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a Dep. of Plant and Soil Sciences, 12F Stockbridge Hall, Univ. of Massachusetts, Amherst, MA 01003
b Soil, Crop, and Atmospheric Sciences, 1021 Bradfield Hall, Cornell Univ., Ithaca, NY 14853
* Corresponding author (sebdon{at}pssci.umass.edu)
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONREFERENCES |
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Abbreviations: AMMI, additive main effect and multiplicative interaction • ANOVA, analysis of variance • GE, genotype by environment • KBG, Kentucky bluegrass • IAS, interaction score • NTEP, national turfgrass evaluation program • PRG, perennial ryegrass • RMS PD, root mean square predictive difference • SS, sum of squares • SVD, singular value decomposition
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONREFERENCES |
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GE interactions are problematic for both the agronomist and breeder because genotype means (averaged over environments) are unreliable for predicting performance of a genotype and, furthermore, genotypes must be targeted to individual locations in order to maximize performance. Also, the interaction contains noise that further complicates cultivar recommendations (Gauch, 1988, 1990; Gauch and Zobel, 1988). Noise often causes more genotypes to win in various environments than would be the case were the variety trial more accurate. Thus, noise causes spurious complexity.
Additive main effect and multiplicative interaction (AMMI) analysis has been shown to be more effective than the conventional two-way fixed effects model with interaction (Zobel et al., 1988) because it achieves several important goals including (i) parsimony, because the model contains relatively few of the interaction degrees of freedom, (ii) effectiveness, because the model contains most of the interaction sum of squares (SS) that is rich in pattern, leaving a residual that is rich in noise with most of the degrees of freedom but small SS, thereby affording greater predictive accuracy and statistical efficiency (frequently 2 replications with AMMI are as accurate as 3 to 6 replications without AMMI) (Gauch, 1992; Gauch and Zobel, 1996a), and (iii) mega-environment analysis (Gauch and Zobel, 1997), which identifies homogenous subregions within a crop's growing region having similar GE interactions. AMMI can simplify cultivar recommendations by reducing the number of winning genotypes through gains in statistical efficiency and accuracy (goals i and ii) (Gauch, 1988, 1990, 1992; Gauch and Zobel, 1988, 1996a) and by combining multiple test site locations into regions having similar cultivar recommendations (goal iii) (Brown et al., 1983; Peterson and Pfeiffer, 1989; Gauch and Zobel, 1997). Mega-environment analysis can be used to increase heritabilities within well defined and predictable environments (Annicchiarico and Perenzin, 1994); target genotypes to appropriate areas (Brown et al., 1983; Peterson and Pfeiffer, 1989); and to direct the allocation of resources in order to increase the efficiency of testing.
The AMMI accuracy gain (over empirical means computed as averages over replicates) has been documented in numerous yield trials (Gauch, 1988, 1990, 1992; Gauch and Zobel, 1988, 1996a), but has not been tested in turf performance trials. Turf performance is a subjective evaluation of turfgrass quality and is based on a visual rating system (NTEP uses a 1 to 9 scale with 9 indicating the highest quality) assessing aesthetic appeal and function. In field trials such as those sponsored by NTEP with G genotypes growing in E environments (location-year combinations), using R replications (NTEP uses 3), the full model (cell means or AMMI-F model) considers as relevant data the R replicates for a particular genotype G growing in environment E and uses their average over replicates as its estimate for turfgrass quality. Conversely, the AMMI model identifies and extracts a reduced model and uses the entire GER observations by fitting a multivariate model to the interaction to estimate turfgrass quality of genotype G growing in environment E. The AMMI model is more predictively accurate because it considers all the data as relevant in predicting future performance. Secondly, AMMI selectively recovers pattern and discards the noise that introduces discrepancies between the estimate Yge and the corresponding true mean µge. Also, mega-environment analysis can be used to improve predictive accuracy because a given location will have good predictive value for other locations in the same mega-environment, but not outside the location's own mega-environment (Annicchiarico, 1992). Multivariate analysis such as AMMI can be more reliable for predicting future performance (Romagosa and Fox, 1993).
