SAMPLE SIZE AND LINEAR RELATIONS IN TRAITS OF CUT AND GRAZING SORGHUM

– The objectives of this work were to determine the sample size (number of plants) needed to estimate the mean of cut and grazing sorghum ( Sorghum bicolor (L.) Moench), Nutribem cultivar, traits and investigate the linear relations among traits. At 86 days after sowing, 110 plants of sorghum were selected at random. The traits evaluated for each plant were: plant height, stem diameter, number of nodes, number of leaves, leaf fresh matter, stem fresh matter, shoot fresh matter, leaf dry matter, stem dry matter, and shoot dry matter. The sample size was determined to estimate the means of the traits, assuming estimation errors equal to 1% (higher precision), 2%, ..., and 20% (lower precision) of the mean. Scatter plots, correlation analysis, and path analysis investigated the relationship among traits. Fourteen plants were needed to estimate the means of plant height, stem diameter, number of nodes, and number of cut and grazing sorghum leaves, with a maximum error of 10% of the mean and a 95% confidence level. With the same precision, to estimate the means of leaf, stem, and shoot fresh and dry matter, 48 plants are needed. Plant height positively correlates with stem and shoots fresh and dry matter. The number of leaves has a positive linear relation with leaf fresh and dry matter.


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In experiments with crops, such as sorghum (Sorghum bicolor (L.) Moench), in its different types, classified according to their purpose of use (grain, silage, sweet, biomass, broom, and cut and grazing), when evaluating a trait it is common to find variability among plants, even among those subjected to the same treatment. Thus, it is essential to size the number of plants that must be evaluated to obtain reliable information on the trait under evaluation.
In field experiments, it is common to measure several plant traits, which may or may not be linearly correlated. For example, in the case of sorghum (Sorghum bicolor (L.) Moench) cultivars used for cutting and grazing, such as the Nutribem cultivar, more significant amounts of fresh and dry matter are desired. Quantifying fresh and dry matter requires destructive sampling, so it is essential to know their relations with other traits, such as plant height, stem diameter, number of nodes, and number of leaves. Once correlations are proven, it is possible to estimate fresh and dry matter yields without needing to harvest plants.
The interrelations among traits can be visualized in scatter plots and investigated using Pearson's linear correlation coefficient (r) and path analysis. The coefficient r measures the intensity or degree of the linear relation between two random variables. The r sign expresses the association direction, while the intensity is represented by a numerical value between -1 and 1. In extreme situations, two traits may have perfect negative (r = -1) or perfect positive (r = 1) linear correlation, or lack of linear relation (r = 0) (Ferreira, 2009;Bussab & Morettin, 2017). Path analysis makes it possible to partition the correlation coefficients into direct and indirect effects of explanatory variables on the main variable and identify if there is a linear association between cause and effect (Cruz et al., 2012(Cruz et al., , 2014Cruz, 2016).
Correlation and path analysis has been used in sorghum (Lombardi et al., 2015;Vendruscolo et al., 2016;Ceccon et al., 2017;Silva et al., 2017;Mengesha et al., 2019;Oliveira et al., 2021) and black oat (Cargnelutti Filho et al., 2015a). These studies have found critical linear associations among the traits used in plant breeding programs. Therefore, it is assumed that the inclusion of studies on sample sizing and association among sorghum traits aggregates essential information to support the planning of experiments with better precision and, consequently, with more excellent reliability in the results.
Thus, the objectives of this work were to determine the sample size (number of plants) needed to estimate the mean of cut and grazing sorghum, Nutribem cultivar, and traits and investigate the linear relations among traits.

