METHODS ARTICLE published: 22 March 2013 doi: 10.3389/fpls.2013.00039 The plant ionome revisited by the nutrient balance concept Serge-Étienne Parent 1, Léon Etienne Parent 1*, Juan José Egozcue2, Danilo-Eduardo Rozane3, Amanda Hernandes4, Line Lapointe5,Valérie Hébert-Gentile5, Kristine Naess6, Sébastien Marchand 1, Jean Lafond 7, Dirceu Mattos Jr.8, Philip Barlow 9 and William Natale4 1 Équipe de Recherche en Sols Agricoles et Miniers, Department of Soils and Agrifood Engineering, Université Laval, Québec, QC, Canada 2 Department of Applied Mathematics III, Universitat Politècnica de Catalunya, Barcelona, Spain 3 Departamento de Agronomia, Universidade Estadual Paulista, Campus de Registro, São Paulo, Brasil 4 Departamento de Solos e Adubos, Universidade Estadual Paulista, Jaboticabal, São Paulo, Brasil 5 Centre d’Étude de la Forêt, Department of Biology, Université Laval, Québec, QC, Canada 6 Centre de Recherches Les Buissons, Pointe-aux-Outardes, QC, Canada 7 Agriculture and Agri-Food Canada, Normandin, QC, Canada 8 Centro de Citricultura Sylvio Moreira (IAC), Cordeirópolis, Säo Paulo, Brazil 9 Bio Soil and Crop Ltd, Tauranga, New Zealand Edited by: Richard A. Jorgensen, University of Arizona, USA Reviewed by: Elizabeth Pilon-Smits, Colorado State University, USA Heiner Goldbach, University of Bonn, Germany *Correspondence: Léon Etienne Parent, Department of Soils and Agrifood Engineering, Paul-Comtois Building, Université Laval, Québec, QC G1V 0A6, Canada. e-mail: leon-etienne.parent@ fsaa.ulaval.ca Tissue analysis is commonly used in ecology and agronomy to portray plant nutrient sig- natures. Nutrient concentration data, or ionomes, belong to the compositional data class, i.e., multivariate data that are proportions of some whole, hence carrying important numer- ical properties. Statistics computed across raw or ordinary log-transformed nutrient data are intrinsically biased, hence possibly leading to wrong inferences. Our objective was to present a sound and robust approach based on a novel nutrient balance concept to classify plant ionomes. We analyzed leaf N, P, K, Ca, and Mg of two wild and six domesticated fruit species from Canada, Brazil, and New Zealand sampled during reproductive stages. Nutri- ent concentrations were (1) analyzed without transformation, (2) ordinary log-transformed as commonly but incorrectly applied in practice, (3) additive log-ratio (alr) transformed as surrogate to stoichiometric rules, and (4) converted to isometric log-ratios (ilr) arranged as sound nutrient balance variables. Raw concentration and ordinary log transformation both led to biased multivariate analysis due to redundancy between interacting nutrients. The alr- and ilr-transformed data provided unbiased discriminant analyses of plant ionomes, where wild and domesticated species formed distinct groups and the ionomes of species and cultivars were differentiated without numerical bias. The ilr nutrient balance concept is preferable to alr, because the ilr technique projects the most important interactions between nutrients into a convenient Euclidean space.This novel numerical approach allows rectifying historical biases and supervising phenotypic plasticity in plant nutrition studies. Keywords: compositional data analysis, ionome classification, nutrient interactions, numerical biases, isometric log-ratio, plant nutrition INTRODUCTION Salt et al. (2008) defined the ionome as the mineral nutrient and trace element composition of an organism that represents the inorganic component of cellular and organismal systems. The need for linking plant ionomes – often referred as plant nutri- ent signatures (Willby et al., 2001) or profiles (Tennakoon et al., 2011) – with genetics (Conn and Gilliham, 2010) and adapta- tion to environmental factors (Chapin, 1989) elevated the study of mineral nutrition of plants as central topic in ecology (Aerts and Chapin, 2000), agronomy (Bergmann, 1988), and genetics (White and Brown, 2010). The plant ionome is a vector of tissue analytical data gener- ally constrained to the dry or fresh matter content. To facilitate the analysis of complex interacting systems such as the concentra- tion vector of plant ionomes, it is often assumed, under the ceteris paribus assumption, that all factors but the ones being varied are equal (Giampietro, 2004). Such assumption denies the principle that components of a whole are inherently related to each other, because changing a proportion inherently affects at least another proportion. In fact, ionome data belong to the class of composi- tional data, i.e., strictly positive data constrained to some whole, that convey only relative information (Aitchison, 1986). Compo- sitional data are intrinsically multivariate: each part cannot be interpreted without being related to the others (Tolosana-Delgado and van den Boogart, 2011). Indeed, statistics computed across compositional data such as nutrient concentrations are inherently biased due to redundancy, scale-dependency, and non-normal distribution (Bacon-Shone, 2011). Compositional data analysis provides unbiased numerical solutions to analyze plant ionomes as self-interactive systems. Plant growth and development depend on a balanced supply of essential elements and this equilibrium is maintained by home- ostatic mechanisms (Williams and Salt, 2009). Dual ratios (Wal- worth and Sumner,1988) and stoichiometric rules (Ingestad,1987; Körner, 2011) have been proposed to reflect nutrient interactions controlling carbon uptake. Agronomists thus developed a large www.frontiersin.org March 2013 | Volume 4 | Article 39 | 1 http://www.frontiersin.org/Plant_Science http://www.frontiersin.org/Plant_Science/editorialboard http://www.frontiersin.org/Plant_Science/editorialboard http://www.frontiersin.org/Plant_Science/editorialboard http://www.frontiersin.org/Plant_Science/about http://www.frontiersin.org/Plant_Nutrition/10.3389/fpls.2013.00039/abstract http://www.frontiersin.org/Community/WhosWhoActivity.aspx?sname=Serge_�tienneParent&UID=48569 http://www.frontiersin.org/Community/WhosWhoActivity.aspx?sname=Leon_EtienneParent&UID=64490 http://www.frontiersin.org/people/Juan_Jos�Egozcue/84344 http://www.frontiersin.org/Community/WhosWhoActivity.aspx?sname=LineLapointe&UID=75282 http://www.frontiersin.org/Community/WhosWhoActivity.aspx?sname=ValerieHebert-Gentile&UID=67532 http://www.frontiersin.org/people/SaraNaess/85233 http://www.frontiersin.org/people/S�bastienMarchand/84787 http://www.frontiersin.org/Community/WhosWhoActivity.aspx?sname=DirceuMattos_Jr&UID=36850 http://www.frontiersin.org/Community/WhosWhoActivity.aspx?sname=PhilipBarlow&UID=67138 http://www.frontiersin.org http://www.frontiersin.org/Plant_Nutrition/archive mailto:leon-etienne.parent@fsaa.ulaval.ca mailto:leon-etienne.parent@fsaa.ulaval.ca Parent et al. Nutrient balance in ionomics spectrum of dual (e.g., N/P) and amalgamated (e.g., K/[Ca+Mg]) ratios for diagnostic purposes (Bergmann, 1988). However, one can generate D(D−1)/2 dual ratios and D(D−1)2/2 amalgamated ratios from a D-part ionome, that actually carries D−1 degrees of freedom (Aitchison and Greenacre, 2002; Egozcue and Pawlowsky- Glahn, 2005). For example, an ionome including 10 elements generates up to 45 dual nutrient ratios such as the K/Ca ratio and up to 405 amalgamated dual ratios such as the K/(Ca+Mg) ratio, but only nine variables are linearly independent. Researchers