The objectives of this research are (i) to compare AMMI estimates of cultivar performance with empirical cell means with respect to predictive accuracy and statistical efficiency using NTEP performance data, (ii) compare cultivar recommendations based on AMMI adjusted means with unadjusted means (raw data) as reported by NTEP, and (iii) identify subregions within the cool-season turfgrass growing region using mega-environment analysis.
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONREFERENCES |
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AMMI Analysis
AMMI analysis of turfgrass performance data was computed with the software MATMODEL (Gauch and Furnas, 1991; Gauch, 1998). For a detailed description of the AMMI model used in the analysis of NTEP data refer to Ebdon and Gauch (2002) and for appropriate statistical fixed-effect models for AMMI and all of its subcases refer to Zobel et al. (1988). The additive effects [grand mean (µ), genotype mean deviations (
g), environment mean deviations (ße)] are fitted with ordinary ANOVA (Snedecor and Cochran, 1989) to partition the total treatment variation sum of squares (SS) into three orthogonal sources: genotype, environment, and GE interaction. The non-additive residual (
ge) or interaction SS is fitted with singular value decomposition (SVD) to obtain the multiplicative part [singular vector for axis n (
n), genotype singular vector for axis n (
gn), environment singular vector for axis n (
en)] and, thus partitions the GE interaction into several orthogonal vectors of interaction axis scores (IAS). The sum, 
n ngnen, of products of genotype interaction scores (0.5ngn) and environment interaction scores (0.5n en) gives the estimated interaction.
NTEP trials were established as randomized complete blocks with three replications. The AMMI results presented here were analyzed as complete randomized designs (as a matter of convenience) so the resultant F-statistic and statistical significance are conservative. But greater emphasis is placed on tests of predictive accuracy, as described next.
AMMI Model Selection
The first step in the AMMI analysis was to identify (diagnose) which model from the AMMI family is most appropriate (accurate for predicting the true means) for each data set. Data splitting and cross-validation procedures (Gauch and Zobel, 1988) were used to identify the most accurate model. The data was split into two subgroups, two randomly chosen observations from each GE combination being put into the data subset for modeling and the remaining data (one observation from each GE combination) put into the subset for validation.
Several models were compared (see Gauch and Zobel, 1988) for evaluating predictive accuracy. The models evaluated included the additive two-way ANOVA model, which retains none of the interaction axes (AMMI-0), AMMI-1 to AMMI-7 models which combine the additive main effects and the GE interaction estimated from 1 to 7 IAS with the remaining IAS discarded, and also the full model (AMMI-F), i.e., the empirical cell means model (means averaged over replicates, or raw data), equivalent to means as reported in NTEP final reports.
The predictive values from each model were compared with the validation set by computing the sum of squared differences, dividing this by the number of validation observations and finally taking the square root to give the root mean square predictive difference (RMS PD). This cross validation/data splitting procedure was repeated 1,000 times and results averaged. The model having the best predictive accuracy (smallest RMSPD) was selected. Research has shown that 10 to 100 randomizations are generally adequate (Crossa et al., 1990) for predictive success especially with rather large data sets such as NTEP. However, because of the efficiency of the software used, we did not feel it necessary to experiment with fewer randomizations.
Mega-Environment Analysis
NTEP data summarized in final reports are organized by cultivar and location (averaged over years). These reports are then used to make cultivar recommendations and to select cultivars adapted to specific locations (not for location–year combinations). In order to subdivide the NTEP locations according to their GE interaction patterns into homogenous subregions, the 125 KBG genotype by 17 location data set and 123 PRG genotype by 19 location data set were subjected to mega-environment analysis according to the methods of Gauch and Zobel (1997). Genotypic correlations were computed between locations to assess the predictive value of a given location for other locations within and outside a location's own mega-environment.
