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Prediction Assessment of Shrinkage Estimators of Multiplicative Models for Multi-Environment Cultivar Trials.

IN AGRICULTURAL RESEARCH, multilocation trials are used to develop recommendations concerning cultivars and agronomic treatments (management practices, fertilizer levels, plant density, etc.) for adoption by farmers. Multiplicative statistical models are useful for studying patterns of yield response across sites (yield stability) and estimating realized cultivar response in specific sites that provide better estimates than do empirical cell means (Crossa and Cornelius, 1993; Cornelius et al., 1993, 1996; Gauch, 1988; Gauch and Zobel, 1996).

Two criteria have been used to determine the optimal number of multiplicative terms to be retained in a multiplicative model: tests of hypotheses concerning the multiplicative terms and cross validation. Hypothesis test procedures which give satisfactory control of Type I error rates are the approximate F tests [F.sub.GH1], [F.sub.GH2], [F.sub.R], Simulation, and Iterated Simulation tests (Cornelius et al., 1993, 1996; Cornelius, 1993). The cross validation method (Gauch, 1988; and Gauch and Zobel, 1988) uses a random data splitting scheme where [N.sub.m] replicates of each cultivar X environment combination are used for modeling, and the remaining [N.sub.v] replicates for validation. The model considered best is the model that gives the smallest mean squared error of prediction (or its square root, denoted as RMSPD). However, the number of terms which results in best prediction computed from a subset of data does not necessarily lead to the best model obtainable from the complete data set. Simulation results (Cornelius, 1993; Cornelius et al., 1996) have indicated that there is little loss in efficiency (and sometimes a gain) if a truncated model is selected on the basis of [F.sub.GH1] or [F.sub.GH2] tests applied to the complete data set rather than by randomly splitting the data and performing cross validation.

Recently, shrinkage estimators of multiplicative models have been suggested for estimating the realized performance level of a cultivar in the testing environments (Cornelius et al., 1993; Cornelius and Crossa, 1995; Cornelius et al., 1996). Simulation studies incidental to the development of simulation tests (Cornelius, 1993) had shown that (i) least squares estimates of the scale parameters (singular values denoted as [Lambda] k) associated with multiplicative interaction components (principal component axes) were always biased upward and (ii) replacement of the least squares estimates with the shrinkage estimates used in the simulation step in computing simulation tests would often reduce the per cell `interaction mean squared error' (IMSE as defined by Cornelius, 1993) below its value for the best (smallest IMSE) truncated model and much below the IMSE for the least squares estimates of the interactions (unpublished results). Further study revealed the IMSE is usually still further reduced with still more shrinkage of the estimates. The shrinkage factor used by Cornelius et al. (1993, 1996) and in this study is max (1 - [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],0), where [F.sub.k] is obtained as the sum of squares due to the kth multiplicative term divided by an estimate of the contribution of the error variance to that sum of squares. Cornelius et al. (1996) presented simulated IMSE results for shrinkage estimates and best (minimum IMSE) truncated AMMI models fitted to simulated 9 x 20 data sets with seven sets of realized "true" [Lambda] k values. The shrinkage estimates resulted in IMSE smaller than IMSE for the best truncated model in every case.

The shrinkage factor 1 - [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is intuitively appealing because it estimates signal/(signal + noise), and shrinkage factors of similar or identical form occur elsewhere in statistical practice, most notably in some cases of empirical best linear unbiased predictors (BLUPs) (equivalent to empirical Bayes estimates under normal priors) (Henderson, 1984; Harville, 1977; Lindley and Smith, 1972; Feinberg, 1972; Robinson, 1991), ridge regression under certain choices of the biasing parameter (Hoerl and Kennard, 1970, 1988), and the James-Stein estimators (James and Stein, 1961; Lemmer, 1988). Piepho (1994) has recently shown examples of data from faba bean (Vicia faba L.) cultivar trials where BLUPs of the cell means gave slightly superior cross validation as compared to the best truncated AMMI models.

Cross-validation results, comparing performance of shrinkage estimators to truncated models (fitted by least squares) and BLUPs (Cornelius et al., 1993, 1996), have suggested that the shrinkage estimators provide estimates of realized cultivar performance at least as good as the better choice of best truncated multiplicative models or BLUPs. These results were based on 10 random data splittings without any adjustment for block differences in the randomized complete block designs. This paper reports more extensive and precise evaluations of performance of shrinkage estimators of multiplicative models applied to actual cultivar yield data. We compare predictive accuracy, as determined by the data splitting, cross validation method of Gauch (1988) and Gauch and Zobel (1988), of (i) truncated models fitted by least squares, (ii) shrinkage estimates of multiplicative models, (iii) BLUPs using a two-way random effects model with interaction, and (iv) empirical cell means. Throughout the paper our usage of the term `BLUP' will be with respect to the two-way random effects model with interaction and not with respect to some other variance-covariance structure. We assume that all effects in all models are random in the sense that they result from some set of stochastic processes. We focus on estimation of the realized yield levels [[micro].sub.ij] and, as in all cases where individual cell values are estimated under a linear model, be it fixed effects or realized values of random effects, the inference space is narrow (Littell et al., 1996).

MATERIALS AND METHODS

Models and Estimators

We write the conventional two-way model with interaction for the mean response of the/th cultivar (genotype) over n replicates at the jth site as

[1] [Y.sub.ij.] = [micro] + [[Tau].sub.i] + [[Delta].sub.j] + [([Tau] [Delta]).sub.ij] + [e.sub.ij]

i = 1,..., g; j = 1,..., e. We assume the [e.sub.ij.] are normally and independently distributed (NID) with mean zero and variance [sigma.sup.2]/n. For each cell, the empirical BLUP of [u.sub.ij] = E([y.sub.ij.]) based on assumptions that [[Tau].sub.i], [[Delta].sub.j], and ([[Tau] [Delta].sub.ij]) are random effects with variances [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] respectively, was computed as

[2] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [S.sub.[Tau]] = max(1 - [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [S.sub.[Delta]] = max(1 - [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) and [So.sub.[Tau][Delta]] = max(1 - [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], [F.sub.[Tau]] = MS(Cultivars)/MS(Error), [F.sub.[Delta]]= MS(Sites)/MS(Error), and [F.sub.[Tau][Delta]] = MS(Cultivars X Sites)/MS(Error); MS(Error) is a pooled error mean square. Eq. [2] is derived in Appendix 1.

The multiplicative model forms AMMI (Additive Main effects and Multiplicative Interactions), GREG (Genotypes Regression), SREG (Sites Regression), SHMM (Shifted Multiplicative), and COMM (Completely Multiplicative) all include a sum of multiplicative terms [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] but differ with respect to which additive components in Eq. [1] are retained and which are deleted; deletion allows such variation to contribute multiplicatively rather than additively to the model. Explicitly, these models are

AMMI: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

GREG: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

SREG: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

SHMM: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

COMM: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In each model, the [[Alpha].sub.ik] and [[Gamma].sub.jk] are subject to orthonormality constraints, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and for k [not equal to] k', [[Sigma].sub.i] [[Alpha].sub.ik] [[Alpha].sub.ik'] = [[Sigma].sub.j] [[Gamma].sub.jk] [[Gamma].sub.k'] = 0. In AMMI and GREG [[Sigma].sub.j] [[Gamma].sub.jk] = 0 and in AMMI and SREG [[Sigma].sub.i] [[Alpha].sub.ik] = 0. The number of multiplicative terms t [is less than or equal to]p where p is the number of terms required to saturate the model if fitted by least squares, namely, min(g-1,e-1), min(g,e-1), min(g-1,e), min[g,e,max(g,e)-1], and min(g,e), for AMMI, GREG, SREG, SHMM, and COMM, respectively.

All of these multiplicative model forms are special cases of the General Linear-Bilinear Model (GLBM) (Cornelius and Seyedsadr, 1997). A GLBM is a Balanced Linear-Bilinear Model (BLBM) if there exists a least squares solution such that estimates of the linear (additive) effects are free of the bilinear (multiplicative) effects. Applied to a balanced data set, AMMI, GREG, SREG, and COMM are BLBMs, but SHMM is not.

For the BLBMs, least squares estimates of the main effects [[Tau].sub.i] and [[Delta].sub.j] in models which retain one or both of these effects are the same as for Model [1], namely, [[Tau].sub.i] = [y.sub.i..] - [y.sub....] and [[Delta].sub.j] = [y.sub..j.] - [y.sub....,] and closed form least squares estimates of [[Lambda].sub.k], [[Alpha].sub.ik] and [[Gamma].sub.jk] are obtained as the kth component of the singular value decomposition (SVD) of the g X e matrix of residuals of the cell means after fitting the additive effects. For SHMM (an unbalanced GLBM) the least squares solution is [Beta] = [y.sub....] - [[Sigma].sub.k] [[Lambda].sub.k] [[Alpha].sub.k] [[Gamma].sub.k] where [[Alpha].sub.k] = [g.sup.-1] [[Sigma].sub.i] [[Alpha].sub.ik], [[Gamma].sub.k] = [e.sup.-1] [[Sigma].sub.j] [[Gamma].sub.jk] with [[Lambda].sub.k], [[Alpha].sub.ik] and [[Gamma].sub.jk] constituting the kth component of the SVD of matrix Z = [[z.sub.ij]], where [z.sub.ij] = [y.sub.ij]. - [Beta]. [Except in special cases, this is not a closed form solution and its computation requires an iterative algorithm (Seyedsadr and Cornelius, 1992).]

