My Multiway Analysis: From Jan de Leeuw to TWPack and Back

This paper is a mixture of my personal experiences of Jan de Leeuw as a supervisor of my master’s and Ph.D. theses, as well as a sketch of how three-way analysis, the subject Jan chose for me, developed over time. The emphasis is on where it is and was applied, and to what extent it stole the hearts of applied researchers in diﬀerent disciplines. Furthermore, the paper contains some musings about how we should go about promoting the use of the techniques, especially in the social and behavioural sciences. Finally, an overview is provided of available software and attention is paid to how (three-way) software may be designed to encourage its use by the scientiﬁc community, as it beﬁts a paper in the Journal of Statistical Software .


The Lion and I
One of the recurrent plots in science fiction is the "What if . . ." question.What would have happened to the universe if there had no been no Starship Enterprise?This is the type of speculation that hits me when I try to contemplate what would have happened to me if there had been no Jan de Leeuw, or John Lion if you prefer.(Hear how to properly pronounce his last name on his website at De Leeuw 2016c.)He certainly kick-started my scientific career, and rather than working on three-way methods I might have ended up writing a thesis on "Exact tests in contingency tables", a noble subject but not as fascinating as my current work.Three-way analysis has brought me into contact with many researchers and many disciplines, such as psychology, education, nursing, medicinal pharmacy, cancer research, and plant breeding concerned with crops such as maize, soya, wheat, adzuki beans, and peanuts, and, of course, the environmental state of affairs on the Great Barrier Reef.Jan was and is a charismatic figure who in ancient times would have been the stuff of legends.A probably apocryphal story goes that one evening after a prolific drinking session he was handed the 460-page book by Warren Torgerson on Theory and Methods of Scaling (Torgerson 1958) with the message that this would probably be something worthwhile for him to read.And sure enough, the next morning after apparently having finished the book (when exactly is still a mystery) he summarised the author's achievements and indicated places where interesting extensions of the procedures described in the book could be added.
Jan is a somewhat unconventional person who does not like formalities much ("undisciplined and wild" as he says in his own CV De Leeuw 2016b).As a full professor in Leiden he seemed to have appeared once at a Ph.D. defense with a red sweater under his gown, while a dark suit with a white shirt and black tie are compulsory; see Figure 1 for an appropriately dressed Jan de Leeuw during my Ph.D. ceremony.
Jan's influence on the world of psychometrics and beyond has been profound, as can be deduced from the papers, published and unpublished, which he wrote on all aspects of what was then already called Data Theory.The contributions in this volume and his bibliography (De Leeuw 2016a) are ample proof of this.He joined his thesis supervisor John P. van de Geer at the Leiden Department of Data Theory.This department became a shining research light in the Faculty of Social and Behavioural Sciences at Leiden University.Jan left Leiden in 1987 for the University of California, Los Angeles, but continued to supervise Dutch Ph.D. students.By 1990 there were fourteen of them, of whom eight went on to become full professors themselves.
