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STATISTICS

Validity and Efficiency of Neighbor Analyses in Comparison with Classical Complete and Incomplete Block Analyses of Field Experiments

^a Nanjing Agricultural University, Nanjing, China
^b Dep. of Plant Science, McGill University, Macdonald Campus, 21,111 Lakeshore Rd., Ste-Anne-de-Bellevue, QC, Canada H9X 3V9

cydp{at}musica.mcgill.ca

Received for publication May 21, 1997.

	ABSTRACT

TOP ABSTRACT INTRODUCTION Materials and methods Results and discussion Conclusions REFERENCES

 
In classical block analysis, the assumption of independence between observations from neighboring plots within a block is violated when spatial autocorrelation is present, resulting in incorrect treatment effect estimates and an inflated error variance estimate. Before recommending any neighbor analysis designed to account for spatial autocorrelation, its validity in estimation and testing and its efficiency must be examined. Our main objective was to assess the validity and efficiency of five neighbor analyses in comparison with complete and incomplete block analyses, using 50 pseudo-experiments for each of 50 data sets derived from two uniformity trials for soybean and wheat. The neighbor models were first difference with (FD-EV) or without (FD) errors in variables, second difference with (SD-EV) or without (SD) errors in variables, and first-order autoregressive errors. Geostatistical analysis based on the normalized semivariance detected spatially structured variation at different degrees in the data sets. The first and second differences were effective in removing spatial trends and autocorrelation. Our simulation study showed that FD-EV provided valid estimates of the error variance in all and SD-EV in most of the data sets. On average, the observed significance level was slightly beyond the theoretical level for both analyses. The other neighbor analyses underestimated the error variance. Excluding SD, neighbor analyses had similar efficiency and were more efficient than complete and incomplete block analyses, with average relative efficiencies of about 300 and 130%, respectively. Relative efficiency tended to increase with decreasing normalized nugget effect (NC₀) and increasing CV. When spatially structured variation is strong (i.e., NC₀ < 10%), highly rectangular plots must be preferred to nearly square plots in incomplete block and neighbor analyses. Considering validity and efficiency, the best method was FD-EV, which is thus a reliable alternative to classical block analysis when spatial autocorrelation is present.

Abbreviations: CB, complete block • FD, first difference • FD-EV, first difference with errors in variables • Emp, average empirical variance • GLS, generalized least squares • IB, incomplete block • Pre, average predicted variance • RCB, randomized complete block • REML, restricted maximum likelihood • SD, second difference • SD-EV, second difference with errors in variables

	INTRODUCTION

TOP ABSTRACT INTRODUCTION Materials and methods Results and discussion Conclusions REFERENCES

 
VALID DIFFERENTIATION between varieties is crucial in plant breeding programs. Such a differentiation must be based on accurate and precise estimation of variety effects, which depends largely on the control of spatial variation among plots. Spatial variation incorporates many factors, such as moisture, pH, and structure of soil and the pressure of weeds. It can be partitioned into two components: purely random noise and structured variation. All approaches towards improving precision of variety effect estimates seek to minimize the latter. These approaches fall into two categories: (i) block designs, including the randomized complete block (RCB) design, and incomplete block (IB) designs, the latter being mainly lattice and generalized lattice designs; and (ii) neighbor analyses, in which trends as well as the correlation between data of neighboring plots can be accounted for.