The shrinkage estimate of the sum of multiplicative terms in the BLBMs is [[Sigma].sub.k] [S.sub.k] [[Lambda].sub.k] [[Alpha].sub.ik] [[Gamma].sub.jk], where [[Lambda].sub.k], [[Alpha].sub.ik] and [[Gamma].sub.jk] are the least squares estimates and [S.sub.k] = max(1- [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Here [F.sub.k] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [s.sub.2] is the pooled error mean square and [u.sub.k] is such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] A justification is given in Appendix 2.

In practice, the [u.sub.k] are unknown and some estimate or approximation must be used. Our approach is to obtain one set of estimates taking [u.sub.k] as the number of parameters in the kth multiplicative term less the number of constraints on those parameters. This choice for [u.sub.k] is the df assumed to be appropriate for the sum of squares [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] when the Gollob (1968) F test is used to test the hypothesis that [[Lambda].sub.k] = 0. Thus, we call it Gollob's df. Gollob's df has been shown to be unsatisfactory for testing the null hypothesis (Cornelius, 1993), but it follows from Goodman and Haberman (1990) that the correct value of [u.sub.k] tends to Gollob's df as n[[Lambda].sub.k] / [Sigma] [right arrow] [infinity]. Note that this is a result that derives under the alternative hypothesis, not under the null hypothesis, and this is precisely the reason why. Gollob's df works well for defining shrinkage estimates, but results in high Type I error rates when used in testing the null hypothesis.

Occasionally, we find [S.sub.k] [[Lambda].sub.k-1] [is greater than] [S.sub.k-1] [[Lambda].sub.k-1], which violates the model constraint [[Lambda].sub.k] [is less than or equal to] [[Lambda].sub.k-1]. When this occurs, we assume [[Lambda].sub.k] = [[Lambda].sub.k-1] and compute a pooled estimate. This rule is applied sequentially and may result in pooled estimates for sequences of more than two singular values.

Once we have a set of shrinkage estimates based on Gollob's df, we use these to initialize a simulation scheme (Cornelius et al., 1996; Cornelius and Crossa, 1995) to get improved estimates of the df ([u.sub.k]) and subsequent new shrinkage estimates based on those df. The scheme can be iterated. This paper reports results based on Gollob's df and on df determined by simulation without iteration and with one iteration.

Shrinkage estimates of [[Tau].sub.i] and [[Delta].sub.j] in model forms which retain one or both of these parameters are taken as [S.sub.[Tau]]]([y.sub.i..] - [y.sub....]) and [S.sub.[Delta]]([y.sub..j.] - [y.sub....]), respectively, where [S.sub.[Tau] and [S.sub.[Delta]] are as in Eq. [2]. The need to so define [S.sub.[Tau]] and [S.sub.[Delta]] rather than to use the BLUPs under the two-way random effects model (which use the interaction mean square rather than the error mean square as denominator of [F.sub.[Tau]] and [F.sub.[Delta]]) to estimate [[Tau].sub.i] and [[Delta].sub.j] is a consequence of the constraints [[Sigma].sub.i] [[Alpha].sub.ik] = 0 in AMMI and SREG and [[Sigma].sub.j] [[Alpha].sub.jk] = 0 in AMMI and GREG. Further explanation of this point is given in a lengthy footnote on page 21.4 of Cornelius et al. (1996).

Our scheme for shrinkage estimation of SHMM is as follows. Given a least squares solution for SHMM with t multiplicative terms, the solution for [Beta] and the [[Lambda].sub.k], k = 1,...,t can be reproduced as intercept and regression coefficients in the least squares multiple regression of the cell means [y.sub.ij] on regressor variables [X.sub.k], where [X.sub.kij] = [[Alpha].sub.ik] [[Gamma].sub.jk]. This allows us to define the partial sum of squares SS([[Lambda].sub.k] explained by the kth multiplicative term and [F.sub.k] = SS([[Lambda].sub.k])/ [u.sub.k] [s.sup.2], where [u.sub.k] is chosen such that ako-2 is the expected contribution of error variance to SS([[Lambda].sub.k]). Then the shrinkage estimate of [[Lambda].sub.k] is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [S.sub.k] = max(1 - [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are obtained as the kth component of the SVD of matrix Z = [[z.sub.ij]], [z.sub.ij] = ([Y.sub.ij] - [Beta]*) and [Beta]* = [y.sub....] - [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. A justification is given in Appendix 3. An iterative algorithm is required because simultaneous solutions for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [Beta]* do not exist in closed form.

For computation of shrinkage estimates of multiplicative models, SAS/IML program `eigsimsh.sas' (for AMMI, GREG, SREG or COMM) and `eigsimsq.sas' (for SHMM) are available from the first author.

Data and Cross Validation

Data from five CIMMYT multi-environment international cultivar trials were used for comparison of estimation efficiency of model forms and estimators. Trial 1 was a wheat trial with 19 durum wheat cultivars, one bread wheat cultivar, and 34 sites. Trials 2 through 5 were maize trials with number of cultivar and sites (g,e) = (16,24), (9,20), (18,20), and (8,59), respectively. For each trial we used random data splitting cross validation to compare performance of shrinkage estimates with BLUPs of cell means and truncated models fitted by least squares with respect to the RMSPD.

Computer programs using the SAS macro language and procedures GLM and IML (SAS Institute, Inc. 1989a,b,c) were written to compute and cross validate shrinkage estimates, ordinary cell means, BLUPs of cell means, and truncated models. One such program on a single run would compute the cross validation for one of the model forms AMMI, GREG, SREG, or COMM. Because SHMM requires a more complicated algorithm, a separate program was written for it. Essentially, these programs embedded the SAS shrinkage estimation programs eigsimsh.sas and eigsimsq.sas, with added capability to compute BLUPs and least squares truncated model estimates, in a loop each execution of which would randomly and independently extract [N.sub.m] = n-1 observations from each of the g X e cultivar X site combinations for building the model, leaving [N.sub.v] = 1 observation from each cell to validate the models.

In computing shrinkage estimates of multiplicative models, n in [S.sub.k] = ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) was replaced with [N.sub.m] and the pooled within-site error mean square from an analysis of the entire data set was used for [s.sup.2]. Shrinkage factors for computing BLUPs of cell means appropriate for [N.sub.m] replications were computed from variance components estimated from the entire data set. Estimates of additive main effects in truncated models that contained such effects (AMMI, GREG, and SREG) were put equal to their estimates in the shrinkage estimation methods, namely, [S.sub.[Tau]]([y.sub.i..] - [y.sub....]) for cultivar main effects and [S.sub.[Delta]]([y.sub..j.] - [y.sub....]) for site main effects. This was done so that any superiority or inferiority of the shrinkage estimation method as compared with model truncation could be attributed to shrinkage of multiplicative terms and not to a difference in estimators of main effects. [Preliminary investigation prior to Cornelius et al. (1993) indicated that the use of shrinkage estimates for main effects in truncated models typically gave slight improvements.]

The root mean squared predictive difference (RMSPD) was computed for each model as the square root of the mean squared difference between the predicted values and their corresponding validation data. These RMSPD values were stored on a data set for later analysis. To provide a stopping rule, a pooled mean squared predictive difference ([PMSPD.sub.m]) on the mth execution of the loop was also computed. Execution of the loop was terminated if max[(|[PMSPD.sub.m] - [PMSPD.sub.m-1]|)/([PMSPD.sub.m-1])] [is less than] 0.001 held, where the maximum was over the models fitted on that computer run. The model with the smallest mean RMSPD is regarded as the best model. The stored RMSPD data from the individual executions of the loop were analyzed to compare models fitted on the same run of the program with respect to mean RMSPD.

For each model form (AMMI, COMM, SREG, GREG, and SHMM), the program was run twice on each set of data, once on the data as they were and once on replication-adjusted (rep-adjusted) data, i.e., the data after applying least squares adjustments for differences among replications (blocks) within sites. For the data not adjusted for replication differences, the error variance estimate used in computing shrinkage factors was put equal to [s.sup.2] + [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [s.sup.2] is the pooled error variance and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] an estimated variance component due to replications within sites. This provides an estimate of the increased error (noise) in the modeling and validation data that occurs as a consequence of ignoring block differences when conducting the random data splitting.