Material and Methods
A uniformity trial (blank experiment) was conducted with cut and grazing sorghum, Nutribem cultivar, in Santa Maria, State of Rio Grande do Sul, Brazil (coordinates 29º42'S, 53º49'W, and 95 m 3 altitudes), in an area of 8 m × 20 m (160 m²). The climate of this site is humid subtropical -Cfa, according to the Köppen-Geiger classification, with hot summers and no dry season (Alvares et al., 2013). The soil is Argissolo Vermelho Distrófico Arênico (Ultisol) (Santos et al., 2018). On October 28, 2020, 35 kg ha -1 of N, 135 kg ha -1 of P 2 O 5 , and 135 kg ha -1 of K 2 O were incorporated into the soil, and sowing was carried out the broadcast, using 15 kg of seeds ha -1 , aiming at 450,000 plants ha -1 .
In the crop's flowering (reproductive stage), that is, at 86 days after sowing, a pilot sample of 110 plants was randomly harvested. The plants were separated into two parts, leaves and stem + inflorescence, considered in this study as leaves and stem, respectively, because For each trait, based on the 110 plants, the sample size (n) was determined for estimation errors (semi-amplitudes of the confidence interval) fixed at 1%, 2%, ..., 20% of the mean (m), that is, 0.01×m (higher precision), 0.02×m, ..., 0.20×m (lower precision), with a 95% confidence level (1-α) by the expression (Ferreira, 2009;Bussab & Morettin, 2017). The t α/2 is the critical value of the Student's t-distribution, whose area on the right is equal to α/2, that is, the value of t, such that P(t>t α/2 )=α/2, with α=5% probability of error and n-1 degrees of freedom among the traits was determined, and Student's t-test was performed at a 5% significance level.
In the correlation matrix among the traits PH, SD, NN, and NL, multicollinearity was diagnosed and interpreted based on the condition number (CN), which is the ratio between the highest and lowest eigenvalue of the correlation matrix. Multicollinearity was classified as weak when CN ≤ 100, moderate to severe when 100 < CN < 1000, and severe when CN ≥ 1000 (Cruz et al., 2014;Cruz, 2016).
After that, path analysis of the main variables (LFM, STFM, SHFM, LDM, STDM, and SHDM) as a function of the explanatory variables (PH, SD, NN, and NL) was performed according to the methodology described in Cruz et al. (2012Cruz et al. ( , 2014, totaling six path analysis. Finally, statistical analyses were conducted using Microsoft Office Excel ® and Genes (Cruz, 2016 grams. These results reveal that, for the same precision, the sample sizes vary among the traits, as observed in sorghum (Silva et al., 2005), maize (Toebe et al., 2014;Wartha et al., 2016), black oat (Cargnelutti Filho et al., 2015b), millet (Kleinpaul et al., 2017) and rye (Bandeira et al., 2018(Bandeira et al., , 2019. It is also possible to infer that it is difficult to obtain estimates of the mean with this higher (1) PH -plant height, in cm; SD -stem diameter, in cm; NN -number of nodes; NL -number of leaves; LFM -leaf fresh matter, in g plant -1 ; STFM -stem fresh matter, in g plant -1 ; SHFM -shoot fresh matter (SHFM = LFM + STFM), in g plant -1 ; LDM -leaf dry matter, in g plant -1 ; STDM -stem dry matter, in g plant -1 ; and SHDM -shoot dry matter (SHDM = LDM + STDM), in g plant -1 .
(3) * Kurtosis differs from three, through the Student's t-test, at a 5% significance level.  When opting for allowing an estimation error of 20%, that is, 0.20×m (lower precision, in this study), and 95% confidence level, the number of plants to be sampled decreases to 2, 4, 2, 4, 12, 11, 10, 12, 12 and 11, respectively,   If the option were to sample 110 plants, as used in this study, the estimation error would be 2. 34, 3.40, 2.45, 3.56, 6.59, 6.08, 5.99, 6.42, 6.37, and 6.13% of the mean estimate (m), respectively, for the traits PH, SD, NN, NL, LFM, STFM, SHFM, LDM, STDM, and SHDM, that is, the maximum estimation error would be 6.59% (Table 2) Table   3), which suggests that the number of leaves would be more strongly associated with the leaf, stem and shoot fresh and dry matter of sorghum. Therefore, path analysis is an appropriate complementary procedure to infer the genuine cause-effect relationships among the traits (Cruz et al., 2012(Cruz et al., , 2014. Furthermore, this analysis makes it   (1) PH -plant height, in cm; SD -stem diameter, in cm; NN -number of nodes; NL -number of leaves; LFM -leaf fresh matter, in g plant -1 ; STFM -stem fresh matter, in g plant -1 ; SHFM -shoot fresh matter (SHFM = LFM + STFM), in g plant -1 ; LDM -leaf dry matter, in g plant -1 ; STDM -stem dry matter, in g plant -1 ; and SHDM -shoot dry matter (SHDM = LDM + STDM), in g plant -1 . * Significant at a 5% error probability by Student's t-test, with 108 degrees of freedom.