INTRODUCTION:
Garlic (Allium sativum L.) is a vegetable of great importance due to its economical, culinary, nutritional, and medicinal values, presenting varieties with great potential for commerce and for industrialization (^{TRANI, 2009}). Several varieties of garlic exist in Brazil, generated by somatic mutations and selections of desirable characteristics for its handling in agriculture (^{SOUZA & MACÊDO, 2009}), necessitating studies regarding its culture and development.
Among these, studies that evaluate the growth trajectory of plants are indispensable for accomplishing the appropriate handling of the plant, because they aid in the preparation of techniques for cultivation, crop, conservation and detection of problems in the development of cultures (^{SOUZA & MACÊDO, 2009}). Nonlinear regression models are appropriate for describing growth curves because their formulations are based on inherent theoretical considerations of the phenomenon under study (^{MAZUCHELI & ACHCAR, 2002}). Nonlinear models have been used in several studies about dry matter accumulation and growth in different plant cultures, like onion (^{PÔRTO et al., 2007}), banana trees (^{MAIA et al., 2009}), cassava (^{SILVA et al., 2014}), and garlic (^{REIS et al., 2014}), which reported that the Logistic model showed better performance.
When studying plant growth curves, there is usually interest in differentiating subsamples with larger or smaller production levels, when adjusting the model. One alternative to describe the relationship between predictor variables at different levels of the distribution of the response variable, without the need of adjustments in subsamples, is the use of quantile regression. This method, different from others that use conditional means (E(YX)) to obtain the functional relationship among variables, uses conditional quantiles (Q(YX)), adjusting the relationship between independent variables and quantiles (percentiles) of the dependent variable. With quantile regression, it is possible to obtain more information, such as in the inferior or superior tails, and it generates more robust models even in the presence of outliers and heterogeneity of errors (^{KOENKER, 2005}; ^{HAO & NAIMAN, 2007}).
Some researchers studied curves of plant growth using quantile regression models, such as ^{MUGGEO et al. (2013}), who proposed a structure for quantile regression using bsplines to estimate growth curves for Posidonia oceanica seagrass, and ^{SORRELL et al. (2012}), who used quantile regression and nonlinear regression to evaluate the growth of three species of plants of humid areas in response to the depth of the water. However, studies regarding nonlinear quantile regression in dry matter accumulation have not been considered in the literature.
The objective of this research was to adjust nonlinear quantile regression models for the study of dry matter accumulation in garlic plants over time, and to classify the garlic accessions based on their growth rate and asymptotic weight.
MATERIALS AND METHODS:
The experiment was carried out in an experimental area belonging to the Plant Science Department of the Universidade Federal de Viçosa (UFV), in the Zona da Mata region of Minas Gerais, Brazil, with geographical coordinates: 20º45’S and 42º51’W, at an altitude of 650m. Thirty garlic accessions were evaluated in the period from March to November.
The experimental units consisted of four longitudinal rows of 1m length, with a planting space of 0.25x0.10m, with a total of 40 plants, from which the plants of the two central rows were considered as useful. The total dry matter of the plant (TDMP), expressed in grams by plant, was evaluated in four periods: the initial period (60 days after planting (DAP)), second period (90 DAP), third period (120 DAP), and the final period (150 DAP). Descriptive statistics of the data are presented in table 1.
DAP  TDMP  SD  Min  Max 
60  0.9730  0.3000  0.4464  1.6037 
90  4.4448  0.9570  2.8650  6.7250 
120  17.6591  3.5927  11.8000  28.7500 
150  22.5479  4.6964  13.0250  36.1250 
The nonlinear regression model used was the Logistic model. Averages of each accession were used for the adjustment. The Logistic model is defined as y _{ i } = β_{ 1 } [1+ β_{ 2 } exp (β_{ 3 } x _{ i } )] ^{ 1 } + е_{ i } , where: y _{ i } is the i^{th} observation of the response variable, that is, the total dry matter of the plant (TDMP) expressed in grams and, considered to be the average of the accession in the considered period; x _{ i } is the predictor variable, which represents the periods of the dry matter evaluation (in DAP); β_{ 1 } is the parameter that represents the asymptotic weight of the accession; β_{ 2 } a location parameter with no biological interpretation; and β_{ 3 } is the maturity rate (growth rate) of the accession. For the random error, the following distribution is assumed: e _{ i } ~N(0, σ^{ 2 } _{ e } ).
