Jérôme Pagès

Multiple factor analysis (AFMULT package)

Abstract Multiple Factor Analysis (MFA) studies several groups of variables (numerical and/or categorical) defined on the same set of individuals. MFA approaches this kind of data according to many points of view already used in others methods as: factor analysis in which groups of variables are weighted, canonical analysis, Procrustes analysis, STATIS, INDSCAL. In MFA, these points of view are considered in a unique framework. This paper presents the different outputs provided by MFA and an example about sensory analysis of wines.

Testing the significance of the RV coefficient

The relationship between two sets of variables defined for the same individuals can be evaluated by the RV coefficient. However, it is impossible to assess by the RV value alone whether or not the two sets of variables are significantly correlated, which is why a test is required. Asymptotic tests do exist but fail in many situations, hence the interest in permutation tests. However, the main drawbacks of the permutation tests are that they are time consuming. It is therefore interesting to approximate the permutation distribution with continuous distributions (without doing any permutation). The current approximations (normal approximation, a log-transformation and Pearson type III approximation) are discussed and a new one is described: an Edgeworth expansion. Finally, these different approximations are compared for both simulations and for a sensory example.

Multiple factor analysis and clustering of a mixture of quantitative, categorical and frequency data

Analysing and clustering units described by a mixture of sets of quantitative, categorical and frequency variables is a relevant challenge. Multiple factor analysis is extended to include these three types of variables in order to balance the influence of the different sets when a global distance between units is computed. Suitable coding is adopted to keep as close as possible to the approach offered by principal axes methods, that is, principal component analysis for quantitative sets, multiple correspondence analysis for categorical sets and correspondence analysis for frequency sets. In addition, the presence of frequency sets poses the problem of selecting the unit weighting, since this is fixed by the user (usually uniform) in principal component analysis and multiple correspondence analysis, but imposed by the table margin in correspondence analysis. The methods main steps are presented and illustrated by an example extracted from a survey that aimed to cluster respondents to a questionnaire that included both closed and open-ended questions.

Which value can be granted to sensory profiles given by consumers? Methodology and results

The introduction of sensory descriptive questions in consumer studies is often criticized. From a data set, provided for the 5th sensometrics meeting, we show that sensory profiles obtained from consumers can have two essential qualities: consensus and reproducibility. This implies that we cannot (a priori) forbid the use of sensory profiles given by consumers. To show this, we use analyses of variance and multiple factor analysis; these methods are useful to obtain a visualization of the data from several panels.

Hierarchical Multiple Factor Analysis: application to the comparison of sensory profiles

An extension of Multiple Factor Analysis (MFA) to the case where the data at hand are organized into a hierarchical structure is discussed. It is called Hierarchical Multiple Factor Analysis (HMFA). This method of analysis balances the role of the groups of variables at each level of the hierarchy and provides outputs that may be interpretable from an overall perspective (overall hierarchical structure) as well as from various perspectives pertaining to the various levels of the hierarchy. It is illustrated on the basis of a real data set which stems from a sensory profiling experiment involving, on the one hand, several trained panels and, on the other hand, an untrained panel. The results of the analysis are reported and with respect to our data they show the extent to which profiles provided by an untrained panel are similar to those provided by trained panels.

Multiple factor analysis combined with PLS path modelling. Application to the analysis of relationships between physicochemical variables, sensory profiles and hedonic judgements

Abstract Multiple Factor Analysis (MFA) highlights the structures common to a set of J groups (or blocks) of variables observed for the same individuals. PLS path modelling allows a search for latent variables, summarising as far as possible one-dimensional blocks of manifest variables while taking account of causal links between the blocks. These two methods can be combined: MFA, as an exploratory analysis, helps to define blocks, being both one-dimensional and as well-correlated as possible, on which PLS path modelling is performed. In this paper, we present MFA in detail and PLS path modelling more briefly. We also mention some links between MFA, PLS path modelling and PLS regression. A detailed presentation of a sensory analysis example will illustrate the proposed methodology.

Multiple imputation in principal component analysis

The available methods to handle missing values in principal component analysis only provide point estimates of the parameters (axes and components) and estimates of the missing values. To take into account the variability due to missing values a multiple imputation method is proposed. First a method to generate multiple imputed data sets from a principal component analysis model is defined. Then, two ways to visualize the uncertainty due to missing values onto the principal component analysis results are described. The first one consists in projecting the imputed data sets onto a reference configuration as supplementary elements to assess the stability of the individuals (respectively of the variables). The second one consists in performing a principal component analysis on each imputed data set and fitting each obtained configuration onto the reference one with Procrustes rotation. The latter strategy allows to assess the variability of the principal component analysis parameters induced by the missing values. The methodology is then evaluated from a real data set.

A principal axes method for comparing contingency tables: MFACT

A new methodology is introduced for comparing the structures of several contingency tables. The latter, built up from different samples or populations, present the same rows and different columns (or vice versa). This methodology combines some aspects of principal axes methods (global maximum dispersion axes), canonical correlation techniques (canonical dispersion axes) and Procrustes analysis (superimposed representations) but takes into account the particularities of contingency tables in order to extend correspondence analysis to multiple contingency tables. Two main problems arise: the differences between the margins of the common dimension and the need for balancing the influence of the different tables in global processing. A study of the four structures induced on Spanish regions by mortality causes (by gender) and by age distribution (by gender), in conjunction, will illustrate the methodology.

A new unsupervised gene clustering algorithm based on the integration of biological knowledge into expression data.

BackgroundGene clustering algorithms are massively used by biologists when analysing omics data. Classical gene clustering strategies are based on the use of expression data only, directly as in Heatmaps, or indirectly as in clustering based on coexpression networks for instance. However, the classical strategies may not be sufficient to bring out all potential relationships amongst genes.ResultsWe propose a new unsupervised gene clustering algorithm based on the integration of external biological knowledge, such as Gene Ontology annotations, into expression data. We introduce a new distance between genes which consists in integrating biological knowledge into the analysis of expression data. Therefore, two genes are close if they have both similar expression profiles and similar functional profiles at once. Then a classical algorithm (e.g. K-means) is used to obtain gene clusters. In addition, we propose an automatic evaluation procedure of gene clusters. This procedure is based on two indicators which measure the global coexpression and biological homogeneity of gene clusters. They are associated with hypothesis testing which allows to complement each indicator with a p-value.Our clustering algorithm is compared to the Heatmap clustering and the clustering based on gene coexpression network, both on simulated and real data. In both cases, it outperforms the other methodologies as it provides the highest proportion of significantly coexpressed and biologically homogeneous gene clusters, which are good candidates for interpretation.ConclusionOur new clustering algorithm provides a higher proportion of good candidates for interpretation. Therefore, we expect the interpretation of these clusters to help biologists to formulate new hypothesis on the relationships amongst genes.

Presentation of Correspondence Analysis and Multiple Correspondence Analysis with the Help of Examples

Correspondence analysis and multiple correspondence analysis are, like principal component analysis, factor analysis methods.