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RESULTS AND DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONREFERENCES |
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The objective in achieving maximum predictive accuracy is to balance underfitting real structure and overfitting spurious noise (Gauch and Zobel, 1988). The PRG AMMI-2 and KBG AMMI-7 models achieve this by providing the smallest RMS PD (Table 1). Alternatively, the most predictively accurate model should capture mostly pattern and little noise. To that end, the AMMI model that comes closest to a residual SS equaling our estimate of noise (33.3 and 40.6%, for KBG and PRG, respectively) is likely to be the most predictively accurate model. The PRG AMMI-2 model leaves a residual SS that is 43.7% of the interaction (Table 3) while the KBG AMMI-7 model has a residual of 36.4% (Table 2), which are quite close to our targets for noise. Thus, fitting additional AMMI axes would be adding interaction terms that derive primarily, if not exclusively, from noise.
This accuracy gain improves cultivar recommendations because estimates of turf performance are more reliable. Figure 1 shows the difference between AMMI-F and the more predictively accurate KBG AMMI-7 estimates for turfgrass quality of 125 Kentucky bluegrass genotypes grown in a typical NTEP location–year combination (environment), namely the North Brunswick, NJ, site in 1991. The North Brunswick location and year represents a typical NTEP interaction environment because it has an IAS-1 score close to zero (score of +0.03). Genotype Midnight ranks first based on the data (AMMI-F model) and the KBG AMMI-7 adjusted means. Genotype Eclipse showed a positive gain in turfgrass quality ranking from 23rd based on the data to 4th based on the KBG AMMI-7 adjusted means. Conversely, genotype Blacksburg showed the largest loss in turfgrass quality from 4th (raw data) to 52nd (KBG AMMI-7 adjusted means). Expected future performance according to KBG AMMI-7 estimate suggest that planting Blacksburg based on the recommendation implied by the raw data (i.e., top ten ranking) would result in a turfgrass quality loss of 1.3 from 6.5 to 5.2 (Fig. 1). Similarly, the PRG genotype Danilo (Fig. 1, Pullman, WA, NTEP site in 1994 with environment IAS-1 score of -0.01) showed a loss in ranking from 8th based on the data to 105th based on the more predictively accurate AMMI-2 adjusted means. Danilo showed the largest loss in turfgrass quality from 7.3 (raw data) to 6.5 (PRG AMMI-2 adjusted means). Planting Danilo PRG based on the recommendation suggested by the raw data (NTEP top ten ranking) would result in a turfgrass quality loss of 0.8.
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Compared with AMMI reduced models (PRG AMMI-2 and KBG AMMI-7), the AMMI-F models identify 2 to 3 times more winners. Specifically, PRG AMMI-2 identified 10 winning genotypes while AMMI-F identified 34 winners in 60 location-year combinations. Similarly, the KBG AMMI-7 model identified 18 winning genotypes while AMMI-F identified 29 winners in 69 location-year combinations. In the KBG trial, AMMI-F and KBG AMMI-7 models pick the same winners in only 19 (27.5%) of the 69 environments and picked different winners in 50 (72.5%) of the 69 environments (location-year combinations). In the perennial ryegrass trial, AMMI-F and PRG AMMI-2 estimates picked the same winning genotypes in only 6 (10%) of the 60 environments and picked different winners in 54 (90%) of the 60 environments.
