Incomplete split-plot designs: Construction and analysis

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Abstract

Split-plot designs are very popular in agricultural experiments where two factors require different sizes of plots. In traditional split-plot designs, main plot treatments are allocated using a randomized complete block design and subplot treatments are allocated within each main pot. However, often it may not be feasible to accommodate all the subplot treatments in each main plot. The purpose of this article is to propose a method of construction of such incomplete split-plot designs. Method of analysis of such incomplete split-plot designs is also presented. Construction and analysis methods are implemented using R language.

Introduction

In traditional split-plot designs, the m levels of a factor A (also known as main plot treatments) are arranged in a randomized complete block (RCB) design and s levels of another factor B (also known as subplot treatments) are arranged in s subplots within each of the m whole plots as RCB design. Such split-plot designs lead to estimation of main effect of factor A with less precision and estimation of main effect of factor B and interaction AB with higher precision. However, there arise experimental situations when one cannot apply a complete split-plot design. In such situations, the number of subplots in each whole plot may be restricted to k<s. Such types of experimental designs are called as incomplete split-plot designs.

Robinson (1967) pioneered the work on incomplete split-plot designs and proposed a method of construction of such designs in which m levels of factor A are arranged in a completely randomized design and s levels of factor B are arranged in a balanced incomplete block (BIB) design within each level of factor A. Robinson (1970) obtained incomplete split-plot designs where levels of both the factors A and B were arranged in BIB designs. Mejza and Mejza (1984) proposed incomplete split-plot designs where m levels of factor A were arranged in a completely randomized design and the levels of factor B were arranged in a connected, proper incomplete block design with block size k<s within each level of factor A, by considering the whole plots as blocks. They also detailed the theory of analysis of data from such experiments. Other works on incomplete split-plot designs include Ozawa et al., 2004, Ozawa and Kuriki, 2006, Kuriki and Nakajima, 2007 and Kristensen (2012).

In this article, we propose incomplete split-plot designs where levels of factor A are arranged in an RCB design and s levels of factor B are arranged in a binary proper connected block design within each whole plot. We also propose methodology of analysis of such designs using the standard fixed effects additive linear model approach. Most of the earlier works adopted analysis of such designs through randomization approach of Nelder (1965).

Section snippets

Construction of incomplete split-plot designs

To construct a design, first arrange the m levels of factor A in an RCB design with b blocks. Now, take a binary connected proper equireplicate incomplete block design D with number of treatments s, number of blocks b and block size k<s. With b occurrence of each level of factor A, associate b blocks of design D. Obtained design is an incomplete split-plot design where blocks are complete with respect to factor A and whole plots are incomplete with respect to factor B. We illustrate the

Analysis of incomplete split-plot designs

The model for incomplete split-plot designs considered in this paper is yjil=μ+ρj+αi+γji+βl+δil+ϵjilwhere yjil denote the observation from the experimental unit in jth block receiving ith level of factor A and lth level of factor B, μ, ρj, αi, γji, βl and δil denote the general mean, the effect of jth block, the main effect of ith level of factor A, the interaction terms between blocks and ith level of factor A, the main effect of lth level of factor B and the interaction effect of ith level of

Application

In literature, many examples can be found where complete split-plot designs are used even when there are many subplot treatments within each whole plot. For example, Duan et al. (2012) considered an experiment with split-plot design with three replications, two main plot treatments (irrigated and non-irrigated) and ten subplot treatments (a control, three standard seeding methods and six cotton waste compost seeding methods). If the whole plot remains homogeneous then a complete split-plot

Concluding remarks

The proposed methods of construction and analysis are implemented as part of an R package ‘ispd’ which is available on https://cran.r-project.org/web/packages/ispd/index.html, (Mandal et al., 2019a) and also as part of a web application available on http://drsr.icar.gov.in/ISPD/Home.jsp, see Mandal et al. (2019b). An important question that is not addressed in this paper is about the allocation of levels of factor B within levels of factor A such that the treatment contrasts of factors A, B and

CRediT authorship contribution statement

B.N. Mandal: Conceptualization, Methodology, Software, Writing - review & editing. Rajender Parsad: Conceptualization, Writing - review & editing. Sukanta Dash: Software.

Acknowledgements

The authors sincerely acknowledge the financial support received in the form of Extra Mural Research Grant by Science and Engineering Research Board (SERB), Department of Science and Technology, India . The authors sincerely thank the associate editor and the anonymous reviewers whose suggestions have improved the presentation of the article.

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