As a student of applied mathematics I wrote my master's thesis under his direction.His direction consisted of giving me a few sheets of orange millimeter graph paper containing a series of formulas outlining the TUCKALS2 algorithm (downloadable as De Leeuw 1975).He added that I should read Tucker's 1966 paper, program the algorithm given on his sheets, apply it to an example, and this would lead to a Master's degree.I came to him because as his student assistant I had been programming a precursor of the present Homals algorithm; a program dumped as soon as it was ready, because Jan had since then devised a better algorithm.In addition, one of the professors at the Department of Mathematics had indicated that I could not seriously expect him to pull a subject out of the bag for every student who had the audacious idea of doing mathematics rather than studying it.I set to work on Jan's algorithm and 3000 IBM punch cards later the computer program was ready.I wrote the thesis on an electric typewriter with four "golf balls" that you could swap to get different fonts.Mistakes were covered up with correction tape; see also Jan's similar experiences on the CV page of his website (De Leeuw 2016b).
The research bulletin based on my master's thesis (Kroonenberg and De Leeuw 1977) was transformed into my first paper (Kroonenberg and De Leeuw 1978), which however was in Dutch.However, a couple of years later Jan wanted to know when the 'real' article for an English-language journal was forthcoming.He had even heard that there was some competition.And had he mentioned that I should really expand the three-way model from my master's thesis with a second, more well-known variant (now known as the Tucker3 model)?'Otherwise it will not do'.So I did some more programming, this time on a real Teletype typewriter linked to an IBM mainframe, so there was no longer a need for punched cards and trips up and down to the Computer Centre.In 1979 I wrote the article that went with it, typed it up in IBM's DisplayWrite3 on a word processor the size of a bookcase, showed it to Jan as the obvious co-author, photocopied it six times, and sent it by post to Psychometrika.
A few months later I received to my surprise a letter from the journal (no email yet at the time) containing two articles about multivariate analysis of three-way data that had been submitted to the journal.The editor wanted to know what I thought about those articles in relation to my own.I went to see Jan, somewhat shell-shocked: "What am I supposed to do with this?"; "Well, answer the questions of course," he said, "and tell them what you think".This is what they call "reviewing an article", but it was the first I'd heard of it.No matter, I rolled up my sleeves and got the job done.
A little after that I received a message to say our article had been accepted, conditional on certain additions and revisions.I set to work once more, but took my time.Until one day, now 35 years ago, an actual telegram arrived from the journal: "Where is your final version?We want to include your article in our March issue together with one of the others that you reviewed".I quickly finished my paper and sent it off.I am proud to say that now, early in 2016, this article (Kroonenberg and De Leeuw 1980) is number four on Jan's list of most cited papers.
Jan continued to play an important part in my academic life.Leiden University came to insist that all its staff should have a Ph.D. degree.Because I intended to continue with three-mode analysis as a research project, the obvious thing to do was to go back to Jan and ask him to be my supervisor, even though he was not even four years my senior.Jan, laid-back as always, approved, and let me draw up my own plan except that he urged me to include a chapter on a subject that I was not familiar with.I did not really want to include this chapter, but out came the graph paper again and Jan outlined the mathematical and algorithmic content for the chapter.Apparently Jan had sufficient faith in me, so that we managed go through the 400-page thesis in only a few sessions.The thesis itself was published by the now-defunct DSWO Press (Kroonenberg 1983).
My next academic encounter with Jan was later in 1983 when he received a request to contribute a paper to the first multi-mode data analysis book (Law, Snyder Jr., Hattie, and McDonald 1984).He put this request on my desk and said that this was typically something for me to do (Kroonenberg 1984).In 2005 he also asked me to contribute to the Encyclopedia of Statistics in Behavioral Science (Kroonenberg 2005).