The precision of block analyses relies on the control of heterogeneity within blocks. Generally, the greater the heterogeneity within blocks, the poorer the precision of variety effect estimates. By nature, there is typically an appreciable correlation among data from neighboring plots, which is termed spatial autocorrelation. Recognizing this, neighbor analyses were proposed as an alternative to block analyses. Spatial autocorrelation can be taken into account by covariate analysis (Papadakis, 1937; Bartlett, 1978; Brownie et al., 1993) or by a trend-plus-error model or a first-difference model with errors in variables (Wilkinson et al., 1983; Green et al., 1985; Besag and Kempton, 1986; Williams, 1986; Gleeson and Cullis, 1987; Baird and Mead, 1991; Grondona and Cressie, 1991). Spatial structure may be in the mean of the variable of interest, and spatial heterogeneity in the mean of trend type (Dutilleul, 1993) can then be modeled by trend surface analysis (e.g., Tamura et al., 1988). All these methods differ in the model used to account for the trend and the assumptions required by the estimation techniques. Recently, considerable interest has been shown in neighbor analyses, because of the noticeable gains in precision that can be achieved in variety effect estimation by using neighbor analysis rather than IB analysis. In a study of 118 wheat variety trials, Kempton and Howes (1981) estimated that the iterated Papadakis procedure (Bartlett, 1978) increased the precision of variety mean differences by 8.7% compared with the IB analysis. In a study of 1019 variety trials over a range of crops, Cullis and Gleeson (1989) showed that the one-dimensional neighbor analysis resulted in a 42% reduction of the variances of varietal yield differences, compared with the RCB analysis, while the IB analysis resulted in a 33% reduction. In a simulation study, Brownie and Gumpertz (1997) showed that model-based methods accounting for spatial variation in terms of correlations between plot errors are robust with respect to validity when spatial autocorrelation is present. Spatial trends were modeled by trend surface analysis and the spatial covariance function used for estimating treatment effects was assumed to be the exponential function used in simulations. However, the neighbor analysis methods based on the differencing technique seem to have never been completely examined for validity and efficiency in comparison with block analyses, after such an examination had been initiated by Besag and Kempton (1986)(Appendix 2) and Baird and Mead (1991).

Because the primary objective of most field experiments is the accurate and precise estimation of treatment contrasts, validity in estimation of the error variance (unbiasedness) and efficiency (precision) should be thoroughly examined before any neighbor method can be recommended as a routine method for field experimentation. Validity in testing should also be assessed by comparing observed and theoretical significance levels. Through randomization, the classical analyses have been proven to be valid in estimation because the treatment mean square and the residual mean square have the same expectation in the absence of treatment effects, but there is no analytical proof that the neighbor methods are valid within such a randomization framework (Baird and Mead, 1991). However, the validity of neighbor methods can be examined empirically by using simulation based on uniformity data (Wilkinson et al., 1983; Besag and Kempton, 1986). Unless neighbor methods are shown to provide unbiased error variance, it is not reasonable to compare the efficiency of neighbor methods relative to block analyses on the basis of average standard errors of pairwise treatment differences.

The objectives of this study therefore are (i) to evaluate the validity and efficiency of neighbor analyses in comparison with classical complete and incomplete block analyses by simulation based on uniformity trials and (ii) to study the effects of spatial dependency, coefficient of variation, block size, plot size, and plot shape on the efficiency of neighbor analyses, using data sets obtained from soybean and wheat uniformity trials on Jiangpu Farm of Nanjing Agricultural University, Nanjing, China (Wu et al., 1995) and on the Aberdeen Substation, Aberdeen, ID (Wiebe, 1935), respectively. In a preliminary step, spatial structure (i.e., trends at large scale and autocorrelation at small scale) is identified in the field experiments by geostatistical analysis.

	Materials and methods

TOP ABSTRACT INTRODUCTION Materials and methods Results and discussion Conclusions REFERENCES

 
Uniformity Trials and Data Collection
Validity and efficiency were evaluated by simulation based on data sets generated from the soybean [Glycine max (L.) Merr.] and wheat (Triticum aestivum L.) uniformity trials described below. The experiment design (including the number of varieties and complete blocks, and the plot size and shape) and the coefficient of variation meet the practical requirements of field variety trials in plant breeding programs (Tables 1 and 2) .

The wheat uniformity trial was conducted on the Aberdeen Substation, Aberdeen, ID (Wiebe, 1935). It consisted of 1500 unit rows of `Federation' wheat (CItr 4734), 0.30 m apart and 4.57 m long. The 1500 unit rows were originally arranged in 12 columns of 125 unit rows each in the field. The crop was grown in the summer of 1927. Grain yield per unit row was determined after harvest (Table 1 in Wiebe, 1935).

Contour maps show that for both soybean and wheat trials, yield in the eastern part was more homogeneous than in the western part of the experiment field (Fig. 1) . The plots were rectangular in shape and were constructed by combining unit rows and columns of the original field layout. The different arrangements of columns and unit rows into plots resulted in a total of 50 data sets, corresponding to various combinations of plot size, plot shape, and number of dummy varieties and complete blocks (Tables 1 and 2).

