Bioinformatics Advance Access originally published online on May 12, 2005
Bioinformatics 2005 21(13):3043-3048; doi:10.1093/bioinformatics/bti476
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ANOVA-simultaneous component analysis (ASCA): a new tool for analyzing designed metabolomics data
1Biosystems Data Analysis, Faculty of Sciences, University of Amsterdam Nieuwe Achtergracht 166, 1018 WV Amsterdam, The Netherlands
2TNO Quality of life PO Box 360, 3700 AJ Zeist, The Netherlands
3Center for Medical Systems Biology, LACDR, Leiden University, Gorleaus Laboratories 2300 RA Leiden, The Netherlands
4Heymans Institute of Psychology, DPMG, University of Groningen Grote Kruisstraat 2/1, 9712 TS Groningen, The Netherlands
*To whom correspondence should be addressed.
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Abstract |
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Motivation: Datasets resulting from metabolomics or metabolic profiling experiments are becoming increasingly complex. Such datasets may contain underlying factors, such as time (time-resolved or longitudinal measurements), doses or combinations thereof. Currently used biostatistics methods do not take the structure of such complex datasets into account. However, incorporating this structure into the data analysis is important for understanding the biological information in these datasets.
Results: We describe ASCA, a new method that can deal with complex multivariate datasets containing an underlying experimental design, such as metabolomics datasets. It is a direct generalization of analysis of variance (ANOVA) for univariate data to the multivariate case. The method allows for easy interpretation of the variation induced by the different factors of the design. The method is illustrated with a dataset from a metabolomics experiment with time and dose factors.
Availability: M-files for MATLAB for the algorithm used in this research are available at: http://www-its.chem.uva.nl/research/pac/Software/ or at http://www.bdagroup.nl
Contact: asmilde{at}science.uva.nl
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INTRODUCTION |
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Recent developments in genomics and human systems biology have shown the importance of metabolomics (Clish et al., 2004; Lindon et al., 2004; van der Greef et al., 2003). This is understandable, since metabolomics is a crucial element in bridging the difference between the genotype and phenotype of an organism (Fiehn, 2002). Considerable effort has gone into the development of instrumental methods for metabolite profiling, since it emerged in the 1960s and 1970s in clinical chemistry (Gates and Sweeley, 1978; Jellum, 2001), especially focusing on inborn errors of metabolism. The combination with mass spectrometry (GC–MS) and in particular chemometrics created the basis of the technology of today in metabolomics for mammalian systems (Gaspari et al., 2001; van der Greef et al., 1983). Nuclear Magnetic Resonance (NMR) spectroscopy has become an important component in the field for screening of biofluids (Bales et al., 1984) and an important step was the combination with chemometrics (Gartland et al., 1990). Moreover, methods for handling metabolomics data are receiving increased attention, with respect to both the preprocessing of metabolomics data (Keun et al., 2003; Vogels et al., 1996) and the analysis of data itself (Antti et al., 2002; Jansen et al., 2004; Keun et al., 2004).
Metabolomics datasets are becoming more and more complex. It is not uncommon to measure a multiple of metabolites in body fluids of several animals at different points in time with an underlying experimental design, e.g. different dose groups (Antti et al., 2004; Keun et al., 2004; Lamers et al., 2003). This calls for data analysis methods specifically suited for time-resolved (or ‘longitudinal’), multigroup, multisubject (containing data of multiple animals) and multivariate data.
When a single variable (e.g. a metabolite) is measured as a function of design factors, analysis of variance (ANOVA) is a well established technique to analyze the data (Searle, 1971). When measuring many metabolites simultaneously, generalizations of ANOVA are necessary. In the statistics literature, the classical generalization of ANOVA to multiple variables is multivariate-ANOVA (MANOVA) (Mardia et al., 1979). For the large number of measured variables in a metabolomics experiment, however, MANOVA breaks down owing to problems of singularity of covariance matrices and assumptions that are not fulfilled (Ståhle and Wold, 1990).
In the data analysis literature, mixtures of multivariate analysis and ANOVA have also been reported. One approach performs a principal component analysis of the entire dataset first, and then uses ANOVA on the component score values to test the effects (Bratchell, 1989). This approach has been criticized, since the separate ANOVAs on the score values are not independent (Jackson, 1991). Moreover, the initial principal component analysis does not necessarily distinguish between the groups in the data. Another approach is suggested by using PLS (partial least squares; a popular regression technique for collinear data) to solve the problem (Ståhle and Wold, 1990). However, different suggestions have been made to implement this method depending on whether the coded design variables are regressors (Martens and Martens, 2001) or regressands (Ståhle and Wold, 1990). Moreover, the exact properties of these methods are unknown, since the criterion that is maximized or minimized is not clear. Redundancy analysis using a coded design matrix seems to be a better alternative than the PLS based approach (Van den Brink and Ter Braak, 1999).
