Statistical Analysis for Correlated Paired-plot Designs

This work is aimed to evaluate different statistical analysis with a randomized complete block design for paired-plot studies of herbicide selectivity in sugarcane experiments. Two procedures were considered: i) the construction of a t-test to assess the hypothesis of the paired-plot mean difference to be zero in each treatment; or ii) the use of an analysis of variance where the treated and paired plots were considered in a split-plot model and the interaction is sliced by treatment. By simulation with normal bivariate distributions, with uniform (zero, 0.11, 0.33, 0.44, 0.67 or 0.89) or Short Research Article Perecin et al.; AJEA, 9(6): 1-7, 2015; Article no.AJEA.20722 2 heterogeneous correlations, the two procedures showed similar performance. The power of the tests increases as the correlation of paired-plot increases.


INTRODUCTION
Classic experiments are conducted in various areas that allow assessing responses in paired comparisons. For example, observing an attribute of the sampled plot before and after a treatment or observing responses to similar parts treated and untreated. The performance of paired comparisons can be more interesting and more realistic to assess the ordering of treatments (or processes of interest) than the simple ordering of responses directly [1][2].
The example discussed in this study was motivated by assessments studies on selectivity of herbicides in sugarcane crops, but it can also be applied to other studies on pests, diseases, weeds, etc. It is also applicable when measurements of the attribute, whether visual or mechanical, need to be calibrated by non-treated plot, preferentially to the side, or with repeated readings after a reference measurement.
For the specific case that motivated this research, consider evaluating the selectivity of five treatments (herbicides) in a sugarcane cultivar and the use of four replications in a randomized complete block design. The allocation can be made with five plots per block, each divided into two plots, applying herbicide in half and leaving the other half as paired control [3] or each block can be formed with 10 plots with allocation in pairs, applying herbicide in one and leaving the other as paired control. Azania et al. [4] showed an analysis where paired-plots were allocated as a supplementary treatment called control. If this method were applied to the example in question, each of the five treatments would have four replicates and control (treatment 6) with 20 replications. The proposed analysis compares the design with the classic case of just one control per plot, but it does not consider the paired comparison that actually exists. There are also other parametric or non-parametric forms to analyze paired-plot experiments [1][2].
This study discussed two forms of statistical analysis for these designs, as well as the effects of different correlations between paired comparisons.

Simple Analysis of Variance per Condition
For an example with five pairs of plots in each of four replications in a randomized complete block design, the analysis of variance scheme, per condition, is that presented in Table 1. A multiple comparisons test can be used with significant F-value (p < 0.05) [5]. Among the alternatives, probably the most interesting for biological fields is the usual t test (LSD = least significance difference), whose minimal significant difference is given by LSD = t x (Mean Square of Error/rep) 1/2 . In this case, we have t = 2.1788 (tabled value, Degree of Freedom (DF) = 12, p<0.05) and rep = 4 replications. When there are many treatments, an interesting alternative to control false significance would be False Discovery Rate (FDR) procedures [6] rather than conservative procedures such as Tukey [7] or Scott and Knott [8].

Analysis of Mean Difference between Treated and Paired Plots
Working with differences (T-P), the analysis in Table 1 allows to accept whether the means of differences (T-P) in each treatment are equal or, using a test for multiple comparisons, which are different.
In practice, however, the interest can be a test for the hypothesis of the mean differences (T-P) in each treatment be zero. Rejecting this hypothesis indicates the treatment effect. The test statistics is for each pairing: TC = dif(T-P)/ [t x (Mean Square of Error/rep) 1/2 ]. In bilateral manner, we have t = 2.1788 (tabled value, p = 0.05) and rep = 4 (replications). Using this t value in the previous formula, we can have the minimum value significant to differ from zero (in absolute value): dif (T-P) = 0 = LSD (0) = | t x (Mean Square of Error/rep) 1/2 |.
The LSD of the previous item, as well as LSD (0) of this item may present low level of significance set per experiment [5], especially for many treatments, but it is good not to greatly increase type II error (fail to detect differences when they exist), according to [9].

Analysis of Variance Using the Splitplot Model
For this analysis (Table 2), each pair of plots (treated and paired) are considered experimental plots.
In this case, F2 statistics (Table 2) tests the overall effect of the pairing, that is, if the overall mean of the treated plots is equal to the control mean, and F3 statistics ( Table 2) tests the interaction treatments x pairing. If it is significant, it is interesting to evaluate the pairing by treatment (Table 3), which is another test for the difference equal to zero in each pairing.