View larger version (69K): [in this window] [in a new window]   Fig. 1 Contour maps of grain yield (g per unit row) in (left) soybean and (right) wheat uniformity trials. For soybean, the contour interval is 30 g; the lightest shade corresponds to a value of 40 g and the darkest shade, 310 g. For wheat, contour interval is 60 g; the extreme shades correspond to values of 375 and 915 g. The scale used is representative of the true physical size of each field. Source of data: Wu et al. (1995) for soybean and Wiebe (1935) for wheat

[1]

where N(h) is the number of pairs of observations (z_i, z_i+h) separated by distance h. It is expected that the difference z_i - z_i+h decreases as h decreases. In practice, as h approaches zero, (h) approaches a positive value C₀ called the nugget effect. A nugget effect equal to the sample variance indicates a purely random noise. This is one of the basic assumptions to be satisfied by the residuals of any classical ANOVA model. Conversely, when C₀ is less than the sample variance, the semivariogram indicates spatial structure. Thus, C₀ can be used as an indicator of the portion of spatial structure in the total variation. To compare the spatially structured portion of variation among data sets, we normalized the semivariance by dividing it by the sample variance; 1.0 minus the normalized semivariance also provided a spatial autocorrelation coefficient. The autocorrelation was assumed to be consistent over columns, so that for each data set the normalized semivariance estimate was computed by

[2]

where m and c are the number of columns and the number of plots per column, respectively; s² is the sample variance of z_i,j; and h' is the number of plots between z_i,j and z_i,j+h'. Then h, the distance between z_i,j and z_i,j+h', is equal to h' times the width of a plot, where h' = 1, 2, ..., H and m(c - H) > 50 because (h) is unreliable for N(h) < 50 (Journel and Huijbregts, 1978). In particular, lags 1 and 2 represent distances equal to one and two plot widths.

To evaluate the effect of blocking and differencing on the spatial structure, different z_i,j's were used in the semivariance analysis: (i) y_i,j - overall mean, (ii) y_i,j - complete block mean, (iii) y_i,j - incomplete block mean, (iv) y_i,j - y_i,j+1, and (v) y_i,j - 2y_i,j+1 + y_i,j+2, where y_i,j is the grain yield for plot j of row i in the uniformity trial. For z_i,j = y_i,j - incomplete block mean in (iii), spatial autocorrelation was assumed to be consistent over blocks, and the number of columns and number of plots per column in Eq. [2] were replaced by the number of blocks and number of plots per block, respectively. The geostatistical analysis was performed with our own computer programs written in the SAS/IML language (SAS Inst., 1989).

Criteria of Evaluation
The criteria of validity and efficiency used here follow Besag and Kempton (1986). For each method of analysis (see below), the average empirical variance (Emp) and the average predicted variance (Pre) of pairwise variety contrasts are defined as

[3]

and

[4]

where _a and _b represent the estimates of the effect of varieties a and b, and v is the number of dummy varieties. In a randomization framework, an analysis is said to provide valid estimates of the error variance if Emp Pre over a series of randomized layouts. When the value of Pre is substantially smaller (or larger) than that of Emp for a particular method, it suggests that the method underestimates (or overestimates) the variance of the estimated differences between varieties. Since there is no real variety effect in pseudo-experiments that originate from a uniformity trial where dummy varieties are randomly relabeled, the true theoretical difference between any pair of varieties is zero. Thus, Emp reflects the efficiency of variety effect estimates; the smaller the value of Emp, the higher the efficiency.

In hypothesis testing, a method is said to be valid at level if the probability that it detects a treatment difference, when in fact there is none, is less than or equal to . Since detection of significant variety effects here is based on F-testing, the valid method should provide a significance level of to the F-test; i.e., the frequency at which the F-statistic exceeds the tabulated critical value F_1- should be (at least approximately) equal to in the absence of real variety effects.

Statistical Methods of Data Analysis
Suppose that an experiment comprises m columns, with c plots each. Let the random variable y_i,j denote the yield of plot j in row i, and Y the corresponding n-vector with n = mc. The classical RCB ANOVA model (where the block factor is considered fixed) can then be written as

[5]

where µ is the grand mean and 1 is the n-vector of ones, is the v-vector of variety effects and D is the n x v variety design matrix, ß is the r-vector of block effects and B is the n x r block design matrix, and the _i,j's are assumed to be independent N deviates. The expected value of Y and its variance–covariance matrix are then given, respectively, by E = 1µ + D + Bß and Var = ² I_n , where I_n is the n x n identity matrix, and Pre = , where r is the number of complete blocks.