In this paper, a new method is presented that can deal with a temporal and/or design structure of complex multivariate datasets, such as those emerging from metabolomics experiments. However, the problems to which ANOVA-simultaneous component analysis (ASCA) can be applied are certainly not limited to metabolomics. Complex multivariate datasets are abundant in the other post-genomic technologies (e.g. transcriptomics, proteomics) and also in many more fields of biological and non-biological research.
ASCA builds upon and generalizes some earlier proposed methods. Two early papers in pomology and botanics realized the importance of distinguishing ‘between’ and ‘within’ factor treatments (Jeffers, 1962; Pearce and Holland, 1960). In the metabolomics literature, the SMART method (Keun et al., 2004) also makes this distinction, but is less general than the method proposed in the current paper. The method proposed in this paper builds on the multilevel component analysis method that was developed in psychometrics (M.E. Timmerman, submitted for publication) and metabolomics (Jansen et al., 2004) and generalizes it for a situation with any design structure underlying the metabolomics data.
The ASCA method will be explained and illustrated with an example from a metabolomics intervention study, where guinea pigs from a strain developing osteoarthritis (OA) were treated with several dosage levels of vitamin C and their urine was analyzed using NMR spectroscopy at several points in time. This study is therefore a typical example of a designed metabolomics experiment. OA is a multifactorial chronic joint disease that is characterized by the progressive destruction of articular cartilage, resulting in impaired movement, pain and ultimately disability (Creamer and Hochberg, 1997). The Hartley outbred strain guinea pig develops spontaneous progressive knee OA starting when they are 
10 months of age, with features similar to the human disease (Bendele, 2001; Huebner et al., 2001). Ascorbic acid has been associated with the slowing of OA progression in guinea pig and human (McAlindon et al., 1996). However, recent studies indicate that vitamin C increases the severity of development of OA in the guinea pig (Kraus et al., 2004). The details of the biological questions regarding this study are published elsewhere (Lamers et al., 2003).
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SYSTEMS AND METHODS |
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Urine samples and data acquisition
This dataset contains information about male Hartley guinea pigs, which develop OA during aging. Beginning at 4 months of age, the guinea pigs are divided randomly into three dose groups to which varying doses of vitamin C are provided: low dose (2.5–3 mg/day), medium dose (30 mg/day) and high dose (150 mg/day). The doses were chosen such that the low dose exceeds the minimum amount to prevent scurvy and the medium dose corresponds to the normal intake of vitamin C. The high dose of vitamin C corresponds to the amount that was shown in previous studies to slow the development of surgically induced OA.
Each dose group consists of six animals. Urine samples are collected at 4, 7, 10 and 12 months, where the samples collected after 4 months are pre-dose samples. Each urine collection was performed for 24 h, to eliminate the influence of diurnal variation of the metabolite composition of the urine. The total dataset consists of 72 samples. These samples were analyzed with NMR spectroscopy and the dataset was prepared as peak listings (NMR spectra) using the standard Varian software (Varian Inc., Palo Alto, CA). The dataset contains spectra with peaks listed at 253 chemical shifts, expressed in parts per million (p.p.m.), that are equal for all spectra. A typical spectrum in this dataset is given in Figure 1.
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The dataset is a subset of a larger dataset. The acquisition of this larger dataset has been described elsewhere (Lamers et al., 2003).
Data analysis
Structure of the dataset
The structure of the dataset is shown in Figure 1. The following indices will be used: j = 1,...,J for the chemical shifts, k = 1,...,K for the time-points at which measurements are taken, h = 1,...,H for the dosage groups (h = 1, low; h = 2, medium and h = 3, high dosage) and ih = 1,...,Ih for the guinea pigs nested within the dosage groups. The guinea pigs the ‘low’ in dosage group are different from those in the other dosage groups (the subindex h on i is used to emphasize this fact). This will be important to realize for the remainder of the paper.