Analysis of Cross Validation Results

After determining the maximum number of data splittings required to meet the program's stopping rule on the computer runs for the five model forms (for each trial, one run for each model form), those model forms for which the program terminated at an earlier stage were rerun to complete a balanced set of RMSPD values for all data splitting X model and method combinations. This was done to facilitate the interpretation of a combined analysis of the RMSPD values for each trial.

The subsequent combined analyses (for each trial) were done on SAS procedure MIXED (SAS Institute, Inc., 1997) with an unstructured covariance matrix in order to allow for heterogeneous variances and covariances among RMSPD values obtained by different model forms and methods. Because RMSPD data for BLUPs of cell means (BLUPCELL) and ordinary cell means (CELLMEAN) from the separate computer runs for the five model forms were identical regenerations of the same results, these were included in the combined RMSPD data set only once and were given unique designations for model and method. Thus, the model forms to be compared were AMMI, GREG, SREG, COMM, SHMM, BLUPCELL, and CELLMEAN. Initially, we defined methods to include the three shrinkage estimators resulting from df of the multiplicative terms determined as Gollob's df (GOLLOB), by simulation (SIM) and by iterated simulation (ITERSIM), the best (smallest RMSPD) truncated model (BTRUNC) for each model form (possibly different number of terms retained for different model forms), BLUPCELL and CELLMEAN. BLUPCELL and CELLMEAN methods each, of course, occurs only in combination with its own corresponding designation as a model form and vice versa. Thus, the structure is an incomplete factorial layout of seven model forms and six methods.

For each trial, the method, and the model form X method combinations, with the smallest mean RMSPD were identified a priori. Then in the analysis by PROC MIXED all comparisons of other methods and model form X method combinations were compared with the best (smallest mean RMSPD) by the Scheffe adjustment of P-values,. Since the Scheffe adjustment controls the experimentwise Type I error rates for the set of all possible linear contrasts, it is conservative for the restricted set of all pairwise differences from the empirically best. The PROC MIXED syntax for these analyses is given in Appendix 4.

Attempts to run this analysis with both SIM and ITERSIM methods included failed because of a singular R matrix. Evidently, data on these two methods we. re so highly correlated that MIXED could not invert the resulting empirical covariance matrix. Consequently for each trial, only the best choice of SIM or ITERSIM over all model forms was included. A separate analysis to compare SIM and ITERSIM was done assuming a simpler covariance structure defining variance components for data splittings, model forms X splittings and methods X splittings.

RESULTS

Trial 1

Cross validation for rep-adjusted data showed shrinkage estimates were the best (all with RMSPD values nearly equal) followed by BLUPs of cell means (Table 1). The best truncated AMMI, SREG, and GREG models, i.e., [AMMI.sub.4], [SREG.sub.5] and [GREG.sub.4], the subscripts denoting number of multiplicative terms retained, showed RMSPDs slightly worse than the BLUPs of cell means, but the best truncated COMM and SHMM ([COMM.sub.5] and [SHMM.sub.5]) were slightly better. Shrinkage estimates, BLUPs of cell means, and best truncated models were substantially better predictors of the validation data than the cell means.

Table 1. Root mean square predictive difference (RMSPD) from cross validation of models fitted to Trial I data adjusted for replicate effects (Rep-adjusted) and unadjusted (Rep-unadjusted) with three observations used for modelling ([N.sub.m] = 3) and one observation used for validation ([N.sub.v] = 1). Italic number is the lowest RMSPD within the column and underlined number is the lowest RMSPD for a truncated model.
Description                     AMMI([dagger])    COMM      SREG

                                            Rep-adjusted
Shrinkage

GOLLOB ([double dagger])            628.55       628.10    628.15
SIM                                 627.82       627.19    627.97
ITERSIM                             627.65       626.92    628.04
BLUPCELL                            637.40       637.40    637.40

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TRUNC0 ([sections])                 698.06         --        --
TRUNC1                              661.61       694.07    670.69
TRUNC2                              653.42       656.43    650.64
TRUNC3                              640.22       653.49    642.55
TRUNC4                              637.45       639.69    641.46
TRUNC5                              645.73       636.40    641.12
TRUNC6                              650.88       645.27    647.98
TRUNC7                              653.82       650.30    651.30
TRUNC8                              655.35       653.38    652.96
TRUNC9                              659.08       656.87    658.01
CELLMEAN                            671.10       671.10    671.10
Sample size ([paragraph])             50           50        50

                                             Rep-unadjusted
Shrinkage

GOLLOB([double dagger])             720.76       719.76    720.17
SIM                                 718.63       717.17    718.88
ITERSIM                             718.54       717.13    719.19
BLUPCELL                            734.77       734.77    734.77

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TRUNC0 ([sections])                 767.13         --        --
TRUNC1                              738.40       763.34    745.39
TRUNC2                              736.37       733.46    731.82
TRUNC3                              729.04       736.07    729.95
TRUNC4                              730.75       729.56    733.39
TRUNC5                              741.39       729.93    737.02
TRUNC6                              750.43       741.01    745.77
TRUNC7                              756.51       750.15    753.14
TRUNC8                              760.95       756.49    757.80
TRUNC9                              766.45       762.57    764.29
CELLMEAN                            786.82       786.82    786.82
Sample size ([paragraph])             47           47        47

Description                         GREG          SHMM

                                      Rep-adjusted
Shrinkage
GOLLOB ([double dagger])            629.17       629.25
SIM                                 628.31       628.47
ITERSIM                             628.17       628.61
BLUPCELL                            637.40       637.40

Truncated

TRUNC0 ([sections])                   --           --
TRUNC1                              691.87       693.59
TRUNC2                              654.70       654.37
TRUNC3                              651.00       651.32
TRUNC4                              638.90       639.89
TRUNC5                              639.31       636.71
TRUNC6                              646.69       645.18
TRUNC7                              650.87       649.79
TRUNC8                              653.72       653.12
TRUNC9                              657.28       657.06
CELLMEAN                            671.10       671.10
Sample size ([paragraph])             50           50

                                       Rep-unadjusted
Shrinkage
GOLLOB([double dagger])             721.31       721.55
SIM                                 718.91       719.09
ITERSIM                             718.89       719.25
BLUPCELL                            734.77       734.77

Truncated

TRUNC0 ([sections])                   --           --
TRUNC1                              762.23       762.86
TRUNC2                              732.93       731.39
TRUNC3                              735.31       734.36
TRUNC4                              729.45       730.21
TRUNC5                              734.75       731.00
TRUNC6                              743.54       741.47
TRUNC7                              751.58       750.40
TRUNC8                              757.26       756.29
TRUNC9                              763.37       762.72
CELLMEAN                            786.82       786.82
Sample size ([paragraph])             47           47


([dagger]) AMMI: Additive Main effect and Multiplicative Interaction; COMM: Completely Multiplicative model; SREG: Sites Regression model; GREG: Genotypes Regression model; SHMM: Shifted Multiplicative model.

([double dagger]) GOLLOB, SIM, ITERSIM: Shrinkage estimates with df ([u.sub.k]) estimated by Gollob's df, simulation and iterated simulation, respectively. BLUPCELL: BLUPs of cell means based on two-way random effects model.

([sections]) TRUNC0, TRUNC1, TRUNC2, ...: zero multiplicative terms, one multiplicative term, two multiplicative terms, ..., etc.; CELLMEAN: cell means.

([paragraph]) Sample size is the number of random data splittings for which RMSPDs were computed and averaged.

With rep-unadjusted data shrinkage estimates were also better predictors of the validation data than the best truncated models ([AMMI.sub.3], [COMM.sub.4], [SREG.sub.3], [GREG.sub.4], and [SHMM.sub.4]) and BLUPs of cell means. However, as opposed to rep-adjusted data, with rep-unadjusted data, RMSPD values of the best truncated models were smaller than those for BLUPs of cell means. Cell means were the worst predictors of the validation data. RMSPDs of rep-unadjusted data were greater than for rep-adjusted data.

Trial 2

For rep-adjusted data, shrinkage estimates of multiplicative models had the lowest RMSPDs; all were very nearly equal (Table 2). BLUPs of cell means were slightly worse. Best truncated models ([AMMI.sub.8], [COMM.sub.8], [SREG.sub.8], [GREG.sub.8], and [SHMM.sub.8]) were substantially worse.

