Partial collection of data on potato yield for experimental planning

Field Crops Research 121 (2011) 286–290

Contents lists available at ScienceDirect

Field Crops Research journal homepage: www.elsevier.com/locate/fcr

Partial collection of data on potato yield for experimental planning Lindolfo Storck ∗ Department of Crop, Federal University of Santa Maria (UFSM), 97105-900 Santa Maria, RS, Brazil

a r t i c l e

i n f o

Article history: Received 16 August 2010 Received in revised form 16 December 2010 Accepted 18 December 2010 Keywords: Solanum tuberosum Experiment planning Plot size Sampled experimental area Bootstrap resampling

a b s t r a c t The aim of this study was to estimate the number of blank experiments (BE) (i.e., a uniformity trial) required to estimate the optimum plot size for use in experiments involving potato crops. The study was based on data on the mass of potato tubers (Solanum tuberosum L.) from 3456 hills (i.e., 24 rows of 144 hills each) obtained from a BE. Using these data, BE of different sizes (i.e., 2 rows of 24, 36, 48 and 72 hills) were planned to estimate optimum plot size. For each BE, 11 plot sizes (X) were planned based on the sum of adjacent hills, and the mean, variance and coefficient of variation (CV) between plots of the same size were calculated. Regression models for CV were adjusted in terms of X to estimate the optimum plot size. For each BE size, a bootstrap resampling method was used to estimate the sufficient number of BE to enable precise estimates of optimum plot size, mean and other statistics. It was found that a sampling potato hill yield of 39% of subdivisions within an experimental area where a potato experiment is to be performed is sufficient to estimate optimum plot size for the experiment. Plots composed of one row of six hills are sufficient to estimate potato yield. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Knowing the variability of agricultural crop yields in an experimental area is important in order to plan experiments effectively. This planning entails determining, for a given number of treatments, the experimental design, plot size and number of replications (Gomez and Gomez, 1984; Steel et al., 1997; Ramalho et al., 2000). When attempting to control the experimental error generated by the heterogeneity of experimental units, researchers must dedicate special attention to certain aspects of estimation. So, it is necessary to perform blank experiments (i.e., a uniformity trial), with the aim of selecting appropriate block design, plot size and shape, the number of replications and treatments, and the level of precision for the chosen experimental area. The best combination for plot size and the number of replications and treatments are the main methods used to address the heterogeneity of crop yield in order to maximize the information obtained from the experimental area (Storck et al., 2006). To understand variability in an experimental area, blank experiments may be performed that are specifically designed to identify the variability or level of heterogeneity for a given characteristic. In this methodology, data are collected on the yields of small plots called “basic units” (BU), and these BU are used to plan plots of different sizes, according to the grouping together adjacent BU.

∗ Tel.: +55 55 3220 8451; fax: +55 55 3220 8899. E-mail address: [email protected] 0378-4290/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.fcr.2010.12.018

For each plot size, the number of plots of a given size together with means, variances and coefficients of variation are calculated. Based on the relationship between basic unit variances and planned plot size, the level of heterogeneity is estimated (Smith, 1938). Meier and Lessman (1971) showed that the optimum plot size can be obtained as the point of maximum curvature in the relationship between the coefficients of variation and the respective planned plot sizes. However, the use of blank experiments for an entire experimental area during planning is time-consuming and expensive. As a result, data obtained from experiments showing the effects of treatments in plots that have been subdivided into block, plot and subplot size or arranged in a lattice design may be used (Koch and Rigney, 1951) to make unbiased estimates of the level of heterogeneity (Hatheway and Williams, 1958). Lin and Binns (1984) discussed the possibility of estimating the level of heterogeneity in an experimental area based on intraclass correlation for experiments using a randomized complete block design in which there are two plot sizes (i.e., blocks and plots). Applications for experiments involving common bean (Phaseolus vulgaris L.) cultivars (Storck et al., 2007) and corn (Zea mays L.) cultivars have also been discussed (Carvalho et al., 2009). Estimates of the level of heterogeneity based on the intraclass coefficient of correlation are not as precise as those obtained from blank experiments using Smith’s (1938) method (Lin and Binns, 1984, 1986) or those obtained from experiments using lattice and subdivided plot arrangements (Koch and Rigney, 1951). However, due to the ease of obtaining estimates of the level of heterogeneity and the coefficients of variation with respect to a large number of experiments over the course of time and in different environments without

