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Finding Treasure Buried in Data

- maps in management: multidimensional scaling

by Cecilio Mar Molinero and Carlos Serrano Cinca

The main characteristics of multivariate data can be made explicit by means of a statistical map. Multidimensional Scaling (MDS) provides a series of techniques to project the main features of the data in the form of maps. We survey the application of MDS in Management and give an example of its use. MDS maps can be augmented with the results of other multivariate statistical techniques.

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A map to describe a treasure island

Tenerife is a treasure island. The treasure is hidden in its weather, the variety of its landscapes, and its flora. The highest mountain in Spain, the Teide, is there. Its main town, Santa Cruz, is an important industrial port. Its university, one of the oldest in Europe, is in the unspoilt town of La Laguna, not far from Santa Cruz. La Laguna is on high ground, on the slopes of the Teide. The South of the island is dry, being not far from the Sahara desert. The North, which faces the Atlantic sea, is wet. Tourism tends to concentrated on the dry South West, in places such as Los Cristianos and Los Gigantes, far from Santa Cruz.

The above paragraph describes Tenerife. We could summarise the above information, and we would be expected to do so, in the form of a map. In fact, one of the first thoughts of anyone who goes to a new place is to ask for a map. By reading a map, one can gather information about proximity between places, difficulty of access, and common characteristics. Figure 1 shows the map of Tenerife. This is not a very sophisticated or accurate map, but it clearly displays relationships of proximity. The map has been augmented with arrows that indicate which way is North and which way is East. This serves to orient the map, and to see extra information in context. For example, if we are considering visiting a hotel between Los Crisitianos and Los Gigantes, we know that we are to expect dry weather, as it would be situated in the South, the dry part of the island. We also know that this is the area preferred by European holidaymakers, so we will be amongst other tourists rather than amongst the local population. Most locals would be conducting their economic activity at the other end of the island.

Figure 1: Map of the Island of Tenerife with orientation

Maps show many dimensions of the data

To produce a map, information has to be collected about, at least, three measurements: latitude, longitude and altitude over the sea. Other characteristics can be added to the map, such as rainfall. Typically, the map will be in two dimensions, as we see in the figure. Lines can be produced to show areas of equal altitude, and colours can be added to indicate rainfall. Such a map would be a four dimensional representation in the sense that every point in the map would have four numbers associated with it. These numbers will typically be reference with respect to a series of scales. For example, latitude and longitude would be read by means of a grid, and altitude would be shown by means of level curves. Thus, a map contains a set of numbers for a series of locations, or points. We can slightly alter our way of talking, talk about observations instead of locations, and talk about variables instead of measurements. The end result is a matrix of variables and observations. This is multivariate data. The techniques to analyse multivariate data have long been available in the form of multivariate statistical analysis.

Multivariate data in management

Multivariate data has become more abundant in recent times, given the ease with which it can be gathered by electronic means and stored in computers. For example, it is now possible for supermarkets to record, at point of sale, the amount of each item that a particular individual has purchased. In this way, a matrix can be built in which each individual would be an observation and each item on sale at the supermarket would be a variable. If a particular individual visits the supermarket on several opportunities, a decision has to be made on how to deal with the information. It would, for example, be possible to treat successive visits as part of the same observation, or to treat each visit as a different observation. Supermarkets would be interested to know which items sell jointly, in identifying groups of customers with similar characteristics in order to segment the market, etc. In other words, they would be interested in doing multivariate analysis.

The techniques of multivariate analysis have long been known to statisticians. There are many textbooks on this subject. When faced with a large data set, one would start by contemplating some data reduction; ie, trying to limit the number of variables to a small set that summarises the main characteristics of the data without much loss of generality. Principal Component Analysis and Factor Analysis come immediately to mind. Groups of individuals can be formed by means of, for example, Cluster analysis. The behaviour of individual customers can be explored using Logit Analysis or Multivariate Linear Regression. There is no intellectual problem. However, the standard difficulty in Statistics is one of communication. It is not a condition of employment for a supermarket manager to have a degree in Statistics. Very sophisticated work can be done, but this can be lost if the user does not understand what it means, or does not feel confident to implement the consequences of the analysis. As an old Spanish proverb puts it: "He who does not understand is like he who does not see". The corollary is clear: we want people to be able to "See the data".

Multidimensional Scaling provides a map to guide us through multivariate data

Geographical maps provide visual representation of geographical multivariate data. The question now emerges: can the relationships that exist in multivariate data be made explicit in the same form? Can we produce a map from a matrix of variables and observations? The answer is positive, this can be done with the techniques of Multidimensional Scaling (MDS).