Table 2. Root mean square predictive difference (RMSPD) from cross validation of models fitted to Trials 2 through 5 adjusted for replicate effects (Rep-adjusted) with three observations used for modelling ([N.sub.m] = 3) and one observation used for validation ([N.sub.v] = 1). Italic number is the lowest RMSPD within the column and trial and underlined number is the lowest RMSPD for a truncated model.
Description                AMMI([dagger])      COMM           SREG

                                              Trial 2
Shrinkage

GOLLOB([double dagger])        1275.57        1274.11       1276.02
SIM                            1273.67        1272.27       1275.44
ITERSIM                        1273.70        1272.25       1275.71
BLUPCELL                       1284.04        1284.04       1284.04

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TRUNC0([sections])             1500.84          --             --
TRUNC1                         1336.09        1498.67       1384.31
TRUNC2                         1334.55        1330.11       1335.37
TRUNC3                         1338.84        1331.32       1328.28
TRUNC4                         1328.88        1331.99       1333.64
TRUNC5                         1324.01        1322.01       1330.79
TRUNC6                         1324.45        1322.39       1325.80
TRUNC7                         1325.16        1320.79       1321.34
TRUNC8                         1322.60        1316.89       1318.83
TRUNC9                         1325.27        1323.86       1324.97
CELLMEAN                       1331.26        1331.26       1331.26
Sample size([paragraph])          60             60             60

                                              Trial 3
Shrinkage

GOLLOB([double dagger])         800.36         799.42        800.48
SIM                             802.12         801.05        801.97
ITERSIM                         802.60         801.34        802.33
BLUPCELL                        817.64         817.64        817.64

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TRUNC0([sections])              902.59           --            --
TRUNC1                          816.41         884.67        816.30
TRUNC2                          820.48         813.39        826.69
TRUNC3                          827.66         814.66        825.57
TRUNC4                          825.37         827.18        823.81
TRUNC5                          831.55         827.79        832.60
TRUNC6                          837.89         833.71        839.90
TRUNC7                          845.92         837.87        845.88
TRUNC8                          849.55         845.74        849.39
CELLMEAN                        849.45         849.45        849.45
Sample size([paragraph])          64             64            64

                                              Trial 4
Shrinkage

GOLLOB([double dagger])         796.69         795.66        798.33
SIM                             796.68         795.64        797.52
ITERSIM                         796.92         795.74        797.51
BLUPCELL                        798.86         798.86        798.86

Truncated

TRUNC0([sections])              918.41           --            --
TRUNC1                          870.14         914.98        923.00
TRUNC2                          843.79         866.14        864.60
TRUNC3                          836.29         839.43        836.07
TRUNC4                          827.17         839.93        837.93
TRUNC5                          828.66         828.39        824.75
TRUNC6                          829.73         829.94        823.79
TRUNC7                          828.17         829.43        824.60
TRUNC8                          828.28         825.81        827.70
TRUNC9                          827.02         826.33        828.60
TRUNC10                         823.57         824.83        827.10
TRUNC11                         826.59         822.29        827.34
TRUNC12                         827.43         824.28        827.45
TRUNC13                         828.30         826.44        827.41
TRUNC14                         828.51         828.14        827.49
TRUNC15                         829.76         829.66        829.60
CELLMEAN                        831.10         831.10        831.10
Sample size([paragraph])          39             39            39

                                              Trial 5
Shrinkage

GOLLOB([double dagger])         668.50         668.14        671.60
SIM                             667.17         666.65        671.80
ITERSIM                         666.65         666.10        671.68
BLUPCELL                        663.95         663.95        663.95

Truncated

TRUNC0([sections])              677.49           --            --
TRUNC1                          685.47         675.79        684.56
TRUNC2                          696.92         684.43        689.08
TRUNC3                          701.89         698.21        698.38
TRUNC4                          702.45         704.23        698.26
TRUNC5                          712.25         702.49        709.79
TRUNC6                          716.24         711.49        715.42
CELLMEAN                        715.84         715.84        715.84
Sample size([paragraph])          58             58            58

Description                      GREG          SHMM

                                       Trial 2
Shrinkage

GOLLOB([double dagger])        1273.87        1275.24
SIM                            1272.21        1275.35
ITERSIM                        1272.32        1276.30
BLUPCELL                       1284.04        1284.04

Truncated

TRUNC0([sections])                --            --
TRUNC1                         1490.19        1499.30
TRUNC2                         1324.90        1330.52
TRUNC3                         1328.95        1332.97
TRUNC4                         1329.87        1331.53
TRUNC5                         1318.52        1322.39
TRUNC6                         1320.30        1324.89
TRUNC7                         1319.59        1320.80
TRUNC8                         1316.28        1315.54
TRUNC9                         1322.72        1324.08
CELLMEAN                       1331.26        1331.26
Sample size([paragraph])          60            60

                                      Trial 3
Shrinkage

GOLLOB([double dagger])          798.24        799.40
SIM                              799.67        800.79
ITERSIM                          800.14        800.87
BLUPCELL                         817.64        817.64

Truncated

TRUNC0([sections])                 --           --
TRUNC1                           872.03       871.68
TRUNC2                           808.03       810.16
TRUNC3                           808.80       811.76
TRUNC4                           821.69       822.83
TRUNC5                           829.87       832.26
TRUNC6                           831.83       835.12
TRUNC7                           840.72       840.27
TRUNC8                           847.13       846.74
CELLMEAN                         849.45       849.45
Sample size([paragraph])           64           64

                                      Trial 4
Shrinkage

GOLLOB([double dagger])          797.38       799.55
SIM                              797.24       800.15
ITERSIM                          797.19       800.61
BLUPCELL                         798.86       798.86

Truncated

TRUNC0([sections])                 --           --
TRUNC1                           915.13       915.55
TRUNC2                           865.62       866.72
TRUNC3                           839.03       839.98
TRUNC4                           838.56       839.95
TRUNC5                           831.39       828.31
TRUNC6                           832.06       830.31
TRUNC7                           831.29       829.97
TRUNC8                           827.16       826.45
TRUNC9                           826.76       827.22
TRUNC10                          824.58       825.88
TRUNC11                          822.53       823.61
TRUNC12                          824.71       825.53
TRUNC13                          827.02       826.98
TRUNC14                          828.02       828.21
TRUNC15                          829.23       829.26
CELLMEAN                         831.10       831.10
Sample size([paragraph])           39           39

                                      Trial 5
Shrinkage

GOLLOB([double dagger])          668.09       671.48
SIM                              666.61       671.86
ITERSIM                          666.09       672.01
BLUPCELL                         663.95       663.95

Truncated

TRUNC0([sections])                 --           --
TRUNC1                           675.08       676.03
TRUNC2                           683.76       684.41
TRUNC3                           697.05       697.76
TRUNC4                           703.09       703.34
TRUNC5                           702.90       703.05
TRUNC6                           711.84       711.93
CELLMEAN                         715.84       715.84
Sample size([paragraph])           58           58


([dagger]) AMMI: Additive Main effect and Multiplicative Interaction; COMM: Completely Multiplicative model; SREG: Sites Regression model; Genotypes Regression model; SHMM: Shifted Multiplicative model.

([double dagger]) GOLLOB, SIM, ITERSIM: Shrinkage estimates with df ([u.sub.k]) estimated by Gollob's df, simulation and iterated simulation, respectively. BLUPCELL: BLUPs of cell means based on two-way random effects model.

([sections]) TRUNC0, TRUNC1, TRUNC2, ...: zero multiplicative terms, one multiplicative term, two multiplicative terms, ..., etc.; CELLMEAN: cell means.

([paragraph]) Sample size is the number of random data splittings for which RMSPDs were computed and averaged.

For rep-unadjusted data, shrinkage estimates were improved by obtaining [u.sub.k] by simulation (as compared with the use of Gollob's degree of freedom for obtaining [u.sub.k], data not shown). Iterated simulation led to still further improvement. Best truncated models for the five model forms ([AMMI.sub.1], [COMM.sub.2], [SREG.sub.1], [GREG.sub.2], and [SHMM.sub.2]) were much more parsimonious than with rep-adjusted data, and were competitive with SIM and ITERSIM shrinkage estimates. BLUPs were inferior, and cell means substantially more inferior, to shrinkage estimates and best truncated models.

Trial 3

Cross validation performed on rep-adjusted data showed RMSPD values for shrinkage; estimates that were nearly equal for all shrinkage methods and multiplicative models forms, and these were all smaller than RMSPDs for BLUPs of cell means and all truncated multiplicative models (Table 2). RMSPDs of best truncated models [AMMI.sub.1], [COMM.sub.2], [SREG.sub.1], [GREG.sub.2], and [SHMM.sub.2] were as good as, or slightly better than, RMSPD of BLUPs of cell means. Cell means were the worst predictors of the validation data.

On rep-unadjusted data RMSPD values were larger than on rep-adjusted data, and shrinkage estimates of multiplicative models were the best predictors of the validation data, followed by the best truncated models [AMMI.sub.1], [COMM.sub.2], [SREG.sub.1], [GREG.sub.2], and [SHMM.sub.2] (data not shown). BLUPs of cell means were worse predictors of the validation data than shrinkage estimates and best truncated models. Again, cell means were the worst predictors of the validation data.