realized the great difficulty of interpreting myriads of ratios and proposed integrative empirical models such as the “Diagnosis and Recommendation Integrated System”(DRIS) (Beaufils, 1973). However, DRIS is noisy (Parent et al., 2012a). Although principal component analysis (PCA) also provided a dimension reduction method for nutrient data (Baxter et al., 2008), PCA does not tackle the numerical biases inherent to compositional data (Aitchison, 1986). Unfortunately, most researchers still use at fault raw con- centration data (Lahner et al., 2003; Conn and Gilliham, 2010; White and Brown, 2010), their ordinary log-transformation (Han et al., 2011), or dual ratio expressions when conducting multi- variate analyses of ionomes. But fortunately, compositional data analysts have developed log-ratio transformations that generate scale-invariant variables, avoid redundancy, and are free to range in real space (Aitchison, 1986; Egozcue et al., 2003). Parent and Dafir (1992) were, to our knowledge, the first to correct numerical biases in DRIS using the row-centered log-ratio (clr) transformation proposed by Aitchison (1986). The clr is com- puted as ln(xi/g (x)), where xi is the ith component (i∈ 1 to D) and g (x) is the geometric mean of the compositional vector. However, matrix singularity occurs in the multivariate analysis, because the D clr values add up to zero. As a result, one clr value must be removed. Other log-ratio transformations can compress a D-part composition into D–1 variables without losing information, hence avoiding singularity problems. Aitchison (1986) proposed using the additive log-ratio (alr) computed as ln(xj/xA) where xj is the jth component (j∈ 1 to D except A) and xA, the common denominator of the composi- tional vector. The alr transformation can reflect the stoichiometric rules used in plant physiology and nutrient management (Inges- tad, 1987). However, alr variables are at an angle of 60˚ between them and are thus geometrically difficult to handle (Egozcue and Pawlowsky-Glahn, 2006). A dual ratio between two nutrients is a dual balance, hence removing one variable while keeping all the relevant informa- tion. An extended balance system can be illustrated by a ternary diagram (Lagatu and Maume, 1934) or by a mobile and its ful- crums built according to an ad hoc scheme for several components (Parent et al., 2012a). In compositional data analysis, balances are expressed as log-ratio contrasts between the geometric means of two parts or groups of parts (Egozcue and Pawlowsky-Glahn, 2005). Assigning orthogonal coefficients to contrasts allows com- puting orthogonal balances as isometric log-ratio (ilr) in the Euclidean space (Egozcue et al., 2003). The ilr technique was found to be the most appropriate to describe natural patterns in geo- chemistry (Buccianti, 2011), plant nutrition (Parent, 2011; Parent et al., 2012c), environmental sciences (Filzmoser et al., 2009a), soil physics (Parent et al., 2012b), chemistry and biochemistry (Par- ent et al., 2012a), and other disciplines (Pawlowsky-Glahn and Buccianti, 2011). Our objective is to present an unbiased balance concept to the plant nutrition community using data sets of fruit species, to illus- trate and provide a robust perspective to solve the important prob- lem of data representation when conducting multivariate analysis of ionomes. We hypothesize that ionomes of wild (low pheno- typic plasticity) and domesticated (high phenotypic plasticity) species differ markedly from each other due to natural adapta- tion or human selection pressure (Chapin, 1989). We expand the nutrient balance concept to species and cultivars. THEORY OF COMPOSITIONAL DATA ANALYSIS SAMPLE SPACE The sample space (e.g., the space of compositional data of plant ionomes reported on dry mass basis) is defined by SD, a posi- tive vector of D components adding up to a constant κ, such as 1 (fractions of some whole), 100% (e.g., N-P-K ternary diagram representing an ionome subcomposition), 1000 g kg−1 (e.g., sum of nutrient concentrations and of the filling value in an ionome), etc. The closure operatorC computes the constant sum assignment as follows (Egozcue and Pawlowsky-Glahn, 2006): SD = C (c1, c2, . . . , cD) = [ c1K∑D i=1 ci , c2K∑D i=1 ci , . . . , cDK∑D i=1 ci ] (1) where κ is the unit or scale of measurement and ci is the ith part of a composition containing D parts. The ionome comprises analyt- ical results as well as, optionally, undetermined concentrations of other elements summarized by the filling value. The filling value is computed by difference between the unit or scale of measurement and the sum of analytical results. The sample space can be subdi- vided into non-overlapping subspaces made of two (dual ratios), three (ternary diagram), or more interacting components where each subspace can be interpreted independently and coherently. NUMERICAL BIASES The redundancy and scale-dependency inherent to compositional data generate spurious correlations (Pearson, 1897; Tanner, 1949; Chayes, 1960) that distort their multivariate analysis (Aitchison, 1986). The multivariate analysis of concentration values or their ordinary log transformation may thus lead to biased and even meaningless results (Filzmoser et al., 2009b). These biases can be avoided using compositional data analysis techniques (Egozcue and Pawlowsky-Glahn, 2006; Mateu-Figueras et al., 2011). Redundancy can be avoided by (1) sacrificing a component for use as common denominator (alr transformation) (Aitchi- son, 1986) or (2) using the principle of contrasts orthogonality whereby the orthogonally arranged balances acquire linear inde- pendence (Rodgers et al., 1984) using ilr transformation (Egozcue et al., 2003). The isometry of ilr variables means that the geome- try is Euclidean, which is the very basic geometry in multivariate analysis (Egozcue and Pawlowsky-Glahn, 2006). Scale invariance assures that data have the same covariance structure no matter the base across which they are scaled, e.g., Frontiers in Plant Science | Plant Nutrition March 2013 | Volume 4 | Article 39 | 2 http://www.frontiersin.org/Plant_Nutrition http://www.frontiersin.org/Plant_Nutrition/archive Parent et al. Nutrient balance in ionomics across wet, dry, organic, mineral or macronutrient basis. Scale invariance is required to provide a coherent interpretation of mul- tivariate analyses of compositional data (Aitchison, 1986; Egozcue and Pawlowsky-Glahn, 2005). Non-normal distribution inherent to compositional data is improved by projecting the constrained space of raw composi- tional data into a real space of log-ratios. Because log-ratios can take any value in the domain ±∞, alrs and ilrs can be mapped in real space, as required under the normality assumption (Egozcue and Pawlowsky-Glahn,2006). By comparison, confidence intervals that may reach values <0 or beyond 100% under the normality assumption have no physical meaning (Weltje, 2002). The log- ratio transformation improves normal distribution compared to raw concentrations or their ordinary log-transformation (Filz- moser et al., 2009a). All in all, the ilr transformation is recom- mended for conducting multivariate analyses of compositional data (Filzmoser and Hron, 2011). THE LOG-RATIO TRANSFORMATIONS The alr transformation Log transforming the P/N, K/N, Ca/N, and Mg/N ratios elaborated by Ingestad (1987) to monitor the plant nutrition