Analyses for multi-site experiments using augmented designs

Walter T. FedererCornell University, Departments of Biometrics and Statistical Sciences434 Warren Hall, Cornell UniversityIthaca, New York, USAwtf1@cornell.edu

Mathew ReynoldsCIMMYT, Department of Plant BreedingMexico D. F., Mexico

Jose CrossaCIMMYT, Biometrics and Statistics UnitMexico D. F., Mexicojcrossa@alphac.cimmyt.mx

1. Introduction

The class of augmented experiment designs (AEDs) contains two kinds of treatments, check orstandard and new. The latter are usually considered to be random and the former as fixed. The newtreatments are usually unreplicated while the checks are replicated to obtain an estimate of the errorand to obtain estimates of blocking effects. An experiment design is selected for the check treatmentsand then the blocks, rows, and/or columns are enlarged to accommodate the new treatments and thisforms the augmented design. AEDs were introduced by Federer (1956) (Also see Federer et al., 1975and Federer, 1995, 1998) as an alternative to the systematically arranged check and new treatments.AEDs have several advantages over the systematic arrangement. They are useful for screening newtreatments such as genotypes, insecticides, herbicides, drugs, etc.

An appropriate response model needs to be selected for the results from each AED. A mixedeffects model with two kinds of treatments needs to be formulated. For multi-site trials, this may meana different experiment design and a different model for each site, thus complicating the combination ofresults. The errors at each site may, and usually do, vary, further complicating combining results.Blocking effects and site effects may be considered to be random effects.

2. Data Sets

The data are yields of 120 new wheat genotypes grown at three sites. The experiment design attwo of the sites was a 15 row by 12 column layout of the 120 new entries included once and twochecks which were included 30 times each. The response models used at these two sites for the checkswere:

Yhij = µ+R1+R2+R4+R8+R10+C1+C2+C3+C4+C6+C8+R1*C1+R1*C2+R1*C3+%hij

and

Yhij = µ+R2+C1+C4+C10+R1*C1+R2*C2+R2*C3+R3*C4+R4*C2+R4*C3+R4*C4+%hij

where Yhij is the yield of the jth check in the hth row and ith column, µ is a general mean effect, Rh andCi are polynomial regressions of row and column effects, respectively, and %hij is a random erroreffect distributed with mean zero and variance 52. A mixed model effect model for the Rh and Ci andfor new genotypes was used to obtain the new treatment means. Considerable reduction in the residualmean squares were obtained for these models over those obtained from a model with only row,column, and entry effects. The designs were also unconnected for these three effects consideredsimultaneously.

The experiment design at the third site was an incomplete block design for v = bk = 120treatments in b = 15 incomplete blocks of size k = 8 in each of two replicates. The response modelselected was

Yield = rep + entry + block within rep + linear gradients within block and rep + random error

A reduction in the error mean square of 21% over the textbook incomplete block design analysis wasobtained by taking the linear gradients within each each incomplete block into account.

3. Methods of Combining Over Sites

Several procedures are available for combining results over sites. Some of these are:

(i) a fixed effects analysis for the selected model(ii) a mixed model analysis for blocking and new treatments random(iii) a fixed effects analysis on site means for entries(iv) a mixed effects model with site and new as random on fixed effect site means(v) a site by entry analysis with REML solutions for site means(vi) a site by entry analysis of site REML means divided by standard errors

The method selected will be to obtain the best estimate of the entry means for each site and to dividethese means by their standard errors. Then a two-way site by entry analysis will be performed on thesemeans. Since the variance will be one, an estimate of the site by entry interaction variance componentmay be obtained. REML solutions for entry means are obtained for sites and entries random. Theselected procedure will be compared with each of the other methods to determine the effect on therankings of entry means over sites.

References

Federer, W. T. (1956). Augmented (or hoonuiaku) designs. Hawaiian Planters’ Record LV(2):191-208.Federer, W. T. (1993). Statistical Design and Analysis for Intercropping Experiments. Volume I. TwoCrops. Springer-Verlag, New York, Heidelberg, Berlin, Chapter 10.Federer, W. T. (1998). Recovery of interblock, intergradient, and intervariety information inincomplete block and lattice rectangle designed experiments. Biometrics 54:471-481.Federer, W. T., Nair, R. C., and Raghavarao, D. (1975). Some augmented row-column designs.Biometrics 31:361-373.

ResumeDessians augmentées étaient developpées par sélection de traitements nouveux en une manièrecapable. Ensembles données des trois emplacements étaient employées illustrer le analyse statistiquepar emplacement chaque et par résultats combiner sur emplacements. Méthodes de modeles mêléesétaient employées.