The multiway world
Jan set me on the path of my life-long fascination with three-way, and later multiway data analysis.In this section I will present a brief introduction into three-way and multiway data analysis and provide a condensed overview what happened in that field over the years1 .

A brief introduction to multiway component analysis
Even though multiway is more than three-way, in this paper I will consider the terms threeway and multiway component analysis as largely synonymous, because the fundamental step is actually from two-way to three-way analysis.The generalisation from three-way to multiway does not introduce any new mathematical or statistical intricacies, but primarily increases the notational and interpretational complexity.Thus, in this section I will concentrate on: "What is three-way component analysis?"Before going into this, a word about the terms way and mode, both employed in this context.The first term, way, indicates the number of dimensions of a data box: one-way refers to a vector, two-way to a matrix, and three-way to an array: data that fit in a box.The second term, mode, refers to the content of a way.Within the social and behavioural sciences a standard two-way data matrix has rows (subjects) and columns (variables), and many threeway data boxes have rows (subjects), columns (variables), and tubes (conditions or time points).Hence, such data are three-way three-mode data.However, a set of correlation matrices is three-way but only two-mode, as the rows and columns both consist of the same entities, mostly variables.Thus, the word 'way' is more general than the word 'mode' and in what follows I will mainly use that.
Two-way principal component analysis is based on the singular value decomposition: a subject space spanned by the subject components, and a variable space spanned by the variable components plus the singular values (the square roots of the eigenvalues).Each subject component is uniquely linked to a variable component, so that they can occupy the same space and the size of the singular values indicates the strengths of the links between the two types of components.
Generalising this idea to the analysis of three-way data, we have to add a condition space spanned by the condition components.Again we can define links, but now for three sets of components; such links are referred to as three-way singular values or as elements of the core array.The three-way component model was introduced by Tucker (1963;1966;1972), who also coined the word core to indicate that the links in the core array play a fundamental role in modelling the relationships between the components of the three ways (see Figure 2).
The other major three-way model is the parallel factors (PARAFAC) model (Harshman 1970;Harshman andLundy 1984b,a, 1994).This model is also referred to as the CP model, an abbreviation of the CANDECOMP-PARAFAC model recognising Carroll and Chang's 1970 work on what they called the canonical decomposition (CANDECOMP) model.Later, another meaning was given to the same CP abbreviation, namely canonical polyadic decomposition.The latter designation goes back to Hitchcock (Hitchcock 1927a,b) and was resurrected by researchers such as De Lathauwer, Comon, and Lek-Heng.In Figure 2 P = Q = R are equal for the PARAFAC model, so that its core array is a cube which in addition is superdiagonal with only P elements.
The major differences between two-way and three-way models centre around the properties of uniqueness and complete decomposability.In the two-way component model the components can be transformed without losing fit of the model to the data.This transformational property is used extensively to enhance and facilitate interpretation.At the same time, the components are not unique, which hampers the theoretical identification of the solution.
As we have seen in three-way component analysis, there are several component models which can claim to be the generalisation of the two-way component model and the singular value decomposition.A very limited number of them have the uniqueness property, i.e., the solution is mathematically identified and cannot be transformed without losing fit of the model to the data.The downside of this is that such models, primarily the PARAFAC model and its descendants (Harshman and Lundy 1984b), cannot fit every single three-way data set and a complete decomposition of the data into its components is not always possible.In contrast, the models proposed by Tucker, such as the Tucker3 model, are not unique but can fully decompose any three-way data set.It is especially the uniqueness property of the PARAFAC models which has been of great value in analytical chemistry, signal processing, and other exact sciences which have physical three-way models describing the patterns underlying their three-way data.

The rise of multiway analysis
To get an idea of what went on in the multiway world we can take a look at what happened with the proposed fundamental models over the years; in particular, what happened since Tucker's introduction of three-mode data analysis in psychology (Tucker 1966) 2 and Harshman's ground-breaking work on the PARAFAC (or CP) model (Harshman 1970).
The curves in Figure 3 represent the number of citations to multiway articles written by authors from the fields of psychology, chemistry, signal detection, and mathematics, respectively.It is clear that there was a strong acceleration from 1992 onwards, and that for the time being there seems no end in sight.We can see that particularly in the field of chemistry the unbridled enthusiasm has continued unabated, but that researchers from other disciplines have become involved as well.
By the way, it is interesting to note that theoretical mathematicians such as Hitchcock came up with three-way models as early as 1927, but that for a long time mathematicians seemed to have forgotten about them.It was not until around 1998 that multiway models attracted their attention once more.It is noteworthy that psychometricians also started to publish in chemometric journals (e.g., Kiers and Harshman 1997;Kiers, Ten Berge, and Bro 1999;Harshman 2001;Kroonenberg and Van Ginkel 2012).This is primarily due to an ever-