of tree seedlings yield D−1 alr variables. The number of degrees of freedom is reduced by using one component as common denominator. The choice of the common denominator has no influence on multi- variate analysis (Aitchison, 1986). In the Ingestad (1987) model, the common basis is N concentration. The jth alr is computed as follows: alrj = ln cj N (2) where cj is the jth nutrient excluding N. If a tissue contains 2.50% N and 0.15% P, the Redfield ratio (Güsewell, 2004) is 16.7 and the corresponding alr [P/N] value is ln(0.15/2.50)=−2.81. However, the alrs are oblique to each other and difficult to rectify (Egozcue and Pawlowsky-Glahn, 2005). The ilr transformation The ilr transformation has the advantage over the alr to be geo- metrically suited to conduct multivariate analysis (Filzmoser et al., 2009a). Another advantage of ilrs is a special device of balances or linearly independent ratios among nutrients called sequential binary partition (SBP). The SBP describes the D−1 orthogonal (geometrically independent) balances between parts and groups of parts. The SBP is a (D−1)×D matrix, in which parts labeled “+1” (group numerator) are balanced with parts labeled “−1” (group denominator). A part labeled “0” is excluded from the bal- ance between parts. The composition is partitioned sequentially into contrasts at every hierarchically ordered row until the (+1) and (−1) groups each contain a single part. To establish contrast orthogonality, it is necessary to imbed subcompositions into larger ones and assign orthogonal coeffi- cients to each log-ratio contrast (Egozcue et al., 2003). The ilr is computed as follows (Egozcue and Pawlowsky-Glahn, 2005): ilrj = √ rj sj rj + sj ln g (c+j ) g (c−j ) (3) where, in the jth row of the SBP, ilrj is the jth isometric log-ratio;√ rj sj/rj + sj is the orthogonal coefficient of the jth balance (or log contrast) designed in the SBP; rj and sj represent the number of parts in the +1 and −1 groups of the jth balance, respectively; g (c+j ) is the geometric mean of components in the+1 group and g (c−j ) is the geometric mean of components in the −1 group. The partition between two components or groups of components is presented as [Sr | Ss]. If a tissue contains 2.50% N and 0.15% P, the Redfield ratio (Güsewell, 2004) is 16.7 and the ilr value is [N |P] = √ 1 2 ln (16.7) = 1.99. NUTRIENT BALANCES Nutrient balances are robustly amenable to myriads of statisti- cal techniques analysis: balances variables are non-redundant and scale-invariant, are mapped in a real space and respect the D−1 degrees of freedom of a compositional vector. The SBP allows the analyst to define orthogonal axes in order to focus upon interpretable balances. In order to compare the ionomes of plant species, a subcomposition of the compositional vector was defined as S5 =C (N, P, K, Ca, Mg). The SBP in Table 1 formalizes bal- ance dendrograms such as the one presented in Figure 1. In a more intuitive but similar approach, nutrient balance could be illustrated by a mobile-and-fulcrums design where nutrient con- centrations in weighing pans are equilibrated according to ad hoc nutrient balances. There are indeed four orthogonal balances in S5 (Figure 1). Our SBP initiator was [N, P, K | Ca, Mg] to reflect sequen- tially the relationships between N, P, and K (Lagatu and Maume, 1934; Wilkinson et al., 2000) in agroecosystems, the Ca and Mg composition that reflects geographical position and soil mineralogy (Walworth and Sumner, 1987), and the Redfield ratio that reflects the balance between two fundamental life processes, protein, and r-RNA synthesis (Loladze and Elser, 2011). THE AITCHISON DISTANCE The Aitchison distance (A) between two D-part compositions is computed as a Euclidean distance across selected ilr coordinates as follows (Egozcue and Pawlowsky-Glahn, 2006): A = √√√√√D−1∑ j=1 ( Ailrj − Bilrj )2 (4) where Ailrj and Bilrj are the jth ilr coordinate of the composition of two rows A and B. If one of the two rows is a null vector, A is called the Aitchison norm. If Euclidean geometry is not valid, arithmetic mean is likely to be a poor estimate of data center (Filzmoser et al., 2009a). Even after ordinary log transforming compositional data, the squared Euclidean distance (ε2) between ordinary log-transformed com- positions x and y, i.e., between ln(x) and ln(y), is always equal to or greater than the squared A distance between the ilrs of com- positions x and y (Eq. 4) as driven by the number of components www.frontiersin.org March 2013 | Volume 4 | Article 39 | 3 http://www.frontiersin.org http://www.frontiersin.org/Plant_Nutrition/archive Parent et al. Nutrient balance in ionomics Table 1 | Sequential binary partition (SBP) elaborated to compute balances between groups of nutrients as isometric log-ratios (ilr ). ilr SBP contrasts Balance designation r† s† Ilr computation‡ N P K Ca Mg Fv ilr1 +1 +1 +1 −1 −1 0 [N, P, K | Ca,Mg] 3 2 √ 3x2 3+2 ln g(cN cP cK ) g(ccacMg) ilr2 +1 +1 −1 0 0 0 [N, P | K] 2 1 √ 1×2 1+2 ln g(cN cP ) g(cK ) ilr3 +1 −1 0 0 0 0 [N | P] 1 1 √ 1×1 1+1 ln g(cN ) g(cP ) ilr4 0 0 0 +1 −1 0 [Ca | Mg] 1 1 √ 1×1 1+1 ln g(cca ) g(cMg) Optional 1 1 1 1 1 −1 [N, P, K, Ca, Mg | F υ ] 5 1 √ 5x1 5+1 ln g(cN cP cK ccacMg) g(cFυ ) An optional balance is a computed as a contrast between the geometric mean of nutrient analyses and the filling value, computed by difference. †r and s are counts of +1 and −1 elements in the balance, respectively, ‡g is geometric mean. FIGURE 1 | Mobile-and-fulcrums at mass equilibration point illustrates four hierarchically nested balances that represent a subcomposition or subspace of nutrients in the ionome. and their geometric means g (x) and g (y), as follows (Lovell et al., 2011): ε2 [ln (x) , ln ( y )] = A2 + D × ( ln ( g (x) g (y) ))2 ≥ A2 (5) Numerical biases can be measured as a positive shift from A to ε. Note that ε2 = A2 only when g (x)= g (y). As a result, comput- ing univariate or multivariate distances across raw or ordinary log-transformed concentration data is geometrically irrelevant (Aitchison, 1986). MATERIALS AND METHODS DATASETS The selected fruit species were either wild (lowbush blueberry and cloudberry) or domesticated to achieve high productivity (other species). Nutrient data were collected for kiwifruit [Actinidia deli- ciosa (A Chev) C F Liang et A R Ferguson var deliciosa] grown in the North Island of New Zealand, guava (Psidium guajava), orange (Citrus sinensis), and mango (Mangifera indica) grown in the state of São Paulo, Brazil, and apple (Malus domestica Borkh.), cranberry (Vaccinium macrocarpon Ait.), lowbush blueberry (Vac- cinium angustifolium Ait.), and cloudberry (Rubus chamaemorus L.) from the province of Quebec, Canada. The number of observations was comparable or less com- pared to other studies on mango (n= 525 collected in a sin- gle year: Schaffer et al., 1988), hazelnut (n= 624 collected over 16 year: Alkoshab et al., 1988), sweet cherry (n= 475 collected over 3 year: Davee et al., 1986), and orange (n= 3161 collected over 21 year: Beverly et al., 1984). Leaf samples from 4 to 32 plants were composited in each plot or orchard area to min- imize between-plant variability, compared to one tree in other studies. Marschner (1995) claimed that physiological age of a plant or plant part is, next to mineral nutrient supply, the most impor- tant factor affecting plant nutrient concentration. Across-season samplings (e.g., Han et al., 2011) thus influence nutrient con- centrations (Bould, 1968) as well as ratios (Güsewell, 2004). The developmental stage for sampling occurs during phases of minimum or indeterminate nutrient changes in the fully devel- oped leaves (Bould, 1968). Therefore sample collection must be completed within a short period of time to minimize sea- sonal variability (Willby et al., 2001). Foliar samplings of fruit- bearing shoots were performed during the reproductive stages either at full bloom (guava, mango), after flowering (kiwifruit), during fruit development (orange, apple), during fruit mat- uration (cranberry, blueberry), or from fruit set to maturity (cloudberry). Two to three of the youngest fully expanded leaves were col- lected from 32 vines (excluding young vines and sick leaves) on the second lateral cane within 0 to 4 weeks in 908 commercial “Hort 16a Gold” and “Hayward” kiwifruit orchards in the North Island of New Zealand during the 2002–2010 period. Fruit yield averaged 31800 kg ha−1. The climate is subtropical humid, and soils are Andisols of volcanic origin. Guava, mango, and orange yields and nutrient data were col- lected in the state of São Paulo, Brazil. Thirty pairs of leaves around each of 25 trees were composited (Quaggio et al., 1997). A survey was conducted on 137 irrigated “Paluma” guava orchards (three cycles per 2 years) during the 2009–2010 and 2010–2011 produc- tion cycles. Fruit yield averaged 56155 kg ha−1. From 2009 to 2011, leaf data were collected in 95 mango orchards where varieties “Espada,”“Palmer,”and“Tommy”were grown. Fruit yield averaged 15700 kg ha−1. Foliar samples were collected between 1978 and Frontiers in Plant Science | Plant Nutrition March 2013 | Volume 4 | Article 39 | 4 http://www.frontiersin.org/Plant_Nutrition http://www.frontiersin.org/Plant_Nutrition/archive Parent et al. Nutrient balance in ionomics 2005 in 104 orange orchards producing the varieties “Valencia,” “Hamlin,”“Pêra,” and “Natal.” Fruit yield averaged 49300 kg ha−1. The climate is subtropical humid, and the soils are Oxisols and Ultisols of basaltic origin. Apple data of the “Morspur McIntosh” variety were obtained from an N, P, K, Ca and Mg fertilizer trial (576 observations) established in southwestern Quebec, Canada (Parent and Granger, 1989). Ten to 30 leaf samples were collected in the middle of the annual growth. Fruit yield averaged 33600 kg ha−1. Climate is temperate humid continental and soils are Spodosols of morainic origin. Cranberry (cv.“Stevens”) yield and nutrient data were collected at five sites in Central Quebec, Canada (Parent and Marchand, 2006) in 2000, 2001, and 2002, for a total of 149 observations. One- hundred leaves from current season stems were sampled randomly in a 1-m2 plot. Berry yield averaged 28200 kg ha−1. The climate is temperate humid continental, and soils are Spodosols of marine origin. Yield and nutrient of lowbush blueberry totaling 345 observa- tions were collected from 2001 to 2006 in eight commercial fields in northern Quebec, Canada (Lafond, 2009). Leaf tissues from 25 randomly selected stems were sampled in the 2001, 2003, and 2005 sprout (vegetative) years in 50-m2 plots and composited. Berry yield averaged 3600 kg ha−1. The climate is cold, and soils are Spodosols developed on deltaic deposits. In 2009, cloudberry leaves were collected in 86 stands of con- trasting productivity along the Lower North Shore of the St. Lawrence River, Quebec, Canada (Hébert-Gentile et al., 2011). Six shoots were randomly selected in 5-m2 plots. The median fruit yield was 35 kg ha−1. The climate is cold, and soils are Histosols developed on ombrotrophic peat lands covered with sphagnum (wetter areas) or lichens (drier areas). TISSUE ANALYSIS Tissue P, K, Ca, and Mg levels in the leaves of kiwifruit were deter- mined by plasma emission spectroscopy after microwave digestion (Blackmore et al., 1987). Total N was determined by dry com- bustion using a Leco CNS-2000 analyzer (Leco, St. Joseph, MI, USA). For guava, orange and mango, tissue N was determined by micro-Kjeldahl and P, K, Ca, and Mg by ICP-OES after diges- tion in a mixture of nitric and perchloric acids (Bataglia et al., 1983; Jones and Case, 1990). For apple, total N was determined by micro-Kjeldahl and other nutrients colometrically (P) or by AA spectrophotometry (K, Ca, Mg). For cranberry, blueberry, and cloudberry leaves, total N was determined by micro-Kjeldahl digestion or by Leco CNS-2000 combustion. Other elements were quantified by colorimetry (P), AA spectrometry, or ICP-OES after digestion in a mixture of perchloric and nitric acids (Jones and Case, 1990). STATISTICAL ANALYSIS Statistical computations were conducted in the R statistical envi- ronment (R Development Core Team, 2011). Compositional data analysis was conducted using the R “compositions” package (van den Boogaart et al., 2011). Multivariate outliers were removed for robust multivariate analysis (Filzmoser et al., 2008) using the Mahalanobis distance at a 0.01 level of significance with the R “mvoutlier” package (Filzmoser and Gschwandtner, 2011). Data distribution was tested with the Anderson–Darling nor- mality test (Thode, 2002) using the “nortest” package (Gross, 2006). Spurious correlations were reported as Pearson correla- tion coefficients. Discriminant analysis (DA) was conducted with the R “ade4” package (Chessel et al., 2011) to compare the clas- sification of plant nutrient signatures of wild and domesticated species. RESULTS DISTRIBUTION, SCALE-DEPENDENCY, AND SPURIOUS CORRELATIONS The Euclidean distance computed across ordinary log- transformed concentration data was higher and more dispersed compared to the Aitchison distance across balances (Figure 2). This discrepancy is a measure of numerical biases in multivariate analysis of compositional data using ordinary log transformations. Moreover, 70% of the ilrs were normally distributed (p- value< 0.01). The [N | P] balance was the most frequently diag- nosed as non-normally distributed. Most other balances (84%) were normally distributed across species. By comparison, only 33 and 35% of the raw or ordinary log-transformed concentrations values, respectively, were normally distributed. Data distributions of nutrient concentrations and balances (ilr) are presented in the form of box plots in Figure 3 for the eight species. The mean of ilr often differed between species as shown by non-overlapping ranges. Besides, correlation coefficients changed in magnitude, sign, or probability level depending on the choice of the scale of nutri- ent expressions (sum of nutrients vs. dry matter basis) (Table 2). Scale-dependency causes a serious problem of interpretation when statistical analyses are based on the covariance or correlation matrix. FIGURE 2 | Numerical biases are illustrated by the inflated Euclidean distance across ln-transformed five nutrient concentrations compared to ilr transformation across four balances. A.d., kiwifruit [Actmidia deliciosa (A Chev) C F Liang et A R Ferguson var deliciosa]; C.s., orange (Citrus sinensis); M.d., apple (Malus domestica Borkh.); M.i., mango (Mangifera indica); P.g., guava (Psidium guajava); R.c., cloudberry (Rubus chamaemorus L.); V.a., lowbush blueberry (Vaccinium angustifolium Ait.); V.m., cranberry (Vaccinium macrocarpon Ait.). www.frontiersin.org March 2013 | Volume 4 | Article 39 | 5 http://www.frontiersin.org