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CROP BREEDING, GENETICS & CYTOLOGY

Efficiency of Spatial Analyses of Field Pea Variety Trials

^a Alberta Agriculture, Food, and Rural Development, Room 300, 7000–113 Street, Edmonton, AB, Canada T6H 5T6, and Dep. of Agricultural, Food, and Nutritional Sci., Univ. of Alberta, Edmonton, AB, Canada T6G 2P5
^b Alberta Agriculture, Food, and Rural Development, Room 300, 7000–113 Street, Edmonton, AB, Canada T6H 5T6
^c Alberta Agriculture, Food, and Rural Development, RR6, 17507 Fort Road, Edmonton, AB, Canada T5B 4K3
^d Alberta Agriculture, Food, and Rural Development, Brooks, AB, Canada T1R 1E6

^* Corresponding author (rong-cai.yang{at}ualberta.ca).

	ABSTRACT

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION REFERENCES

 
Several spatial analyses of neighboring plots are now available for improving the precision of variety trials. The objective of this study was to evaluate the efficiency of three commonly used spatial analyses, a nearest neighbor adjustment (NNA), a least squares smoothing (LSS), and a first-order autoregressive model (AR1), in removing field trends from 157 field pea (Pisum sativum L.) variety trials tested in different growing zones across Alberta, Canada, during 1997 to 2001. All trials were conducted with a randomized complete block (RCB) design with three or four replications. A complete replication (block) was planted in a single field tier. Yield data from each of the 157 trials were subject to the conventional RCB analysis and the three spatial analyses. The LSS, NNA, and AR1 analyses removed an average of 22, 16, and 7% residual variation compared with the RCB analysis, respectively, but the amount of removal by the three analyses varied considerably among the trials. Each spatial analysis achieved more error reduction in 1997 and 1998, where trials contained larger block sizes than in 1999 to 2001, where trials contained smaller block sizes. The efficiency in spatial variation removal was great with large block sizes that involved large numbers of varieties. Furthermore, the LSS and NNA analyses were more effective in such removal than the AR1 analysis.

Abbreviations: AR1, first-order autoregressive model • LDF, loss of degrees of freedom • LSS, least squares smoothing • NNA, nearest neighbor adjustment • RCB, randomized complete block • REML, restricted maximum likelihood

	INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION REFERENCES

 
VARIETY TRIALS evaluate performance of different genotypes in crop improvement programs and test large numbers of selections for possible release as new varieties. With current move toward diversification of crops in North America and elsewhere (Blade et al., 2002), some variety trials are set out to test registered varieties of nontraditional crops in new environments. In any case, accurate estimates of variety means or variety differences require control of error variation that is left unaccounted for by variety effects, either by the use of appropriate experimental design or by statistical analysis. The RCB design, because of its simplicity, continues to be a popular choice for many variety trials. The validity and efficiency of the RCB analysis depends on whether or not plots within each block would have relatively homogeneous growing conditions (e.g., soil fertility and moisture). However, spatial homogeneity within blocks of more than 8 to 12 plots seldom occurs in field trials (Stroup et al., 1994). Thus, efficiency of the RCB design is often poor in variety trials involving a large number of entries. An incomplete block design such as a lattice or an -design can have smaller blocks but spatial heterogeneity may persist in small blocks. Evidently, such design-based control of the error variation alone may not be sufficient to remove all spatial trends in the variety trials.

Different model-based analyses that exploit the information on plot positions have been developed and applied to estimate and correct for spatial variation within and among blocks. These spatial analyses include NNA (Bartlett, 1978; Wilkinson et al., 1983; Zimmerman and Harville, 1991), LSS (Green et al., 1985), and modeling of spatial autocorrelation such as the AR1 model (Gleeson and Cullis, 1987; Gilmour et al., 1997). Efficiency of different spatial analyses relative to the analysis of the block designs has been frequently evaluated (Ball et al., 1993; Brownie et al., 1993; Clarke et al., 1994; Stroup et al., 1994; Grondona et al., 1996; Wu et al., 1998; Helms et al., 1999), but such evaluations are usually based on a limited number of field trials. On the other hand, other studies (e.g., Kempton and Howes, 1981; Cullis and Gleeson, 1989; Kempton et al., 1994) have used a large number of trials, but focused on the comparison of one spatial analysis with the conventional RCB analysis. Variety trials are often performed in a large number of test sites across many years. Patterns and extent of spatial variability may vary greatly among environments, suggesting that different spatial analyses may differ in their ability to remove spatial heterogeneity in different environments. It would be desirable to evaluate the efficiency of different spatial analyses across a large number of trials encompassing different environments.