The equation for the IB ANOVA model with recovery of interblock information (where the block factor is considered random; see, for example, Yates, 1939; Cochran and Cox, 1957) is similar in writing to Eq. [5], except that E = 1µ + D and Var = ²_ß BB^T + ² I_n = V , where ²_ß denotes the variance component associated with the random block factor and the T superscript is the transpose matrix operator (Graybill, 1983); other notations are as in Eq. [5]. Variance components ²_ß and ² were estimated by restricted maximum likelihood (REML) (Patterson and Thompson, 1971). The generalized least-squares (GLS) estimate of , which takes into account the interblock information and optimally benefits from unbiasedness and minimum variance (Searle, 1971), is

[6]

where ²_ß and ² are replaced by their REML estimates in V and the -1 superscript is the inverse matrix operator (Graybill, 1983). Pre can then be computed by

[7]

where trace() and sum() denote the sum of diagonal elements and the sum of all elements of a matrix, respectively (Graybill, 1983).

In this study, we considered only one-dimensional spatial trends, which usually suffices in practice (Kempton, 1985; Besag and Kempton, 1986). The first-difference model with errors in variables (FD-EV) for the n-vector of yields Y defined above can be written as

[8]

where _(-v) is a (v-1)-vector of variety effects (note that the effect of the vth variety is derived from the sum of the v-1 others below) and D is the corresponding n x (v - 1) variety design matrix; is an n-vector of spatial trend effects whose first differences ₁ are assumed to be independent and identically distributed N, where ₁ is the first-difference operator; and is an n-vector of random errors assumed to be independent N. The second-difference model with errors in variables (SD-EV) can be written similarly, with the same notations and assumptions except that the second differences ₂ are assumed to be independent and identically distributed N, where ₂ is the second-difference operator. Operators ₁ and ₂ can be written in matrix notation as follows:

and

where is the Kronecker or direct product (Graybill, 1983).

Hereafter, the estimation methods are presented jointly for the first-difference and the second-difference models, using the notation _d (d = 1, 2). Assuming the difference _d is independent of the error (i.e., cov(_d, ) = 0 for d = 1, 2), and denoting _dY by W and _dD by X, we have E = X and Var = ² I_n + ² _d^T_d = V_d . Variance components ² and ² were estimated by REML and the vector of variety effects _(-v) was estimated by GLS as follows:

[9]

where ² and ² are replaced by their REML estimates in V_d, so the vth variety effect estimate is given by

[10]

The variance–covariance matrix of vector _(-v) is given by the upper left x submatrix of ^-1; the variance of _v is equal to the sum of all elements of this submatrix; and Cov(_a, _v) is equal to minus the sum of elements of the ath row of the same submatrix (a = 1, ..., v - 1). Thus,

[11]

To assess the goodness-of-fit of neighbor models for field experiments, the first-difference (FD) and second-difference (SD) models without errors in variables and the RCB ANOVA model with first-order autoregressive or AR(1) errors were also considered. The model of the former is

[12]

with E = X and Var = ² I_n . The computation of Pre is the same as for Eq. [11], with similar notations and assumptions (except ). Under the RCB ANOVA model with AR(1) errors, the n-vector Y satisfies Eq. [5] where the variance–covariance structure of the n-vector of errors is assumed to be first-order autoregressive; that is,

with

The variance component ² and the autocorrelation parameter are estimated by REML. The GLS estimate of and the computation of Pre are similar to those of the IB analysis (see Eq. [6] and [7]).

Simulation Study
For each of the 50 data sets (Table 1), the methods of analysis were applied to 50 pseudo-experiments, which were produced by randomly arranging dummy variety labels within blocks according to a lattice, generalized lattice or complete block design (Table 2). For each method, the sample means of Emp and Pre and the frequency at which the null hypothesis was falsely rejected (Type I error rate) were computed over the 50 pseudo-experiments. Pre is said to be significantly different from Emp if

where the variance s²_D is calculated on the 50 differences Emp - Pre. The efficiency of a given method relative to the classical RCB ANOVA was estimated by

[13]

Depending on the method, the computation of Emp, Pre and the observed significance level was performed with PROC MIXED of SAS version 6.10 (SAS Inst., 1995) in combination with our own computer programs written in the SAS/IML language (SAS Inst., 1989). PROC MIXED was used for the REML estimation of variance components for all methods except RCB ANOVA and for the estimation of treatment effects only for IB; we used PROC IML for the GLS estimation of treatment effects for the neighbor analyses. For the block analyses (RCB ANOVA and IB), the F-test was based on the variety mean square over error mean square ratio, while for the neighbor analyses it was given by the Emp against Pre ratio. Only for the RCB ANOVA are the two F-ratios equivalent.

