Contribution biplots

In order to interpret the biplot it is necessary to know which points usually variables are the ones that are important contributors to the solution, and this information is available separately as part of the biplot s numerical results. We propose a new scaling of the display, called the contribution biplot, which incorporates this diagnostic directly into the graphical display, showing visually the important contributors and thus facilitating the biplot interpretation and often simplifying the graphical representation considerably. The contribution biplot can be applied to a wide variety of analyses such as correspondence analysis, principal component analysis, log-ratio analysis and the graphical results of a discriminant analysis/MANOVA, in fact to any method based on the singular-value decomposition. In the contribution biplot one set of points, usually the rows of the data matrix, optimally represent the spatial positions of the cases or sample units, according to some distance measure that usually incorporates some form of standardization unless all data are comparable in scale. The other set of points, usually the columns, is represented by vectors that are related to their contributions to the low-dimensional solution. A fringe benefit is that usually only one common scale for row and column points is needed on the principal axes, thus avoiding the problem of enlarging or contracting the scale of one set of points to make the biplot legible. Furthermore, this version of the biplot also solves the problem in correspondence analysis of low-frequency categories that are located on the periphery of the map, giving the false impression that they are important, when they are in fact contributing minimally to the solution.

Correspondence analysis of raw data

Correspondence analysis has found extensive use in ecology, archeology, linguisticsand the social sciences as a method for visualizing the patterns of association in a table offrequencies or nonnegative ratio-scale data. Inherent to the method is the expression of the datain each row or each column relative to their respective totals, and it is these sets of relativevalues (called profiles) that are visualized. This relativization of the data makes perfect sensewhen the margins of the table represent samples from sub-populations of inherently differentsizes. But in some ecological applications sampling is performed on equal areas or equalvolumes so that the absolute levels of the observed occurrences may be of relevance, in whichcase relativization may not be required. In this paper we define the correspondence analysis ofthe raw unrelativized data and discuss its properties, comparing this new method to regularcorrespondence analysis and to a related variant of non-symmetric correspondence analysis.

Distributional equivalence and subcompositional coherence in the analysis of contingency tables, ratio-scale measurements and compositional data

We consider two fundamental properties in the analysis of two-way tables of positive data: the principle of distributional equivalence, one of the cornerstones of correspondence analysis of contingency tables, and the principle of subcompositional coherence, which forms the basis of compositional data analysis. For an analysis to be subcompositionally coherent, it suffices to analyse the ratios of the data values. The usual approach to dimension reduction in compositional data analysis is to perform principal component analysis on the logarithms of ratios, but this method does not obey the principle of distributional equivalence. We show that by introducing weights for the rows and columns, the method achieves this desirable property. This weighted log-ratio analysis is theoretically equivalent to spectral mapping , a multivariate method developed almost 30 years ago for displaying ratio-scale data from biological activity spectra. The close relationship between spectral mapping and correspondence analysis is also explained, as well as their connection with association modelling. The weighted log-ratio methodology is applied here to frequency data in linguistics and to chemical compositional data in archaeology.

Weighted metric multidimensional scaling

This paper establishes a general framework for metric scaling of any distance measure between individuals based on a rectangular individuals-by-variables data matrix. The method allows visualization of both individuals and variables as well as preserving all the good properties of principal axis methods such as principal components and correspondence analysis, based on the singular-value decomposition, including the decomposition of variance into components along principal axes which provide the numerical diagnostics known as contributions. The idea is inspired from the chi-square distance in correspondence analysis which weights each coordinate by an amount calculated from the margins of the data table. In weighted metric multidimensional scaling (WMDS) we allow these weights to be unknown parameters which are estimated from the data to maximize the fit to the original distances. Once this extra weight-estimation step is accomplished, the procedure follows the classical path in decomposing a matrix and displaying its rows and columns in biplots.

Biplots of fuzzy coded data

A biplot, which is the multivariate generalization of the two-variable scatterplot, can be used to visualize the results of many multivariate techniques, especially those that are based on the singular value decomposition. We consider data sets consisting of continuous-scale measurements, their fuzzy coding and the biplots that visualize them, using a fuzzy version of multiple correspondence analysis. Of special interest is the way quality of fit of the biplot is measured, since it is well-known that regular (i.e., crisp) multiple correspondence analysis seriously under-estimates this measure. We show how the results of fuzzy multiple correspondence analysis can be defuzzified to obtain estimated values of the original data, and prove that this implies an orthogonal decomposition of variance. This permits a measure of fit to be calculated in the familiar form of a percentage of explained variance, which is directly comparable to the corresponding fit measure used in principal component analysis of the original data. The approach is motivated initially by its application to a simulated data set, showing how the fuzzy approach can lead to diagnosing nonlinear relationships, and finally it is applied to a real set of meteorological data.