The average gain in Kentucky bluegrass turfgrass quality from selecting KBG AMMI-7 winners over AMMI-F winners was 0.3. The largest individual gain in turfgrass quality for Kentucky bluegrass occurred in Adelphia, NJ, in 1992 where the data (AMMI-F model) picked genotype Unique but KBG AMMI-7 model picked genotype Midnight. Based on KBG AMMI-7 adjusted means, the turfgrass quality rating for Midnight (KBG AMMI-7 winner) was 7.9 and the KBG AMMI-7 estimate for Unique (AMMI-F winner) was 6.6. This difference between AMMI-F and KBG AMMI-7 winners equates to a turfgrass quality gain of 1.3. For the 1990 perennial ryegrass trial the average gain in turfgrass quality by giving priority to PRG AMMI-2 winners over AMMI-F winners was 0.2. The largest individual gain in turfgrass quality by selecting PRG AMMI-2 winners over AMMI-F winners was 0.8 and occurred in Carbondale, IL, in 1994 where the AMMI-F model picked genotype Prelude II but PRG AMMI-2 model picked genotype Prizm. Conversely, giving priority to winners suggested by the raw data (NTEP means) over AMMI winners equated to an average loss in turfgrass quality of 0.4 for both the Kentucky bluegrass and perennial rye-grass trials. This loss in turf quality with AMMI-F winners is illustrated by the genotype Lynx (Fig. 1). Lynx perennial ryegrass (AMMI-F winner) had a quality rating of 7.9 based on the raw data but had an adjusted quality rating of 7.5 based on PRG AMMI-2 estimates.
Figure 1 illustrates differences in rank order between NTEP means and AMMI adjusted means for typical NTEP location-year combinations. For some genotypes no difference in rank were observed between AMMI-F and AMMI-adjusted means (Midnight, Fig. 1) while for other genotypes substantial differences in rank were detected. For example, the perennial ryegrass genotype Morning Star growing in Richmond Hill, ON, Canada in 1991 was identified by AMMI-F as the winning genotype. The PRG AMMI-2 model picked CLP 39 as the winning genotype and ranked Morning Star 112th based on the PRG AMMI-2 adjusted means, resulting in a rank difference of 111. Similarly, the Kentucky bluegrass genotype SR-2000 growing in Carbondale, IL, in 1991 was identified by AMMI-F as the winning genotype but the more predictively accurate KBG AMMI-7 model picked SR-2100 as the winning genotype and ranked SR-2000 80th based on the KBG AMMI-7 adjusted means.
There are also considerable differences among NTEP locations in mean rank between AMMI-F means and the more predictively accurate AMMI reduced model adjusted means (computed as |AMMI-F-AMMI adjusted|, averaged over genotype). The average rank difference for the 17 NTEP locations from the 1990 Kentucky bluegrass trial ranged from 5.4 (North Brunswick, NJ) to 27.4 (Pullman, WA) and for the 19 NTEP locations from the perennial ryegrass trial the average rank difference between models ranged from 6.0 (North Brunswick, NJ) to 30.0 (Richmond Hill, ON, Canada). If AMMI reduce models reported here based on validation studies are to be trusted more than the raw data in predicting future performance, then these differences in cultivar rankings between AMMI-F (NTEP means) and AMMI adjusted means suggest the potential to improve the predictive accuracy and reliability of cultivar recommendations by giving priority to AMMI adjusted means.
Mega-Environment Analysis
Gauch and Zobel (1997) introduced interaction plots specifically to identify mega-environments. Figure 2
shows the AMMI-1 model predicted turfgrass quality for KBG and PRG genotypes as a function of environment (location) interaction scores. Scores for the locations are indicated as triangles along the abscissa in Fig. 2. Predicted turfgrass quality is derived from the AMMI-1 model where ße = 0 (the environment deviation is zero). The environment deviations are ignored in mega-environment analysis because they do not alter cultivar rankings (but they can be included for estimating turfgrass quality). Environments alter ranking only through their interaction with genotypes. But genotypes alter cultivar ranking through both main effect and interaction with environments. In Fig. 2, a line represents each genotype. The line is derived mathematically using genotype and environment interaction scores and genotype main effect means according to the methods described by Gauch and Zobel (1997). Briefly, for any line in Fig. 2, the general equation is yij = ai + bixj where yij is AMMI-1 predicted turfgrass quality for genotype i growing in environment (location) j, ai is the genotype i main effect mean, bi is genotype i IAS-1 score, and xj is environment j IAS-1 score. Genotype interaction scores determine differences in the directions of the slopes of the lines; positive scores result in a positive slope (increasing from left to right) and negative scores result in a negative slope. Genotype main effects alter displacement of genotype lines along the ordinate in Fig. 2.