	Results and discussion

TOP ABSTRACT INTRODUCTION Materials and methods Results and discussion Conclusions REFERENCES

 
Geostatistical Analysis
To assess the spatially structured variation of the residuals from block analyses, the normalized nugget effect NC₀ was estimated by linear regression using the three first points (h, *(h)) in the normalized semivariogram (Table 3) . The NC₀ values for IB(6) (incomplete block analysis with blocks of size 6) are not given in Table 3, as the semivariograms were then not smooth enough to allow reliable linear regression. NC₀ values less than unity or 100% indicate spatially structured variation; the smaller the NC₀, the stronger the spatial structure. The normalized nugget effects for yields (y) minus overall mean are less than those for yields minus CB mean and yields minus IB(10) mean. This shows that the block analyses, and especially the IB, may be useful, to some degree, for removing or taking into account the spatial structure. Given the coefficients of variation of Table 1, spatial structure tends to be weaker for data sets with lower heterogeneity (e.g., wheat B1, CV = 9.6%, and wheat B2, CV = 9.1%, for v = 60). With these two exceptions, only 4 to 16% of the sample variance is random in nature for y - CB mean, based on NC₀ (Table 3). Therefore, much of the spatially structured variation is not explained by the ANOVA. For most data sets, NC₀ is about 40% for y - IB(10) mean; despite the use of small blocks, structured variation thus still exists, so the precision could be further improved.

View this table: [in this window] [in a new window]   Table 3 Normalized nugget effects (NC0) for data (y) centered to the overall mean, to the means of complete blocks (CB), and to the means of incomplete blocks of size 10 [IB(10)] for data sets with v = 60.

View this table: [in this window] [in a new window]   Table 4 Empirical lag-1 autocorrelation coefficients (r1) for data (y) centered to the overall mean, to the means of complete blocks (CB) and to the means of incomplete blocks of size 10 [IB(10)] and size 6 [IB(6)] for data sets with v = 60.

View larger version (35K): [in this window] [in a new window]   Fig. 2 Normalized semivariograms for the residuals of the randomized complete block model (dashed line) and for the first differences (solid line) and second differences (dotted line) of the raw data, by using soybean data sets with v = 60 (no. of dummy varieties). See Tables 1 and 2 for the data set codes and characteristics

View larger version (37K): [in this window] [in a new window]   Fig. 3 Normalized semivariograms for the residuals of the randomized complete block model (dashed line) and for the first differences (solid line) and second differences (dotted line) of the raw data, by using wheat data sets with v = 60 (no. of dummy varieties). See Tables 1 and 2 for the data set codes and characteristics

View larger version (21K): [in this window] [in a new window]   Fig. 4 Spatial autocorrelations for the residuals of the randomized complete block model (dashed line) and for the first differences (solid line) and second differences (dotted line) of the raw data for (a) soybean A1, (b) soybean B5, (c) wheat B2, and (d) wheat A3. See Tables 1 and 2 for the data set codes and characteristics

View this table: [in this window] [in a new window]   Table 5 Sample means of average empirical variance (Emp) and average predicted variance (Pre) of pairwise variety contrasts computed over 50 pseudo-experiments from data sets with v = 60. Values reported are percentages relative to the sample means of Emp for the RCB ANOVA.

View this table: [in this window] [in a new window]   Table 6 Sample means of average empirical variance (Emp) and average predicted variance (Pre) of pairwise variety contrasts computed over 50 pseudo-experiments from data sets with v = 25. Values reported are percentages relative to the sample means of Emp for the RCB ANOVA.#

View this table: [in this window] [in a new window]   Table 7 Sample means of average empirical variance (Emp) and average predicted variance (Pre) of pairwise variety contrasts computed over 50 pseudo-experiments from data sets with v = 100. Values reported are percentages relative to the sample means of Emp for the RCB ANOVA.