http://www.frontiersin.org/Plant_Nutrition/archive Parent et al. Nutrient balance in ionomics A B FIGURE 3 | Boxplots of ionomes of eight fruit plant species (A) across nutrient concentrations and (B) across ilr balances. A.d., kiwifruit [Actmidia deliciosa (A Chev) C F Liang et A R Ferguson var deliciosa]; C.s., orange (Citrus sinensis); M.d., apple (Malus domestica Borkh.); M.i., mango (Mangifera indica); P.g., guava (Psidium guajava); R.c., cloudberry (Rubus chamaemorus L.); V.a., lowbush blueberry (Vaccinium angustifolium Ait.); V.m., cranberry (Vaccinium macrocarpon Ait.). Table 2 | Correlation matrices of nutrient data of Malus domestica computed across two scales: dry matter content and N-P-K-Ca-Mg. Scale Nutrients N P K Ca Mg Data scaled on dry matter content (common expression) N 1 0.023 0.068ns 0.232** 0.271** P 1 −0.003ns 0.138** 0.220** K 1 −2.38** −0.205** Ca 1 0.080ns Mg 1 Data scaled on the sum (N+P+K+Ca+Mg) N 1 −0.029ns −0.591** −0.245** 0.293** P 1 −0.219** −0.003ns 0.200** K 1 −0.574** −0.455** Ca 1 −0.017ns Mg 1 ns, *, **: non-significant and significant at the 0.05 and 0.01 levels, respectively. DISCRIMINANT ANALYSIS The DAs returned different schemes whether nutrients were expressed as raw concentrations or their ordinary log transfor- mations (Figures 4A,B). Large semitransparent ellipses enclosing swarms of data points represent regions that include 95% of the theoretical distribution of canonical scores for each ionome. The swarms of wild and domesticated species (large ellipses) overlapped using raw concentration data, but were separated using ordinary log transformation of concentrations. The smaller plain white ellipses represent the confidence regions about the mean of canonical scores at the 95% confidence level. Plain white ellipses related to A.d. (kiwifruit) and V.a. (lowbush blueberry) were too small to be visible. Mean nutrient signatures differed significantly between species because the white ellipses did not Frontiers in Plant Science | Plant Nutrition March 2013 | Volume 4 | Article 39 | 6 http://www.frontiersin.org/Plant_Nutrition http://www.frontiersin.org/Plant_Nutrition/archive Parent et al. Nutrient balance in ionomics A B C D FIGURE 4 | Discriminant analysis of ionomes by species using (A) raw concentrations, (B) ln-transformed concentration values, (C) additive log-ratios, and (D) isometric log-ratio balances. Large semitransparent ellipses that enclose swarms of data points represent regions that include 95% of the theoretical distribution of canonical scores for each species. Smaller plain white ellipses represent confidence regions about means of canonical scores at 95% confidence level. Empty ellipses represent data swarms for wild and domesticated species, respectively. A.d., kiwifruit [Actmidia deliciosa (A Chev) C F Liang et A R Ferguson var deliciosa]; C.s., orange (Citrus sinensis); M.d., apple (Malus domestica Borkh.); M.i., mango (Mangifera indica); P.g., guava (Psidium guajava); R.c., cloudberry (Rubus chamaemorus L.); V.a., lowbush blueberry (Vaccinium angustifolium Ait.); V.m., cranberry (Vaccinium macrocarpon Ait.). overlap, indicating plant-specific ionomes. Eigen vectors were sim- ilar between raw and ordinary log-transformed data, where K and Ca loaded most on the first axis. While DAs of the ordinary log, alr and ilr representations led to a separation between wild and domesticated species, the swarms of species were positioned differently in the Euclidean space (Figures 4C,D). The unbiased alr- and ilr-based DAs were almost identical. Differences are imputed to different outlier detec- tion results caused by the different geometries of alr and ilr. Figures 4C,D showed that some ionome distributions (semi- transparent gray ellipses) overlapped, while the confidence regions about means (white ellipses) differed significantly between species. In the alr-based DA, [Mg | N] and [P | N] loaded the most on the first axis. In the ilr-based DA, [N, P | K], related to nutrient man- agement in agroecosystems, and [Ca | Mg], related to geographical position as well as soil liming in agroecosystems, loaded the most on the first axis. Although the way nutrient balances are arranged into alr or ilr variables produced almost identical DAs, interpre- tation of results depended on data representation as log ratios and this emphasizes the importance of sound data representations when conducting multivariate analysis of compositional data. On the other hand, the small ellipses of species were at sig- nificant distance from each other (p< 0.05). In a finer analysis at cultivar level, DA of the ionomes that were averaged across cultivars of orange and mango (Figure 5) indicated significant differences (p< 0.05) between means of discriminant scores of orange cv. “Hamlin” and others, and between mango cvs,“Palmer,” and “Tommy,” indicating genotypic differences or phenotypic adjustment of each species to local factors. DISCUSSION UNBIASED ANALYSIS OF PLANT IONOMES The DAs performed using unstructured raw or ordinary log- transformed concentration data posed serious interpretation problems in the multivariate analysis of plant ionomes. First, the normality assumption is violated intrinsically by the constrained compositional space. Second, as a result of scale-dependency, the multivariate analysis can differ by simply changing the dry mass basis for another denominator such as the wet mass (Walworth and Sumner, 1988) or the sum of nutrients. Third, one may con- clude that Ca and K concentrations are the most discriminant variables, but K concentration is inherently connected to Ca in plant nutrition (Wilkinson et al., 2000). Indeed, K bears redun- dant information about Ca because K and Ca interact in the plant and are thus inherently correlated to each other: indeed, Ca may decrease as K concentration increases in the confined compositional space as driven by K antagonism or luxury con- sumption (Marschner, 1995). Compositional data analysis avoids www.frontiersin.org March 2013 | Volume 4 | Article 39 | 7 http://www.frontiersin.org http://www.frontiersin.org/Plant_Nutrition/archive Parent et al. Nutrient balance in ionomics FIGURE 5 | Discriminant analysis of ionomes by cultivar using isometric log-ratio balances. Large semitransparent ellipses that enclose swarms of data points represent regions that include 95% of the theoretical distribution of canonical scores for each cultivar. Smaller plain white ellipses represent confidence regions about means of canonical scores at 95% confidence level. Empty ellipses represent data swarms for mango and orange species, respectively. Orange (Citrus sinensis): H, Hamlin; N, Natal; P, Pera; V, Valencia. Mango (Mangifera indica); P, Palmer; T, Tommy. redundancy by relating Ca to K in linearly independent (i.e., orthogonally arranged) log-ratios. In addition to the numerical advantages of ilr discussed above, nutrient balances also reflect nutrient interactions, which are gen- erally neglected in the multivariate analysis of plant ionomes. The balance concept (1) relates nutrients to each other, hence cap- turing nutrient interactions, (2) avoids the need for the ceteris paribus assumption of other nutrients being equal by adjusting any nutrient or group of nutrients to others, and (3) provides a more holistic stand-alone approach illustrated by a pan balance design (Figure 1) and synthesized by an Aitchison or