discussed. It is indicated that the former, as a mixed model, is in fact an extension of the general Gauss-Markov model. Thus, the analysis based on the randomization-derived model requires a more extended methodical approach. The present paper has been inspired by one of the last papers of Professor Wiktor Oktaba.},

author = {Tadeusz Caliński},

journal = {Biometrical Letters},

keywords = {block designs; estimation; general Gauss-Markov model; hypothesis testing; mixed model; randomization-derived model},

language = {eng},

number = {1},

pages = {1-36},

title = {On the General Gauss-Markov Model for Experiments in Block Designs},

url = {http://eudml.org/doc/268753},

volume = {49},

year = {2012},

}

TY - JOUR

AU - Tadeusz Caliński

TI - On the General Gauss-Markov Model for Experiments in Block Designs

JO - Biometrical Letters

PY - 2012

VL - 49

IS - 1

SP - 1

EP - 36

AB - The main estimation and hypothesis testing results related to the Gauss- Markov model, in its general form, are recalled and the application of these results to the analysis of experiments in block designs is considered. Special attention is given to the randomization-derived model for a general block design, and for a proper block design in particular. The question whether the randomization-derived model can be considered as a particular general Gauss-Markov model is discussed. It is indicated that the former, as a mixed model, is in fact an extension of the general Gauss-Markov model. Thus, the analysis based on the randomization-derived model requires a more extended methodical approach. The present paper has been inspired by one of the last papers of Professor Wiktor Oktaba.

LA - eng

KW - block designs; estimation; general Gauss-Markov model; hypothesis testing; mixed model; randomization-derived model