L. Storck / Field Crops Research 121 (2011) 286–290

287

24 very large BE (2 x 72 hills)

… or 36 large BE (2 x 48 hills)

… or 48 medium BE (2 x 36 hills)

… or 72 small BE (2 x 24 hills)

Fig. 1. Different sizes of blank experiments (BE) planned in the same experimental area.

additional costs. So, experimental precision may be monitored and improved using other methodologies, such as that proposed by Hatheway (1961), by varying the number of replications, treatments and plot size. For some species cultivated in largely independent rows, such as corn (Zea mays L.), sorghum (Sorghum bicolor), sunflower (Helianthus annuus L.), cassava (Manihot esculenta) and potato (Solanum tuberosum), the possibility of using data from certain rows or groups of rows of a given length taken randomly from across the entire experimental area is a feasible alternative according to Lorentz et al. (2010). From this sample of data representing part of the experimental area, estimates may be obtained of the parameters necessary for experimental planning without loss of precision and at a lower cost in terms of financial and human resources. As such, the aim of this work was to estimate the number of blank experiments required to estimate optimum plot size for use in experiments involving potato crops.

large BE, Ne = 36 for large BE, Ne = 48 for medium BE and Ne = 72 for small BE) for each BE size, i.e., N = 3456/(X × Ne). For each of the 180 BE and for each plot size X, the following information was determined: the variance (V(x)) between plots of size X; the variance per BU (VU(X)), calculated between plots of size X as VU(x) = V(x)/X2 ; the coefficient of variation (CV(x)) between plots of X BU size; the mean (M(x)) of plots of size X; and the mean (M1 ) of plots of one BU. Based on these calculations, for each of the 180 BE, the level of yield heterogeneity (b) was estimated using the relationship identified by Smith (1938), that is, VU(x) = V1 /Xb . Parameters V1 and b were estimated based on the logarithmic transformation of the function and weighted by degrees of freedom (DF = N − 1) (Steel et al., 1997). The coefficient of determination (R2 ) was also estimated. Similarly, estimates were made of parameters A and B for the function CV(x) = A/XB , which was used to calculate the optimum plot size (X0 ) following the modified maximum curvature method (Meier and Lessman, 1971), with

2. Materials and methods

2.3. A sufficient sample size for bootstrapping

2.1. Experimental data

For each BE size, the Ne estimates of the M1 , V1 , b, A, B, R2 and X0 statistics were used to estimate the sufficient number of BE or the adequate fraction of the experimental area with a total of 3456 hills. Considering that these statistics generally do not follow a known probability distribution, a bootstrap resampling methodology (Efron, 1979; Ferreira, 2005; Confalonieri et al., 2006) was adopted. Following this method, Ne observations for each statistic were used to generate 2000 samples with replacement (i.e., resampling), and the mean was calculated for each resample. These 2000 mean figures were ordered to identify the minimum (Min) and maximum (Max) values. The Min and Max values were considered to be an estimate for a confidence interval with a zero error rate. The same procedure involving 2000 samples with replacement values was performed for different sample sizes, that is, BE number (k = 3, 4, . . ., Ne − 1). For each k value, the 2000 mean figures were ordered to identify the 0.025 quantile as the lower limit (LL(k)) and the 0.975 quantile as the upper limit (UL(k)). The LL(k) and UL(k) were used as estimates for a confidence interval with an error rate of 5% for Ne = k. The BE number was considered sufficient when LL(k) > Min and UL(k) < Max with an error rate of no more than 5%. This was used to determine the lowest value of k sufficient (k ) to sample the population of Ne blank experiments. Accordingly, for each BE size and statistic evaluated, the sufficient fraction of the experimental area to be evaluated was p(k ) = k /Ne. To make the calculations, an Excel spreadsheet was used together with the SAEG (2007) application and an application created in Pascal language for bootstrap simulation.