MDS has been in existence since the 1960's when Kruskal developed a remarkable algorithm. Given a measure of proximity between pairs of points, or dissimilarity, the algorithm derived their position in the space. The algorithm is based on the preservation of ordinal relationships; i.e., if the dissimilarity between two points is high, these will be located far apart in the space, and if it is low they will be located next to each other. Kruskals algorithm relies only on orderings, and these can be easily be created. Consider the following situation. We would venture to say that we like a particular type of drink more than another type, but we would not be willing to attach a particular number to how much we like each particular drink. Take another situation, it would be reasonable to say that Peter loves Mary more than he loves Ann, but it would not make sense to say that Peter loves Mary 68 and that he loves Ann 39. Thus, we would expect that in a graphical representation of the relationship between Peter, Mary and Ann, Peter and Mary would be close to each other, while Ann would be looking at them from a distance. A good introduction MDS can be found in the monograph by Kruskal and Wish (1978).

MDS owes most of its early developments to Psychometrics. It is in Psychology where most research has taken place using MDS, and where the tools were developed and refined. This pedigree is reflected in the language used within the technique; for example, one does not talk about goodness of fit, but about stress. MDS soon found its way to other areas of knowledge. The way in which committees vote - Rockness and Nikolai (1977), the order in which burial sites were dug - Kendall (1975), oil pollution at the sea - Mantoura et al (1982), have been analysed by means of MDS.

The concept of dissimilarity

The usual way of proceeding in MDS is to find measurements on a set of variables for a number of observations. From these measurements, a measure of dissimilarity between entities - which could be either observations or variables - is developed, and the map is derived from the table of dissimilarities. We can think of this process as the reverse of what we actually do with a geographical map, where we estimate the distances from the position of the points in the map. In MDS we estimate distances first, and from the distances we created the map. We do not talk about distances, since the concept of distance implies a measurement, and MDS does not require exact measurements. It only requires relationships of order to be observed; ie, it requires a knowledge of how far apart two points are, not the exact distance between them. The process of producing a map can be observed in Figure 2.

Figure 2: Steps in the construction of a MDS map

MDS in management

Marketing is the field in which MDS has produced a most extensive crop, and for which it is ideally suited. An example of the use of MDS in this way can be found in Mar Molinero and Mao Qing (1990). Early applications of MDS in Management and Accounting relied on counts or judgements for data. For example, Rockness and Nikolai (1977) used the technique to explore voting patterns in the Accounting Principles Board of the United States. Belkaoui (1980) reasoned that Accounting is a language and as such a shaper of the environment, and argued that there would be cultural differences between the various groups that form the Accounting community; he explored the similarities and differences using MDS. Following the same line of thought, Bailey et al (1983) explored the language used in audit reports by means of MDS. In the area of Management, the work of Pat Rivett deserves mention, particularly his use of MDS to derive statements of preference from data based on statements of indifference, an approach that gives new light on some multicriteria problems - Rivett (1980). All these studies have a common characteristic, the data is ordinal or nominative, or just counts. It is data which is not easily amenable to analysis with standard statistical techniques. MDS was used because it only needs statements of ordering in order to provide a pictorial representation of the data.

There is, of course, no reason why data which is measured on an interval or ratio scale, the kind of data on which it is right and proper to apply standard multivariate analysis techniques, should not be analysed with MDS. In fact, there are good reasons to do so. It can be proved that the MDS is more general than techniques based on the multivariate normal distribution, and that both methods produce the same results when the distributional conditions are satisfied - Mar Molinero (1991). MDS has the advantage of accessibility: no prior knowledge is needed to understand the output of an MDS analysis. Examples of the use of MDS in this context are abundant. Mar Molinero and Ezzamel (1991) examined the path that companies follow on their way to bankruptcy. Mar Molinero, Serrano Cinca and Apellaniz Gomez (1996) interpreted the information on which bond ratings are based. Mar Molinero and Serrano Cinca (2000) produced a graphical representation of the banking crisis of the 1970s in Spain.

An example from management

The data set available in this case relates to individuals in an university department. The question is whether, on the basis of objective information, one could deduce something about groups of influence. In a sense, we are interested in treating members of staff as soap powders and segmenting the market into different groups of products.

The data was obtained from an academic department, which is known to the authors, but will remain nameless in this case. Members of staff have also been kept anonymous by the use of numbers, thus rather than say "Charles" we have preferred to say, "observation 17".