UR - http://eudml.org/doc/268753

ER -

## References

top- Bailey R.A. (1981): A unified approach to design of experiments. Journal of the Royal Statistical Society, Series A 144: 214-223. Zbl0469.62053
- Bailey R.A. (1991): Strata for randomized experiments. Journal of the Royal Statistical Society, Series B 53: 27-78. Zbl0800.62477
- Baksalary J.K., Kala R. (1983): On equalities between BLUEs, WLSEs, and SLSEs. The Canadian Journal of Statistics 11: 119-123. Zbl0522.62047
- Baksalary J.K., Puntanen S. (1990): Characterizations of the best linear unbiased estimator in the general Gauss-Markov model with the use of matrix partial orderings. Linear Algebra and its Applications 127: 363-370.[Crossref] Zbl0695.62152
- Caliński T. (1996): The basic contrasts of a block design with special reference to the recovery of inter-block information. In: A. Pázman and V. Witkovský (eds.), Tatra Mountains Mathematical Publications, Vol. 7: PROBASTAT' 94 Smolenice. Mathematical Institute, Bratislava, pp. 23-37. Zbl0919.62080
- Caliński T., Kageyama S. (2000): Block Designs: A Randomization Approach, Volume I: Analysis. Lecture Notes in Statistics, Volume 150, Springer, New York. Zbl0963.62071
- Demidenko E. (2004): Mixed Models: Theory and Applications. Wiley, Hoboken, New Jersey. Zbl1055.62086
- Fisher R.A. (1925): Statistical Methods for Research Workers. Oliver Boyd, Edinburgh. Zbl51.0414.08
- Gauss C.F. (1809): Theoria Motus Corporum Coelestium. Perthes and Besser, Hamburg.
- Gauss C.F. (1855): Méthode des Moindres Carrés. Mallet-Bachelier, Paris.
- Houtman A.M., Speed, T.P. (2008): Balance in designed experiments with orthogonal block structure Annals of Statistics 11: 1069-1085. Zbl0566.62065
- Hinkelmann K., Kempthorne O. (2008): Design and Analysis of Experiments Volume I: Introduction to Experimental Design 2nd ed. Wiley, Hoboken, New Jersey. Zbl1146.62054
- John J.A. (1987): Cyclic Designs. Chapman and Hall, London. Zbl0731.05001
- Jones R.M. (1959): On a property of incomplete blocks. Journal of the Royal Statistical Society, Series B 21: 172-179. Zbl0117.14308
- Kala R. (1991): Elements of the randomization theory. III. Randomization in block experiments. Listy Biometryczne|Biometrical Letters 28: 3-23 (in Polish).
- Kempthorne O. (1952): The Design and Analysis of Experiments. Wiley, New York. Zbl0049.09901
- Marko A.A. (1900): Wahrscheinlichkeitsrechnung. Telner, Leipzig.
- Martin F.B., Zyskind G. (1966): On combinability of information from uncorrelated linear models by simple weighting. Annals of Mathematical Statistics 37: 1338-1347.[Crossref] Zbl0149.15607
- Nelder J.A. (1954): The interpretation of negative components of variance. Biometrika 41: 544-548. Zbl0056.12703
- Nelder J.A. (1965): The analysis of randomized experiments with orthogonal block structure. Proceedings of the Royal Society, Series A 283: 147-178. Zbl0124.10703
- Nelder J.A. (1968): The combination of information in generally balanced designs. Journal of the Royal Statistical Society, Series B 30: 303-311. Zbl0164.49101
- Neyman J. (with cooperation of K. Iwaszkiewicz and S. Ko lodziejczyk) (1935): Statistical problems in agricultural experimentation (with discussion). Journal of the Royal Statistical Society, Supplement 2: 107-180.
- Ogawa J. (1961): The e ect of randomization on the analysis of randomized block design. Annals of the Institute of Statistical Mathematics 13: 105-117. Zbl0124.10904
- Ogawa J. (1963): On the null-distribution of the F-statistic in a randomized balanced incomplete block design under the Neyman model. Annals of Mathematical Statistics 34: 1558-1568.[Crossref] Zbl0117.36605
- Oktaba W. (1984): Tests of hypotheses for the general Gauss-Markov model. Biometrical Journal 26: 415-424.[Crossref] Zbl0565.62036
- Oktaba W. (1989): F-tests for hypotheses with block matrices and under conditions of orthogonality in the general multivariate Gauss-Markov model. Biometrical Journal 31: 317-323.[Crossref] Zbl0712.62056
- Oktaba W. (1996): Asymptotically normal distributions in the multivariate Gauss- Markov model. Listy Biometryczne|Biometrical letters 33: 25-31. Zbl1147.62309
- Oktaba W. (1998): Characterization of the multivariate Gauss-Markov model with singular covariance matrix. Applications of Mathematics 43: 119-131.[Crossref] Zbl0937.62067
- Oktaba W. (2003): The general multivariate Gauss-Markov model of the incomplete block design. Australian & New Zealand Journal of Statistics 45: 195-205.[Crossref] Zbl1064.62079
- Oktaba W., Kornacki A., Wawrzosek J. (1986): Estimation of missing values in the general Gauss-Markov model. Statistics 17: 167-177.[Crossref] Zbl0601.62090
- Oktaba W., Kornacki A., Wawrzosek J. (1988): Invariant linearly suficient transformations of the general Gauss-Marko model. Estimation and testing. Scandinavian Journal of Statistics 17: 117-124. Zbl0682.62041
- Patterson H.D., Thompson R. (1971): Recovery of inter-block information when block sizes are unequal. Biometrika 58: 545-554.[Crossref] Zbl0228.62046
- Pearce S.C. (1983): The Agricultural Field Experiment: A Statistical Examination of Theory and Practice. Wiley, Chichester.
- Pearce S.C., Caliński T., Marshall T.F. de C. (1974): The basic contrasts of an experimental design with special reference to the analysis of data. Biometrika 61: 449-460.[Crossref] Zbl0292.62052
- Raghavarao D., Padgett L.V. (2005): Block Designs: Analysis, Combinatorics and Applications. World Scientific Publishing Co., Singapore. Zbl1102.62080
- Rao C.R. (1959): Expected values of mean squares in the analysis of incomplete block experiments and some comments based on them. Sankhy~a 21: 327-336. Zbl0090.11303
- Rao C.R. (1971): Unified theory of linear estimation. Sankhy~a, Series A 33: 371-394. Zbl0236.62048
- Rao C.R. (1973): Linear Statistical Inference and Its Applications 2nd ed. Wiley, New York. Zbl0256.62002
- Rao C.R. (1974): Projectors, generalized inverses and the BLUEs. Journal of the Royal Statistical Society, Series B 36: 442-448. Zbl0291.62077
- Rao C.R., Kle e J. (1988): Estimation of Variance Components and Applications. North-Holland, Amsterdam.
- Rao C.R., Mitra S. K. (1971): Generalized Inverse of Matrices and Its Applications. Wiley, New York. Zbl0236.15004
- Scheffe H. (1959): The Analysis of Variance. Wiley, New York. Zbl0086.34603
- Seber G.A.F. (1980): The Linear Hypothesis: A General Theory. Charles Grif- fin, London.
- Shah K.R. (1992): Recovery of interblock information: An update. Journal of Statistical Planning and Inference 30: 163-172.[Crossref] Zbl0850.62615
- White R.F. (1975): Randomization in the analysis of variance. Biometrics 31: 555-571.[PubMed][Crossref] Zbl0307.62054
- Yates F. (1939): The recovery of inter-block information in variety trials arranged in three-dimensional lattices. Annals of Eugenics 9: 136-156.[Crossref]
- Yates F. (1940): The recovery of inter-block information in balanced incomplete block designs. Annals of Eugenics 10: 317-325.[Crossref]
- Zyskind G. (1967): On canonical forms, non-negative covariance matrices and best and simple least squares linear estimators in linear models. Annals of Mathematical Statistics 38: 1092-1109. [Crossref] Zbl0171.17103

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.