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a Dep. of Biometrics, 434 Warren Hall, Cornell Univ., Ithaca, NY 14853
b Wheat Program, CIMMYT, Lisboa 27, Apdo. Postal 6-641, Mexico D.F., Mexico
c Biometrics and Statistics Unit, CIMMYT, Lisboa 27, Apdo. Postal 6-641, 06600 Mexico D.F., Mexico
Corresponding author (wtf1{at}cornell.edu)
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTSCONCLUSIONSREFERENCES |
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Abbreviations: RCBD, randomized complete block • ICBD, an incomplete block • ANOVA, analysis of variance • REML, restricted maximum likelihood • smean, standardized means
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSCONCLUSIONSREFERENCES |
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An augmented experiment design is useful for screening new treatments such as genotypes, insecticides, herbicides, drugs, etc. The number of new treatments, n, may be large, even in the hundreds and thousands. An augmented experiment design is constructed by selecting an experiment design for the check treatments. The experiment design could be a randomized complete block (RCBD), an incomplete block (ICBD), a row-column design, or some other design. If the selected design for check treatments is an ICBD, then the rb blocks (b incomplete blocks per complete block or replicate) are enlarged to accommodate n/rb new treatments per incomplete block. If a row-column design is selected, the numbers of rows and columns are increased to include the n new treatments. If new treatments are included more than once, the analysis must take this into account. The augmented design analyses described herein are for one replicate of each new treatment. Responses may be obtained for all or for only a fraction of the new (augmented) treatments as only the replicated check treatment responses are used for obtaining solutions for blocking effects and an estimate of the error variance.
Using an appropriate response model for each site, analyses are presented for experiments at the different sites. The selected model needs to account for the spatial variation present in the experiment. Standard textbook models may be inappropriate, and a model needs to be selected from a class of plausible models (see, e.g., Federer, 1998). The standard procedure is to use a fixed effect analysis to select a response model and blocking parameters. Then, for the selected model, a mixed effect analysis of fixed checks and random blocking and new treatment effects is performed. The check means are adjusted for the information in the random blocking effects, i.e., recovery of inter-block, inter-regression (when regression is used as a blocking variable to explain trends in spatial variation), inter-row, and/or inter-column information. The new treatment means are adjusted by recovering inter-regression, inter-block, and/or inter-variety information. Recovery of random effect information results in adjusted means with smaller variances; it shrinks the size of fixed effect estimates by taking account of the random distribution of effects. Computer programs are available for both fixed and mixed effects analyses (Wolfinger et al., 1997). For each site, the most appropriate response model and best estimate of the new treatment means should be utilized. Utilizing the adjusted means for each site, it is demonstrated how to combine the results over sites. Two methods for combining results over sites are presented.
The main objective of the experiment used for an illustrative example in this paper, was to measure genetic diversity for physiological performance within a family of spring wheat (Triticum aestivum L.) lines derived from a cross between a heat-tolerant and a heat-sensitive parent (Reynolds et al., 1998). Quantitative traits such as those associated with heat tolerance may show considerable interaction with the environment. Therefore, the testing of new lines requires extensive evaluation in different locations (environments) to establish their genetic potential. Improved methods for identifying superior lines can result in significant savings. One approach is to reduce the number of replications used in a single field trial, assuming that performance can still be evaluated accurately. Augmented designs were developed for such situations.
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSCONCLUSIONSREFERENCES |
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The experiment design at the third site was an incomplete block design of v = 120 treatments (the new treatments at the above two sites) in incomplete blocks of size k = 8. Check varieties were not included at this site. The 15 incomplete blocks in each of r = 2 replicates were laid out adjacent to each other effectively making an 8 column by 15 row layout in each replicate. The spatial layout of the experiment indicates that the design of the experiment should be considered as a resolvable 15 row by 8 column design rather than as an incomplete block design.
At Sites 1 and 2, polynomial functions of the rows and column effects were used to determine which functions of rows and columns describe the variation present in the experiment. Polynomial regressors use centered values of the powers of the independent variable. Hence, any of the regression coefficients may be large or small, depending on the nature of the spatial variation. The SAS/GLM procedure (SAS Inst., Cary, NC, 1996) was used to obtain the analysis of variance (ANOVA). Polynomial functions for rows, Rj, and columns, Ci, were fitted, up to 12th degree for rows (j = 1,..., 12) and up to 10th degree (i = 1,..., 10) for columns (see Federer and Wolfinger, 1998) as the rank of the normal equations was two less than needed for a row, column, treatment ANOVA. Fitting row effects is equivalent to using a 14th degree polynomial. Using the procedure described by Bozivich et al. (1956) and Federer et al. (1998), all terms in the polynomials with F values less than the 25% level were relegated to the residual category. The remaining regressors were treated as random blocking effects for explaining the spatial variation present in the experiment. This rule results in a slight change in the Type I error probability levels (Bozivich et al., 1956). Since the gradients in a field are not always in the same direction as the row-column layout, interactions of polynomial regressors such as row-linear x column-linear, row-linear x column-quadratic, etc. were fitted to account for this type of variation. The interactions Ci x Rj fitted were from i = 1 to 4 and j = 1 to 4. Again, the Bozivich et al. (1956) procedure was used to determine which of the 16 interactions should be pooled with the residual and which should be retained to account for the spatial variation present in the experimental area. This selection procedure provides a safeguard against over-parameterization. For the third site, differential linear gradients within the incomplete blocks accounted for the spatial variability present in the experiment.