The nonlinear quantile regression y _{ i } = β_{ 1 } (τ)[1+ β_{ 2 } (τ) exp(β_{ 3 } (τ)X _{ i } )]1 + e _{ 1 } (τ) was adjusted at the quantiles τ =0.25, τ =0.5 and τ =0.75, where τ refers to the assumed quantile (τ ( [0,1]). This model was adjusted by an Interior Point Algorithm, proposed by ^{KOENKER & PARK (1996}), which has the purpose of computing estimates of quantile regression for cases in which the response function is nonlinear in the parameters. A model with the method of ordinary least squares, using GaussNewton’s iterative process, was also adjusted, in order to compare it to the QR model.
The likelihood ratio test was applied to the estimated parameters at the three different quantiles of the quantile regression model (τ = 0.25, τ = 0.5, and τ = 0.75) in order to test whether significant difference exists among these quantiles. For this purpose, a variation of the likelihood ratio test was used (based on the chisquared distribution), as proposed by ^{KOENKER & MACHADO (1999}). This test is based on the L_{1} regression, which minimizes the absolute value of the sum of the deviations, and differs from the traditional least squares method, in which the square of this sum is minimized. The considered hypotheses were:
H _{ 0 } ^{ (1) } : β_{ 1 } (τ = 0.25 = β_{ 1 } (τ = 0.5)= β_{ 1 } (τ = 0.75) vs H _{ 1 } ^{ (1) }
not all β _{ 1 } are equal;
H _{ 0 } ^{ (2) } : β_{ 2 } (τ = 0.25 = β_{ 2 } (τ = 0.5)= β_{ 2 } (τ = 0.75) vs H _{ 1 } ^{ (2) }
not all β2 are equal;
H _{ 0 } ^{ (3) } : β_{ 3 } (τ = 0.25 = β_{ 3 } (τ = 0.5)= β_{ 3 } (τ = 0.75) vs H _{ 1 } ^{ (3) }
not all β3 are equal.
Once the models are adjusted, it is desirable to classify the accessions according to their different growth patterns: accessions with different weights and growth rates possess a varied nutritional demand, which makes it more advantageous to distinguish their treatments (^{SOUZA & MACÊDO, 2009}). The distances between the observed values of each accession and the predicted values at each one of the three quantiles of the quantile regression model were calculated. Euclidean distance was used as the dissimilarity measure, calculated as
The analyses were implemented in the statistical software R, version 3.2.1 (^{R DEVELOPMENT CORE TEAM, 2017}). For the adjustment of the nonlinear model with the least squares method, the function nls was used. Adjustment of the nonlinear quantile model was accomplished with the function nlrq of the quantreg package (^{KOENKER, 2016}).
RESULTS AND DISCUSSION:
Logistic models were adjusted using the methodology of quantile regression at the quantiles τ = 0.25, τ = 0.5, and τ = 0.75, and a model was also adjusted using the ordinary least squares method for comparative ends (Figure 1). All the curves presented a welldefined sigmoid shape, characteristic in growth curves of plants (^{PÔRTO et al., 2007}; ^{MAIA et al., 2009}; ^{SILVA et al., 2014}; ^{REIS et al., 2014}).
Estimates of the parameters of the adjusted models are shown in table 2, as well as the mean, standard deviation and coefficient of variation for these estimates. The likelihood ratio test allows us to conclude that the estimates of the parameters for the three levels of the quantile regression model differ significantly (P <0.001), indicating that the garlic accessions have a different impact on the dry matter accumulation of the plant.
Parameters  
Models 



OLS  23.1320 (0.6991)  8694.29 (6733.11)  0.0852 (0.0076) 
QR (τ = 0.25)  20.0051 (0.7189)  8739.27 (5606.66)  0.0849 (0.0072) 
QR (τ = 0.5 )  23.3309 (1.4246)  5077.55 (3209.59)  0.0787 (0.0074) 
QR ( τ = 0.75)  27.0572 (1.7074)  2434.28 (899.06)  0.0725 (0.0047) 
Mean  23.3813  6236.3475  0.0803 
Deviation  2.8853  3060.7532  0.0060 
CV  0.1234  0.4908  0.0749 
The estimates for the β_{1} parameter, which represents the asymptotic weight, were close for the ordinary least square minima (OLS) model and for the median quantile regression one (QR (τ =0.5))with masses of 23.1320g and 23.3309g, respectively. The
The
The 30 garlic accessions were classified based on the quantile of the QR model that presented the lowest distance between observed values and predicted values (Table 3). Classifications were: of lesser interest for planting (τ = 0.25), of intermediate interest for planting (τ = 0.5), and of greater interest for planting (τ = 0.75). In practical terms, the total dry matter accumulated represents the productive potential of the plant, and is also a qualitative factor for producing bulbs of greater market value (^{DIRIBASHIFERAW, 2016}). Therefore, accessions with higher weight at the harvest are more economically promising.