Trial 4

Cross validation with rep-adjusted data showed all shrinkage estimates of all multiplicative model forms, as well as BLUPs of cell means, to be almost equally good (Table 2). RMSPDs for truncated models, with increasing number of terms, appeared to decrease to a minimum or near minimum at 10 multiplicative terms for AMMI, 11 multiplicative terms for COMM, GREG, and SHMM, and six multiplicative terms for SREG. Cell means were inferior to the best truncated models, but all of these were definitely inferior to BLUPs and shrinkage estimates.

For rep-unadjusted data, RMSPDs were greater than for rep-adjusted, but comparisons were qualitatively similar. BLUPs and shrinkage estimates of multiplicative models were superior to the best truncated models ([AMMI.sub.4], [COMM.sub.5], [SREG.sub.5], [GREG.sub.3], and [SHMM.sub.3]), which, nevertheless, were superior to cell means (data not shown).

Trial 5

Cross validation performed on rep-adjusted data gave RMSPD values for BLUPs of cell means that were slightly superior to shrinkage estimates of AMMI, COMM, and GREG and slightly more superior to shrinkage estimates of SREG and SHMM. For AMMI, COMM and GREG, SIM and ITERSIM shrinkage estimates were very slightly superior to GOLLOB. RMSPD for BLUPs of cell means and all shrinkage estimates of multiplicative models were smaller than RMSPDs for all truncated models (Table 2). The best truncated models for the five model forms were [AMMI.sub.0], [COMM.sub.1], [SREG.sub.1], [GREG.sub.1], and [SHMM.sub.1]. Cell means were generally worse predictors of the validation data than BLUPs, shrinkage estimates of multiplicative models, or the best truncated multiplicative models.

On rep-unadjusted data, RMSPD values were substantially larger than on rep-adjusted data. BLUPs of cell means, shrinkage estimates of AMMI, COMM, and GREG and truncated models [AMMI.sub.0], [COMM.sub.1], [GREG.sub.1], and [SHMM.sub.1] gave RMSPDs very nearly equal and smaller than RMSPD for the other models and methods (data not shown). As with rep-adjusted, cell means were the poorest predictors of validation data.

Summary of Results

Results of the separate analysis for comparing SIM and ITERSIM done by defining variance components for data splittings, model forms X splittings, and methods X splittings showed that SIM and ITERSIM methods were not significantly different across models for Trials 1 through 4 and significantly different (P [is less than] 0.001) for Trial 5, but for practical purposes, the difference for Trial 5 was negligible (mean RMSPD 668.81 kg [ha.sup.-1] and 668.51 kg [ha.sup.-1] for SIM and INTERSIM, respectively).

For all trials, the BTRUNC method for the five model forms was always inferior to the shrinkage estimates (Table 3). For Trials 1 through 3 the RMSPD of BLUPCELL were significantly (P [is less than] 0.001) greater than the RMSPD of the best methods (ITERSIM, SIM, and GOLLOB for Trials 1-3, respectively) and also significantly greater than the RMSPD of the best model and method combination [Trial 1: (COMM, ITERSIM); Trial 2: (GREG, SIM); Trial 3: (GREG, GOLLOB)]. For Trial 4, the RMSPD of BLUPCELL was not significantly (P [is greater than] 0.001) greater than the RMSPD of the best method (SIM) or the RMSPD of the best model and method combination (COMM and SIM). For Trial 5, BLUPCELL was both the best method and best model and method combination. Significantly greater RMSPDs resulted from all other methods, and all other model and method combinations except (COMM, ITERSIM) and (GREG, ITERSIM).

Table 3. Root mean square predictive difference (RMSPD) for estimation methods and for model form and estimation method combinations in cross validation analyses of data from five multisite cultivar trials adjusted for replicate effects. Italic number is the lowest root mean square predictive difference for a model and method combination. Underlined number is the lowest root mean squared predictive difference for a method. Number in parenthesis is the number of multiplicative terms in the best truncated model. "ITERSIM or SIM" method is ITERSIM for Trials 1 and 5 and is SIM for Trials 2-4.
                                METHOD([dagger])
MODEL([double
 dagger])              BTRUNC                    GOLLOB
                                  Trial 1

AMMI                   637.45 (4)([sections])    628.55
COMM                   636.40 (5)([sections])    628.10
SREG                   641.12 (5)([sections])    628.15
GREG                   638.90 (4)([sections])    629.17
SHMM                   636.71 (5)([sections])    629.25
LINEAR([paragraph])      --                        --

Mean                   638.12(#)                 628.65
                                  Trial 2

AMMI                   1322.60 (8)([sections])   1275.57
COMM                   1316.89 (8)([sections])   1274.11
SREG                   1318.83 (8)([sections])   1276.02
GREG                   1316.28 (8)([sections])   1273.87
SHMM                   1315.54 (8)([sections])   1275.24
LINEAR([paragraph])       --                       --

Mean                   1318.31(#)                1274.96
                                  Trial 3

AMMI                   816.41 (1)([sections])    800.36
COMM                   813.39 (2)([sections])    799.42
SREG                   816.30 (1)([sections])    800.48
GREG                   808.03 (2)([sections])    798.24
SHMM                   810.16 (2)([sections])    799.40
LINEAR([paragraph])      --                        --

Mean                   812.86(#)                 799.58
                                  Trial 4

AMMI                   823.57 (10)([sections])   796.69
COMM                   822.29 (11)([sections])   795.66
SREG                   823.79 (6)([sections])    798.33
GREG                   822.53 (11)([sections])   797.38
SHMM                   823.61 (11)([sections])   799.55
LINEAR([paragraph])      --                        --

Mean                   823.16(#)                 797.52
                                  Trial 5

AMMI                   677.49 (0)([sections])    668.50([sections])
COMM                   675.79 (1)([sections])    668.14([sections])
SREG                   684.56 (1)([sections])    671.60([sections])
GREG                   675.08 (1)([sections])    668.09([sections])
SHMM                   676.03 (1)([sections])    671.48([sections])
LINEAR([paragraph])      --                        --

Mean                   677.79(#)                 669.56(#)

                                METHOD([dagger])
MODEL([double
 dagger])                ITERSIM or SIM         BLUPCELL
                                   Trial 1

AMMI                     627.65                 --
COMM                     626.92                 --
SREG                     628.04                 --
GREG                     628.17                 --
SHMM                     628.61                 --
LINEAR([paragraph])       --                    637.40([sections])

Mean                     627.88                 637.40(#)
                                   Trial 2

AMMI                     1273.67                --
COMM                     1272.27                --
SREG                     1275.44                --
GREG                     1272.21                --
SHMM                     1275.35                --
LINEAR([paragraph])       --                    1284.04([sections])

Mean                     1273.79                1284.04(#)
                                   Trial 3

AMMI                     802.12                 --
COMM                     801.05                 --
SREG                     801.97                 --
GREG                     799.67                 --
SHMM                     800.79                 --
LINEAR([paragraph])       --                    817.64([SECTIONS])

Mean                     801.12                 817.64(#)
                                   Trial 4

AMMI                     796.68                 --
COMM                     795.64                 --
SREG                     797.52                 --
GREG                     797.24                 --
SHMM                     800.15                 --
LINEAR([paragraph])       --                    798.86

Mean                     797.45                 798.86
                                   Trial 5

AMMI                     666.51([sections])     --
COMM                     666.10                 --
SREG                     671.68([sections])     --
GREG                     666.09                 --
SHMM                     672.01([sections])     --
LINEAR([paragraph])       --                    663.95

Mean                     668.51(#)              663.95

                         METHOD([dagger])
MODEL([double
 dagger])                    CELLMEAN
                             Trial 1

AMMI                          --
COMM                          --
SREG                          --
GREG                          --
SHMM                          --
LINEAR([paragraph])           671.10([sections])

Mean                         671.10(#)]
                             Trial 2

AMMI                          --
COMM                          --
SREG                          --
GREG                          --
SHMM                          --
LINEAR([paragraph])          1331.26([sections])

Mean                         1331.26(#)
                             Trial 3

AMMI                          --
COMM                          --
SREG                          --
GREG                          --
SHMM                          --
LINEAR([paragraph])          849.45([sections])

Mean                         849.45(#)
                             Trial 4

AMMI                          --
COMM                          --
SREG                          --
GREG                          --
SHMM                          --
LINEAR([paragraph])            831.10([sections])

Mean                         831.10(#)
                             Trial 5

AMMI                          --
COMM                          --
SREG                          --
GREG                          --
SHMM                          --
LINEAR([paragraph])          751.84([sections])
Mean                         715.84(#)


([dagger]) BTRUNC: best truncated model; GOLLOB, SIM, ITERSIM: shrinkage estimates with df ([u.sub.k]) estimated by Gollob's df, simulation and iterated simulation, respectively; BLUPCELL: BLUP cell means; CELLMEAN: cell means.

([double dagger]) AMMI: Additive Main effect and Multiplicative Interaction; COMM: Completely Multiplicative model; SREG: Sites Regression model; GREG: Genotypes Regression model; SHMM: Shifted Multiplicative model.