This paper reports an evaluation of the efficiency of three spatial analyses (NNA, LSS, and AR1) relative to conventional RCB analysis based on 157 field pea variety trials tested in different growing zones across Alberta, Canada, during 1997 to 2001. These trials were part of the Alberta Field Pea Regional Variety Test Program that was established in 1987 to carry out multiyear and multisite tests for recommending registered varieties to local pea producers across the province (Park and Lopetinsky, 1999).

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION REFERENCES

 
Description of Variety Trials
The 157 field pea variety trials used for this study were performed across 5 yr: 11 trials in 1997, 20 in 1998, 44 in 1999, 39 in 2000, and 43 in 2001 (Table 1). The same varieties were included in all trials within a year but different varieties except for check varieties were usually used in different years either due to a turnover to newly registered varieties or to unavailability of pedigree seed of older varieties. Two types of field pea varieties, green and yellow, were grown in the same trials in 1997 and 1998, but in separate trials at the same test sites in 1999 to 2001. The increase in number of trials in the last 3 yr corresponded largely to the decrease in number of varieties per trial (Table 1). These trials were grown at various test sites in southern Alberta, east-central Alberta, west-central Alberta, and Peace River Region (Park and Lopetinsky, 1999). The southern Alberta region was further divided into irrigated and nonirrigated areas. The Peace River Region included some neighboring sites in British Columbia.

Spatial Analyses
Yield data for each trial were analyzed with a model that allowed for the incorporation of spatial variation (Brownie et al., 1993):

[1]

where Y_ij is the observed yield (kg ha^–1) in the jth plot within the ith block or plot ij, the term µ + _k(ij) represents the mean performance of variety k in plot ij, T_ij is the trend effect representing systematic spatial variation in this plot, and _ij is the random residual. Different spatial analyses were performed by modeling T_ij and _ij in Eq. [1]. The baseline analysis was the RCB analysis of variance. In this case, the trend effects T_ij were assumed to be constant for all plots within the same block, that is, T_ij = ß_i (the effect of the ith block).

For the NNA analysis, we used the iterative one-dimensional modification of Papadakis procedure (Wilkinson et al., 1983) to calculate a trend index from the neighbors on either side of each plot but the block effect (ß_i) was preserved. Thus, T_ij + _ij in Eq. [1] became ß_i + bX_ij + e_ij in the NNA analysis, where X_ij = (e_i,j–1 + e_i,j+1)/2, b = the regression coefficient associated with the covariate X_ij, and e_ij = Y_ij – _ij with _ij being the variety mean in plot ij. For border plots at either end of a block, X_ij was calculated as the residual for the one neighbor. Each iteration started with a new trend index that was the difference between the observed and adjusted variety means from the previous iteration. The iteration continued until the difference between the adjusted means in the two successive iterations was negligible.

For the LSS analysis (Green et al., 1985), trend effects, T_ij, were estimated with the constraint that their second differences were zero (T_{i,j – 1} – 2T_ij + T_{i,j + 1} = 0). It was further assumed that residuals (_ij) were uncorrelated with each other and with the trend effects. An appropriate tuning constant () must be chosen to ensure uncorrelated residuals between neighboring plots while maintaining a smooth trend. Following Clarke et al. (1994), we searched for a value that would give a near-zero serial correlation in the estimated residuals. The searching process was stopped when the absolute value of the serial correlation was <0.02 or when reached a preset upper limit of 10⁶, even though the serial correlation remained to be away from zero.

The third analysis was the direct fitting of field trends (Zimmerman and Harville, 1991; Gilmour et al., 1997) rather than by differencing or use of neighbor residuals as in the first two analyses. In this approach, the residuals (_ij) were assumed to be distributed according to spatial correlation models. A commonly used spatial correlation model is the AR1. The covariance between _ij and _ij' under the AR1 model is given by

[2]

where ² is the residual variance that would be estimated with the RCB analysis (no spatial trend). The presence of spatial trend would suggest that neighboring plots tend to be more alike than those farther apart ( > 0). We only considered spatial relationships within blocks (a one-dimensional AR1 model) because a full replication (block) was in a single field tier. In the presence of complicated covariance structure for the residuals such as AR1, the likelihood-based mixed model analysis was needed to model and estimate the covariance structure and adjusted variety means.