Mahalanobis distance to facilitate interpreting nutrient inter-relationships as looked after in the concluding remarks of recent studies (Han et al., 2011). PLANT NUTRIENT SIGNATURES At ecosystem level, the soil substrate influences the distribution of terrestrial plants while genotype adaptation and genetic manipu- lation greatly improved crop performance in nutritionally diverse habitats (Epstein and Bloom, 2005). Wild and domesticated fruit species acquire: allocate nutrients differently and this must impact on the way plant nutrition experiments are designed and nutrients are diagnosed and managed in terms of nutrient requirements and timeframe for observable effects of nutrient supply on ionomes. Wild species have lesser and slower response to nutrient supply compared to domesticated species (Lafond, 2009; Hébert-Gentile et al., 2011). Despite similar fundamental physiological mecha- nisms involved in nutrient acquisition, wild and domesticated species differ markedly in nutrient allocation between roots, stems, and the harvested part (Chapin, 1980; Jackson and Koch, 1997). The main adaptation of wild species to infertile soils appears to be to constrain growth rate to the resources available with- out apparent dysfunction (Chapin, 1980). Low nutrient absorp- tion rates allow wild species to survive in nutrient-limiting, slow ion-diffusing, and stressful environments where low phenotypic plasticity maintains high root-to-shoot ratios despite occasional nutrient flushes. Domesticated species have been selected for desir- able traits under conditions of high soil fertility and thus often respond to low nutrient availability with very low concentrations and visual deficiency symptoms (Chapin, 1989). Domesticated species are most often bred for high productivity under relatively luxurious environments, where there is little selective advantage in efficient nutrient use, leading to high phenotypic plasticity (Chapin, 1980, 1989). Nutrient balances are more meaningful measures of nutrient signature in this context, because nutrient imbalance caused by shortage of certain nutrients or luxury con- sumption of others can be detected as large multivariate distance from a landmark composition. As expected, the alr- and ilr-based DA showed two broad categories of ionomes, the wild and the domesticated ionomes. However, this classification could be interpreted as an effect of environmental conditions or sampling protocols rather than selec- tion pressure, but our results did not support such hypothesis. Lowbush blueberry is a wild species growing in Spodosols. Cran- berry is also grown in Spodosols and fertilized similarly to lowbush blueberry, but is much more productive due to domestication. Both species were sampled at the same developmental stage. Cran- berry is being selected for commercially viable traits since 1835 and across the twentieth century (Roper and Vorsa, 1997). As a result, the cranberry showed more acquaintance with domesticated than wild species, as confirmed by the ilr-based DA. On the other hand, although cranberry, apple, orange, mango, guava, and kiwifruit were grown in very contrasting environments, their large ellipses overlapped and were neatly separated from cloudberry and low bush blueberry ellipses, indicating human vs. natural selection pressure, respectively. At cultivar level, the ionomes of cultivars of orange and mango grown in Ultisols and Oxisols in the state of São Paulo were differentiated by the balance model, but the cause of these dif- ferences could not be established with the present data set. In case of high phenotypic plasticity of genotypes to nutrient supply, recent research in agronomy showed that plant ionomes can be tightly supervised by balance response models and critical hyper- ellipsoids in the Euclidean space (Hernandes et al., 2012; Parent et al., 2012a; Marchand et al., 2013). CONCLUSION This paper presents a novel numerical solution to conduct unbi- ased multivariate analyses of plant ionomes. The ilrs are orthog- onally arranged log contrasts that rectify nutrient interactions of interest. The use of ilr balances avoids distortion due to the important properties of compositional data such as redundancy, non-normal distribution, and scale-dependency. As shown in this paper, ignoring these properties and related spurious correlations may lead to biased multivariate analyses of plant ionomes. Our finding is fundamental to plant nutritionists, physiologists, ecol- ogists, and agronomists who attempt to classify or diagnose the ionomes of wild and domesticated species. There is a need for paradigm shift in future research. The concept of growth-limiting nutrient concentrations, supported by the “Law of minimum” and illustrated by Liebig’s barrel, should be replaced by a concept Frontiers in Plant Science | Plant Nutrition March 2013 | Volume 4 | Article 39 | 8 http://www.frontiersin.org/Plant_Nutrition http://www.frontiersin.org/Plant_Nutrition/archive Parent et al. Nutrient balance in ionomics of growth-limiting nutrient balances illustrated by a pan bal- ance design, where groups of elements are balanced optimally in weighing pans. This robust nutrient balance concept pro- vides a structured and holistic approach to the classification of plant ionomes. Developing other suitable nutrient balances in plant nutrition studies is challenging. Obviously, many studies conducted so far in plant ionomics should be revisited. Future eco- logical and agronomic applications of ilr compositional models appear to be numerous. ACKNOWLEDGMENTS This project was funded by the Natural Sciences and Engineering Council of Canada (CG-2254 and CRDPJ 385199–09), the Fun- dação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), the Brazilian Coordinação de Aperfeiçoamento de Pessoal de Nivel Superior (CAPES), the Spanish Ministry of Education and Sci- ence (MTM2009-13272 and CSD2006-00032), and the Agència de Gestió d’Ajuts Universitaris i de Recerca of the Generalitat de Catalunya (2009SGR424). REFERENCES Aerts, R., and Chapin, F. S. (2000). The mineral nutrition of wild plants revisited: a re-evaluation of processes and patterns. Adv. Ecol. Res. 30, 1–67. Aitchison, J. (1986). The Statistical Analysis of Compositional Data. Lon- don: Chapman and Hall. Aitchison, J., and Greenacre, M. (2002). Biplots of compositional data. J. R. Stat. Soc. Ser. C Appl. Stat. 51, 375–392. Alkoshab, O., Righetti, T. L., and Dixon, A. R. (1988). Evaluation of DRIS for judging the nutritional status of hazelnuts. J. Am. Soc. Hortic. Sci. 113, 643–647. Bacon-Shone, J. (2011). “A short his- tory of compositional data analy- sis,” in Compositional Data Analy- sis: Theory and Applications, eds V. Pawlowsky-Glahn and A. Buccianti (New York: John Wiley and Sons), 3–11. Bataglia, O. C., Furlani, A. M. C., Teix- eira, J. P. F., Furlani, P. R., and Gallo, J. R. (1983). Métodos de Análise Química de Plantas. Camp- inas: Instituto Agronômico. Baxter, I. R., Vitek, O., Lahner, B., Muthukumar, B., Borghi, M., Mor- rissey, J., et al. (2008). The leaf ionome as a multivariable system to detect a plant’s physiological sta- tus. Proc. Natl. Acad. Sci. U.S.A. 105, 12081–12086. Beaufils, E. R. (1973). “Diagnosis and recommendation integrated system (DRIS),” in Soil Science, Bulletin, 1 (Pietermaritzburg: University of Natal), 1–132. Bergmann, W. (1988). Ernährung sstörungen bei Kulturpflanzen, 2. Auflage. Stuttgart: Gustav Fisher Verlag. Beverly, R. B., Stark, J. C., Ojala, J. C., and Embleton, T. W. (1984). Nutri- ent diagnosis of “Valencia” oranges by DRIS. J. Am. Soc. Hortic. Sci. 109, 649–654. Blackmore, L. C., Searle, P. L., and Daly, B. K. (1987). Methods for Chemical Analysis of Soils. New Zealand Soil Bureau, DSIR, Scientific Report 80. Auckland. Bould, C. (1968). Leaf analysis as a diagnostic method and advisory aid in crop production. Exp. Agric. 