([sections]) Significantly greater than the best model and method combination (P < 0.001) by Scheffe procedure.

([paragraph]) Non-multiplicative model.

(#) Significantly greater than the best method (P < 0.001) by Scheffe procedure.

DISCUSSION

For all trials, results show that cross validation conducted on data not adjusted by replicate differences within a site gives larger values of RMSPD than cross validation done on rep-adjusted data.

The number of multiplicative terms in the best truncated model obtained by cross validating the rep-adjusted data tended to be equal to or greater than the number obtained when the cross validation was done on rep-unadjusted data. This was most evident in Trials 2 and 4 and was expected because there is more noise in the rep-unadjusted modeling and validation data than in the rep-adjusted data. The main reason for cross validating rep-adjusted data is that ultimately the chosen model will be fitted to the cell means from the complete data, and differences among such cell means within a site are free of the replication differences. Consequently, the modeling and validation data subsets should also be free of the replicate differences.

These results suggest that a model chosen by cross validation using data not adjusted for rep differences, when fitted to the complete data, may not be as predictively accurate as a model chosen by cross validation using data adjusted for rep differences, and very possibly not the most predictively accurate model that could be obtained. In lieu of rep-adjustment, one could use Piepho's (1994) restricted random splitting method in which n-1 complete replicates within each site are used for modeling and the remaining complete replicate is used for validation; this splitting is done independently for each site.

With only a few exceptions, when using rep-unadjusted data for any model form in any of the data sets presented, with or without adjustment for replication differences, shrinkage estimates of multiplicative models were superior to the best truncated model.

Researchers need a set of predicted values computed from the entire set of data. In choosing a truncated model by cross validation, the best model is subsequently fitted to the complete data to obtain the predictors. With shrinkage estimation, shrinkage factors and resulting shrinkage estimates are computed from the complete data set. Thus, the advantage of shrinkage estimation over truncated models chosen by cross validation is likely to be greater than cross validation will demonstrate. Moreover, for shrinkage estimation, cross validation is irrelevant, unless one wishes to use it for selecting a model form (AMMI, GREG, SREG, COMM or SHMM). The cross validation results reported here suggest that there is little difference between model forms for prediction of cell means if shrinkage estimates are used.

The question of BLUP methodology or shrinkage estimation of multiplicative models relates to the question of transferability. An underlying assumption for BLUPs based on the conventional two-way random effects model is that the interactions are NID(0,[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]). This implies that the interactions contain no transferable information, i.e., that the interaction in a particular cell tells us nothing about the interaction likely to occur in another cell. However, because prediction using AMMI shrinkage estimates will be identical to BLUPs of cell means if we put the AMMI shrinkage factors [S.sub.k] all equal to the BLUP shrinkage factor for the interaction, use of BLUPs of cell means will not result in a complete loss of all transferable information. Shrinkage estimates were clearly superior to BLUPs of cell means in Trials 1, 2, and 3, and slightly so in the case of rep-unadjusted data in Trial 4. Where shrinkage estimates were not distinctly better than BLUPs of cell means, there was essentially no difference. This shows that data structures can and will occur in which the equal shrinkage used by BLUPs is a sub-optimal strategy.

In the only trial where BLUPs were consistently, but only slightly, superior (Trial 5), the best truncated AMMI model was the additive model ([AMMI.sub.0]), suggesting that the interactions in Trial 5 contain little transferable information. This is consistent with the assumptions for the two-way random effects model, in particular that interactions are independently distributed. This inference is further supported by PRESS (prediction sum of squares by delete-one-cell jackknifing) statistics (Cornelius et al., 1996; Cornelius and Seyesadr, 1997) which, among truncated AMMI models, was smallest for [AMMI.sub.0] (data not shown). Among all models, PRESS was the smallest for [COMM.sub.1], but was only slightly larger for [SHMM.sub.1]. This suggests that there is a small amount of transferable information in the interactions, but it is contained essentially entirely in a 'concurrence' component (Mandel, 1961). AMMI is known to be rather insensitive to a concurrence component of interaction (Seyedsadr and Cornelius, 1992; Johnson and Hegemann, 1976). Although the concurrence component is statistically significant, its sum of squares is only 1.7% of the interaction sum of squares.

The results of this study indicate that shrinkage estimators are generally as good as the better choice of BLUPs of cell means or truncated multiplicative models. These results agree with findings of Cornelius et al. (1993, 1996) and Cornelius and Crossa (1995) in the sense that shrinkage estimation sometimes yields better estimates of cultivar performance and eliminates the need for either cross validation or tests of hypotheses as criteria for determining the number of multiplicative terms to be retained in a model.

Unlike comparisons of theoretical variances or standard errors which depend on underlying model assumptions, comparisons of the RMSPD in cross validation is independent of model assumptions; thus, differences between methods with respect to predictive accuracy should be solely a consequence of differing estimation efficiency. However, cross validation does not take into consideration differences between methods with respect to how much improvement in estimation efficiency will occur when applied to the complete data set as compared with its performance on a subset. Truncated models may be considered shrinkage estimates with shrinkage factors of unity for the terms retained and zero for the terms deleted. These factors are locked in once the model has been chosen by cross validation. But shrinkage factors for shrinkage estimators and BLUPs are not locked in, but will tend to be self correcting for the lower noise level found in cell means from the complete data set. Thus, the superiority of shrinkage estimates over model truncation found may underestimate the superiority of an estimate obtained from the complete data.

Shrinkage estimates putting [u.sub.k] equal to the degrees of freedom that Gollob assigned to the multiplicative terms were usually as good as shrinkage estimates using degrees of freedom ([u.sup.k]) determined by simulation or iterated simulation (an exception here was Trial 2 without adjustment for replication differences). If Gollob's degrees of freedom are used, shrinkage estimates of AMMI, COMM, SREG, or GREG are as easy to compute as BLUPs of cell means.

Under the postulated random model, the proposed shrinkage estimates of multiplicative components in AMMI, COMM, SREG, and GREG should have nearly optimal properties, particularly in large data sets, since we used [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as an estimate of its own expectation and its reliability as such should increase as the size of the data set increases.

While we postulate a random model, it is interesting to consider the appropriateness and performance of the shrinkage estimates under fixed effects assumptions. Apparently, under a fixed effects model, the exchangeability assumptions are not valid. Without the exchangeability assumptions, it is difficult to argue convincingly for validity of the assumption E([[Eta].sub.jk][[Lambda].sub.m][[Alpha].sub.im][[Gamma].sub.jm]) = 0, although it seems clear from Goodman and Haberman (1990) that this will hold asymptotically as the [[Lambda].sub.k] values and their differences become large relative to the variance of a cell mean. The assumption, of course, must hold trivially for cases with [Lambda]k = 0. Thus, it is not surprising that simulation studies (Cornelius et al., 1996) have demonstrated that the shrinkage estimates perform very well even when applied to cases of specific fixed sets of [Lambda]k values with no random effects assumptions except for the residual error. Our shrinkage estimation method is essentially an empirical Bayes method and it is not uncommon for Bayesian estimators to perform well when investigated from a purely frequentist perspective (Carlin and Louis, 1996).

Noting the equivalence of empirical BLUPs of realized values of random effects in linear models and empirical Bayes estimates, Robinson (1991) drew a distinction between an `objective' prior distribution of random effects used to describe variation (from which BLUPs derive) and a `subjective' prior distribution, from which `fixed effects might be considered to have come', used to describe uncertainty (from which Bayes estimates derive). In his last paragraph (before comments and discussion), Robinson states, "Within the Bayesian paradigm, there is little reason for distinguishing between fixed and random effects. All effects are treated as random in the sense that probability distributions used to describe uncertainty are not treated any differently from probability distributions used to describe variation."

In discussing Robinson's paper, Harville (1991) suggested that the distinction between variation and uncertainty is not so clear cut. In practical situations, the distinction is surely moot when the two points of view lead to the same prior distributions and the same estimators. The fundamental issue then becomes not whether the effects are random or fixed, but whether the assumed prior distribution is or is not reasonable (or, if unreasonable, what the consequential impact on the performance of the estimators may be).

Concerning the empirical Bayes method developed for SHMM in Appendix 3, the only subjective feature is the choice of a definition for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which is both intuitively reasonable and empirically (i.e., objectively) estimable. Interestingly, if the logic used in deriving the empirical Bayes method we used for SHMM is applied to the other model forms, the estimators obtained are identical to those previously derived (see Material and Methods and Appendix 2) by modeling variation. Consequently, it is not surprising that the shrinkage estimators of SHMM perform essentially as well as the shrinkage estimators of the other model forms. Note that this implies that, for the model forms other than SHMM, the empirical Bayes estimates of the multiplicative terms which maximize the posterior likelihood under normal priors on the [Lambda]k are shrinkage estimators of the form for which we derived optimum solutions in the Materials and Methods and Appendix 2.