View this table: [in this window] [in a new window]   Table 8 Observed significance levels for the F-test for variety effects, as computed over 50 pseudo-experiments from soybean data sets with v = 60 and for theoretical significance levels of 0.05 and 0.01.

View this table: [in this window] [in a new window]   Table 9 Empirical lag-1 autocorrelation coefficients (r1) for first (1) and second (2) differences, and empirical lag-2 autocorrelation coefficient (r2) for second difference for data sets with v = 60.

Efficiency Analysis
In pseudo-experiments, under the assumption of treatment additivity, a method of analysis is said to be more efficient than another if it has a smaller average empirical variance Emp for pairwise treatment contrasts (Baird and Mead, 1991, p. 1474; see also Besag and Kempton, 1986). The outstanding feature of Tables 5, 6, and 7 is the consistent reduction in Emp from the RCB ANOVA to the IB analysis and the neighbor analyses AR(1), FD, FD-EV, and SD-EV, in degrees that may vary with the number of dummy varieties (v). The neighbor analyses, especially the FD-EV and SD-EV, take the spatial autocorrelation adequately into account. Thus, they are more efficient than the IB analysis, even when there is no remaining autocorrelation in incomplete blocks of size 6 for data sets such as soybean A1, A2, and A3 (Table 4). This indicates neighbor analyses not only account for autocorrelation in errors, but also fit the spatial trend. Four of the five neighbor analyses (i.e., AR(1), FD, FD-EV, and SD-EV) provided approximately the same Emp values (Tables 5 and 6). Evidently, they removed a similar amount of structured variation from the error term. Because the difference in efficiency among these four neighbor analyses was slight, only the FD-EV was retained for discussion below. On the other hand, the SD analysis was not studied further after its poor performance with respect to validity.

The efficiency of the neighbor and IB analyses relative to the RCB ANOVA is influenced by several factors, among which is the CV from the RCB ANOVA. According to Fig. 5 , which shows the relationships between CV and NC₀ and the RE for FD-EV and IB(6), relative efficiency tends to be higher for data with higher CV. This shows that precision can be improved substantially by using small blocks or neighbor analysis in case of strong heterogeneity among plots within blocks. For example, for soybean A1 with v = 60 and CV = 15.7%, Emp is equal to 19.1% for the FD-EV analysis and 24.8% for IB(6), which corresponds to RE values of 524 and 403%, respectively. On the other hand, since low CV indicates relatively weak heterogeneity within blocks, sizeable improvement of precision is not likely to arise from data with lower CV. For example, for wheat A1 with v = 25 and CV = 6.2%, RE is equal to 142 and 114% for FD-EV and IB, respectively. Furthermore, relative efficiency tends to increase with decreasing NC₀—that is, with increasing portion of spatially structured variation explained by the method. Moreover, the efficiency for data sets in Group A is higher than that for the corresponding data sets in Group B. This is due to the higher variation among plots and the higher portion of spatially structured variation in Group A (Fig. 1). In view of Fig. 5, the relationship between RE and NC₀ is stronger than between RE and CV for FD-EV, and the relationship between RE and NC₀ is stronger for FD-EV than for IB. This suggests that NC₀ is more reliable than CV as an indicator of efficiency for the neighbor analyses.

View larger version (21K): [in this window] [in a new window]   Fig. 5 Relative efficiency in variety effect estimation in relation to the CV and the normalized nugget effect for (a) soybean (v = 60) and the FD-EV model, (b) wheat (v = 25 and 60) and the FD-EV model, (c) soybean (v = 60) and the IB(6) model, and (d) wheat (v = 25 and 60) and the IB(6) or IB(5) model

Another factor that may affect efficiency is plot shape, the ratio of length to width. To address this point, we have calculated the relative efficiency of FD-EV and IB(6) for three pairs of data sets with the same plot size but different plot shapes, namely data sets 2 and 3, 4 and 5, and 6 and 7 from the western (Group A) and eastern (Group B) parts of the soybean uniformity trial. For both one-dimensional analyses, efficiency is higher for the data set with narrower plots in Group A. For example, soybean A3 presents an RE of 401% for FD-EV, whereas the same method provides an RE of 526% for soybean A2. In contrast, both types of plot shape have similar efficiency for data sets in Group B. Since there is stronger spatial structure in Group A characterized by higher CV (Table 1) and lower NC₀ (Table 2), it can be concluded that when the spatial structure is strong (i.e., NC₀ < 10%), the neighbor and IB analyses with narrower plots are more effective than those with wider plots in fitting trends and accounting for autocorrelation. Based on Tables 5, 6, and 7, where plot size increases from data set A1(B1) to A8(B8) for soybean and from data set A1(B1) to A4(B4) for wheat, there is no apparent relationship between plot size and RE. Finally, from Tables 1, 3, 5, 6, and 7, it does not seem necessary to use plots larger than 8 m² for soybean and larger than 4 m² for wheat to reduce the error variance and ensure good precision in variety effect estimation.