Multiway in the social and behavioural sciences
From Figure 3 one might get the impression that the increase in citations to psychometric three-way articles reflects an upsurge in their use in the social and behavioural sciences.Unfortunately, nothing is further from the truth.Figure 4 shows a very different picture.It is based on references to the work of the founding fathers Tucker and Harshman, split according to whether the citing author works in the natural sciences or in the social and behavioural sciences.We see that the increase in citations to the work of the two psychometricians is exclusively due to researchers from the natural sciences, rather than to researchers from their own discipline.
Why is this?It could be that describing patterns in data does not particularly appeal to social and behavioural scientists, but that they prefer testing new and well-founded theories about human behaviour.Such testing is done through inferential rather than descriptive statistics.It could also be that many social scientists focus on finding the causes of phenomena and predicting future behaviour.This is something three-way methods were not originally designed for, but the first steps in this direction have been made.Another factor that could play a part is a justifiable desire to keep analyses as simple as possible in order to prevent conflicts with journal editors and their reviewers.After all, the latter are not necessarily well-versed in advanced statistical methods.A further important aspect could be that multi-variate descriptive techniques, and three-way techniques in particular, are mainly intended to model individual differences rather than test group characteristics such as averages, standard deviations, correlations, etc.
Moreover, social and behavioural scientists are missing something that in natural sciences such as chemistry is the motor for much three-way work: substantive theories that take the form of the statistical models used in three-way analysis.If such models are available, three-way techniques directly provide the estimates for the parameters of the formal physical models, and there is no need to wonder afterwards how parameters should be interpreted -which happens not infrequently in the social and behavioural sciences.This difference in modelling is discussed in, for instance, Hand's presidential address to the Royal Statistical Society in 2009 under the heading "iconic models" versus "empirical models".The physical sciences typically deal with the former, while the social and behavioural sciences have to make do with the latter.However, it should be understood that empirical modelling occurs in some way or another in nearly every kind of science, physical science included, as Hand convincingly demonstrates.

A cornucopia of applications
In this section I provide a brief and non-exhaustive overview of the breadth of topics that have been researched with multiway methods from domains other than the social and behavioural sciences (which are treated in a separate section).Lieven De Lathauwer mentioned, for example, that tensors, the mathematical elements that form the basis for many multi-way models, are the golden tools for "signal processing, array processing, data mining, machine learning, system modelling, scientific computing, statistics, wireless communication, audio and image processing, biomedical applications, bio-informatics and so forth" (De Lathauwer 2014), not even mentioning fields such as the social and behavioural sciences, environmental studies, and agriculture.Kroonenberg's 2008 book contains examples from several of these last ones.

Chemistry
Besides physical models, chemistry also employs 'linked' measuring techniques that work together to produce three-way data such as gas chromatography coupled with mass spectrometry.As a result, the chemometricians need techniques to analyse such data, and three-way and multiway methods have come to the rescue.This also resulted in three-way models for online process control in industry.A view of the breadth of applications in chemistry can be found in the review papers by Bro (2006), Açar and Yener (2009), Kolda and Bader (2009), the edited book by Muñoz de la Peña, Goicoechea, Escandar, and Olivieri (2015), and the books by Smilde, Bro, and Geladi (2004) and Olivieri and Escandar (2014).

Agriculture and food industry
In the field of agriculture, plant breeders want specific varieties of crops, say peanuts with specific attributes to suit specific purposes.They also want to know which crops can best be grown in which locations.Analysis of individual variation is of prime importance here: the industry wants peanuts that can easily be turned into peanut butter, while consumers want king-size peanuts that feel right in the mouth.Bro's 1998 thesis provides a background to both chemical and food industry applications (see also Smilde et al. 2004), while Basford and Kroonenberg discuss applications in agriculture, especially plant breeding, in various papers (see, e.g., Basford, Kroonenberg, and Cooper 1996).

Signal detection
Three-way analysis is of prime importance for signal detection, particularly for keeping apart signals from different devices -blind source separation.Are there nowadays people who do not have a device on them that constantly detects signals, whether it is a mobile phone, tablet or sat-nav?It is in this field that there is a great upsurge of applications.Much about these developments can be found in the Handbook of Blind Source Separation (Comon and Jutten 2010, Chapter 13), the review paper by Cichocki, Mandic, De Lathauwer, Zhou, Zhao, Caiafa, and Phan (2015), and the book by Cichocki, Zdunek, Phan, and Amari (2009).