This study was based on data on the mass of potato tubers (Solanum tuberosum L.) (g hill−1 ) obtained from a blank experiment involving the Macaca cultivar that was conducted in the potato-growing area at the State Agricultural Research Foundation (Fundac¸ão Estadual de Pesquisa Agropecuária) in Júlio de Castilhos, Rio Grande do Sul, Brazil (29◦ 12 S, 53◦ 41 W; altitude 490 m). The area received uniform crop management and phytosanitary treatment (Pereira and Daniels, 2003). In all, 3456 hills were cultivated and distributed into 24 rows (width) of 144 hills (length). The rows were spaced 0.80 m apart, with 0.30 m space between hills. The crops were harvested and evaluated separately for all hills, which were the basic units (BU). 2.2. Sizes of plot and blank experiments Using these data, plans were made for 72 small blank experiments (BE) (i.e., 2 rows of 24 hills), 48 medium BE (i.e., 2 rows of 36 hills), 36 large BE (i.e., 2 rows of 48 hills) and 24 very large BE (i.e., 2 rows of 72 hills), all using a total number of 3456 hills in the experimental area (Fig. 1). For each of the 180 BE (i.e., 72 small BE, 48 medium BE, 36 large BE and 24 very large BE), 11 plots were proposed, varying the number of rows (X1) and the number of adjacent hills (X2) with the size of a plot (X) was calculated according to X = X1 × X2. The proposed plots (X1;X2) were (1;1), (1;2), (1;3), (1;4), (1;6), (1;8), (1;12), (2;4), (2;6), (2;8) and (2;12). The number of replications (N) for each X was limited in accordance with the number of BE (Ne; Ne = 24 for very

X0 = [A2 B2 (2B + 1)/(B + 2)]

1/(2B+2)

.

288

L. Storck / Field Crops Research 121 (2011) 286–290

Table 1 Mean M1 potato tuber mass in g hill−1 , estimates of VU(x) = V1 /Xb and CV(x) = A/XB , coefficient of determination (R2 ) and optimum plot size (X0 ) obtained for 24 blank experiments (BE) of very large size (i.e., 2 rows of 72 hills). BE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Mean SD CV k LL UL

M1 704.4 703.5 678.3 913.9 679.1 703.8 821.5 747.4 770.8 733.6 868.4 862.1 899.8 821.6 975.2 969.6 1065.3 1132.9 812.0 805.5 903.2 1017.4 871.4 857.6 846.6 123.8 14.6 9 770.5 930.8

V1 103,633 123,332 80,119 98,554 67,581 96,605 50,487 123,982 76,832 85,268 37,791 64,209 41,459 44,175 59,899 123,877 52,088 67,581 59,369 87,453 38,685 54,281 65,922 137,474 76,694a 29,639 38.6 9 58,908 96,147

b

A

B

0.494 0.418 0.674 0.511 0.812 0.808 0.910 0.721 0.636 0.376 0.667 0.536 0.804 0.992 1.067 0.752 0.804 0.994 1.021 0.541 1.042 0.448 0.961 0.352

45.7 49.9 41.7 34.3 38.3 44.2 27.3 47.1 35.9 39.8 22.4 29.4 22.6 25.6 25.1 36.3 21.4 22.9 30.0 36.7 21.8 22.9 29.4 43.2