All required calculations and analyses described above were performed with SAS/IML Workshop Version 2.0 for Release 8.2 of the SAS System (SAS Institute, 2002). One of the new features in SAS/IML Workshop Version 2.0 is its ability to call SAS procedures within the context of an IMLPlus program. For example, we performed the restricted maximum likelihood (REML) analysis of the AR1 model through calling and executing SAS PROC MIXED (SAS/STAT Release 8.2) from the IMLPlus program we developed. To carry out the mixed model analysis of the AR1 model, we included block effects in the RANDOM statement of PROC MIXED, and specified SUBJECT = BLOCK and TYPE = AR(1) as options for the REPEATED statement. The SAS source code for all calculations and analyses is available upon request.

Calculation of Efficiency
In evaluating the relative efficiency of the NNA analysis to the RCB analysis in removing field trends, Stroup et al. (1994) and Agrobase (1999) have defined the NNA weight as 1 – E_a/E_u, where E_a and E_u are the error mean squares from the NNA and RCB analyses, respectively. The NNA weight of >0.3 is used to indicate the presence of moderate to large field trends (Stroup et al., 1994; Helms et al., 1999). However, it remains controversial what would be the appropriate df to the residual mean square from the NNA analysis (Binns, 1987). Even if, in some cases, the reduction in the residual df is negligible due to a large residual df from the unadjusted data, the NNA weight as defined is not necessarily bounded within the range of 0 to 1 as claimed in Agrobase (1999)(p. 281). The NNA adjustment with the analysis of covariance loses 1df in E_a compared with E_u. Thus, while the adjusted error sum of squares (SSE_a) is always less than or equal to the unadjusted error sum of squares (SSE_u), the condition of E_a E_u is not guaranteed. For example, the condition of E_a E_u did not hold in 48 of the 157 trials used in our study, leading to a negative NNA weight. For these reasons, we proposed to measure the efficiency of the NNA or LSS analysis relative to the RCB analysis by 1 – SSE_a/SSU_u.

A different measure of the efficiency of the AR1 analysis was needed, as the REML analysis of the AR1 model directly estimated the residual variance without the need to explicitly consider df. Because the SEDs of adjusted means derived from the AR1 analysis varied among different pairs of varieties, an average SED was calculated as the square root of the average of squares of SEDs across all pairs of adjusted means [SED_AR1]. Similarly, the likelihood-based analysis of the RCB design gave the estimate of SED_RCBD that was simply equal to (2²/b)^1/2, with b being the number of blocks. Thus, the relative efficiency of the AR1 model to the RCB analysis was calculated as 1 – SED_AR1/SED_RCBD.

	RESULTS AND DISCUSSION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS AND DISCUSSION REFERENCES

 
Estimated efficiency of the NNA, LSS, and AR1 procedures in removing spatial trends varied considerably among the 157 field pea variety trials (Table 2, Fig. 1). These procedures did not detect any trend (i.e., zero or near-zero percentage removal of residual variation due to the trends) in some trials but were able to remove as much as 85% of the residual variation due to the trends in other trials. The RCB analysis showed significant (P < 0.05) block effects in 104 of the 157 trials. For the remaining 53 trials with insignificant RCB block effects, the more insignificant (higher probabilities from the F test) the RCB block effects, the more effective the NNA analysis in removing intrablock spatial variation (r = 0.305, P = 0.026, n = 53). Similar correlation between the insignificance of RCB block effects and the efficiency of the AR1 analysis was also observed but was not significant (r = 0.201, P = 0.158, n = 53). The LSS analysis did not separate the RCB block effects and spatial trends and thus there was little relationship between the insignificance of RCB block effects and the efficiency of the LSS analysis.

View larger version (18K): [in this window] [in a new window]   Fig. 1. Distributions of the efficiency of (A) the nearest neighbor adjustment (NNA), (B) least squares smoothing (LSS), and (C) first-order autoregressive model analyses (AR1) in removing spatial trends from the residual variances in 157 field pea variety trials tested in Alberta, Canada, during 1997 to 2001. SSENNA, SSELSS, and SSERCBD are the error sums of squares from the NNA, LSS, and randomized complete block design analyses. SEDAR1 and SEDRCBD are the standard errors of difference from the AR1 and RCBD analyses.

