Field variation in a large field trial could cause a significant issue in statistical analysis and conclusions. Therefore, spatial variation control has become a crucial step to screen large-scale plant genotypes in a large field trial. Conventional blocking designs (like randomized complete block, split plots, Latin square, and incomplete block designs, etc.) [1] have been commonly used for field variation control since Fisher proposed his “The Arrangement of Field Experiments” [2] . Fisher’s experimental design principles (1926) provide some measure of protection against spatial variation, many statisticians agree that additional layers of spatial control are beneficial [3] - [5] . Interestingly, though the spatial analysis was proposed by Papadakis just 11 years after Fisher’s blocking design [6] , its applications to field trial analysis have been unpopular compared to Fisher’s blocking designs. A possible reason is the difficulty of integrating spatial analysis into the conventional field trial data process. Therefore, investigating spatial variation control methods featured with efficiency and integrability will be helpful to enhance field trial analysis.

Fisher’s blocking design (1926) is ANOVA (analysis of variance) based thus simplicity is one of the major features. Under the principle of a linear model, Fisher’s blocking design can be extended to multi-dimension or hierarchical blocking designs. Examples include Latin square and alpha-lattice designs [1] [7] . An efficient blocking design should be assumed that within-block variability is minimized while between-block variability being maximized [8] . Because uniformity within a large block is unlikely in most field trials, conventional blocking designs would be rarely satisfactory. On the other hand, it is more likely that plot-to-plot variation within a block may be affected by competition between genotypes with the trial [9] - [11] by soil variation and fertility and previous land use [12] [13] .

Removing spatial variation from a field trial has been a crucial goal for many statisticians [14] . Over 80 years ago Papadakis [6] believed that traditional blocking in field experiments might not adequately represent soil fertility patterns and he instead proposed adjusting the yield of each plot by the performance of the neighboring plots. According to Bartlett [15] , the adjustment method suggested by Papadakis (1937) was approximately valid and sometimes useful in experiments where a block size is large. Unfortunately, Papadakis’s method did not receive as serious attention as Fisher’s blocking designs until the 1970’s [16] - [18] . The nearest neighbor adjustment (NNA) method was claimed to be a relatively simple yet effective method for accounting for within-block variation [18] [19] . Later, a random field approach [20] was proposed as an improvement on the NNA methods. The key point of using the NNA methods is to develop covariates to adjust the original data [21] . Stroup et al. (1994) compared RCB analysis with the NNA methods in a multi-environment winter-wheat trial data and concluded that the NNA approach was better as evidenced by a lower coefficient of variation and greater ability to distinguish the differences among cultivars. A recent study showed that modified Papadakis covariates incorporating spatial positions could enhance the adjustment efficiency [21] . However, such a modified covariate may require more information that many data lack and thus it could reduce the application of this method.

Above mentioned covariates are based on the residuals obtained from a particular experimental design (equivalent to a particular statistical model). In other words, the calculation of these covariates is model-dependent. Recently, a moving grid adjustment (MGA) method was proposed by Technow [22] and a covariate is also required to adjust field spatial patterns. Compared to Papadakis’s covariate, the MGA covariate has two noticeable differences. The first one is that the MGA covariate is based on phenotypic data rather than residuals, implying that the MGA adjustment can stand alone. The second one is that each MGA covariate has the flexibility to catch different directions of spatial patterns. In addition, the corresponding R package for the MGA method mvngGrAd is publicly available [22] . Numerical evaluation regarding the adjustment efficiency via Monte Carlo simulations will reveal more statistical properties under the conventional RCB design.

Therefore, there were two objectives to be targeted in this study. Our first objective was to numerically validate the efficacy of the MGA methods via simulated data generated from a commonly used RCB design with one-way and two-way spatial patterns. Correlation coefficients between adjusted and control data were used to determine the efficiency of the MGA method. The second objective was to apply the MGA method to analyze a cotton trial data set with emphasis on cotton yield (sensitive to field spatial pattern) and lint percentage (non-sensitive to field spatial pattern). The main purpose of this study is to provide a useful method to enhance breeding data analysis when spatial patterns are significant.