The resulting fixed effect response models selected for the three sites were:
Site 1 Yield = Entry C1 C2 C3 C4 C6 C8 R1 R2 R4 R8 R10 C1xR1 C2xR1 C3xR1
Site 2 Yield = Entry C1 C4 C10 R2 C1xR1 C1xR3 C2xR2 C2xR4 C3xR2 C3xR4 C4xR3 C4xR4
Site 3 Yield = Replicate entry block(replicate) C1 x Block(replicate)
where Ci and Rj represent the orthogonal polynomial regression coefficients of degree i for columns and degree j for rows, respectively. The models are written in the form required by the SAS/GLM procedure. All elements in the above response models for Sites 1 and 2 except for the check treatments are considered to be random effects. Since the 120 new genotypes are untested, they are considered to be random effects (see Federer, 1998). The goal of model selection is to find the response model that explains the spatial variation present in the experiment. Higher degree polynomials and interactions are required to account for nonuniform gradients as opposed to uniform ones. The ordering of regression degree is irrelevant; no explanation is needed beyond that it accounts for the spatial variation present in the experiment.
For Site 3, only the 120 new treatments were included and it is still possible to consider the new treatment effects as random using the SAS/MIXED procedure. The following is the SAS code for the Site 3 analysis:
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Notice that no effects are listed in the MODEL statement since only fixed effects can be included here. With the SOLUTION option in the MODEL statement and ENTRY listed in the RANDOM statement, one needs to add the intercept value to the solution for the entry effect to obtain the mean (called REML or BLUP mean). In the SAS/MIXED procedure, one may use the default option, restricted maximum likelihood (REML), or one of the other options such as, e.g., ANOVA solutions for the variance components. This is done using the statement
PARMS (A) (B) (C) /NOITER
where X in (X) is the value for a variance component. "NOITER" denotes no iterations are to be used. This statement appears after the RANDOM statement. Also, the word NOPROFILE needs to be added in the PROC MIXED statement. This procedure is useful when the analyst wishes to use ANOVA instead of REML solutions for the variance components. ANOVA solutions are unbiased and do not depend on the normality assumption.
Pertinent references for combining results from a series of experiments in a different context are Cochran (1939), Yates and Cochran (1938), Federer (1951), Cochran and Cox (1957), references on analysis of crop rotations, and references on meta-analysis. Combined analyses using means per site is simple and straightforward. As suggested by Cullis et al. (1996) and Frensham et al. (1997) for the case of replicated experiments, heterogeneity of within-site error variance can be considered by using the reciprocal of the standard errors of the means. Crossa and Cornelius (1997) used this transformation in the multiplicative site regression model with the purpose of identifying subsets of sites without genotypic crossover genotype x environment interaction. Piepho (1999) suggested a weighted two-stage analysis across sites where, in the first stage, the treatment-site means and their associated error variances are obtained considering sites as fixed effects. The second stage considers a mixed model with sites and genotypes x site interaction as random effects and where weights of the reciprocal of the standard error of a mean are used.
Several procedures are available for combining results over sites. Some of these are: (i) a fixed effects analysis of rows, columns, sites, and entries; (ii) a mixed model analysis of random effects sites, rows, columns, and new genotypes with checks as fixed effects and perhaps adding row x column interaction as random; (iii) fixed effects analysis on fixed effects means; (iv) a mixed model analysis with site and new as random using the fixed effects means; (v) a regression-site-new random effects analysis over sites with checks as fixed; or (vi) a bootstrap analysis of the yields in Tables 1 and 2 or of those in Table 3.
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Analysis (i) is easily accomplished using a software procedure like SAS/GLM, but it does not make use of inter-regression and inter-variety information, and hence is inefficient. Analysis (ii) could be used using SAS/MIXED, say, as the REML solutions obtained depend on the normal distribution and not on the over-parameterization or the connectedness of the design. Standard errors for over-parameterized models with REML will be inflated. Analysis (iii) is straightforward but does not make use of the random effects information. Analysis (iv) performs a fixed effect analysis at the individual sites and then considers sites and new entries as random effects using only the least squares fixed effect means for the new genotypes. Analysis (v) requires the blocking (trend) effects to be the same at all sites, and hence is not usable here.