4, 17–27. Buccianti, A. (2011). “Natural laws gov- erning the distribution of the ele- ments in geochemistry: the role of the log-ratio approach,” in Com- positional Data Analysis: Theory and Applications, eds V. Pawlowsky- Glahn and A. Buccianti (New York: John Wiley and Sons), 255–266. Chapin, S. III. (1980). The mineral nutrition of wild plants. Ann. Rev. Ecol. Syst. 11, 233–260. Chapin, S. III. (1989). Ecological aspects of plant nutrition. Adv. Plant Nutr. 3, 161–191. Chayes, F. (1960). On correlation between variables of constant sum. J. Geophys. Res. 65, 4185–4193. Chessel, D., Dufour, A. B., and Dray, S. (2011). ade4: Analysis of Ecolog- ical Data: Exploratory and Euclid- ean methods in Environmental sci- ences. R package version 1.4– 17. Available at: http://CRAN.R- project.org/package=ade4 Conn, S., and Gilliham, M. (2010). Comparative physiology of elemen- tal distributions in plants. Ann. Bot. 105, 1081–1102. Davee, D. E., Righetti, T. L., Fallahi, E., and Robbins, S. (1986). An evalua- tion of the DRIS approach for iden- tifying mineral limitations on yield in “Napolean” sweet cherry. J. Am. Soc. Hortic. Sci. 111, 988–993. Egozcue, J. J., and Pawlowsky-Glahn, V. (2005). Groups of parts and their balances in compositional data analysis. Math. Geol. 37, 795–828. Egozcue, J. J., and Pawlowsky-Glahn, V. (2006). “Simplicial geometry for compositional data,” in Composi- tional Data Analysis: Theory and Applications, eds V. Pawlowsky- Glahn, G. Mateu-Figueras, and A. Buccianti (London: Geological Soci- ety of London), 145–160. Egozcue, J. J., Pawlowsky-Glahn, V., Mateu-Figueras, G., and Barceló- Vidal, C. (2003). Isometric log- ratio transformations for composi- tional data analysis. Math. Geol. 35, 279–300. Epstein, W., and Bloom, A. J. (2005). Mineral Nutrition of Plants: Princi- ples and Perspectives, 2nd Edn. Sun- derland, MA: Sinauer Associates. Filzmoser, P., and Gschwandtner, M. (2011). mvoutlier: Multivari- ate Outlier Detection Based on Robust Methods. R package version 1.9.4. Available at: http://CRAN.R- project.org/package=mvoutlier Filzmoser, P., and Hron, K. (2011). “Robust statistical analysis,” in Com- positional Data Analysis: Theory and Applications, eds V. Pawlowsky- Glahn and A. Buccianti (New York: John Wiley and Sons), 57–72. Filzmoser, P., Hron, K., and Reimann, C. (2009a). Univariate statistical analysis of environmental (compo- sitional) data: problems and pos- sibilities. Sci. Total Environ. 407, 6100–6108. Filzmoser, P., Hron, K., and Reimann, C. (2009b). Principal component analysis for compositional data with outliers. Environmetrics 20, 621–632. Filzmoser, P., Maronna, R., and Werner, M. (2008). Outlier identification in high dimensions. Comput. Stat. Data Anal. 52, 1694–1711. Giampietro, M. (2004). Multi-Scale Integrated Analysis of Agroecosystems. Boca Raton, FL: CRC Press. Gross, J. (2006). nortest: Five Omnibus Tests for the Composite Hypothe- sis of Normality. R package version 1.0. Available at: http://CRAN.R- project.org/package=nortest Güsewell, S. (2004). N:P ratios in ter- restrial plants: variation and func- tional significance. New Phytol. 164, 243–266. Han, W. X., Fang, J. Y., Reich, P. B., Woodward, F. I., and Wang, Z. H. (2011). Biogeography and variabil- ity of eleven mineral elements in plant leaves across gradients of cli- mate, soil and plant functional type in China. Ecol. Lett. 14, 788–796. Hébert-Gentile, V., Naess, S. K., Par- ent, L. E., and Lapointe, L. (2011). Organo-mineral fertilization in nat- ural peatlands of the Quebec North- Shore, Canada: dispersion in soil and effects on cloudberry growth and fruit yield. Acta Agric. Scand. B Soil Plant Sci. 61(Suppl. 1), 8–17. Hernandes, A., Parent, S. E., Natale, W., and Parent, L. E. (2012). Balanc- ing guava nutrition with fertiliza- tion and liming. Rev. Bras. Frutic. 34, 224–1234. Ingestad, T. (1987). New concepts on soil fertility and plant nutrition as illustrated by research on for- est trees and stands. Geoderma 40, 237–252. Jackson, L. E., and Koch, G. W. (1997). “The ecophysiology of crops and their wild relatives,” in Ecology in Agriculture, ed. L. E. Jackson (San Diego: Academic Press), 3–37. Jones, J. B. Jr., and Case, V. W. (1990). “Sampling, handling, and analyzing plant tissue samples,” in Soil Test- ing and Plant Analysis, 3rd Edn, ed. R. L. Westerman (Madison, WI: Soil Science Society of America, Inc.), 389–427. Körner, C. (2011). The grand challenges in functional plant ecology. Front. Plant Sci. 2:1. doi:10.3389/fpls.2011.00001 Lafond, J. (2009). Optimum leaf nutri- ent concentrations of wild lowbush blueberry in Quebec. Can. J. Plant Sci. 89, 341–347. Lagatu, H., and Maume, L. (1934). Le diagnostic foliaire de la pomme de terre. Ann. Éc. Natl. Agron. Montpel- lier (France) 22, 50–158. Lahner, B., Gong, J., Mahmoudian, M., Smith, E. L., Abid, K. B., Rogers, E. E., et al. (2003). Genomic scale profiling of nutrient and trace ele- ments in Arabidopsis thaliana. Nat. Biotechnol. 21, 1215–1221. Loladze, I., and Elser, J. J. (2011). The origins of the Redfield nitrogen-to- phosphorus ratio are in a homoeo- static protein-to-rRNA ratio. Ecol. Lett. 14, 244–250. Lovell, D., Müller, W., Tayler, J., Zwart, A., and Helliwell, C. (2011). “Pro- portions, percentages, ppm: do the molecular biosciences treat compo- sitional data right?”in Compositional Data Analysis: Theory and Applica- tions, eds V. Pawlowsky-Glahn and A. Buccianti (New York: John Wiley and Sons), 193–207. www.frontiersin.org March 2013 | Volume 4 | Article 39 | 9 http://CRAN.R-project.org/package$=$ade4 http://CRAN.R-project.org/package$=$ade4 http://CRAN.R-project.org/package$=$mvoutlier http://CRAN.R-project.org/package$=$mvoutlier http://CRAN.R-project.org/package$=$nortest http://CRAN.R-project.org/package$=$nortest http://dx.doi.org/10.3389/fpls.2011.00001 http://www.frontiersin.org http://www.frontiersin.org/Plant_Nutrition/archive Parent et al. Nutrient balance in ionomics Marchand, S., Parent, S.-É., Deland, J. P., and Parent, L. E. (2013). Nutrient signature of Quebec (Canada) cran- berry (Vaccinium macrocarpon Ait.). Rev. Bras. Frutic. 35. (in press). Marschner, H. (1995). Mineral Nutri- tion of Higher Plants. New York: Academic Press. Mateu-Figueras, G., Pawlowsky-Glahn, V., and Egozcue, J. J. (2011). “The principle of working on coordi- nates,” in Compositional Data Analy- sis: Theory and Applications, eds V. Pawlowsky-Glahn, and A. Buccianti (New York: John Wiley and Sons), 31–42. Parent, L. E. (2011). Diagnosis of the nutrient compositional space of fruit crops. Rev. Bras. Frutic. 33, 321–334. Parent, L. E., and Dafir, M. (1992). A theoretical concept of composi- tional nutrient diagnosis. J. Am. Soc. Hortic. Sci. 117, 239–242. Parent, L. E., and Granger, R. L. (1989). Derivation of DRIS norms from a high density apple orchard established in Quebec Appalachi- ans. J. Am. Soc. Hortic. Sci. 114, 915–919. Parent, L. E., and Marchand, S. (2006). Response to phosphorus of cran- berry on high phosphorus testing acid sandy soils. Soil Sci. Soc. Am. J. 70, 1914–1921. Parent, S.