Comparison between Statistics of Items 2.2 and 2.3
There are statistical differences in the construction of the tests. In the TC case (item 2.2), we use the differences, pair-to-pair, and the respective residual variance of these differences.
In the split-plot design, responses of treated and non-treated are used, building up contrasts between pairing means. There are differences in residual variances, depending on the correlation or covariance between the paired-plots and associated DF (12, in example in Table 1) of TC statistics of item 2.2 and 15 (  (2) heterogeneous correlations (different combinations from the above) according to replications (in this case, blocks) of each of the treatments.
In the first situation, to estimate the power of the test (ability to detect differences) using the means of treatments (treated and paired, in Table 4), uniform standard deviation 4.5 for both treated and paired, compatible covariances with stipulated correlations (Cases 1 and 2, described in the previous paragraph) and fixed effects of blocks (-10, -5, 5, 10) are used. In a second situation, to estimate the rate of type I error (rejecting equalities, when they exist), the treated and paired were equal (using A = 130, B = 110, C = 120, D = 120, E = 120), effects of covariance and blocks, as in previous cases.  Table  3) statistics, p-values were obtained the (probability of rejection of no difference between treated and paired). The simulation study was performed in the computing environment R, version 3.1.1 [10]. It was used in the R environment, besides the basic distribution packages, the package "MASS" [11] to generate bivariate normal distributions by means of the function "mvrnorm". Further information on the use of the R environment can be found at Matloff [12].

RESULTS AND DISCUSSION
For one of the simulations of responses for sugarcane production (tons per hectare), the results could be those presented in Table 4. Treatment A increases the mean response of 15.00 (ton. ha -1 ), treatment B decreases the same 15.00 (ton. ha -1 ), C and D decrease 9.00 (ton. ha -1 ) and 10.50 (ton. ha -1 ) and E increases only 1.50 (ton. ha -1 ), that is, a situation with different effects of treatments is simulated to evaluate the power of the test.

Simple Analysis of Variance per Condition
With this analysis, we observe among the treated (T), treatment A shows the greatest response. Paired control (P) are equal (p > 0.05) and the differences (T-P) show treatment A with the highest positive difference, differing from treatment E, which differed from other three showing negative differences and are similar (same letter c).

Analysis of the Mean Difference between Treated and Paired Plots
For a hypothesis test of difference (T-P) to be zero, with the data of the example, we have:

Analysis of Variance Using the Splitplot
For this analysis, the plots of each pair (treated and paired) are considered experimental plots. F3 statistics, calculated as shown in Table 2, indicates that the significance of pairing x treatment interaction and pairing per treatment was evaluated (

Comparison between Statistics of Items 2.2 and 2.3
To summarize the simulation results, Tables 6  and 7 show the means and standard deviations of p-values of TC (item 2.2) and F (item 2.3, Table 3) statistics, as well as the frequencies of p-values that exceed 0.05 (rejection of no treatment effect).
In the case of differences (T-P), the power of the test is similar for the two procedures, in all treatments (Table 6). In treatment E, with the smallest difference (T-P), the difference increases with increasing correlation between paired plots, in other words, the increase of this correlation makes detection of the paired treatment effect easier. The same fact also occurs for the other treatments with greater difference (T-P), however, it is not observed in Table 6, because the power of the test proved to be virtually 100%. In the case of heterogeneous covariance, the power of the tests was similar to that of the covariance mean (or correlation). In summary, for all cases, frequencies of hypothesis rejection of zero differences are similar in both procedures, TC (item 2.2) and F (item 2.3, Table 3) statistics. The presence of positive correlation between paired plots may occur in practice, mainly through the spatial proximity of adjacent plots, and it makes detection of the paired treatment effect easier, mainly if the difference (T-P) is smaller, which is desirable for the experimental standpoint.
With respect to type I errors (Table 7), the two procedures are similar with means (M) of p-values around expected (0.5) and the same occurs for rejection frequency of the hypothesis of no difference between means of treated and paired plots (around 50 in 1000). The standard deviations (SD) of p-values are similar to the two statistics, in Tables 6 and 7.

CONCLUSION
The two procedures show similar performance for the evaluation of experiments with paired plots: i) a t test to test whether, in each treatment, the means of paired differences can be zero; or ii) an analysis of variance in which treated and paired plots are considered in a splitplot model and the unfolding of the interaction for treatment is done. With increasing correlation between paired plots, the power of tests to detect differences actually existing increases.