Randomization of neighbour balanced generalized Youden designs

https://doi.org/10.1016/j.jspi.2006.03.010Get rights and content

Abstract

If a crossover design with more than two treatments is carryover balanced, then the usual randomization of experimental units and periods would destroy the neighbour structure of the design. As an alternative, Bailey [1985. Restricted randomization for neighbour-balanced designs. Statist. Decisions Suppl. 2, 237–248] considered randomization of experimental units and of treatment labels, which leaves the neighbour structure intact. She has shown that, if there are no carryover effects, this randomization validates the row–column model, provided the starting design is a generalized Latin square. We extend this result to generalized Youden designs where either the number of experimental units is a multiple of the number of treatments or the number of periods is equal to the number of treatments. For the situation when there are carryover effects we show for so-called totally balanced designs that the variance of the estimates of treatment differences does not change in the presence of carryover effects, while the estimated variance of this estimate becomes conservative.

Introduction

In crossover designs, experimental units are exposed to a series of treatments, such that the effect of treatments can be compared independently of differences between the units. Generally, there are b experimental units and t treatments. In each of k periods each experimental unit receives one of the t treatments and a variable of interest, y, is measured on this unit. In what follows, we restrict attention to the case kt.

An important instance of crossover designs are sensory experiments. Here, a group of b assessors evaluates t products, such that each assessor gives k assessments, one after the other. Clearly, assessors correspond to experimental units and products correspond to treatments. The number of assessments by each assessor is the number of periods. We can describe this as a row–column experiment with periods as rows and assessors as columns. The discussion of the paper is presented in the context of sensory experiments. It should be pointed out, however, that our results are valid for all row–column experiments with possible carryover effects in columns.

For the analysis of sensory experiments, it is usually assumed that there are no carryover effects. This is at least partially justified by the use of washout periods and other methods that should reduce the effect of a given product on those that will be tasted afterwards.

However, in recent years, the sensometrics literature has become increasingly interested in the liability of carryover effects. The general approach seems to be to analyse using a model without carryover effects, but to use designs that are carryover balanced. In a carryover balanced design, each product is preceded by every other product equally often. Many authors, see e.g. MacFie et al. (1989), Schlich (1993) and Wakeling and MacFie (1995) discussed the construction of carryover balanced designs, while Ball (1997) and Périnel and Pagès (2004) proposed designs that are “near” such designs. Relatively little is said about the analysis of carryover balanced designs.

We consider the analysis of carryover balanced designs under a randomization viewpoint. Clearly, randomization of the order of presentation of the products is not possible for such designs. Kunert (1998) used simulations to demonstrate two randomization methods that keep the neighbour structure intact, but justify simple analyses. One of these was proposed by Azaïs (1987) and allows an analysis in the simple block model. If, however, the experimenter wishes to balance the number of appearances of the products in each period, the only option is to randomize assessors and product labels. It was shown by Bailey (1985) that this randomization justifies an analysis with the row–column model, provided the starting design is a generalized Latin square and there are no carryover effects.

However, these designs are only possible in cases where each assessor rates each product and the number of assessors is a multiple of the number of products. In Section 2, we extend this result to generalized Youden designs (GYDs) where k can be less than t, provided b is divisible by t, or where b does not have to be a multiple of t, provided k=t.

The purpose of carryover balanced designs is to provide protection against carryover. Indeed, these designs do provide some robustness as was shown by several authors, see e.g. Azaïs and Druilhet (1997). They minimize the bias due to carryover of the estimated product differences. It should be pointed out, however, that even if the design is carryover balanced the non-corrected estimates are biased to some extent.

In Section 3, we consider the effect of unsuspected carryover under a randomization viewpoint. We determine the bias due to carryover effects under the randomization of assessors and product labels for GYDs. For special GYDs, so-called totally balanced designs, we show that the variance of the estimates of product differences does not change in the presence of carryover effects, while the estimated variance of this estimate becomes conservative.

It is not possible to extend our proofs to the designs by Ball (1997) or Périnel and Pagès (2004). In a companion paper, Kunert and Sailer (2006), we demonstrate with the help of simulations that the results are very similar, at least for reasonable designs. For designs that are near the carryover balanced GYDs, the increase of the variance estimates appears to remain reasonably small.

Section snippets

Validity of the row–column model if there are no carryover effects

The products in a sensory experiment are assigned to the assessors via a design δ. Formally, the design δ is a mapping from {1,,k}×{1,,b} to {1,,t}, where k, b and t are as in Section 1. In period i, assessor j receives product δ(i,j).

Definition 1

If a design δ fulfils the conditions (GYD1)–(GYD4), it is called a GYD, cf. Kiefer, 1958, Kiefer, 1975.

  • (GYD1)

    The number of periods is not greater than the number of products and each assessor receives each product at most once.

  • (GYD2)

    For any pair of products the number of

Influence of carryover effects on the estimates

Introducing carryover effects into the randomization model (1), we getyij=ηij+τδ(i,j)+ρδ(i-1,j),where ρδ(i-1,j) is the carryover effect from the product tasted in the preceding period. The carryover effect in period 1 is set to 0. If, for the ηij in (9), the row–column model is assumed, then it can be shown that designs which are balanced for carryover effects have excellent optimality properties, see e.g. Kunert (1984).

Again, we would like to validate the row–column model (2) for the ηij in

Discussion

Our results are for the case that each assessor evaluates each product at most once. The proof of Proposition 6 needs that each Hdi=[Ik,0]Δi, where Δi is a permutation matrix. This is not the case if there are repeated evaluations by the same assessor. Therefore, our results cannot be extended to the case when the number of periods is larger than the number of products. In that case, however, the simple row–column model should not be used anyway. In sensory trials, there will almost always be

Acknowledgements

Support by the Deutsche Forschungsgemeinschaft (SFB 475, Komplexitätsreduktion in multivariaten Datenstrukturen) is gratefully acknowledged. The authors would also like to thank the associate editor and the referees for their careful reading and thoughtful comments.

References (19)

There are more references available in the full text version of this article.

Cited by (2)

  • Product selection for liking studies: The sensory informed design

    2015, Food Quality and Preference
    Citation Excerpt :

    Deppe et al. (2001) provided a procedure for constructing nested incomplete-block designs. Kunert and Sailer (2007) discussed the development of generalized Youden designs, where the experimental units are randomized. In sensory analysis, the experimental units are either consumers or sensory assessors and the treatments are food products.

  • Pseudo generalized Youden designs

    2018, Journal of Combinatorial Designs
View full text