0.247 0.209 0.337 0.255 0.406 0.404 0.455 0.360 0.318 0.188 0.333 0.268 0.402 0.496 0.533 0.376 0.402 0.497 0.510 0.271 0.521 0.224 0.480 0.176

0.723 0.226 31.3 10 0.591 0.860

33.1 9.2 27.7 10 27.8 38.9

0.361 0.113 31.3 9 0.293 0.436

R2

X0

0.95 0.95 0.98 0.96 0.83 0.64 0.95 0.60 0.96 0.97 0.93 0.76 0.97 0.98 0.93 0.95 0.97 0.98 0.97 0.94 0.98 0.98 0.98 0.90 0.92a 0.10 11.4 11 0.85 0.97

5.93 5.79 6.38 4.81 6.36 7.04 5.19 7.14 5.57 4.48 3.98 4.37 4.36 5.07 5.08 5.98 4.20 4.72 5.67 5.23 4.61 3.19 5.53 4.59 5.22 0.96 18.5 9 4.64 5.88

Mean, standard deviation (SD), coefficient of variation (CV), estimate of sufficient number of BE (k ) and lower and upper confidence interval limits for k BE (LL and UL , respectively) with p = 0.95. a Variable with non-normal distribution (5%).

3. Results and discussion The coefficients of variation (CV) for the 7 statistics (M1 , V1 , b, A, B, R2 and X0 ) for the 24 BE of very large size (i.e., 2 rows of 72 hills) are heterogeneous (Table 1), indicating that these variables do not have the same probability distribution. By the Lilliefors test

and using the SAEG (2007) application, only M1 , b, A, B and X0 statistics follow a normal probability distribution (p > 0.05). Considering the 7 statistics, the sufficient number k of blank experiments for a very large BE varies between 9 and 11. However, under the normal distribution, the ideal sample size of 10%, according to estimates obtained using the SAEG (2007) application, is equal to 9, 64, 42,

Table 2 Mean, minimum value (Min), maximum value (Max), and lower and upper confidence interval limits (LL and UL, respectively) for p = 0.95 obtained from a simulation of 2000 resamples of size k and sufficient (yes/no) number of blank experiments (BE) using results for optimum plot size (X0 ) for 24 BE with 2 rows of 72 hills. k

Mean

Min

Max

LL

UL

Sufficient

2 3 4 5 6 7 8 9 = k 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

5.22 5.23 5.21 5.21 5.21 5.21 5.22 5.23 5.22 5.24 5.22 5.22 5.22 5.22 5.23 5.23 5.22 5.22 5.22 5.22 5.22 5.21 5.22

3.19 3.19 3.74 3.91 3.86 3.84 3.76 4.14 4.20 4.25 4.20 4.38 4.35 4.20 4.47 4.48 4.55 4.43 4.58 4.52 4.61 4.61 4.59 = Min

7.14 7.07 6.80 6.68 6.66 6.56 6.42 6.44 6.28 6.17 6.10 6.16 6.12 6.18 6.02 6.01 6.04 6.13 6.00 5.96 5.98 5.87 5.99 = Max

3.98 4.22 4.29 4.40 4.49 4.50 4.58 4.64 = LL 4.63 4.67 4.70 4.71 4.74 4.75 4.76 4.78 4.80 4.79 4.81 4.83 4.83 4.83 4.84

6.54 6.27 6.13 6.08 5.96 5.92 5.87 5.88 = UL 5.83 5.80 5.77 5.76 5.73 5.69 5.70 5.68 5.65 5.63 5.64 5.64 5.61 5.60 5.58

No No No No No No No Yesa Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes

a

For the optimum number of BE (k = 9), we present the confidence interval LL and UL .