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In mega-environment analysis the emphasis is placed on crossover interactions among winning genotypes. In the NTEP variety trials having 125 KBG and 123 PRG genotypes, if equal attention was given to all possible crossovers including those among losers, this would increase the number of mega-environments (subregions) unnecessarily and detract from the primary objective of identifying which genotype tends to win where.
In Fig. 2, three KBG genotypes that are predicted winners over some locations (a genotype's winning region) are identified by thick lines while others that are never predicted to win are represented by thin lines. The genotype Midnight is the predicted winner in 14 locations (identified as subregion 2), Ba 73-382 is predicted to win in one location (Haymarket, VA), and the genotype Glade predicted winning region includes two locations (East Lansing, MI and Carbondale, IL). The smallest region (with one location, Haymarket, VA) was reassigned to the next closest region with similar interactions (East Lansing, MI and Carbondale, IL) to form subregion 1.
Five PRG genotypes that won over some locations are also shown in Fig. 2. Genotypes Prizm, Prelude II, Assure, Cowboy II, and CLP 39 are predicted to win in 4, 7, 4, 2, and 2 locations, respectively. The two smallest regions (each with two locations) were combined to form a single mega-environment (subregion 4), in which Cowboy II and CLP 39 are nearly equal in predicted quality.
Tables 4 and 5 show the correlation matrix for turfgrass quality for KBG and PRG variety trials, respectively, with NTEP locations subdivided according to mega-environment analysis recommended subdivisions (Fig. 2). Locations (and subregions) in Tables 4 and 5 are ordered according to their corresponding environment interaction scores (interaction patterns). Turfgrass quality ratings for locations within the same subregion (mega-environment) have better predictive value for other locations within the same mega-environment (indicated by a correlation coefficient with a P 
0.001). The possible exception is subregion 4 (Table 5) where the predictive value among locations is poor. Generally, locations have poorer predictability for NTEP sites outside its own mega-environment. Additionally, locations have poorer predictability for NTEP sites with opposite and distinctly different interaction patterns (environment scores). Consequently, cultivar recommendations can be simplified because the same cultivars can be targeted to each location from the same subregion since cultivar performance is similar within the same mega-environment.
The environment scores shown in Fig. 2 (and Tables 4 and 5) were correlated with maintenance practices. Specifically, the environment scores shown in Fig. 2 were correlated with mowing height of cut (r = -0.60, P 0.01) in the maintenance of KBG. Kentucky bluegrass NTEP locations shown in Fig 2 and Table 4 with large negative IAS-1 scores were mown at 5 to 7.5 cm height of cut, while locations with a large positive score were mown at 2.5 to 3.75 cm. The environment scores shown for PRG locations in Fig. 2 and Table 5 were correlated with annual nitrogen applied (r = -0.67, P 0.01) in the maintenance of NTEP trials. Perennial ryegrass locations in Fig 2 and Table 5 with large negative IAS-1 scores were fertilized with nitrogen at 245 to 294 kg ha-1, while locations with a large positive score were fertilized at 98 to 147 kg ha-1. Consequently, the subregions (mega-environments) follow a cultural intensity gradient and therefore can be altered by mowing height and nitrogen fertilization. For example, if the number of NTEP locations for KBG was expanded, then increasing the number of subregion 1 locations would be recommended because this mega-environment has only three locations (Haymarket, VA, East Lansing, MI, and Carbondale, IL). Locations for this mega-environment would be encouraged to maintain sites at cultural intensity levels according to Haymarket, VA and East Lansing, MI by mowing at the high end of the recommended cutting height range (5.0 to 7.5 cm). Conversely, if the number of KBG NTEP sites was reduced, then NTEP administrators could target any of the 14 locations from subregion 2. Thus, allocation of NTEP resources can be based on results from mega-environment analysis.