Several possibilities exist to test for genotype x site interaction. Analysis (i) could be used in some situations. Also, an average effective error of the means or solutions could be obtained and used as an error term to test for site x genotype interaction in analyses (iii), (iv), and (v). In many situations, a test for genotype x site interaction may be of minor interest. Instead, the plant breeder selects those entries having the highest yields over all sites and environments. It may be that selections will be made for specific environments. Since it is known that a genotype x environment interaction more than likely exists for the entries in this study, there is little interest in testing for it. For random sites, the genotype x site interaction mean square is the appropriate error mean square for testing genotype effects over sites. The new x site mean square has a different expectation than the check x site mean square. A comparison of new with check is the Behrens (also called the Behrens-Fisher) situation of unequal variances.
To overcome the flaws listed above, we shall use the following two methods for combining results from experiments over sites:
Method 1. Best estimates of new treatment means are obtained for each site. Then a two-way analysis of random site effects and fixed entry effects is performed. An estimate of the error variance over sites is obtained by computing the average effective error variance at each site and then averaging these over sites. This is a method suggested by Cochran and Cox (1957). Their procedure is modified here to treat the new treatments as random effects.
Method 2. Best estimates of new treatment means are obtained for each site. These means are divided by their standard errors. These are called the standardized means and will have an expected error variance of unity. This means that estimates of variance components from a site x entry analysis will be ratios of the variance component to the error variance component.
These methods do not require the same experiment design at each site, do not require the same check treatments at each site, do not require the same response model at each site, and do not require homogeneity of error variances from site to site. These two methods are applied to the data from the three sites for means and for standardized means (smean). Also, since the two replicate means at Site 3 were quite different, they were also treated as two sites. There was a heavy disease infection in one of the replicates. The unadjusted data from individual plots were used. A randomized complete block design variance was used to obtain the standard error for the data from the two replicates at Site 3. Owing to the large residual (error) variance, this resulted in a lower weight being given to the data from Site 3. This is counter-balanced to some extent by using the replicates as two sites. The new treatments are considered as random effects in this case. Four sites were used for the third (means) and fourth (smeans) analyses of the data.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSCONCLUSIONSREFERENCES |
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The means in Table 1 were divided by their standard errors to obtain the standardized means. The standard error for Site 1 was taken to be an average of the REML solution standard errors divided by the square root of two. The Site 1 means were divided by 30.06, Site 2 means by 49.50, and Site 3 means by 154.00 to obtain the standardized means, denoted as smeans. Since the data for sites denoted as 4 and 5 were unadjusted values, they were considered as coming from a randomized complete block design. The standard error for a single plot was 546.63. Dividing the data in columns 5 and 6 of Table 1 by 546.63 results in the standardized means for these columns. When data are standardized in this manner, the expected population error variance is unity, as in the unit normal distribution. This theoretical variance is the parameter and has infinite degrees of freedom.
Sites were considered as random effects in computing the REML means over sites. The 120 entry means and standardized means for Sites 1, 2, and 3 and for Sites 1, 2, 4, and 5 are given in Table 2. Each of the four sets of means averaged over sites has been ranked in descending order. The highest 15 means for Sites 1, 2, and 3 were the following entries (Table 2): 90, 108, 29, 94, 93, 11, 91, 4, 96, 31, 8, 43, 89, 62, 33
For the smeans of Sites 1, 2, and 3 in column four, 8 of the above 15 appeared in the top 15 smeans. For the means averaged over Sites 1, 2, 4, and 5, 10 of the top 15 were in agreement. For the standardized means over Sites 1, 2, 4, and 5, entries 108, 11, 90, 120, 93, 95, 99, 114, 13, 60, 2, and 53 appeared in the top 15 smeans for Sites 1, 2, and 3. The large error variance for the randomized complete block design analysis at Site 3 resulted in small smeans and a shorter range of means at Sites 4 and 5. Thus, more weight is given to the data from Sites 1 and 2. Similar comparisons may be made for the lowest yielding group of entries. For example, entries 44, 69, and 74 appeared at the bottom for means. The same entries appeared at the bottom or near the bottom for the other three methods of combining results.
Analyses of variance (ANOVAs) were obtained for the four sets of means and are presented in Table 3. The site or environment means squares are relatively large compared with the site x entry (genotype x environment) interaction. This is probably explained by the fact that the three sites differed in planting date as well as location. For the experiment at Tlaltizapan, Site 1 was planted in November and Site 2 in February and both were controlled for disease. For the third site, Obregon, the experiment was sown in February and had a heavy stem rust infection. The later the sowing date the greater the decrease in grain yield in these environments. Environmental factors influenced by the location and sowing date included temperature, photo-period, and soil physical and chemical properties. Yields generally decline at the higher temperatures, mainly because the growing cycle is truncated. All of the factors are likely to interact with genotype. Since such environmental differences exist, there is the question as to whether some real explainable effect is present and should be utilized in the interpretation of the results.