L. Storck / Field Crops Research 121 (2011) 286–290

289

Table 3 Mean, coefficient of variation (CV), estimate of the sufficient number of blank experiments (k ) and upper and lower confidence interval limits at a 0.95 confidence level (LL and UL , respectively) for k BE for potato tuber mass (M1 in g hill−1 ). Size and number (Ne) of BE

M1 Mean CV k p(k )a LL UL V1 Mean CV k p(k ) LL UL b Mean CV k p(k ) LL UL A Mean CV k p(k ) LL UL B Mean CV k p(k ) LL UL R2 Mean CV k p(k ) LL UL X0 Mean CV k p(k ) LL UL

Mean

Very large (Ne = 24)

Large (Ne = 36)

Medium (Ne = 48)

Small (Ne = 72)

846.6 14.6 9 0.38 770.5 930.8

846.6 16.3 14 0.39 775.8 915.2

846.6 16.9 17 0.35 777.1 912.9

846.6 17.4 29 0.40 793.2 899.5

129,151 103.3 9 0.38 58,908 96,147

74,175 43.5 15 0.42 59,371 91,026

71,907 47.4 21 0.44 58,933 87,548

75,455 54.7 31 0.43 62,439 90,235

87,672 62.2

0.844 56.2 10 0.42 0.591 0.860

0.821 30.9 14 0.39 0.700 0.957

0.768 29.7 25 0.52 0.681 0.862

0.933 34.4 22 0.31 0.806 1.064

0.842 37.8

39.9 51.5 10 0.42 27.87 38.88

32.76 33.2 12 0.33 27.43 39.15

32.18 33.8 17 0.35 27.58 37.48

32.98 38.8 30 0.42 28.68 37.48

34.46 39.3

0.422 56.2 9 0.38 0.293 0.436

0.411 30.9 17 0.47 0.356 0.470

0.376 30.3 17 0.35 0.324 0.429

0.466 34.4 32 0.44 0.412 0.522

0.419 40.0

0.80 12.6 11 0.46 0.85 0.97

0.90 13.0 15 0.42 0.83 0.95

0.74 17.6 18 0.38 0.68 0.79

0.83 18.0 28 0.39 0.77 0.88

0.82 15.3

5.57 40.4 9 0.38 4.64 5.88

5.43 23.2 15 0.42 4.87 6.12

5.22 25.4 17 0.35 4.64 5.88

5.56 27.5 29 0.40 5.00 6.13

5.45 29.1

846.6 16.3 0.38

0.42

0.41

0.38

0.41

0.41

0.39

Estimates of parameters for VU(x) = V1 /Xb and CV(x) = A/XB , coefficient of determination (R2 ) and optimum plot size (X0 ) for different sizes of blank experiments (BE). a p(k ) = k /Ne = proportion of blank experiments.

32, 42, 6 and 14 BE, respectively, for M1 , V1 , b, A, B, R2 and X0 . This divergence in results, which is due to differences in probability distributions, is sufficient to justify the use of bootstrap resampling methodology for point and interval estimates as well as to estimate the sufficient number of BE. With resampling, estimates for each interval are independent of each variable’s probability distribution (Efron, 1979; Ferreira, 2005). This methodology has been used in studies to determine sufficient sample size in the context of identifying rice plant and soil characteristics (Confalonieri et al., 2009). The results from optimizing plot size (X0 , Table 1) were use to determine BE (Table 2). The use of k = 9 BE is sufficient to estimate X0 , as it meets the criterion that LL > Min and UL < Max. Consequently, in an experimental area subdivided into 24 BE composed of 2 rows of 72 hills, only 9 of these BE (38%) must be sampled in order to estimate the optimum plot size for experiments to be conducted in this area at a 95% confidence level.