Systems biology
A very special area of application is systems biology -a discipline that aims to study organisms as a whole, with specialisms such as metabolomics (the systematic study of the unique chemical characteristics of specific cell processes) and proteomics (the large-scale study of proteins, particularly their structure and function).Examples of the use of multiway methods in these areas can be found in, for instance, Yener, Açar, Aguis, Bennett, Vandenberg, and Plopper (2008); Rubingh, Bijlsma, Jellema, Overkamp, Van der Werf, and Smilde (2009); and Açar, Lawaetz, Rasmussen, and Bro (2013).

Applications in the social and behavioural sciences
As shown above, three-way analysis is not commonly used within the social and behavioural sciences, although there are many kinds of projects that would benefit from its power and the insights derived from three-way techniques.If we want to explore the use of three-way analysis in social and behavioural scientific research in more depth, a crucial question is: "Is descriptive analysis, and three-way analysis in particular, actually useful in the social and behavioural sciences?".Despite the relatively modest growth in three-way applications, I maintain that this is definitely the case, but to my mind this applies to all descriptive analytical techniques.

Three-way data types in social and behavioural research
To answer the question of the usefulness of three-way methods in the social and behavioural sciences in detail, we must first establish if the data collected within this discipline actually call for three-way techniques.In order to eliminate any doubt on this point, I here provide a non-exhaustive overview of major types of three-way data encountered in the social and behavioural sciences; for applications see, e.g., Kroonenberg (2008).

• Three-way profile data
Persons have scores on a number of variables measured under several conditions or on several occasions.
Example: It is recorded whether children seek proximity to strangers, observe them with distrust, etc. when child, parent, and stranger are in the same room; when the child is alone with the stranger; and when the stranger enters after the child has been left alone in the room.
• Repeated measures data A number of participants are measured several times on the same variables.
Example: Each year in a sample of primary school children their reading ability is assessed on several different measures.
• Three-way rating data A number of subjects judge various emotions in different situations leading to a kind of stimulus-response data.
Example 1: Children with asthma indicate on several scales how they would react in different situations, for example playing alone in the street, walking alone in the woods, playing with their friends in the school grounds, going on a cycling trip.For each of these situations the children are asked to imagine how afraid they would be to have an asthmatic attack, how lonely they would feel, or how exhilarated they would be.
Example 2: Individuals are presented with a number of concepts pertaining to their dayto-day life which have to be judged with a set of scales describing the characteristics of these concepts.A well-known representative of this kind of data is Osgood's semantic differential technique.Therapists presented a woman with a multiple personality with concepts regarding her daily life (state of mind, mental illness, her doctor, her spouse, her job, etc.) and asked her to rate the concepts on scales such as good versus bad, calm versus vehement, strong versus weak (see Osgood and Luria 1954). •

Sets of correlation or covariance matrices
Example: The correlations or covariances between subtests of an intelligence test are computed for samples of children at different ages, or from different countries.The question is whether the intelligence test has the same structure over time and/or in different languages.
• Sets of similarity matrices Example: Children are asked to indicate to what extent they prefer peanut butter sandwiches over jam sandwiches, cheese, kippers, truffles, etc.An additional three-way profile data set with the characteristics of these food stuffs can be useful for explicitly linking the judged preferences between the food stuffs to their perceived characteristics. •