As addressed in Section 2, we define $y_0$ representing phenotypic data without a spatial pattern thus $y_0$ can be considered as the control group; $y_1$ representing phenotypic data with a spatial pattern for each of the four settings ( Table 1 ). Both $a{y}_0$ and $a{y}_1$ are the adjusted data for $y_0$ and $y_1$ by the MGA method, respectively. For simplification, we only report three correlation coefficients as $r_{01}$, $a{r}_{01}$, and $aa{r}_{01}$, where, $r_{01}$ between $y_0$ and $y_1$ indicates the impact of spatial patterns on phenotypic data; $a{r}_{01}$ is between $y_0$ and $a{y}_1$ indicating the adjustment efficiency of the MGA method on the data with spatial patterns; $aa{r}_{01}$ is between $a{y}_0$ and $a{y}_1$ indicating the similarity between adjusted data from data with and without a spatial pattern. The correlation coefficients are summarized (minimum, maximum, mean values) from 100 simulations for each of the four settings and are provided in Tables 2 - 5 , separately.

The high correlation coefficients, $r_{01}$, $a{r}_{01}$, and $aa{r}_{01}$, showed that the data for both $y_0$ and $y_1$ and $a{y}_0$ and $a{y}_1$ are identical or nearly identical when there

^†: $y_0$ = simulated data without spatial pattern (control); $y_1$ = simulated data with spatial pattern; $a{y}_0$ = adjusted data for $y_0$; and $a{y}_1$ = adjusted data for $y_1$. ^‡: $r_{01}$ = correlation between $y_0$ and $y_1$; $aa{r}_{01}$ = correlation between $a{y}_0$ and $a{y}_1$; $a{r}_{01}$ = correlation between $y_0$ and $a{y}_1$.

^†: $y_0$ = simulated data without spatial pattern (control); $y_1$ = simulated data with spatial pattern; $a{y}_0$ = adjusted data for $y_0$; and $a{y}_1$ = adjusted data for $y_1$. ^‡: $r_{01}$ = correlation between $y_0$ and $y_1$; $aa{r}_{01}$ = correlation between $a{y}_0$ and $a{y}_1$; $a{r}_{01}$ = correlation between $y_0$ and $a{y}_1$.

^†: $y_0$ = simulated data without spatial pattern (control); $y_1$ = simulated data with spatial pattern; $a{y}_0$ = adjusted data for $y_0$; and $a{y}_1$ = adjusted data for $y_1$. ^‡: $r_{01}$ = correlation between $y_0$ and $y_1$; $aa{r}_{01}$ = correlation between $a{y}_0$ and $a{y}_1$; $a{r}_{01}$ = correlation between $y_0$ and $a{y}_1$.

^†: $y_0$ = simulated data without spatial pattern (control); $y_1$ = simulated data with spatial pattern; $a{y}_0$ = adjusted data for $y_0$; and $a{y}_1$ = adjusted data for $y_1$. ^‡: $r_{01}$ = correlation between $y_0$ and $y_1$; $aa{r}_{01}$ = correlation between $a{y}_0$ and $a{y}_1$; $a{r}_{01}$ = correlation between $y_0$ and $a{y}_1$.

is no spatial pattern (sp = 0) (setting 1, Table 2 ). When a spatial pattern exists in both directions (setting 2), the mean correlation coefficient ( $r_{01}$) between $y_0$ and $y_1$ was 0.779 while the mean correlation coefficients between $a{y}_0$ and $a{y}_1$ ( $aa{r}_{01}$) or $a{y}_1$ and $y_0$ ( $a{r}_{01}$) were 0.969 and 0.975, respectively ( Table 3 ). These two correlation coefficients were significantly higher than that between $y_0$ and $y_1$. Similar results can be observed for settings 3 and 4 ( Table 4 and Table 5 ).

Figures 1 - 3 showed the heatmaps for one set of simulated data of $y_0$, sp, $y_1$,

and $a{y}_1$ for settings 2 to 4, respectively. These three figures showed that the heatmap for the adjusted data $a{y}_1$ is very similar to that for the control data $y_0$. The heatmaps from these three different settings are consistent with the correlation analysis. Therefore, according to the simulation results ( Tables 2 - 5 and Figures 1 - 3 ), we can conclude that 1) the MGA method can effectively remove field spatial variation if it does exist and 2) the MGA method will not change phenotype results if field spatial variation does not exist.