F tests could be performed but their relevance in this context is in question. Instead some multiple comparisons procedure like subset selection appears more appropriate. If tables of Dunnett's comparison with a control were available for more than 20 entries (see Bechhofer and Dunnett, 1988), one could use this procedure. To illustrate, let us suppose that the tabled value is 3.00 for 120 entries with 238 degrees of freedom for the entry error variance and that a 90% coverage is desired. Then, for the entry with the highest mean over Sites 1, 2, and 3, number 90, the interval is computed as
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Thus, all means lower than 1126.6 are considered to be significantly different from entry 90 and all those higher than 1126.6 are not. The site by entry mean square, 33663, was used as the error term since the sites are considered to be a random sample of sites. Instead of the above, a multiple range test could be utilized for comparing means. For the means from Sites 1, 2, and 3, a 95% studentized multiple range test is computed as
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Owing to the large genotype x environment interaction, a study needs to be made of the results at each of the three sites. The ANOVAs for the three sites are given in Table 4. The F values indicate large entry differences at each site. The 15 top yielding entries at each of the three sites are
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Site 2 18, 103, 37, 72, 85, 65, 26, 11, 79, 94, 110, 15, 25, 20, 95
Site 3 90, 29, 31, 4, 93, 94, 108, 62, 96, 91, 8, 43, 32, 33, 11
The 15 lowest yielding entries at each of the three sites are
Site 1 66, 5, 107, 55, 42, 56, 28, 17, 6, 51, 52, 43, 44, 81, 50
Site 2 29, 69, 36, 83, 31, 107, 73, 19, 117, 35, 74, 118, 56, 34, 32
Site 3 66, 10, 102, 18, 110, 111, 49, 113, 109, 98, 44, 71, 69, 100, 74
There is considerable disagreement in the ranking at the three sites: e.g., genotype 90 ranked 15th in Site 1 and 1st in Site 3. Entry 29 ranked 2nd in Site 3 and 106th in Site 2. Entry 110 ranked 11th in Site 2 but 106th in Site 3. Many of these genotypes appear to be site-specific for yield.
Variance component estimates for genotype x environment interaction as well as site are often desired. To do this, it is necessary to obtain an estimate of the pooled error mean square. One procedure is to average the error variances at the three sites as
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(Note that there were 
replicates of the 120 entries at Site 3). This gives equal weight to each site. Another procedure is to weight the error variances by their degrees of freedom as follows
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Owing to the relatively large spatial variations at Site 3, and to some extent at Site 2, there is some genotype x environment interaction in the residual mean square. The site x entry interaction has an expected value of error variance component plus site x entry variance component. The site x entry variance component for means at Sites 1, 2, and 3 would be 
. The ratio of the site x entry variance component to the error variance component would be 
. Note that 10113 may be used in place of 11263 if equal site weights are desired.
For the ANOVA on smeans for Sites 1, 2, and 3, the expected value of the site x entry mean square is 1 + the site x entry variance component divided by the error variance component. Thus, only a ratio of the two variance components can be obtained from the ANOVA on smeans. Ratios of variance components to the error variance component are used in measures of genetic advance (see, e.g., Sprague and Federer, 1951). For this example for smeans for Sites 1, 2, and 3, this ratio is 
, or roughly equal to that obtained above.
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CONCLUSIONS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSCONCLUSIONSREFERENCES |
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A large genotype x environment interaction for yield of these 120 wheat genotypes was found. It should be noted that each variable may have its own response model at a given site as they may be affected by different spatial variation patterns during the course of the experiment. A response model should be selected for each characteristic measured in an experiment. For yield, the large difference in variation at each site would indicate that some genotype x environment interaction was occurring at some sites. For Site 3, a large difference in replicate means occurred. For Site 2, there was considerable variation in yields from one part of the row-column design to another. Since this represents a change in environment, an interaction is possible and suspected.
Apart from the savings on land and management costs, reducing the number of repetitions at a site and increasing the number of sites, improves the efficiency with which certain traits are evaluated by reducing the time it takes to characterize a family of lines. An example would be where trait evaluation is sensitive to changes in environmental conditions such as temperature and radiation intensity, as is the case of canopy temperature measurement (Reynolds et al., 1998).
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ACKNOWLEDGMENTS |
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Received for publication October 19, 1999.
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REFERENCES |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSCONCLUSIONSREFERENCES |
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