Accession  Quantile  Accession  Quantile  
τ = 0.25  τ = 0.5  τ = 0.75  τ = 0.25  τ = 0.5  τ = 0.75  
1  0.1948  3.4757  8.0140  16  11.6559  8.4215  5.3862 
2  4.5794  1.5676  4.1193  17  7.7016  4.3567  1.7319 
3  7.3814  4.0787  1.7136  18  6.0881  3.2317  2.9424 
4  11.8375  8.4373  4.1483  19  3.3912  5.7568  10.1586 
5  21.7391  18.3983  13.9788  20  2.0914  1.8220  6.2217 
6  11.8081  8.8136  5.2493  21  2.3492  5.6329  10.1570 
7  1.9261  4.5881  8.7603  22  6.7786  10.1103  14.4170 
8  2.7858  2.7378  6.3400  23  7.1110  4.9067  4.5826 
9  10.0389  6.6763  2.5681  24  1.4718  4.6810  9.0747 
10  4.2713  0.9383  3.6931  25  1.4514  4.0328  8.5464 
11  6.1041  3.1517  2.8669  26  8.3523  5.0403  1.4490 
12  0.9330  3.9746  8.4087  27  3.3446  6.6795  11.1190 
13  0.9832  3.4485  7.8404  28  2.0965  2.4819  6.5687 
14  8.2210  5.3319  2.5678  29  4.3203  2.2406  4.5070 
15  1.5968  2.4477  6.7124  30  4.9340  1.7828  3.1753 
The classification of each accession is shown in table 4. Of the 30 accessions, 12 showed values closer to those estimated at the τ = 0.25 quantile, 6 were closer to the estimates at the τ = 0.5 quantile, and the remaining 12 were closer to the τ = 0.75 quantile. The accessions classified as A (of lesser interest for planting) have lower weights at the final stage than the other accessions, with observations that vary from 13.0250g to 20.7000g. The accession with the lowest weight is in this group (unidentified (14)). The accessions classified as B (of intermediate interest) have, in general, final weights between the other two classifications, from 20.0000g to 23.7500g. Accessions classified as C (of greater interest for planting) showed the highest values of asymptotic weight and the lowest values of growth rate. The final weights observed in the accessions of this classification are between 22.4750g and 36.1250g. The accession with the highest weight is in this group (Patos de Minas). The groups formed are consistent with those in the work of ^{REIS et al. (2014}), who adjusted nonlinear regression models to describe the dry matter accumulation in clusters of garlic provided from the same experiment.
N°  Accession  Class.  N°  Accession  Class. 
1  Barbado de Rio Grande  A  16  Unidentified (9)  C 
2  Branco de Dourados  B  17  Sapé (2)  C 
3  Montes Claros  C  18  Unidentified (10)  C 
4  Sacaia de Guanhães  C  19  Unidentified (11)  A 
5  Patos de Minas  C  20  Unidentified (12)  B 
6  Cateto Roxo (1)  C  21  Unidentified (13)  A 
7  Unidentified (1)  A  22  Unidentified (14)  A 
8  Unidentified (2)  B  23  Cultura de tecidos  C 
9  Unidentified (3)  C  24  Amarante  A 
10  Unidentified (4)  B  25  Caturra  A 
11  Unidentified (5)  C  26  Cateto Roxo (2)  C 
12  Unidentified (6)  A  27  Amarante Novo Cruzeiro  A 
13  Unidentified (7)  A  28  Cateto Roxão  A 
14  Unidentified (8)  C  29  Chinês  1  B 
15  Sapé (1)  A  30  Chinês  3  B 
CONCLUSION:
It was possible to adjust a nonlinear quantile regression model to distinguish garlic accessions based on different levels of dry matter accumulation over time.
The 30 garlic accessions were grouped according to the quantile of closest estimates. Twelve were classified as of lesser interest for planting (lower value of asymptotic weight, but higher growth rate); six were classified as intermediate; and 12 were classified as of greater interest for planting, possessing a lower growth rate and higher asymptotic weight.