Results of fMRI studies
Example: The results of brain scans measured in various locations in the brain, under different conditions, in different people.
• Three-way interactions Example 1: Three-way interactions resulting from analysis of variance.If the three (or more) factors in the anova have only few categories, analysing the three-way interaction with three-way component methods is hardly profitable.This changes, however, if there are many levels.In a study on the perceived reality of television films for children (the response variable) there were eight categories of films, five age groups, and eleven perceived reality aspects.This leads to an 8 × 5 × 11 three-way interaction, and next to other effects, it is this interaction that was the focus of the study (Kroonenberg and Van der Voort 1987).A technical discussion can be found in Van Eeuwijk and Kroonenberg (1998) and an example from nursing in Gilbert, Sutherland, and Kroonenberg (2000).
Example 2: Large three-way interaction tables also occur in the case of large contingency tables with three variables with many categories.As an example, suppose that every month the behaviours between mothers and their babies are observed using a number of behavioural categories; see, e.g., the study by Van den Boom and Hoeksma (1994), analysed by Carlier and Kroonenberg (1996).
• Large binary three-way data sets Example: In a study 54 persons were asked to indicate yes or no whether they displayed any of 15 hostile behaviors in 23 frustrating situations.This resulted in a binary 23 x 15 x 54 data array.Such an array is clearly too large to analyse for many classical techniques, but three-way binary component analysis, called three-way hierarchical classes analysis, is specifically designed to handle this type of large binary contingency tables (Ceulemans, Van Mechelen, and Leenen 2003;Ceulemans and Van Mechelen 2005).
The last two data types deserve further mention because they point to areas where threeway analysis within various disciplines has been underused.Categorical and binary data are fairly common in the social and behavioural sciences, as both criterion variables and predictor variables.For designs with one or more continuous variables and many levels in the categorical predictors, analysis of variance is used in many cases, and more than two factors with many levels quickly lead to large interaction tables.Similarly, large three-way contingency tables have large three-way interaction tables.Indicating that these interactions are significant hardly answers the research questions about the major underlying patterns.Bro and Jakobsen (2002) developed the GEMANOVA procedure for handling higherdimensional interactions with multiway methods; one example is Romano, Naes, and Brockhoff (2015).Three-way interactions for contingency tables have been tackled with three-way correspondence analysis by Carlier andKroonenberg (1996, 1998).In their book Beh and Lombardo (2015, Chapter 11) provide a contemporary overview of several variants of multiway correspondence analysis.
From the above it should be clear that research in the social and behavioural sciences produces many different kinds of data which can seriously benefit from three-way and multiway analysis.
At the same time it is clear that large-scale descriptive analyses are not everybody's cup of tea, and that there will always be interpretational issues, because understanding the patterns in three different component spaces and the way they are linked is not something that comes easily.On the other hand, the techniques allow for answers to complicated questions about interactions, although the answers will not necessarily be straightforward.

Three-way and multiway software
During my academic career I have been working on several programs for three-way analysis plus one four-way program.These programs have been used in virtually all my published applied papers.In this section I will set out some considerations that went into the development of these programs and the interface binding them together in a single integrated it enabled the individual programs to also function as stand-alone programs with a complete query-and-answer input system as well as a file input system.This in turn allowed for their direct inclusion in simulation programs and all kinds of different interfaces.
The interface was designed in such a way that firstly, menu systems create the necessary parameter files for the analysis programs; see Figure 5 for the structure of the interface.The interface then calls the desired analysis program, which reads data files and the created parameter files for the job at hand.The analysis program produces output files containing plain ascii and browser-readable html.In addition, the program itself creates a parameter file for postprocessing the output, as well as a command file with instructions for the graphics program gnuplot (Williams and Kelley 2016).Postprocessing consists of rotating the components and/or the core array, analysing and displaying residuals, as well as creating further graphics.
A browser internal to the interface displays the output in a form which is reminiscent of the standard SPSS, as it also has a sidebar for navigation.The navigability is a great plus given the large amount of output.
For all of this to work an interface needs to have a memory.In the SPSS statistical package the memory consists of the system file (*.sav), which contains the data and their definitions such as variable names, value names, format, etc. as well as information about missing data, which variables should be analysed as a set, etc. SPSS saves the information about previous analyses in a syntax file (*.sps), thus making rerunning analyses possible.By default the SPSS syntax is also listed in the output and can be copied into the syntax file.The format of the SPSS system file is a proprietary one, but can be read by several other statistical packages such SAS and EQS.
For each data set and its analysis the TWPack interface has one single memory or job file containing a basic description of the data, the locations of the relevant parameter files, and the details of the previous analysis, all of which should reside in the data directory.A data file has its own companions in a label file and a missing data file (if missing data are present).Job files are editable by a basic ascii editor.In order to keep track of what has been done to or with the data, it is necessary to have a separate directory for each data set.The great disadvantage of this system is that after a series of analyses one has a large number of files floating around in the data directory (see Figure 5).However, all files related to a particular job have the same job name as part of their file name.
Unlike the TWPack interface, SPSS (and probably many other statistical programs) does not do reverse engineering, i.e., given a syntax or command file the program cannot set the menu system to correspond with the syntax file.For really complicated syntax files reverse engineering would be a great boon for users, because they do not necessarily have sufficient knowledge of the intricacies of the command language to easily make the same modifications they can make via the menu system.
An attractive aspect of separating interface and analysis programs is that if operation systems change the analysis programs one can just rbe ecompiled, and if other interface building software becomes available or the current one becomes obsolete, one can build and rebuild this separately.Ideally one should be able to build interfaces in interpretational software such as R or MATLAB and still make use of the compiled analysis programs for fast executions, but such interfaces have not been built yet.The idea of separating the two types of programs is even extended in such a way that via relatively minor changes in the interface, external analysis programs with their own built-in interfaces can be called from within TWPack.