-É., Parent, L. E., Rozane, D. E., Hernandes, A., and Natale, W. (2012a). “Nutrient balance as paradigm of plant and soil chemo- metrics,” in Soil Fertility, ed. R. N. Issaka (New York: InTech Publica- tions), 83–114. Available at: http:// www.intechopen.com/books/export /citation/BibTex/soil-fertility/nutri ent-balance-as-paradigm-of-plant- and-soil-chemometricsnutrient- balance-as-paradigm-of-soil-and- Parent, L. E., Parent, S. É., Rozane, D. E., Amorim, D. A., Hernandes, A., and Natale, W. (2012b). “Unbiased approach to diagnose the nutrient status of guava,” in Proceedings of the 3rd International Symposium on Guava and other Myrtaceae ISHS 2012, Vol. 959, eds C. A. F. Santos, S. K. Mitra, and J. L. Griffis (Acta Horticulture), 145–159. Parent, L. E., de Almeida, C. X., Her- nandes, A., Egozcue, J. J., Gülser, C., Bolinder, M. A., et al. (2012c). Compositional analysis for an unbi- ased measure of soil aggregation. Geoderma 179–180, 123–131. Pawlowsky-Glahn, V., and Buccianti, A. (2011). Compositional Data Analysis: Theory and Applications. New York: John Wiley and Sons. Pearson, K. (1897). Mathematical con- tributions to the theory of evolution. On a form of spurious correlation which may arise when indices are used in the measurement of organs. Philos. Trans. R. Soc. Lond. B Biol. Sci. 60, 489–498. Quaggio, J. A., Van Raij, B., and Piza, C. T. Jr. (1997). “Frutíferas” in Recomendações de adubação e calagem para o Estado de São Paulo, 2nd Edn. rev., eds B. van Raij, H. Cantarella, J. A. Quag- gio, and A. M. C. Furlani Bole- tim Técnic #100 (Campinas: Insti- tuto Agronômico/Fundação IAC), 121–125. R Development Core Team. (2011). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing. Version 2.13.1. Available at: http://www.R- project.org Rodgers, J. L., Nicewander, W. A., and Toothaker, L. (1984). Lin- early independent, orthogonal, and uncorrelated variables. Am. Stat. 38, 133–134. Roper, T., and Vorsa, N. (1997). Cran- berry: botany and horticulture. Hor- tic. Rev. (Am. Soc. Hortic. Sci.) 21, 215–249. Salt, E. D., Baxter, I., and Lahner, B. (2008). Ionomics and the study of the plant ionome. Annu. Rev. Plant Biol. 59, 709–733. Schaffer, B., Larson, K. D., Sny- der, G. H., and Sanchez, C. A. (1988). Identification of mineral deficiencies associated with mango decline by DRIS. HortScience 23, 617–618. Tanner, J. (1949). Fallacy of per-weight and per-surface area standards, and their relation to spurious correla- tion. J. Phys. 2, 1–15. Tennakoon, K. U., Chak, W. H., and Bolin, J. F. (2011). Nutritional and isotopic relationships of selected Bornean tropical mistletoe–host associations in Brunei Darus- salam. Funct. Plant Biol. 38, 505–513. Thode, H. C. Jr. (2002). Testing for Nor- mality. New York: Marcel Dekker. Tolosana-Delgado, R., and van den Boogart, K. G. (2011). “Linear models with compositions in R” in Compositional Data Analysis: Theory and Applications, eds V. Pawlowsky-Glahn and A. Buccianti (New York: John Wiley and Sons), 356–371. van den Boogaart, K. G., Tolosana- Delgado, R., and Bren, R. (2011). Compositions: Compositional Data Analysis. R package version 1.10- 2. Available at: http://CRAN.R- project.org/package=compositions Walworth, J. L., and Sumner, M. E. (1987). The diagnosis and recommendation integrated sys- tem (DRIS). Adv. Soil Sci. 6, 149–188. Walworth, J. L., and Sumner, M. E. (1988). Foliar diagnosis: a review. Adv. Plant Nutr. 3, 193–240. Weltje, G. J. (2002). Quantitative analy- sis of dentrial modes: statistically rigourous confidence regions in ternary diagrams and their use in sedimentatry petrology. Earth Sci. Rev. 57, 211–253. White, P. J., and Brown, P. H. (2010). Plant nutrition for sustainable devel- opment and global health. Ann. Bot. 105, 1073–1083. Wilkinson, S. R., Grunes, D. L., and Sumner, M. E. (2000). “Nutrient interactions in soil and plant nutri- tion” in Handbook of Soil Science, ed. M. E. Sumner (Boca Raton, FL: CRC Press), D-89–D-112. Willby, N. J., Pulford, I. D., and Flow- ers, T. H. (2001). Tissue nutri- ent signatures predict herbaceous- wetland community responses to nutrient availability. New Phytol. 152, 463–481. Williams, L., and Salt, D. E. (2009). The plant ionome coming into focus. Curr. Opin. Plant Biol. 12, 247–249. Conflict of Interest Statement: The authors declare that the research was conducted in the absence of any com- mercial or financial relationships that could be construed as a potential con- flict of interest. Received: 07 September 2012; accepted: 13 February 2013; published online: 22 March 2013. Citation: Parent S-É, Parent LE, Egozcue JJ, Rozane D-E, Hernandes A, Lapointe L, Hébert-Gentile V, Naess K, Marc- hand S, Lafond J, Mattos D Jr, Bar- low P and Natale W (2013) The plant ionome revisited by the nutrient bal- ance concept. Front. Plant Sci. 4:39. doi: 10.3389/fpls.2013.00039 This article was submitted to Frontiers in Plant Nutrition, a specialty of Frontiers in Plant Science. Copyright © 2013 Parent , Parent , Egozcue, Rozane, Hernandes, Lapointe, Hébert-Gentile, Naess, Marchand, Lafond, Mattos Jr, Barlow and Natale. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in other forums, provided the original authors and source are credited and subject to any copyright notices concerning any third-party graphics etc. Frontiers in Plant Science | Plant Nutrition March 2013 | Volume 4 | Article 39 | 10 http://www.intechopen.com/books/export/citation/BibTex/soil-fertility/nutrient-balance-as-paradigm-of-plant-and-soil-chemometricsnutrient-balance-as-paradigm-of-soil-and- http://www.intechopen.com/books/export/citation/BibTex/soil-fertility/nutrient-balance-as-paradigm-of-plant-and-soil-chemometricsnutrient-balance-as-paradigm-of-soil-and- http://www.intechopen.com/books/export/citation/BibTex/soil-fertility/nutrient-balance-as-paradigm-of-plant-and-soil-chemometricsnutrient-balance-as-paradigm-of-soil-and- http://www.intechopen.com/books/export/citation/BibTex/soil-fertility/nutrient-balance-as-paradigm-of-plant-and-soil-chemometricsnutrient-balance-as-paradigm-of-soil-and- http://www.intechopen.com/books/export/citation/BibTex/soil-fertility/nutrient-balance-as-paradigm-of-plant-and-soil-chemometricsnutrient-balance-as-paradigm-of-soil-and- http://www.intechopen.com/books/export/citation/BibTex/soil-fertility/nutrient-balance-as-paradigm-of-plant-and-soil-chemometricsnutrient-balance-as-paradigm-of-soil-and- http://www.R-project.org http://www.R-project.org http://CRAN.R-project.org/package$=$compositions http://CRAN.R-project.org/package$=$compositions http://dx.doi.org/10.3389/fpls.2013.00039 http://creativecommons.org/licenses/by/3.0/ http://creativecommons.org/licenses/by/3.0/ http://www.frontiersin.org/Plant_Nutrition http://www.frontiersin.org/Plant_Nutrition/archive The plant ionome revisited by the nutrient balance concept Introduction Theory of compositional data analysis Sample space Numerical biases The log-ratio transformations The alr transformation The ilr transformation Nutrient balances The aitchison distance Materials and methods Datasets Tissue analysis Statistical analysis Results Distribution, scale-dependency, and spurious correlations Discriminant analysis Discussion Unbiased analysis of plant ionomes Plant nutrient signatures Conclusion Acknowledgments References