INTRODUCTION:

Many response variables are frequently observed in the plot from the same experimental design. Thus, all response variables can be simultaneously subjected to the same statistical analysis (multivariate analysis). The continuous development of computer technology, which was almost unimaginable a few decades ago, has facilitated its application and attracted a growing interest in the use of multivariate techniques (^{FERREIRA, 2008}). Multivariate analysis of variance identifies the phenomenon of mutual dependence between response variables. Such information is not covered by the univariate analysis of variance (^{HAIR et al., 2007}).

Multivariate analysis of variance is only the initial stage in the evaluation of a data set. When mean vectors of treatment have significant variance, deriving complete inferences with multivariate techniques is necessary. These can be classified in two main purposes: discrimination of treatments and contrasts of mean vectors of interest. The discrimination techniques are used to classify treatments, in order to predict, explore, and describe possible hypotheses (^{AFIFI, MAY & CLARK, 2003}). In addition, the second analysis contributes to test the significance of hypotheses of interest, and can be aided by investigation of results.

Canonical discriminant analysis is an example of multivariate technique that represents the multivariate dispersion of qualitative explanatory variables, e.g., pure lines of beans. Transformation of original variables in new abstract variables that form a canonical discriminant linear function is the principle of the technique. The experimental design is considered in the analysis, partitioning the matrix information according to the existing treatments (^{SIVAKUMAR et al., 2013}). The technique aims to reduce the number of response variables facilitating inferences to be made and incorporating the maximum variations in the first canonical discriminant linear function. The technique is much applied in various fields of science to explore hypotheses (^{ZHAO & MCLEAN, 2000}; ^{ROSÁRIO et al., 2007}).

The multivariate contrast technique allows testing hypotheses, not to reduce data dimension, but to work with all existing variations and compare specific treatments. Technique partitions the treatment matrix (H) only for the elements under comparison, and calculates a value with multivariate distribution, which can also be done by different statistical methods such as (Wilks), maximum root (Roy), and Trace (Pillai; Hotteling-Lawley). By using the standardized canonical coefficients, the contribution of response variables can be determined in each contrast (^{RENCHER, 2002}). Although application of the multivariate contrasts is simple, they are not commonly used by the scientific community in Brazil.

Assays with pure lines of beans are frequently performed to select and obtain new productive cultivars. Discrimination of pure lines starting from multivariate techniques allows to detect the phenotypic variability for all the characters of agronomic interest (^{STÄHLEIN et al., 2011}). Thus, the agronomic ideotype can be obtained by simultaneous selection of grain yield, characters related to architecture of the ideal plant, resistance to abiotic and biotic stress factors, and nutritional quality of the final product.

The objective of this study was to apply multivariate techniques, canonical discriminant analysis, and multivariate contrasts, and indicate the most advantageous inferences in the evaluation of pure lines of beans.

MATERIALS AND METHODS:

This study was conducted in the experimental area of the Institute for Breeding and Molecular Genetics (UDESC, IMEGEM; Lages, SC, Brazil; 27º 48' S and 50º 19' W). This site is characterized by a wet mesothermal climate and mild summer. The altitude in Lages is 930 meters. The soil is classified as aluminic Humic Cambisol. The relief is undulating and the soil texture is clayey, with mean density of about 1.35g cm^{-3}.

The data are from an experiment formed by 24 pure lines of beans, which were evaluated in the farm crop of 2009/10. The experimental unit consisted of four 4-m long rows. Lines had a 0.45-m floor space between them. The seeding density was 15 seeds per linear meter (spaced evenly). Cultivation treatments were applied according to technical recommendations for bean crop. The useful area consisted of two central lines of the parcel, excluding 50cm in the border between plants. Five plants were collected from the useful area to measure the agronomic characters: plant height (in cm), numbers of pods and grains per plant, and stem diameter (in mm). A randomized block design was used with four replications.