It has been a common practice that large-scare entries are evaluated for either early screening of breeding lines or phenotyping for association mapping study. However, field spatial variation is a common phenomenon in a field trial, especially for a large field trial. Such field spatial variation could greatly impact the conclusions such as inflation/deflation of differences among entries being evaluated. Field spatial variation for a large trial normally results in inflated residual variance and also can be visualized via heatmap. Effectively removing field variation is highly needed and can be indicated by reduced variance and/or depatterned heatmap for residuals.

There are two major ways to control field spatial variation. One major way is to apply various experimental designs such as conventional block designs and augmented experimental designs [1] [32] - [34] . If field conditions are known prior to a field trial being executed, appropriate blocking designs would help control field variation well for a small block size. However, the assumption of within-block uniformity is easily violated when block size is large due to many uncontrollable factors [8] , including the row-column incomplete block designs Among various blocking designs, row-column designs (sometimes called rectangular design) [35] - [37] were more efficient than the conventional RCB designs for different trials and experiment sizes [38] . With our limited number of field trial data processing, we also noticed that the rectangular provided equally well or better field variation control compared to the other blocking designs [39] - [41] . Without spatial analysis, the rectangular design is a recommended option over the conventional RCB designs. The other major approach is to apply spatial statistical analysis which is NNA (near-neighbor adjustment) based. Though NNA based spatial analysis was initially proposed in 1937 (Papadis 1937), the application of NNA based method has been very few compared to Fisher’s based blocking designs.

The efficiency of the MGA method was numerically evaluated by Monte Carlo simulation. The simulation results showed that the MGA method can be used to effectively remove a field spatial pattern in one-way or two-way direction as evidenced by the heatmaps ( Figures 1 - 3 ) and high coefficient of correlation between control data and adjusted data ( Tables 2 - 5 ). The application of MGA method to an actual cotton trial data also showed the similar conclusion ( Figure 4 & Figure 5 ; Table 6 & Table 7 ). Our results also suggested that the MGA method can effectively remove block effects ( Table 6 ). In one of our previous studies, we applied several augmented designs to analyze the same lint yield and lint percentage data. We found that the rectangular blocking design (row-column) was better than other blocking designs for lint yield because both row and column variations were significantly captured. Similar results were found among different experimental designs for lint percentage because no significant row or column effects were detected [39] .

The estimated genotype and residual variance components for lint yield from the adjusted data by the MGA method in this study ( Table 6 ) were compared with the estimated variance components subject to the rectangular (row-column) model reported in one of our previous studies [39] ( Table 4 ). Interestingly, the results were comparable between two studies (15,421 vs 16,400 and 21,482 vs 20,400, equivalent to 6.0% and 5.3% in difference for genotype and residual variance components, respectively). The results suggest that both the MGA method and the row-column augmented blocking design can effectively remove the two-way field spatial variation and thus they both function equally well regarding this data. However, it will be interesting to compare these two methods with more complicated field spatial patterns.

Data adjustment by the MGA can stand alone. Therefore, this adjustment method has a great flexibility to integrate with other statistical analyses including various genetic model analyses and association mapping study [28] [42] - [44] . Many crop trials are repeated over locations or years. It could be statistically challenging to analyze multi-environment data with a GGE (genotype and genotype × environment interaction) model simultaneously when field spatial patterns vary among environments. However, spatial analysis model is usually fitted for each environment [45] - [48] to produce a spatially adjusted genotype means for each environment. With application of the MGA method and other spatial analysis methods, the spatially adjusted data are then combined for the second stage for an across-environment analysis [49] [50] .

In conclusion, both our simulation study and actual data analysis showed that the MGA method has a desirable adjustment efficiency when a field spatial pattern exists in single or both directions. In addition, because the R package for the MGA method has been publicly available [22] , it will be flexible to integrate with other statistical methods under the RStudio platform [30] for more complicated statistical analysis. Therefore, this method can be a great addition to enhance field trial data analyses with potential field variation.

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