Analysis of variance
An NMR signal at one particular chemical shift j, for one-time point k and for one guinea pig ih (in dosage group h) will be denoted by the scalar xhkihj. Collecting such values xhkihj in matrices Xhih of size (K x J) will be convenient. The construction of these matrices is shown in Figure 1. Considering only an NMR signal at one chemical shift (and therefore dropping the index j for convenience) a reasonable ANOVA model would be
 | (1) |
k represents the effect of the factor ‘time’ that is common for all guinea pigs; (ß)hk represents the interaction of ‘time’ and ‘dose’; (ß
)hkih represents the guinea pig specific contribution. Of these effects, (ß)hk is most important for the biological interpretation. It represents the effect of the dosage measured as deviations from the common time effect k. The contributions (ß)hkih represent the variations on the lowest (individual animal-specific) level, and can be used for significance testing. Classical ANOVA techniques can now be used to estimate the factor effects and test significance. Equation (1) shows a division of variation on several factors. This is the basic idea of ANOVA; variation is separated and assigned to factors. The factor effects can be estimated and tested. ANOVA is capable of doing this by splitting the variations in orthogonal and independent parts (Searle, 1971). This division of the variation into orthogonal contributions is the goal of ASCA also (see below).
Simultaneous component analysis
When analyzing the simultaneous underlying variation in several related datasets, simultaneous component analysis is a useful tool. This method was developed in psychometrics (Ten Berge et al., 1992), but extensions also find their use now in metabolomics (Jansen et al., 2004; M. E. Timmerman, submitted for publication).
Let us suppose that data matrices Xi (Ki x J) are available where measurements on J identical chemical shifts are available at Ki time-points on I animals (the subdivision of the individuals i into different dose groups h is omitted from the explanation of SCA for simplicity; therefore, the used indices are simpler in this section compared with the other sections of this paper). Note that the number of measurement time points for individual i, denoted by Ki, can differ between animals in SCA. Then a reasonable model for simultaneously analyzing these data matrices is
 | (2) |
The model parameters in Equation (2) can be found by solving
 | (3) |
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ANOVA-simultaneous component analysis (ASCA)
The ASCA analog for model (1) is
 | (4) |
In other words, Equation (4) means that the matrix Xhih is separated into contributions from the overall mean (1mT), one SCA model 
describing the overall effect of the factor time, an SCA model 
describing the interaction of dose with time and another SCA model 
describing the interaction of dose, time and guinea pig, which is the contribution to the variation of each individual guinea pig. This is a direct multivariate generalization of Equation (1). Note that since the number of components in each part of this model is low (which is the basic idea: dimension reduction) there is a residual matrix Ehih in Equation (4) that contains the information that is not described by any of the ASCA submodels, whereas such a residual term is not present in Equation (1). To illustrate the basic idea behind ASCA, a toy example is given in Appendix 1 of the supplementary data.
Properties of ASCA
The different parts of the model in Equation (4) are orthogonal to each other when proper constraints are imposed on them. These constraints are
 | (5) |
ASCA-algorithms
The parameters of the ASCA model can be calculated by solving the following least squares problem
 | (6) |
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RESULTS |
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Split-up of variation
An impression of the amount of variation related to the design factors can be obtained by separating this variation into contributions from the different factors. Table 1 shows this separation and it is clear that the dominant part of the variation is at the lowest level (guinea pig-specific contributions). This shows the biological variation between the animals used in the study. Note that Table 1 reports sums of squared deviations from the overall mean and not variances.
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Factor ‘time’
The scores of the first component of the factor ‘time’ are given in Figure 2 [the TK values of Equation (4)]. This first component explains 72% of the variation on the factor ‘time’. The maximum number of components that can be fitted for the factor ‘time’ submodel (and therefore the rank of the factor time variation) is three, since the dataset contains four measurement time-points. Hence, only one component is used to illustrate the variation on this level.
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The scores in Figure 2 indicate that all guinea pigs in the data show an initial increasing and a subsequent decreasing behavior. This trajectory is consistent with the biology of growth for the Hartley guinea pig strain (Huebner et al., 2001). From 4 to 7 months the metabolism of the guinea pigs changes, because during this time they are in the growing phase. Between 7 and 10 months, the guinea pigs are full-grown, which is shown by the leveling off in the time profile from 7 and 10 months. From 10 months onward, the guinea pigs develop OA. The decrease in score of the 12 months samples is supposed to reflect this effect (Lamers et al., 2003).