The statistical model (matrix notation) is represented by: Y = Xβ + Zu + ɛ, where Y corresponds to the measurement vector of the four variables analyzed in the parcels; X is the fixed-factor incidence matrix adopted in the design; β is the fixed-parameter vector, i.e., the effect of pure lines of beans; Z is the random-factor incidence matrix adopted in the design; *u* is the random-parameter vector that refers to the block effect; ɛ is the experimental-error vector (^{RENCHER, 2001}).

The data were subjected to analysis of multivariate variance using the Wilks' Lambda statistics with 5%-error probability. Subsequently, the canonical discriminant analysis was performed to evaluate the pure lines of beans. The parallel discrimination rate (PDR) was calculated from the product of standardized canonical coefficients and canonical correlation (^{BARETTA et al., 2008}). One canonical mean score was calculated for each treatment (^{KHATTREE & NAIK, 2000}). The mean scores of the first canonical discriminant linear function were compared by the Scott-Knott test at 5%-error probability (^{CRUZ- CASTILHO }et al., 1994). Graphic dispersion analysis was performed from the canonical scores of the first two canonical discriminant linear functions. The canonical discriminant analysis was performed using the statistical SAS 9.2 software and the CANDISC (^{SAS Institute, 2009}) procedure.

The experimental data were subjected to multivariate contrasts analysis for comparison between pure lines or groups of pure lines of beans. The hypotheses of multivariate contrasts were tested by the Wilks' Lambda statistics with 5%-error probability. In each contrast, standardized canonical coefficients were calculated, informing the contribution of each response variable. The multivariate contrasts were carried out in the SAS 9.2 software using the GLM (^{SAS Institute, 2009}) procedure.

RESULTS AND DISCUSSION:

The multivariate analysis of variance resulted in a significant variation (P0.05) for the pure line factor, when applied to the Wilks' Lambda test (Table 1). Thus, difference was observed between at least two pure lines of beans. Rejection of the null hypothesis suggested questions about the origin of difference between pure lines and which agronomic characters contributed to this result (^{FERREIRA, 2006}). Therefore, use of more accurate multivariate techniques is necessary for a full interpretation of results.

^{*}Significant at 5%-error probability; ^{ns}Non significant at 5%-error probability; NDF: Numerator degrees of freedom; DDF: Denominator degrees of freedom.

The first canonical discriminant linear function captured 81% of the accumulated variance in the eigenvalues, being significant at 5%-error probability in the F test (Table 1). According to ^{CRUZ et al. (2012}), satisfactory interpretations of the variability shown by the treatments are obtained with 80% of the cumulative variance. Thus, only one dimension should identify the contribution of agronomic characters and discriminate the pure lines of beans. The plant height character showed a value of 1.42 for the TDP, exhibiting the highest discriminant potential in the pure lines of beans (^{MALUCHE-BARETTA et al., 2006}).

The 24 pure lines of beans were grouped by the first canonical score using the Scott-Knott test at 5%-error probability. This test showed differences between the pure line 10 in relation to the others (Table 2). The univariate evaluation of the first canonical score is performed when the cumulative variance is greater than 80%, and the multivariate difference between the pure lines of beans can be then tested (^{CRUZ- CASTILHO et al., 1994}; ^{BARETTA et al., 2008}). However, simplification of the matrix information can be questioned regarding its efficiency to test hypotheses, because exploring hypotheses of interest is the only purpose of the use this multivariate technique (^{AFIFI et al., 2003}).

The multivariate dispersion of the 24 pure lines of beans is illustrated by the scores for the first and second canonical discriminant linear function, which together inform 93% of total variance (Figure 1). It is possible to visually perceive the discrepancy between pure line 10 and others, and this statement is confirmed by the Scott-Knott test (Table 2). However, it is possible to raise new hypotheses about pure lines if additional information is provided. For example, the values for the pure lines 15 and 11 are close if only the first canonical discriminant is considered, but they are distant by the second canonical score coordinates (Table 2; Figure 1). The canonical discriminant analysis with 81% of the variation resulted in incomplete results, and this can change discrimination of pure lines of beans.