Varying the size and number of BE, estimates for the mean, coefficient of variation, optimum number of blank experiments (k ) and confidence interval limits at a 95% confidence level for k for M1 , V1 , b, A, B, R2 and X0 are presented in Table 3. No variation in M1 in relation to the size of BE was observed, given that the same measuring unit was used (g hill−1 ). The values for the CV of M1 varied little in relation to BE size. However, the sufficient number of blank experiments k increased as the size of BE decreased, thus maintaining similar proportions of the areas to be evaluated p(k ) similar. As a result, in terms of the amount of work involved in collecting results, sampling 9 very large BE (p(k ) = 0.38) is equivalent to sampling 29 small BE (p(k ) = 0.40). In terms of M1 , the size of the confidence interval (UL − LL ) for k BE was found to decline as BE size fell. The same behavior was not observed for X0 . The mean for statistics A and B, used to express the estimate of X0 , did not rise as BE size fell, and they had very little influence on the magnitude of X0 . Consequently, an

290

L. Storck / Field Crops Research 121 (2011) 286–290

interpretation of the estimate of the yield heterogeneity rate (b), which is equivalent to B = b/2, is not related with the size of BE. However, as the optimum plot size used in experiment planning varied little with changes in BE size, the estimated value of X0 for any size of BE can in fact be considered reliable. Optimum plot size varied between 5.22 and 5.57 hills (both round up to 6 hills), regardless of the size or number of blank experiments. In order to estimate optimum plot size, it was necessary to evaluate around 39% of hills in the experimental area based on blank experiments of sizes ranging from small (i.e., 2 rows of 24 hills) to very large (i.e., 2 rows of 72 hills). The possibility of using a sufficient number of small BE is promising for the study of the effect of crop treatments or management techniques on plot size. Accordingly, Martin et al. (2005) implemented a BE by using grain yields from 4 plots (i.e., 8 rows of 12 m) for each corn genotype. They concluded that there are differences between genotypes in terms of optimum plot size. A similar study, which used 16 plots (i.e., 4 rows of 6m) for each corn genotype and employed the Jackknife resampling method (Confalonieri et al., 2007), concluded that sampling 12.5% of the experimental area is sufficient to estimate plot size and yield for corn grain (Storck et al., 2010). Based on an experiment involving different densities and spatial arrangements of grain sorghum, Lopes et al. (2005) used plots of 12 BU to conclude the following. First, higher numbers of plants per row do not lead to increased grain yields, although they do result in higher quality experiments. Second, the estimated optimum plot size depends on the number of plants used in the BU. Third, spacing between rows does not influence the estimates of optimum plot size. Lorentz et al. (2010) used a sample of 12 blocks (with 1 block equal to 2 rows of 48 BU) in a much larger experimental area (1 ha) for 2 consecutive harvests of sunflower, subject to variations in weather, management and plant density. As there was little variation in X0 among the 24 blocks, it was considered that the use of a sample of 12 blocks (12 BE) is sufficient for the majority of environments in which sunflower is grown. However, the number of blocks necessary or sufficient to estimate X0 was not identified. Also, Confalonieri et al. (2006) analyzed the influence of experimental factors (namely, artificially induced variability) on rice sample size. The obtained sample sizes were compared to investigate the influence of each factor, while keeping the others constant. Their results highlight the influence of experimental factors and the stage of development on within-plot variability and, therefore, the importance of preliminary sampling for determining sample size. In summary, a sampling potato hill yield of 39% of the subdivisions within the experimental area in which a potato experiment is to be performed is sufficient to plan the experiment in terms of optimum plot size. Plots composed of one row of six hills are sufficient to estimate potato yield.

