Skip to main content
SpringerLink
Log in

A New Look at Combining Information from Stratum Submodels

  • Conference paper
  • First Online:
Matrices, Statistics and Big Data (IWMS 2016)

Part of the book series: Contributions to Statistics ((CONTRIB.STAT.))

Included in the following conference series:

Abstract

If an experiment possesses the property of orthogonal block structure, the information contained in the experimental observations can be divided into uncorrelated parts represented by separate stratum submodels. The paper shows that this property is sufficient to obtain the results identical with those established by Nelder (J R Stat Soc Ser B 30:303–311, 1968) under the additional property of general balance. The approach proposed here is direct, quite general, and mainly geometrical.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Anderson, T. W. (1948). On the theory of testing serial correlation. Skandinavisk Actuarietidskrift, 31, 88–116.

    MathSciNet  MATH  Google Scholar 

  2. Bailey, R. A. (1981). A unified approach to design of experiments. Journal of the Royal Statistical Society Series A, 144, 214–223.

    Article  MathSciNet  Google Scholar 

  3. Bailey, R. A. (1991). Strata for randomized experiments. Journal of the Royal Statistical Society Series B, 53, 27–78.

    MathSciNet  MATH  Google Scholar 

  4. Bailey, R. A. (1994). General balance: artificial theory or practical relevance. In T. Caliński & R. Kala (Eds.), Proceedings of the of the International Conference on Linear Statistical Inference LINSTAT’93 (pp. 171–184). Dordecht: Kluwer.

    Chapter  Google Scholar 

  5. Baksalary, J. K., & Kala, R. (1983). On equalities between BLUEs, WLSEs, and SLSEs. The Canadian Journal of Statistics, 11, 119–123.

    Article  MathSciNet  Google Scholar 

  6. Baksalary, J. K., Puntanen, S., & Styan, G. P. H. (1990). On T. W. Anderson’s contributions to solving the problem of when the ordinary least-squares estimator is best linear unbiased and to characterizing rank additivity of matrices. In G. P. H. Styan (Ed.) The Collected Papers of T. W. Anderson: 1943–1985 (pp. 1579–1591). New York, NY: Wiley.

    Google Scholar 

  7. Bose, R. C. (1944). The fundamental theorem on linear estimation. Proceedings of the Thirty-First Indian Science Congress, 4, III, 2–3.

    Google Scholar 

  8. Caliński, T. (1997). Recovery of inter-block information when the experiment is in nested block design. Listy Biometryczne-Biometrical Letters. 34, 9–26.

    MATH  Google Scholar 

  9. Caliński, T., & Kageyama, S. (1996). The randomization model for experiments in bock designs and the recovery of inter-block information. Journal of Statistical Planning and Inference, 52, 359–374.

    Article  MathSciNet  Google Scholar 

  10. Caliński, T., & Kageyama, S. (2000). Block designs: A randomization approach, Vol. I: Analysis. Lecture Notes in Statistics (Vol. 150). New York, NY: Springer.

    Book  Google Scholar 

  11. Caliński, T., & Łacka, A. (2014). On combining information in generally balanced nested block designs. Communications in Statistics-Theory and Methods, 43, 954–974.

    Article  MathSciNet  Google Scholar 

  12. Houtman A. M., & Speed, T. P. (1983). Balance in designed experiments with orthogonal block structure. Annals of Mathematical Statistics, 11, 1069–1085.

    Article  MathSciNet  Google Scholar 

  13. Nelder, J. A. (1965). The analysis of randomized experiments with orthogonal block structure. I. Block structure and the null analysis of variance. Proceedings of the Royal Society of London Series A, 283, 147–162.

    Google Scholar 

  14. Nelder, J. A. (1965). The analysis of randomized experiments with orthogonal block structure. II. Treatment structure and the general analysis of variance. Proceedings of the Royal Society of London Series A, 283, 163–178.

    Google Scholar 

  15. Nelder, J. A. (1968). The combination of information in generally balanced designs. Journal of the Royal Statistical Society: Series B, 30, 303–311.

    Article  MathSciNet  Google Scholar 

  16. Patterson, H. D., & Thompson, R. (1971). Recovery of inter-block information when the block sizes are unequal. Biometrica, 58, 545–554.

    Article  MathSciNet  Google Scholar 

  17. Rao, C. R. (1974). Projectors, generalized inverses and the BLUEs. Journal of the Royal Statistical Society: Series B, 36, 442–448.

    MathSciNet  MATH  Google Scholar 

  18. Rao, C. R., & Rao, M. B. (2004). Matrix algebra and its applications to statistics and econometrics. Singapore: World Scientific.

    MATH  Google Scholar 

  19. Styan, G. P. H. (1973). When does least squares give the best linear unbiased estimate? In: D. G Kabe & R. P. Gupta (Eds.), Multivariate statistical inference (pp. 241–246). Amsterdam: North-Holland.

    Google Scholar 

  20. Zyskind, G. (1962). On conditions for equality of best and simple linear squares estimators. Annals of Mathematical Statistics, 33, 1502–1503.

    Google Scholar 

  21. 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.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical and Statistical Methods, Poznań University of Life Sciences, Poznań, Poland

    Radosław Kala

Authors
  1. Radosław Kala
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Radosław Kala .

Editor information

Editors and Affiliations

  1. Department of Mathematics and Statistics, Brock University, St. Catharines, ON, Canada

    S. Ejaz Ahmed

  2. Departmental Unit of Mathematics and Physics, Polytechnic Institute of Tomar, Tomar, Portugal

    Francisco Carvalho

  3. Faculty of Natural Sciences, University of Tampere, Tampere, Finland

    Simo Puntanen

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Kala, R. (2019). A New Look at Combining Information from Stratum Submodels. In: Ahmed, S., Carvalho, F., Puntanen, S. (eds) Matrices, Statistics and Big Data. IWMS 2016. Contributions to Statistics. Springer, Cham. https://doi.org/10.1007/978-3-030-17519-1_3

Download citation

Publish with us

Policies and ethics

Search

Navigation