Sub-dividing NTEP locations into mega-environments is useful if they are relatively stable (if new genotypes or locations or years do not cause major changes in interaction patterns). Research from yield trials indicate long-term stability of mega-environments across locations (Delacy et al., 1994) and years (Annicchiarico and Perenzin, 1994). Romagosa and Fox (1993) suggest that multivariate analysis such as AMMI can provide reliable predictions for future years based on merely 1 yr of wide testing. In NTEP trials where genotypes have been evaluated over several years of wide testing, the mega-environments reported here are most likely stable. However, occasional review and refinement of megaenvironments is recommended especially with a changing roster of genotypes that can over time induce different interaction patterns (Delacy et al., 1994).
The statistical model used here, AMMI with its axes truncated after they begin to reduce predictive accuracy, was selected for its suitability to serve two purposes with a single model: understanding GE interactions and gaining accuracy. A conceptual and computational economy results from one analysis for both purposes. Yan et al. (2000) proposed another method for mega-environment analysis, but it does not have the clean separation of main and interaction effects affecting wide and narrow adaptations that AMMI uniquely has. Also, its mega-environment assignments can be expected to be virtually the same as those from AMMI-1 analysis, but no extension to more dimensions has been given, whereas when the interaction is large and complex so that the AMMI approach would use the AMMI-2 or higher model.
Cornelius and Crossa (1999) present a magisterial study of multivariate models and fitting procedures for gaining accuracy. They find model choice makes little difference, so AMMI is as suitable as anything else; but fitting procedures can make significant differences. For four of five representative data sets, truncation AMMI models (as used here) were nearly as accurate as anything else. And for one case, the shrinkage AMMI model was decidedly better still, though the larger matter remains that either truncation or shrinkage AMMI models were far superior to standard practice, basing estimates on averages over replicates (perhaps adjusted for block or spatial effects). Some researchers may want to undertake the additional computational effort of fitting AMMI or related models by several fitting procedures in hopes of achieving additional marginal gains. However, shrinkage models are high dimensional models, unlike truncation models retaining just an axis or two, so they cannot be used simultaneously to produce biplots for mega-environment analysis. Accordingly, some researchers may prefer to use a truncated AMMI model for mega-environment analysis, and to keep statistical efforts at an acceptable level and to have the conceptual economy that results from using a single model to achieve several purposes, they may prefer to use this same model for gaining accuracy.
Our studies suggest that priority should be given to AMMI adjusted means over unadjusted means (averaged over replicates) because AMMI estimates are more reliable (accurate) in predicting future performance. NTEP data contains substantial noise that reduces predictive accuracy. Increasing the number of replicates of NTEP raw data by a factor of 2.05 (for KBG) to 5.6 (for PRG) would be required to achieve the same accuracy as AMMI adjusted means. Giving priority to AMMI predictions (winning genotypes) over those implied by NTEP raw means equate to a gain in turfgrass quality as much as 0.8 (for PRG) to 1.3 (for KBG) on a rating scale of 1 to 9. AMMI adjusted means simplify cultivar recommendations by reducing the number of winning genotypes by as much as 1/2 to 2/3 compared with the raw data. Mega-environment analysis identified several subregions (each representing several NTEP locations) within KBG and PRG growing areas having similar interaction patterns and cultivar rankings. Such homogenous subregions may allow the same genotypes to be targeted to all locations from the same subregion, further simplifying cultivar recommendations.
To validate the results from these statistical studies, future research should include field studies comparing winning genotypes suggested by the more predictively accurate AMMI models with AMMI-F winners (suggested by the raw data) in order to determine the significance of the turfgrass quality gains with AMMI predictions. With especially complex interactions, the raw data becomes quickly buried by noise; noise causes imperfect correlation between phenotypes and genotypes (Federer, 1951). Accordingly, turfgrass breeders and agronomists may benefit from AMMI accuracy gain because significant relationships are often detected between AMMI predictions and other genetic or environmental factors which are absent in the raw data (Gauch and Zobel, 1996a).
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ACKNOWLEDGMENTS |
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Received for publication April 2, 2001.
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTS AND DISCUSSIONREFERENCES |
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