The most appropriate technique for situation depends mainly on the research objectives. In general, the purpose of multivariate analysis is to simplify, i.e., summarize a large data set by means of a few parameters that describe the data set properly. The study of ^{AMARANTE et al. (2006}) was clearly an exploratory analysis that was conducted to identify the response variables (nutritional attributes) that contributed most to discriminate the bitter pit severity degrees in apple culture. However, testing the severity levels of bitter pit by averaging the first canonical score was not the goal of this study. In addition, they refined a technique that possibly was not created to be applied in such a way.

An evaluation performed by a scalar (canonical score) from the vector product may be a fearful simplification. In addition, it does not adequately indicate the true causes of intrinsic variations in the pure lines of beans. This fact can break the basic principle of multivariate analysis and valuable information present in the set of agronomic characters may be lost. The canonical discriminant analysis transforms a large number of correlated original characteristics in linear combinations (canonical variables) so that the correlation between these variables becomes null (^{KHATTREE & NAIK, 2000}). Thus, comparing coefficients that are supposedly independent is inconsistent.

The multivariate contrasts shown above were strategically chosen to exemplify the additional information of this technique, and revealed differences within the group formed by the Scott-Knott test (Table 3). This is contrary to what was observed in the univariate contrasts, where significant differences were not found at 5%-error probability. Therefore, these results can be explained by the set of agronomic characters whose information was not incorporated in the canonical scores. The use of mean comparison tests with the first canonical score may have caused the loss of valuable information in discriminating pure lines of beans.

^{*}Significant at 5%-error probability; ^{ns}Non significant at 5%-error probability; G_{1}: group composed by pure lines 02, 05, 08, 11, 16, 17, 20, 21, 22, 23, and 24; G_{2}: group composed by pure lines 01, 03, 04, 06, 07, 09, 12, 13, 14, 15, and 18.

After a careful examination of the canonical scores in the second canonical discriminant linear function (Figure 1), it is possible to suggest that two new groups of pure lines are formed. The upper points (2, 5, 8, 11, 16, 17, 20, 21, 22, 23, and 24) may characterize group 1 (G1) and the lower ones (1, 3, 4, 6, 7, 9, 12, 13, 14, 15, and 18) group 2 (G2). In fact, a significant difference was detected between the groups when this hypothesis was tested. This can be explained by the contribution of the number of pods per plant (3.25), i.e., this is the essentially discriminating character between the groups of pure lines tested (Table 3).

Canonical discriminant analysis was not advantageous for hypotheses testing in the evaluation of pure lines of beans. This analysis allows to suggest hypotheses about the pure lines which will be tested. Evaluation of distances based on the graph coordinates is subjective (STHÄLEIN et al., 2011). E.g., pure line 6 shows a significant difference in relation to pure line 23 when analyzed by the multivariate contrasts. Conversely, pure line 6 do not show a significant difference in relation to pure line 5, although the distance is equal to that used in the previous comparison (Figure 1; Table 3). Therefore, each multivariate technique has specific principles that should be taken into account for a proper application. Knowing these principles is crucial for a researcher who intends to apply multivariate analysis techniques.

The multivariate contrasts technique was supposedly made to test hypotheses, and it can be an excellent alternative to supplement the canonical discriminant analysis. The multivariate contrasts technique is available to researchers, being flexible enough to test any combination between pure lines of beans according to their interests (^{RENCHER, 2002}). The use of canonical coefficients allows inferring about the contribution of each agronomic character for contrast. However, this information cannot be obtained using the Scott-Knott test.

These techniques can be used, for example, in value for cultivation and use (VCU) trials. The purpose of these trials is to compare each pure line with one or more commercial controls (usually more than one). Therefore, the multivariate contrast can be used before launching a cultivar for simultaneous comparison between a pure line and commercial controls, taking into account a set of agronomic characteristics of interest such as grain yield, cycle, plant diseases, nutritional quality of the grain, and any other crucial character.

CONCLUSION:

The use of canonical discriminant analysis to test hypotheses is not justifiable when other appropriate methods such as the multivariate contrasts can be used. The canonical discriminant analysis and multivariate contrasts are excellent techniques to be combined in a multivariate evaluation, allowing to detect the origin of significant variations and contribution of response variables.