Acknowledgements The author thanks the National Council for Scientific and Technological Development (CNPq) for its financial support. References Carvalho, M.P., Lopes, S.J., Cargnelutti Filho, A., Storck, L., 2009. Variation in heterogeneity of the experimental area and experimental plan for corn genotype evaluation. Maydica 54, 39–45. Confalonieri, R., Stroppiana, D., Boschetti, M., Gusberti, D., Bocchi, S., Acutis, M., 2006. Analysis of rice sample size variability due to development stage, nitrogen fertilization, sowing technique and variety using the visual jackknife. Field Crops Res. 97, 135–141. Confalonieri, R., Acutis, M., Bellocchi, G., Genovese, G., 2007. Resampling-based software for estimating optimal sample size. Environ. Model. Softw. 22, 1796–1800. Confalonieri, R., Perego, A., Chiodini, M.E., Scaglia, B., Rosenmund, A.S., Acutis, M., 2009. Analysis of sample size for variables related to plant, soil, and soil microbial respiration in a paddy rice field. Field Crops Res. 113, 125–130. Efron, B., 1979. Bootstrap method: another look at the jackknife. Ann. Stat. 7, 1–26. Ferreira, D.F., 2005. Estatística Básica. UFLA, Lavras. Gomez, K.A., Gomez, A.A., 1984. Statistical Procedures for Agricultural Research, 2nd ed. John Wiley, New York. Hatheway, W.H., 1961. Convenient plot size. Agron. J. 53, 279–280. Hatheway, W.H., Williams, E.J., 1958. Efficient estimation of the relationship between plot size and the variability of crop yields. Biometrics 14, 207–222. Koch, E.J., Rigney, H.J., 1951. A method of estimating optimum plot size from experimental data. Agron. J. 43, 17–21. Lin, C.S., Binns, M.R., 1984. Working rules for determining the plot size and numbers of plots per block in field experiments. J. Agric. Sci. 103, 11–15. Lin, C.S., Binns, M.R., 1986. Relative efficiency of two randomized block designs having different plot size and numbers of replications and of plots per block. Agron. J. 78, 531–534. Lopes, S.J., Storck, L., Lúcio, A.D., Lorentz, L.H., Lovato, C., Dias, V.O., 2005. Tamanho de parcela para produtividade de grãos de sorgo granífero em diferentes densidades de plantas. Pesq. Agropec. Bras. 40, 525–530. Lorentz, L.H., Boligon, A.A., Storck, L., Lúcio, A.D., 2010. Plot size and precision in sunflower experiments. Sci. Agric. 67, 408–413. Martin, T.N., Storck, L., Lúcio, A.D., Carvalho, M.P., Santos, P.M., 2005. Bases genéticas de milho e alterac¸ões no plano experimental. Pesq. Agropec. Bras. 40, 35–40. Meier, V.D., Lessman, K.J., 1971. Estimation of optimum field plot shape and size testing yield in Crambe abyssinica hordnt. Crop Sci. 11, 648–650. Pereira, A.S., Daniels, J., 2003. O Cultivo da Batata na Região Sul do Brasil. Embrapa Informac¸ão Tecnológica, Brasília. Ramalho, M.A.P., Ferreira, D.F., Oliveira, A.C., 2000. A Experimentac¸ão em Genética e Melhoramento de Plantas. EDUFLA, Lavras. SAEG-Sistema para Análises Estatísticas, 2007. Versão 9.1, Fundac¸ão Arthur Bernardes, Vic¸osa. Smith, H.F., 1938. An empirical law describing heterogeneity in the yields of agricultural crops. J. Agric. Sci. 28, 1–23. Steel, R.G.D., Torrie, J.H., Dickey, D.A., 1997. Principles and Procedures of Statistics: A Biometrical Approach, 3rd ed. McGraw Hill Book, New York. Storck, L., Bisognin, D.A., Oliveira, S.J.R., 2006. Dimensões dos ensaios e estimativas do tamanho ótimo de parcela em batata. Pesq. Agropec. Bras. 41, 903–909. Storck, L., Cargnelutti Filho, A., Lopes, S.J., Toebe, M., Silveira, T.R., 2010. Experimental plan for single, double and triple hybrid corn. Maydica 55, 27–32. Storck, L., Ribeiro, N.D., Lopes, S.J., Cargnelutti Filho, A., Carvalho, M.P., Jost, E., 2007. Persistência do plano experimental em ensaios de avaliac¸ão de germoplasma elite de feijão. Ciência Rural 37, 549–1553.