Abdi, H., Williams, L.J. and Valentin, D. (2013). Multiple Factors Analysis: Principal Component Analysis for multitable and multiblock data sets. WIREs Comput. Stat 2013, doi:10.1002/wics.1246
Comments by Tormod Næs (November 2025)
Background
In sensory science, both in designed experiments and in survey studies, there is a need for proper statistical handling and analysis of data. This holds for simple data sets with only a few variables to be related to each other as well as for large multidimensional data. In some cases, the focus is on overall consensus aspects (‘averaged’ over assessors) and in other cases main attention is on individual differences among assessors. In most cases one is interested in differences and similarities between the objects/samples tested. The statistical methodology called multiple factor analysis (MFA), described in the paper highlighted here (Abdi et al., 2013) is useful for both these aspects simultaneously. This is a method with general applicability, also far outside sensory, but has for several reasons become particularly important and influential in the analysis of sensory data. This holds both for descriptive panel data and for consumer data. A major reason for this is that it is very versatile and quite easy to understand and use and can be used in connection with a number of sensory methodologies, for instance QDA, Projective mapping, CATA, RATA, PSP (Varela and Ares, 2012, Næs et al., 2018). It is hard to find a publication in these areas without at least a reference to MFA. It has become a cornerstone in sensory science.
The basic idea is relatively straightforward; MFA is simply based on a Principal Component Analysis (PCA) using as input the (horizontally) concatenated data table for all assessors. The rows in the concatenated data set usually correspond to samples and the columns to the different variable/assessor combinations. This technique is known under different names in statistics, for instance Tucker-1 (Tucker, 1966) and SUM-PCA (Smilde et al, 2003), but MFA is unique in the way it weighs the contribution of the different original data tables. In MFA, each data table is simply divided by the first singular value of the data table before concatenation. This number depends on the spread of the data points within the table in the same way as standard deviation describes variability of one single variable. Variables within a table may also be standardized before concatenation if that is natural from the context, like in for instance PCA. After calculation of the consensus configuration of scores, the data from each individual data matrix can be projected onto it for interpretation of individual differences.
Why important?
The paper by Abdi et al (2013) is of special importance in this context. It gives a broad and instructive presentation of MFA, both conceptually and technically. MFA is motivated very well and described in detail, without drowning into too many technicalities. It is readable both for newcomers as well as experienced researchers in the area. It is recommended also for those who use the method regularly if they want to obtain a deeper understanding of the method itself and its potential in sensory science. The paper requires some statistical knowledge, but can be read also without going into all technical details.
Design and results
The paper is formulated as an overview paper with focus on both how to obtain consensus information as well as information about individual differences, i.e. differences between the individual data tables.
The method is presented in equivalent ways and also related to other similar methods in the area. Our main preference lies in the description of MFA based directly on standard PCA. An example from expert sensory tasting of a number of products is presented and analysed step by step in order to demonstrate the different stages that the user has to go through. This is done in a transparent way using graphical illustrations and tables.
The major contribution of the paper does not lie in its technical novelty, but in the pedagogical aspects and the completeness in the presentation.
What followed
Since the paper was published in 2013 more than 600 publications have referred to it. This clearly shows that the paper has obtained attention and is used as a standard reference in the area. We recommend strongly the paper for all scientists in sensory science.
P.S. The paper is written in collaboration between European, Canadian and American scientists.
References
Abdi, H., Williams, L.J. and Valentin, D. (2013). Multiple factors analysis: principal component analysis for multitable and multiblock data sets. WIREs Comput. Stat. 2013, doi:10.1002/wics.1246
Naes, T., Varela, P., & Berget, I. (2018). Individual differences in sensory and consumer science. (pp. 1–14). Woodhead Publ Ltd. https://doi.org/10.1016/B978-0-08-101000-6.00001-9
Smilde, A.K., Timmerman, M.E. and de Jong, S (2003). A framework for sequential multiblock component methods. Journal of Chemometrics, 17, 323—337. https://doi.org/10.1002/cem.811
Tucker, L.R. (1966) Some mathematical notes on three-mode factor analysis. Psychometrika 31: 79–311 https://doi.org/10.1007/BF02289464
Varela, P., & Ares, G. (2012). Sensory profiling, the blurred line between sensory and consumer science. A review of novel methods for product characterization. Food Research International, 48(2), 893–908. https://doi.org/10.1016/j.foodres.2012.06.037