	Conclusions

TOP ABSTRACT INTRODUCTION Materials and methods Results and discussion Conclusions REFERENCES

 
Based on the foregoing, the following steps are useful for obtaining high precision in the analysis of field experiments, especially in trials with a large number of varieties or treatments.

1. Select the experiment design based on field soil variation. If the heterogeneity is relatively high (i.e., high CV in previous trials), an incomplete block design with highly rectangular plots should be used instead of a complete block design. However, blocking is not required if the spatial structure cannot be predicted or the field productivity is known to be relatively homogeneous.
   
2. Analyze the data using classical block analysis, when applicable, and the neighbor FD-EV method, where variance components are estimated by REML and treatment effects by GLS.
   
3. Compare the efficiency of the different methods. If the efficiency of any (complete or incomplete) block analysis is similar to that of the neighbor FD-EV, results from the block analysis will be more appropriate. Otherwise, if the efficiency of FD-EV is higher, proceed to the next step.
   
4. Test the goodness-of-fit of the FD-EV model. If the lag-1 autocorrelation coefficient of first-difference deviates is negative and significant but the lag-2 autocorrelation coefficient is not significant, the FD-EV model is appropriate. Otherwise, results from the FD-EV analysis may not be appropriate (in particular, the Type I error rate may be exaggerated in significance testing), and we suggest that classical block analyses be used, or other models be tried. The SD-EV analysis will not be necessarily more appropriate, as REML does not always converge for this method.
   

Geostatistical analysis showed that there was a spatial structure in all of the 50 data sets derived from the two uniformity trials. First and second differencing were revealed to be powerful tools to fit large-scale trends and to remove small-scale autocorrelation from errors, and were more effective than incomplete blocking in these respects. In particular, geostatistical analysis of differencing deviates helped explicate the effect of differencing on spatial structure. Incomplete blocking was generally more effective in reducing the unexplained structured variation in comparison with complete blocking. The first-difference method with errors in variables was identified as empirically valid, and the second-difference model with errors in variables was also valid for most of the data sets. Three other neighbor methods (AR(1), FD and SD) underestimated the error variance, leading to inflated rejection rate in hypothesis testing of variety effects compared with the theoretical significance level. The lag-1 and lag-2 autocorrelation coefficients can be used to assess the goodness-of-fit of the difference models. Four of the neighbor methods (AR(1), FD, FD-EV, and SD-EV) were more efficient than complete and incomplete block analyses. For most data sets, the efficiency of neighbor analyses was higher than that of RCB ANOVA and IB by 200 and 30%, respectively. The relative efficiency tended to increase with decreasing normalized nugget effect and, to a lesser degree, with increasing CV.

When spatially structured variation is strong and one-dimensional, highly rectangular plots must be preferred to near-square plots in FD-EV and IB analyses. In terms of validity and efficiency, the first-difference model with errors in variables, in which variance components and treatment effects are estimated respectively by REML and GLS, is the best method. On that basis, FD-EV can be expected to become a routine method for the analysis of field experiments with a large number of varieties or treatments. The computer program implementing the FD-EV method is available from the corresponding author upon request.SAS Institute 1989; SAS Institute 1995

	ACKNOWLEDGMENTS

 
During her scientific visit at McGill University, the senior author was financially supported through the NSERC and FCAR Nouveaux Chercheurs operating grants of the second author and the NSERC operating grant of Dr. Diane Mather (Plant Science). The authors are grateful to three anonymous reviewers whose comments helped improve the presentation of this paper.

	REFERENCES

TOP ABSTRACT INTRODUCTION Materials and methods Results and discussion Conclusions REFERENCES

 

• Baird D., Mead R. The empirical efficiency and validity of two neighbour models. Biometrics 1991;47:1473-1487.
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