Biplots of compositional data

The singular value decomposition and its interpretation as alinear biplot has proved to be a powerful tool for analysing many formsof multivariate data. Here we adapt biplot methodology to the specifficcase of compositional data consisting of positive vectors each of whichis constrained to have unit sum. These relative variation biplots haveproperties relating to special features of compositional data: the studyof ratios, subcompositions and models of compositional relationships. Themethodology is demonstrated on a data set consisting of six-part colourcompositions in 22 abstract paintings, showing how the singular valuedecomposition can achieve an accurate biplot of the colour ratios and howpossible models interrelating the colours can be diagnosed.

Tying up the loose ends in simple correspondence analysis

Although correspondence analysis is now widely available in statistical software packages and applied in a variety of contexts, notably the social and environmental sciences, there are still some misconceptions about this method as well as unresolved issues which remain controversial to this day. In this paper we hope to settle these matters, namely (i) the way CA measures variance in a two-way table and how to compare variances between tables of different sizes, (ii) the influence, or rather lack of influence, of outliers in the usual CA maps, (iii) the scaling issue and the biplot interpretation of maps,(iv) whether or not to rotate a solution, and (v) statistical significance of results.

Weighted Euclidean biplots

We construct a weighted Euclidean distance that approximates any distance or dissimilarity measure between individuals that is based on a rectangular cases-by-variables data matrix. In contrast to regular multidimensional scaling methods for dissimilarity data, the method leads to biplots of individuals and variables while preserving all the good properties of dimension-reduction methods that are based on the singular-value decomposition. The main benefits are the decomposition of variance into components along principal axes, which provide the numerical diagnostics known as contributions, and the estimation of nonnegative weights for each variable. The idea is inspired by the distance functions used in correspondence analysis and in principal component analysis of standardized data, where the normalizations inherent in the distances can be considered as differential weighting of the variables. In weighted Euclidean biplots we allow these weights to be unknown parameters, which are estimated from the data to maximize the fit to the chosen distances or dissimilarities. These weights are estimated using a majorization algorithm. Once this extra weight-estimation step is accomplished, the procedure follows the classical path in decomposing the matrix and displaying its rows and columns in biplots.

Emprego da análise AMMI na avaliação da estabilidade produtiva em soja

O objetivo deste trabalho foi avaliar a influência da interação de genótipos com ambientes (GxA) na produtividade de grãos de um conjunto de linhagens de soja (Glycine max L.). Foram utilizados dados de 11 experimentos (ambientes) realizados no Estado de Goiás. Em cada experimento foram avaliados 18 genótipos, sendo quatro cultivares comerciais como testemunhas. O método de análise da interação foi o procedimento AMMI (modelo de efeitos principais aditivos e interação multiplicativa). O padrão significativo das interações GxA foi captado apenas pelo primeiro eixo principal AMMI, o qual explicou 36% da soma de quadrados GxA original, sugerindo contaminação da matriz de interações clássica por ruídos que prejudicam a qualidade das predições de respostas fenotípicas obtidas pelos métodos tradicionais. Quanto à estabilidade de comportamento, a maioria das linhagens experimentais destacou-se (com menores interações com ambientes) em relação às cultivares testemunhas. Estas, no entanto, foram relativamente mais produtivas, sobretudo a cultivar Conquista. Entre as novas linhagens, os genótipos L-16, L-13 e L-14 mostraram ser os mais promissores para fins de recomendação como cultivares.

Key locations for soybean genotype assessment in Central Brazil

The objective of this work was to identify key locations for the establishment of soybean (Glycine max) genetic breeding programs, in the Central Region of Brazil. Grain yield data of three maturity groups of soybean genotypes, from regional trials conducted over three years, at 18 locations in Brazilian Cerrado were used. A key location for the early phases of the breeding program was defined as the site that best classifies the winning genotypes in the region. Key locations for the final phases were defined as those sites that best represent each environmental stratum, in relation to the adaptability of the respective winning genotype. This adaptability was estimated by additive main effects and multiplicative interaction (AMMI) model analysis, using the distance between the score of each location in a stratum and the score of the winning genotype, which characterizes such stratum in an AMMI biplot. The locations that best classified the winning genotypes over space and time were Mineiros, Placas and Rio Verde. For the final phases of genotype selection, with data from the three maturity group, the recommended locations were: Buritis, Chapadão do Céu, Iraí, Pamplona, Placas, Planaltina, Rio Verde, Sacramento, Senador Canedo, Uberaba, and Uberlândia.

Genotype by environment interaction for seed yield per plant in rapeseed using AMMI model

The objective of this study was to assess genotype by environment interaction for seed yield per plant in rapeseed cultivars grown in Northern Serbia by the AMMI (additive main effects and multiplicative interaction) model. The study comprised 19 rapeseed genotypes, analyzed in seven years through field trials arranged in a randomized complete block design, with three replicates. Seed yield per plant of the tested cultivars varied from 1.82 to 19.47 g throughout the seven seasons, with an average of 7.41 g. In the variance analysis, 72.49% of the total yield variation was explained by environment, 7.71% by differences between genotypes, and 19.09% by genotype by environment interaction. On the biplot, cultivars with high yield genetic potential had positive correlation with the seasons with optimal growing conditions, while the cultivars with lower yield potential were correlated to the years with unfavorable conditions. Seed yield per plant is highly influenced by environmental factors, which indicates the adaptability of specific genotypes to specific seasons.