The loadings P1 belonging to the first component are given in Figure 3. These loadings show the chemical shifts and thus the compounds that are corresponding to the behavior observed in the scores, and can be used for biological interpretation. However, urine is a biological fluid that is used by the body for the excretion of waste products and therefore its contents are difficult to trace back to biology. Nevertheless, metabolites like creatinine (
3.04 and 4.05 p.p.m.), creatine ( 3.04 and 3.95 p.p.m.), glucose ( 3.27, 3.53, 3.60, 3.78, 3.82 and 3.94 p.p.m.), alpha-hydroxybutyrate ( 1.36 p.p.m.), lactate ( 1.32 and 1.34 p.p.m.), glycine ( 3.56 p.p.m.) and acetate ( 1.92 p.p.m.) that change in time may point at altered energy metabolism owing to growth and disease development. These observations are consistent with results that were described previously (Lamers et al., 2003).
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Interaction time x dose
The rank required for this submodel is determined using a scree-test (Cattell, 1966). From this test, the rank of this submodel is determined to be two. A model containing two components explains 65% of the variation corresponding to the factor ‘interaction time x dose’. The first component explains 50% of this variation and the second component explains 15%.
The scores of this submodel should be interpreted as the deviation of each dose group from the ‘time’ factor (
or k in the ANOVA model). The ‘interaction time x dose’ scores for the first component are given in Figure 4. On this component no trend related to vitamin C dose is visible: none of the four measurement time-points show an increasing or decreasing score value for differing vitamin C doses. Also the scores of the second component do not show such a quantitative trend.
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The model results show that the potential of vitamin C in affecting the development of OA is questionable. According to our results, vitamin C has no effect on disease development. Neither the association of vitamin C with the slowing of OA progression in guinea pig and humans (McAlindon et al., 1996) nor the observation that vitamin C could increase the severity of development of OA in the guinea pig (Kraus et al., 2004) can be corroborated with the results of ASCA on the described dataset. However, the results of the ASCA model of this dataset were in agreement with additional clinical measurements that were performed after the experiment on the guinea pigs used in the study. The severity of OA was determined using ‘histology scores’ on the Mankin grading system (Mankin et al., 1971). These scores did not differ between dose groups, which indicates an equal development of OA.
Individual guinea pig contributions
The rank required for the ‘Individual guinea pig contributions’ submodel can also be determined by a scree test. From this test, two components are defined for this submodel. This submodel describes 57% of the variation corresponding to the ‘Individual guinea pig contributions’. The first component explains 45% and the second component explains 12%. The scores of this submodel must be seen as a deviation of each individual from the dose–time interaction.
The scores for the first component of the ‘Individual guinea pig contributions’ submodel are given in Figure 5 for the low, medium and high dose groups. From this figure it is clear that the deviation of the individual profile from the group average profile is largest at 4 and 12 months, i.e. at the start and the end of the experiment. The NMR signals that correspond to this behavior are, amongst others, lactate ( 1.32 and 1.34 p.p.m.), acetate ( 1.92 p.p.m.) and glycine ( 3.56 p.p.m.) that increase and creatinine ( 3.04 and 4.05 p.p.m.) that decrease. The larger interindividual differences at 4 and 12 months may be explained by the fact that at these time-points growth and disease development occur.
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CONCLUSIONS |
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TOPAbstractINTRODUCTIONSYSTEMS AND METHODSRESULTSCONCLUSIONSSUPPLEMENTARY DATAREFERENCES |
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In metabolomics, an increasing amount of datasets becomes available with an underlying design in factors. Currently, no methods are available to analyze such data. The method proposed in this paper called ANOVA-SCA or ASCA for short, fills this gap. It works by separating the variation in the total dataset by parts that can be assigned to contributions of the different factors and interactions thereof.
The ASCA method is illustrated by a real example of an intervention study examining the effect of vitamin C on the development of OA in guinea pigs. This shows how the method works and that interpretation of the resulting components works in the same way as in an ordinary PCA. In the case of a balanced design, the algorithm is simple and comes down to performing PCAs on properly centered and rearranged data. In the case of unbalanced data, a more elaborate algorithm is necessary, but available.
In a follow-up research, questions regarding validation using resampling methods and significance testing of effects will be treated. This will allow not only for the estimation of factor effects but also for judging their reliability and testing their significance.
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SUPPLEMENTARY DATA |
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TOPAbstractINTRODUCTIONSYSTEMS AND METHODSRESULTSCONCLUSIONSSUPPLEMENTARY DATAREFERENCES |
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Supplementary data for this paper is available on Bioinformatics online.
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Acknowledgments |
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The authors would like to acknowledge Jeroen de Groot of TNO Prevention and Health in Leiden, the Netherlands and Virginia B. Kraus of Duke University Medical Center in Durham, NC for providing the guinea pig dataset.
Received on December 23, 2004; revised on April 21, 2005; accepted on April 27, 2005
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