CONCLUSIONS

Shrinkage estimates of multiplicative models were always better than models based on truncated models fitted by ordinary least squares on the basis of data splitting and cross validation. Furthermore, cross validation does not compensate if more multiplicative terms are needed when estimates are computed from the complete data set. Shrinkage estimates of multiplicative terms are generally as good as or better than BLUPs of cell means based on a two-way random effects model with random uncorrelated interactions; however, this superiority is often very small and does not always occur.

As estimates of realized cultivar performance levels, among the estimation methods examined here, the estimates most commonly reported in published yield trial results, namely ordinary cell means, tend to be the worst.

Shrinkage estimates using Gollob's degrees of freedom are nearly as good as when the degrees of freedom are determined by simulation or by iterated simulation methods. From a practical viewpoint, shrinkage factors based on Gollob's degrees of freedom are simpler and less computer intensive than those obtained by simulation and iterated simulation methods.

Abbreviations: AMMI, Additive Main effects and Multiplicative Interaction Model; BLUP, Best Linear Unbiased Prediction; BLUPCELL, BLUP of cell mean; BTRUNC, Best truncated model; CELLMEAN, Empirical cell mean; COMM, Completely Multiplicative Model; (GOLLOB, SIM, ITERSIM), Shrinkage estimates with [sup.u]k obtained as Gollob's df, simulation and iteration simulation, respectively; GREG, Genotypes Regression Model; NID, Normally and Independently Distributed; SHMM, Shifted Multiplicative Model; SREG, Sites Regression Model; SVD, Singular Value Decomposition.

ACKNOWLEDGMENTS

The authors thank the numerous cooperators in national maize and wheat research programs that carried out the experimental cultivar trials providing the data analyzed in this paper.

REFERENCES

Carlin, B.P., and T.A. Louis. 1996. Bayes and empirical Bayes methods for data analysis. Chapman and Hall, London.

Cornelius, P.L. 1993. Statistical tests and retention of terms in the additive main effects and multiplicative interaction model for cultivar trials. Crop Sci. 33:1186-1193.

Cornelius, P.L., J. Crossa, and M.S. Seyedsadr. 1993. Tests and estimators of multiplicative models for variety trials, p. 156-166. In Proceedings of Annual Kansas State University Conference on Applied Statistics in Agriculture, 5th. Manhattan, KS. 25-27 April 1993. Dep. of Statistics, Kansas State Univ., Manhattan, KS.

Cornelius, P.L., and J. Crossa. 1995. Shrinkage estimators of multiplicative models for crop cultivar trials. Technical Report No. 352, University of Kentucky, Dep. of Statistics, Lexington.

Cornelius, P.L., J. Crossa, and M.S. Seyedsadr. 1996. Statistical tests and estimators of multiplicative models for genotype-by-environment interaction, p. 199-234. In M.S. Kang and H.G. Gauch (ed.) Genotype-by-environment interaction, CRC Press, Boca Raton, FL.

Cornelius, P.L., and M.S. Seyedsadr. 1997. Estimation of general linear-bilinear models for two-way tables. J. Statist. Computing Simulation 58:287-322.

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APPENDIX 1

BLUPs of Cell Means

Under the two-way random effects model with interaction (Eq. [1]), the BLUP of a cell mean is BLUP([[micro]ij]) = y ... + BLUP([[Tau].sub.i]) + BLUP([[[Delta].sub.j]) + BLUP[[([Tau][Alpha].sub.)ij]] where, in a balanced data set,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The BLUP of a cell mean is then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The three functions of variance components in this expression are estimated by [S.sub.[Tau]], [S.sub.[Delta]], and [S.sub.[Tau][Delta]], respectively, as defined in Eq. [2]. Thus, Eq. [2] is the empirical BLUP of [[micro].sub.ij].

APPENDIX 2

Derivation of Shrinkage Estimators in AMMI, GREG, SREG, and COMM

First we consider the AMMI model. We assume that any set of [[Alpha].sub.ik] and [[Gamma].sub.jk] values that satisfy the model constraints has some nonzero probability of being the realized set, and if such sets are not all equally likely, at least they are such that we may assume that the [[Alpha].sub.ik] for given k are exchangeable and make a similar assumption with respect to the [[Gamma].sub.jk]. A set of random variables are `exchangeable' if the joint distribution of any subset consisting of m of those variables is the same for any such subset (Galambos, 1982).

Thus we assume that each empirical least squares bilinear component [[Lambda].sub.k][[Alpha].sub.ik][[Gamma].sub.jk] is the sum of some `true' realized outcome [[Lambda].sub.k][[Alpha].sub.ik][[Gamma].sub.jk] plus some perturbation [[Eta].sub.ijk] = [[Lambda].sub.k][[Alpha].sub.ik][[Gamma].sub.jk] - [[Lambda].sub.k][[Alpha].sub.ik][[Gamma].sub.jk] that results as a consequence of the combination of the realized values of the random bilinear effects with the residual errors. Note that for AMMI, GREG, and SREG, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. As a consequence of the exchangeability assumption, [[Eta].sub.ijk] and [[Lambda].sub.k][[Alpha].sub.ik][[Gamma].sub.jk] are uncorrelated and [[Eta].sub.ijk] is also uncorrelated with [[Lambda].sub.m][[Alpha].sub.im][[Gamma].sub.jm] for all m [not equal to] k. It then follows that E([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) = E([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If we consider an estimator of the form [[Sigma].sub.k][S.sub.k][[Lambda].sub.k][[Alpha].sub.ik][[Gamma].sub.jk] for estimation of the contribution [[Sigma].sub.k][[Lambda].sub.k][[Alpha].sub.ik][[Gamma].sub.jk] to the realized cell mean [[micro].sub.ij], the per cell mean squared error of estimation (for an AMMI model, the IMSE), E[[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]]/ge is minimized if [S.sub.k] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Note that [S.sub.k] is the expected regression of the quantities [[Sigma].sub.m][[Lambda].sub.m][[Alpha].sub.im][[Gamma].sub.jm] on the kth least squares component [[Lambda].sub.k][[Alpha].sub.ik][[Gamma].sub.jk], and is the "cell-optimal" shrinkage factor (Cornelius et al., 1996, p. 217) under our random effects model assumptions if optimization is over the population.

Now define [u.sup.k] such that E([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then, E([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) and, consequently, E([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) We use [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to estimate E([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) and the pooled error mean square [s.sup.2] to estimate [[Alpha].sup.2], giving [S.sub.k] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = 1 - [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [F.sub.k] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as previously defined, provided that this gives [S.sub.k] [is greater than] 0 (i.e., provided [F.sub.k] [is greater than] 1); otherwise put [S.sub.k] = 0. Using [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to estimate [Sigma]([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) is easily justified by noting that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the scale parameter for the set of model effects [[Lambda].sub.k][[Alpha].sub.ik][[Gamma].sub.jk] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the sum of squares due to this set of effects. Thus, this usage is consistent with the standard practice of using analysis of variance sums of squares, or mean squares, as estimates of their own expectations.

In deriving [S.sub.k] the only necessary assumptions were E([[Eta].sub.ijk]) = E ([[Eta].sub.ijk][[Lambda].sub.k][[Alpha].sub.ik][[Gamma].sub.jk]) = 0 for all k and m (including k = m), i.e., the constraints [[Sigma].sub.i][[Lambda].sub.k][[Alpha].sub.ik][[Gamma].sub.jk] = 0 and/or [[Sigma].sub.j][[Lambda].sub.k][[Alpha].sub.ik][[Gamma].sub.jk] = 0 are not necessary to the argument. Consequently, the resulting shrinkage estimation procedure is valid for COMM as well as for AMMI, GREG, and SREG.

APPENDIX 3

Derivation of Shrinkage Estimators in SHMM

For the shifted multiplicative model (SHMM), [y.sub.ij] = [Beta] + [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the least squares solution is [Beta] = [y.sub....] - [[Sigma].sub.k][[Lambda].sub.k][[Alpha].sub.k][[Gamma].sub.k], where [[Alpha].sub.k] = [g.sup.-1][[Sigma].sub.i][[Alpha].sub.ik][[Gamma].sub.jk] = [e.sup.-1] [[Sigma].sub.k][[Gamma].sub.jk] and [[Lambda].sub.k], [[Alpha].sub.k] and [[Gamma].sub.k] are obtained as the kth component of the singular value decomposition of the g X e matrix of values [z.sub.ij] = [y.sub.ij] - [Beta]. Given the values of [[Alpha].sub.ik] and [[Gamma].sub.k], the least squares solutions for [Beta] and [[Lambda].sub.k], k = 1,2 ..., t can be reproduced as the least squares solution for the multiple regression problem [y.sub.ij] = [Beta] + [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] + [e.sub.ij] where the regressor variables [X.sub.kij] = [[Alpha].sub.ik][[Gamma].sub.jk].