Different methods to assess yield temporal stability in rubber

The objective of this work was to assess, during six years, the temporal stability of natural rubber yield of 25 superior Hevea brasiliensis genotypes, using the Wricke, Eberhart & Russell, Lin & Binns, additive main effect and multiplicative interaction (AMMI) analysis, and harmonic mean of the relative performance of the genetic values (HMRPGV) methods. The IAC 40 and IAC 300 genotypes were identified as stable and high yielding by the Eberhart & Russell, Lin & Binns, HMRPGV, and AMMI Biplot methods. The ranking of the other more stable genotypes identified by these analyses was altered. The observed results in the AMMI Biplot agreed with those observed in the Wricke method for identifying stable, but lower yielding genotypes. The simultaneous use of different methods allows a more accurate indication of stable genotypes. Stability analyses based on different principles show agreement in indicating stable genotypes.

Identificación de mega‑ambientes para potenciar el uso de genotipos superiores de arroz en Panamá

El objetivo de este trabajo fue evaluar tres métodos para identificar mega‑ambientes, para optimizar el uso del potencial genético de los cultivares de arroz, durante el proceso de selección, y para hacer recomendaciones sobre siembras comerciales en Panamá. Los datos experimentales fueron obtenidos de los ensayos de productividad de cultivares precoces realizados entre 2006 y 2008. Para lograr la estratificación de los ambientes y definir los mega‑ambientes, se utilizaron los métodos del genotipo vencedor mediante el modelo AMMI1, el modelo biplot GGE y el de conglomerado por el método de Ward, complementado con el biplot GGE. Los tres métodos utilizados identificaron dos mega‑ambientes, donde los cultivares sobresalientes fueron Fedearroz 473 e Idiap 145‑05. Hubo una coincidencia de 100% en el agrupamiento del conglomerado x el biplot GGE, mientras que entre conglomerado x AMMI1 y biplot GGE x AMMI1 fue de 95,2%. El genotipo más estable, en ambos mega‑ambientes, fue el cultivar Idiap 145‑05, lo que indica capacidad de adaptación amplia y específica. La capacidad adaptativa de los genotipos superiores y no las condiciones agroclimáticas de las localidades evaluadas fue responsable de la definición de los mega‑ambientes.

Adaptabilidade e estabilidade de cultivares de trigo em diferentes épocas de semeadura, no Paraná

O objetivo deste trabalho foi identificar as melhores épocas de semeadura e avaliar a adaptabilidade e a estabilidade de cultivares de trigo, em duas regiões tritícolas do Paraná. Avaliou-se a produtividade de grãos de sete cultivares, em Guarapuava, e de nove, em Palotina, em quatro épocas de semeadura, em 2006, 2007 e 2008. Utilizou-se o delineamento experimental de blocos ao acaso, com quatro repetições em Guarapuava, e três em Palotina. Foram utilizadas a metodologia REML/BLUP e a dos efeitos principais dos genótipos e da interação genótipo x ambiente (GGE biplot) para a avaliação da adaptabilidade e da estabilidade das cultivares, e o métodoAMMI para a identificação das melhores épocas de semeadura. Semeaduras em julho, em Guarapuava, e em abril, em Palotina, maximizam a produtividade de grãos. As cultivares Safira, em Guarapuava, e CD 113, em Palotina, são estáveis, amplamente adaptadas e apresentam alta produtividade de grãos.

Épocas de semeadura e desempenho agronômico de cultivares de soja

O objetivo deste trabalho foi avaliar o efeito de épocas de semeadura no desempenho agronômico de cultivares de soja em São Domingos, SC, e indicar as cultivares mais estáveis e adaptadas a cada época. O experimento foi conduzido durante dois anos agrícolas (2008/2009 e 2009/2010), com seis cultivares e quatro épocas de semeadura (15/10, 15/11, 15/12 e 15/1), em delineamento experimental de blocos ao acaso, com três repetições e parcelas com área útil de 3,6 m². A metodologia AMMI (modelos de efeitos principais aditivos com interação multiplicativa) foi utilizada para avaliar o desempenho produtivo das cultivares, e a GGE (genótipo e interação genótipo x ambiente) para avaliar a adaptabilidade e a estabilidade das cultivares nas diferentes épocas de semeadura. Em ambos os anos agrícolas, as semeaduras em 15/10 e 15/11 maximizaram o número de vagens por planta, o número de grãos por vagem, a estatura das plantas, o número de ramos, a massa de mil sementes e, consequentemente, a produtividade de grãos. As cultivares de ciclo médio ou precoce com porte elevado são mais adequadas para semeaduras tardias.