Blocking (statistics)

In the statistical theory of the design of experiments, blocking is the arranging of experimental units in groups (blocks) that are similar to one another. For example, an experiment is designed to test a new drug on patients. There are two levels of the treatment, drug, and placebo, administered to male and female patients in a double blind trial. The sex of the patient is a blocking factor accounting for treatment variability between males and females. This reduces sources of variability and thus leads to greater precision.

Suppose we have invented a process intended to make the soles of shoes last longer, and we plan to conduct a field trial. Given a group of n volunteers, one possible design would be to give n/2 of them shoes with the new soles and n/2 of them shoes with the ordinary soles, randomizing the assignment of the two kinds of soles. This type of experiment is a completely randomized design. We ask both groups to use their shoes for a period of time, and then measure the degree of wear of the soles. This is a workable experimental design, but purely from the point of view of statistical accuracy (ignoring any other factors), a better design would be to give each person one regular sole and one new sole, randomly assigning the two types to the left and right shoe of each volunteer. Such a design is called a randomized complete block design. This design will be more sensitive than the first, because each person is acting as their own control and thus the control group is more closely matched to the treatment group.

The theoretical basis of blocking is the following mathematical result. Given random variables, X and Y

$\operatorname{Var}(X-Y)= \operatorname{Var}(X) + \operatorname{Var}(Y) - 2\operatorname{Cov}(X,Y).$

The difference between the treatment and the control can thus be given minimum variance (i.e. maximum precision) by maximising the covariance (or the correlation) between X and Y.

References

• Addelman, Sidney (Oct. 1969). "The Generalized Randomized Block Design". The American Statistician 23 (4): 35–36. doi:10.2307/2681737. JSTOR 2681737. 

• Addelman, Sidney (Sep. 1970). "Variability of Treatments and Experimental Units in the Design and Analysis of Experiments". Journal of the American Statistical Association 65 (331): 1095–1108. doi:10.2307/2284277. JSTOR 2284277. 

• Bailey, R. A (2008). Design of Comparative Experiments. Cambridge University Press. ISBN 978-0-521-68357-9. http://www.maths.qmul.ac.uk/~rab/DOEbook/.  Pre-publication chapters are available on-line.

• Caliński, Tadeusz and Kageyama, Sanpei (2000). Block designs: A Randomization approach, Volume I: Analysis. Lecture Notes in Statistics. 150. New York: Springer-Verlag. ISBN 0-387-98578-6. 

• Gates, Charles E. (Nov. 1995). "What Really Is Experimental Error in Block Designs?". The American Statistician 49 (4): 362–363. doi:10.2307/2684574. JSTOR 2684574. 

• Kempthorne, Oscar (1979). The Design and Analysis of Experiments (Corrected reprint of (1952) Wiley ed.). Robert E. Krieger. ISBN 0-88275-105-0. 
• Hinkelmann, Klaus and Kempthorne, Oscar (2008). Design and Analysis of Experiments. I and II (Second ed.). Wiley. ISBN 978-0-470-38551-7. 
  • Hinkelmann, Klaus and Kempthorne, Oscar (2008). Design and Analysis of Experiments, Volume I: Introduction to Experimental Design (Second ed.). Wiley. ISBN 978-0-471-72756-9. 
  • Hinkelmann, Klaus and Kempthorne, Oscar (2005). Design and Analysis of Experiments, Volume 2: Advanced Experimental Design (First ed.). Wiley. ISBN 978-0-471-55177-5. 

• Lentner, Marvin; Thomas Bishop (1993). Experimental design and analysis (Second ed.). P.O. Box 884, Blacksburg, VA 24063: Valley Book Company. pp. 225–226. ISBN 0-9616255-2-X. 

• Wilk, M. B. (June 1955). "The Randomization Analysis of a Generalized Randomized Block Design". Biometrika 42 (1–2): 70–79. JSTOR 2333423. 

• Zyskind, George (Dec. 1963). "Some Consequences of randomization in a Generalization of the Balanced Incomplete Block Design". The Annals of Mathematical Statistics 34 (4): 1569–1581. doi:10.1214/aoms/1177703889. JSTOR 2238364. 

See also

• Block design
• Paired difference test
• Randomized block design