The partial sum of squares due to [[Lambda].sub.k] (i.e., due to regression on [X.sub.k]) is SS([[Lambda].sub.k]) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where

[A1] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is the sum of squares of [X.sub.k] adjusted for the mean and for its multiple regression on the analogous variables [X.sub.m] = [[Alpha].sub.im][[Gamma].sub.jm], m [not equal to] k. If the `true' realized values of the [[Alpha].sub.ik], [Gamma].sub.jk] and [[micro].sub.ij] are known, [Beta] and [[Lambda].sub.k], k = 1,2, ... ,t, can be determined as the multiple regression [[micro].sub.ij] = [Beta] + [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [X.sub.kij] = [[Alpha].sub.ik][[Gamma].sub.jk]. (The residual term is omitted on the presumption that the equation models the realized [[micro].sub.ij] values exactly; this is always true if the model is saturated, i.e., if t = p.) The partial sum of squares due to regression on [X.sub.k] is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [q.sub.k] is obtained from Eq.[A1] with [[Alpha].sub.m] and [[Gamma].sub.m] replaced by the true realized values [[Alpha].sub.m] and [[Gamma].sub.m].

Define [[micro].sub.k] = nE([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and, hence,

[A2] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Now let us consider an empirical Bayes method with a "subjective prior to describe uncertainty" as opposed to an "objective prior to describe variation" (Robinson, 1991). For the set of realized values of [[Alpha].sub.ik] and [[Lambda].sub.jk], we adopt a non-informative prior (all possible sets of values satisfying the model constraints considered equally likely), and, for [Beta] and [[Lambda].sub.k], we adopt normal priors N(0,[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) and N(0,[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]), respectively, along the lines of Viele and S rinivasan (1998).

For convenience in the sequel define [[Phi].sub.[Beta]] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then the Bayes estimates [Beta]*, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which maximize the posterior likelihood (posterior mode) are [Beta]* = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] constitute the kth component of the singular value decomposition of the g X e matrix of values [z.sub.ij] = [y.sub.ij] - [Beta]* [and, thus [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]]. Clearly, the result will be highly sensitive to the choices of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In the Bayesian paradigm, the variances [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are "hyperparameters", values of which may possibly be chosen to reflect the analyst's confidence (or lack thereof) that the parameters [Beta] and [[Lambda].sub.k] are indeed equal to the means of their respective prior distributions (in this case, zero). A small variance will imply a high degree of confidence in the prior mean as the value of the parameter. If the analyst has little confidence in the prior mean, the variance is set to a large value. Empirical Bayes estimation seeks to obtain information from the data concerning values for hyperparameters (and thus inject more, if not total, objectivity into the methodology).

We will put [[Phi].sub.[Beta]] = 1 (equivalent to regarding [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to be essentially infinite, thus rendering the prior on [3 essentially non-informative). The resulting estimate of [Beta] is

[A3] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

An equivalent formula is

[A4] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The choice of [[Phi].sub.[Lambda]k] needs to be carefully considered. Notice that [[Phi].sub.[Lambda]k] = [([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]).sup.-1] where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a noise to signal ratio. It is tempting to put [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], but we know from experience that this will not result in enough shrinkage of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to result in an efficient estimator. The problem here is that [[Lambda].sub.k] contributes to estimates of the realized cell means only through the product [[Lambda].sub.k][[Alpha].sub.ik][[Gamma].sub.jk], and is, in fact, the scale parameter for those terms (having been included in the model solely for the purpose of scaling the [[Alpha].sub.k][[Gamma].sub.k] products). Consequently, an optimal (or nearly optimal) shrinkage factor should be [(1 + [R.sub.k]).sup.-1] where [R.sub.k] is an estimate of the expected ratio, variance due to noise divided by variance due to signal, for all the terms [[Lambda].sub.k][[Alpha].sub.ik][[Gamma].sub.jk] for a given k. It is the partial sum of squares due to the [k.sup.th] multiplicative term which measures the total variation (noise + signal) due to the kth component free of the mean and the other multiplicative components.

Let superscript (h) on the symbol for a parameter denote the estimate in the [h.sup.th] iteration of an algorithm for computing a simultaneous shrinkage solution for the model parameters. Define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by [A1] with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] substituted for [[Alpha].sub.m] and [[Gamma].sub.m]. The partial sum of squares due to regression on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in the [h.sup.th] iteration will be SS([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with expectation ([A2]) [u.sub.k][[[Alpha].sup.2] + E([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]). The ratio [R.sub.k] of noise to signal is the ratio of the two terms in the expected mean square, i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This implies that we should put [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] resulting in [[Phi].sub.[Lambda]k] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which can be estimated by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = 1 - [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [F.sub.k] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. If [F.sub.k] [is less than] 1 then put [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = 0. Thus, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] will be the estimate of [[Phi].sub.[Lambda]k] in the [h.sup.th] iteration. Note that the expression for [F.sub.k] suggests an interpretation of [u.sub.k] as "degrees of freedom" for SS([[Lambda].sup.(h)]).

The [u.sub.k] values can be estimated by computer simulation, but degrees of freedom analogous to Gollob's degrees of freedom for the other forms, giving [u.sup.k] = g + e + 1 - 2k for k [is less than or equal to] min(g - 1, e - 1) and [u.sup.k] = g + e - 2k for k = min(g, e), may be accurate enough to make the shrinkage estimators perform well in practice.

As in the least squares solution, the estimates [Beta]* and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are dependent on one another, so that if one wants the estimates to be consistent with one another, one needs to iterate the estimation scheme until the estimates converge. Our strategy is to compute such a solution, first using Gollob's degrees of freedom as the [u.sup.k] values. We follow this with a simulation step to determine new [u.sup.k] values, after which we compute the new empirical Bayes (i.e., shrinkage) estimates. One can, of course, iterate the simulation and estimation steps as often as one wishes.

To obtain simulated values of [u.sup.k] without having to compute iterative SHMM solutions for the simulated data sets, we employ the following strategy which gives approximations that have been accurate enough to perform well in practice. From the supposed "true" singular values and vectors, compute [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] once and for all. To obtain simulated results, generate a g X e matrix of random normal noise. To this matrix add the supposed "true" values of the multiplicative terms divided by s/[square root of]n. Obtain the singular value decomposition of this matrix and use its singular values and vectors to compute simulated values of SS([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) for k = 1,2, ... ,p. Accumulate the values of SS([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) for a large number of such simulations. Finally obtain [u.sup.k] by subtracting the "true" [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] from the mean of the simulated SS([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) values. If [[Sigma].sub.k][u.sub.k] [is greater than] ge, multiply all [u.sup.k] by ge/[[Sigma].sub.k][u.sub.k].

Further hints on computation, including details of a Newton-Raphson algorithm for obtaining the shrinkage estimates, given the [u.sub.k] values, and how to impose the ordering constraints [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on the solution are given in Cornelius and Crossa (1995).

APPENDIX 4

Following is the syntax for the PROC MIXED analysis of the RMSPD values from the incomplete 7 x 6 factorial arrangement of model forms (MOD) and methods (METH).
PROC MIXED;
CLASS MOD METH SPLIT;
MODEL RMSPD = MODIMETH/DDFM = SATTERTH;
REPEATED MOD*METH/SUBJECT-- SPLIT
  TYPE=UN;
LSMEANS METH/
DIFF=CONTROLU(`bestmethod')
ADJUST=SCHEFFE OM BYLEVEL;
LSMEANS MOD*METH/DIFF = CONTROLU
  (`bestmodel' `bestmethod')
ADJUST=SCHEFFE OM SLICE(MOD METH);


where, in the LSMEANS METH statement the method with the smallest mean RMSPD is substituted for "bestmethod" and in the LSMEANS MOD*METH statement, the "bestmodel" and "bestmethod" are given as the model and method combination with smallest mean RMSPD (which was determined a priori). Use of the OM and BYLEVEL options on LSMEANS for METH allowed BLUPCELL and CELLMEAN to be compared to the other methods averaged over model forms. Syntax for the separate analyses of RMSPD data on SIM and ITERSIM replaced the REPEATED statement with

RANDOM SPLIT MOD*SPLIT METH*SPLIT;.

Paul L. Cornelius and Jose Crossa(*)

P.L. Cornelius, Dep. of Agronomy and Dep. of Statistics, Univ. of Kentucky, Lexington, KY 40546-0091; J. Crossa, Biometrics and Statistics Unit, CIMMYT, Lisboa 27, Apdo. Postal 6-641, 06600 Mexico D.F., Mexico. This paper is published as journal article 97-06-93 of the Kentucky Agric. Exp. Stn. and is published with the approval of the Director. Received 29 April 1998. (*) Corresponding author (JCROSSA@CIMMYT.MX).

Published in Crop Sci. 39:998-1009 (1999).
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