The efficiency of the NNA analysis (Fig. 1A) showed that about 18% (29/157) of the 157 trials had a >30% removal of the residual variation due to spatial heterogeneity, a criterion used to indicate moderate to large field trends revealed by the NNA analysis (Stroup et al., 1994; Agrobase, 1999; Helms et al., 1999). Out of those 29 trials, 11 had 30 to 40%, seven had 40 to 50%, and 11 had 50 to 85% reduction of residual variation. With the same criterion, 22% (35/157) of the 157 trials had a >30% reduction of residual variation by the LSS analysis (Fig. 1B) but only 4% (7/157) of the trials had a >30% reduction of residual variation by the AR1 analysis (Fig. 1C). The LSS analysis was slightly better than the NNA analysis in error reduction, which is consistent with the observations of Green et al. (1985) and Clarke et al. (1994). The AR1 analysis was the least effective in error reduction. The analysis based on the two-dimensional AR1 model showed no further improvement of efficiency over the one-dimensional AR1 analysis reported here, probably because each block in our RCB design occupied just a single row or column in a single field tier.

The loss of degrees of freedom (LDF) for the residual mean square from the NNA analysis is long recognized but it remains largely a controversial issue (Binns, 1987; Helms et al., 1999). On the other hand, the LSS analysis (Green et al., 1985) provides a means of estimating LDF. The estimates of LDF for the 157 field pea trials were the differences between degrees of freedom for the residual mean square from the RCB analysis and those from the LSS analysis (Table 2). On average, LDF was larger in 1997 (10.2) and 1998 (11.2) than in 1999 to 2001 (<5), suggesting again the presence of stronger spatial trends in the first 2 yr than in the latter 3 yr. The likelihood-based AR1 analysis directly estimated the residual variance without the need to explicitly consider the corresponding degrees of freedom.

While the amount of error reduction varied among the three spatial analyses (NNA, LSS, and AR1), the reduction by these analyses was consistent across the trials, as evident from high correlations among NNA, LSS, and AR1 for all 157 trials and for the 73 trials with the absolute values of serial correlation being within 0.02 (Table 3). Since LDF was indicative of error reduction, it was also highly correlated with the three analyses. The trials with large numbers of varieties per block also had high CV values as indicated by the high correlation between NVAR and CV (55.4% for all the trials and 70.7% for the 73 trials). Moderate to high correlations of the three analyses with NVAR or CV pointed to the obvious expectation that these methods should be more effective for larger blocks with greater residual variation (higher CV). Our results support the view that efficiency in removing spatial variation by spatial analyses is great when block sizes and number of varieties within blocks are large (e.g., Bartlett, 1978; Cullis and Gleeson, 1989).

View larger version (22K): [in this window] [in a new window]   Fig. 2. Decomposition of yield data into variety effect, spatial trend, and residual by the least squares smoothing analysis for two field pea variety trials with high (A) and low (B) coefficients of variation.

Reliable identification of desirable varieties depends on whether or not the yields of check varieties show the same pattern of response across a trial as test varieties. With different yield responses due to the spatial heterogeneity, the use of checks would increase rather than decrease the error of variety comparisons (Kempton and Howes, 1981). Indirect evidence appeared to suggest the existence of such different yield responses in our field pea variety trials. For example, the average differences between RCB and NNA mean yields of check varieties across all the 157 trials were 17.3 kg ha^–1 in 1997, 10.1 kg ha^–1 in 1998, –3.9 kg ha^–1 in 1999, 4.1 kg ha^–1 in 2000, and 4.9 kg ha^–1 in 2001, but there were wide ranges of such differences in different years. Thus, the spatial analyses homogenized yield responses between check and test varieties in individual trials, thereby allowing for more accurate variety comparisons particularly in 1997 and 1998.

	ACKNOWLEDGMENTS

 
We thank two anonymous reviewers for valuable comments. All participants of Alberta Field Pea Regional Variety Test (AFPRVT) Program are thanked for their continued cooperation for field experiments and data collection. The AFPRVT program has been supported in part by Alberta Agriculture, Food, and Rural Development (AAFRD) and Alberta Pulse Growers Commission. This research was supported in part by AAFRD Industry Development Sector New Initiative Fund.

Received for publication January 31, 2003.

	REFERENCES

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This Article Abstract Figures Only Full Text (PDF) Free Alert me when this article is cited Alert me if a correction is posted Services Similar articles in this journal Similar articles in ISI Web of Science Alert me to new issues of the journal Download to citation manager Citing Articles Citing Articles via HighWire Citing Articles via ISI Web of Science (6) Citing Articles via Google Scholar Google Scholar Articles by Yang, R.-C. Articles by Bandara, M. Search for Related Content PubMed Articles by Yang, R.-C. Articles by Bandara, M. Agricola Articles by Yang, R.-C. Articles by Bandara, M. Related Collections Other Legumes Crop Genetics Statistics