Cited by 1,104Optimal Experimental Design MethodsAgriculture, Soil, Plant ScienceAdvanced Combinatorial Mathematics

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Experimental Designs Balanced for the Estimation of Residual Effects of Treatments

Williams, EJ·1949·Australian Journal of Scientific Research Series A Physical Sciences

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Abstract

Where an experiment can be carried out by applying different treatments in succession to the same unit of experimental material, accurate comparisons can be made between the effects of different treatments. To allow for the residual effect of previous treatments on the result obtained for any given treatment, it is desirable to adjust the results for such effects. Methods of constructing balanced designs for the estimation of these residual effects are described in this paper, and are summarized as follows. Designs balanced for effect of single preceding treatment: When n, the number of treatments, is even, a balanced design is possible with n replications ; when n is odd, 2n replications are required.Designs balanced for the effects of any number of preceding treatments, ignoring the interaction of residual effects: When n is a prime or a power of a prime, a balanced design is possible in n(n-1) replications, which may be set out as a set of n-1 mutually orthogonal Latin squares, with the same first columns. Designs which are not expressible as mutually orthogonal Latin squares are also possible. Designs balanced for the effect of the two preceding treatments and their interactions : Adesign can be developed from a set of n-l mutually orthogonal Latin squares obeying certain restrictions. The method of analysis of designs of this type is set out in detail, together with a numerical example. Direct effects of treatments are shown to be only slightly confounded, the maximum confounding being 4 per cent., when there are three treatments. These designs have wide applicability wherever successive treatments can be applied to the same unit of experimental material.

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Self-citation
A Class of Experimental Designs Balanced for First Residuals
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Authors: Williams, E.
Corresponding author: Williams, EJ
Venue: Australian Journal of Scientific Research Series A Physical Sciences
Publisher: CSIRO Publishing
Published: 1949-06-01
Volume: 2
Issue: 2
Pages: 149-168
ISSN: 0365-3676 (ISSN-L)
DOI: 10.1071/ch9490149
OpenAlex: W2033748464
Cited by: 1,104
References: 0
Citations by year: 2026: 20 · 2025: 46 · 2024: 52 · 2023: 50 · 2022: 62 · 2021: 49 · 2020: 67 · 2019: 65 · 2018: 54 · 2017: 43 · 2016: 41 · 2015: 56 · 2014: 42 · 2013: 37 · 2012: 33
FWCI: 1.74
Type: journal article
Language: en
Open access: closed
Licence: doi.org/10.1071/journalslicense (vor, 27706 d)
Locations: Australian Journal of Scientific Research Series A Physical Sciences (publishedVersion)
Topics: Social Sciences › Decision Sciences › Management Science and Operations Research › Optimal Experimental Design Methods
Life Sciences › Agricultural and Biological Sciences › Agronomy and Crop Science › Agriculture, Soil, Plant Science
Physical Sciences › Mathematics › Discrete Mathematics and Combinatorics › Advanced Combinatorial Mathematics
Keywords: residual effects, residual confounding, orthogonal Latin squares, confounding, residual interaction, design construction

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Experimental Designs Balanced for the Estimation of Residual Effects of Treatments

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