Three-way and multiway software in the world at large
Naturally, there are several other collections of programs which are geared to performing three-way and multiway analysis, of which the MATLAB N -way Toolbox is probably the widest range of programs.

Assignments for three-way researchers
Part of the blame for the limited use of three-way methods, especially in the social and behavioural sciences, lies with the psychometricians themselves.As lecturers, I think we have neglected to stop and consider that the use of statistical methods is dictated in part by what we teach and how we present these methods to our colleagues and students.In the future we will have to take into account that academics are interested in their own fields, not in our methods.Thus, if we want to show them that we have something to offer, the burden is on us to demonstrate this.In particular, we have to show that three-way methods have an added value compared to the standard statistical toolkit and can deliver results that cannot be obtained any other way.In other words, we have to argue that three-way analysis is like Heineken (see Figure 6).

Statistical packages: Calculator and adviser in one
In concrete terms I would like to encourage methodologists (including myself) to concentrate, more than they have done so far, on: (1) showing the added value of the techniques in applied articles in non-methodological journals; (2) presenting relevant analyses in collaboration with researchers working in the field, at the latter's own, discipline-oriented, conferences; (3) training young academics specialising in these subjects in the use of three-way techniques, by organising courses that offer plenty of opportunity to practise with their own data; (4) producing guides for performing analyses, not just in print but also through intelligent and self-explanatory computer programs with easily accessible input and output.
Generally, statistical programs for three-way analysis are not half as simple to use as for instance SPSS with its extensive graphical interface.Quite apart from this, I would like programs, including my own, to follow Leland Wilkinson's example in his SkyTree Adviser statistical suite.He describes his system as follows: "SkyTree Adviser is an expert system for machine learning.It not only chooses appropriate models, but it also evaluates them with unique algorithms for detecting miss-specifications, outliers, and other anomalies (Wilkinson 2014)."Thus, after performing a basic analysis the program will evaluate the outcomes and advise the user on the status of the analysis.In a sense, it talks to the researcher.It will also print off a paragraph in which the results are described and the technical implications of the results are explained.In this way, the program functions as an 'analyser' and personal adviser in one.Various multivariate statistics textbooks give examples of how the results of a certain analysis can be interpreted, but they always deal with the authors', not the researcher's data.It would be ideal if programs for statistical analysis explain the meaning of the results provided in their output.Until now, user-friendliness of computer programs has been almost exclusively discussed in terms of input rather than output and interpretation.Creating threeway analysis programs with user-friendly output seems like a perfect, rewarding project for a retired professor.

Figure 1 :
Figure 1: Committee for the Ph.D. defense of Pieter Kroonenberg.

Figure 2 :
Figure 2: The Tucker3 model for the three-way data array X.A = Subject space with P components; B = Variable space with Q components; C = Condition space with R components; G = Core array with the P × Q × R links between the components from the three ways.

Figure 3 :
Figure 3: Citations to three-way and multiway papers by researchers from different disciplines."Multiway 88; Book 1989" refers to the Multiway conference in Rome 1988; Coppi and Bolasco (1989) contains the proceedings.

Figure 4 :
Figure 4: Citations to the work by L. R Tucker and R. A. Harshman in the social and behavioural, and in the physical sciences

Figure 6 :
Figure 6: Power to three-way analysis.