All the information that was collected was of a binary nature. If the individual satisfies the condition, the variable takes the value one, if he/she does not satisfy the condition, it takes the value zero. The variables are listed in groups according the the following criteria: seniority, external activities, administration experience, status, research, participation in social activities, personal characteristics. Many other criteria could have been used, some of them redundant, and this would not be in any sense a burden, as the modelling procedure automatically engages in data reduction.

Each criteria is made up of a series of attributes as follows:

Seniority

  1. Holds a PhD.
  2. Permanent contract.
  3. Professor.
  4. Lecturer.
  5. Temporary lecturer.
  6. Part-time staff.

External activities

  1. Industrial experience.
  2. Member of relevant professional association.
  3. Chartered professional.

Administration experience

  1. Has been in senior departmental position.
  2. Has been chair of influential committee.
  3. Directs a Master programme.

Status

  1. Shares an office.
  2. Has an office near the departmental office.
  3. Has an office with air conditioning.

Research

  1. Acknowledged as an expert in the discipline.
  2. Has published in international journals.
  3. Teaches core subjects.
  4. Has up to date computing equipment.

Participation in social activities

  1. Attends departmental social events.
  2. Has coffee breaks in the senior common room.
  3. Has coffee breaks in the nearby café.

Personal characteristics

  1. Single
  2. Married

There are other characteristics that have not been included in the data set such as, for example, gender. By leaving them outside the data set that is used to create the map, it is possible to assess if they are relevant to the interpretation of the map, in this way we can observe if, for example, there are separate groups of men and women, something that could point towards the presence of sex discrimination.

There are 41 members of staff in the department, The data set is, therefore, a matrix with 41 rows (observations) and 24 columns (variables). It is this matrix that we are trying to represent and whose internal structure we are trying to explore.

The first step is the production of a measure of dissimilarity. There are no fixed rules on how to do it. In this case we have just counted the number of characteristics that two individuals have in common. For example, if two individuals are single, hold PhDs and do not attend departmental social events, they have three characteristics in common. A special programme was written to calculate the dissimilarity matrix, which contains 41 rows and 41 columns, as we are interested in dissimilarities between cases and not dissimilarities between variables.

Using standard MDS techniques, it appears that a very good representation is obtained in six dimensions, and the map was obtained in six dimensions, although here we will only consider the first two dimensions, which are the ones that count for the largest proportion of the variance. The map has been augmented with ovals, which indicate the proximity of the points in the space, and which were obtained by means of Cluster Analysis. The interpretation is as follows. Two individuals who are inside the same oval have more in common than two individuals who are inside different ovals. The projection of the six dimensional map in the first two dimensions can be seen in Figure 3.


Figure 3: MDS representation between members of an academic department

We can see in Figure 3 that the data splits nicely into two clusters, one towards the left of the figure and one towards the right hand side. Examination of the two clusters indicates that the one on the right contains staff with permanent contracts, while the one on the left contains staff with temporary contracts. In other words, the data separates neatly the core of the department and all those members of staff whose role is to support the teaching activity. Within the cluster of permanent members of staff, two groups are apparent, a smaller one towards the top right hand side, which is made up of professors and senior members of staff who hold, or have held positions of responsibility, and a larger one made up of the bulk of permanent staff, mainly young lecturers at the start of their career. In the same way, temporary members of staff divide neatly between PhD students who teach in the department as a way of acquiring experience and supplementing their income, and individuals who are hired to teach some course on which they have a particular expertise. It is clear that the first dimension relates to type of contract, with the most senior individuals towards the right hand side, and those who contribute only casually to the department towards the left and side. The second characteristic which is apparent in the map is power, or influence, within the department, although this would be a line that would move from the bottom left towards the bottom right. It is interesting to note the position of individual number 1, who holds a senior position but does not participate in the administration or the social life of the department, he also appears isolated in the cluster of senior staff.

An example from finance

We have chosen as another example a well studied data set. The data, which has been published by Rahimian et al (1993), relates to 129 USA firms, 65 of which failed. Five financial ratios are available for each one of the firms. The purpose of the study is to find out if there are differences between continuing and failed firms on the basis of these five ratios. This data set was studied by Odom and Sharda (1993) who used Linear Discriminant Analysis, a technique based on the multivariate normal distribution. Rahiman et al (1993) used a neural network approach on the same data. Serrano Cinca (1996) went one step further by using self organising neural networks on the same data set. The five ratios are the same ones published in a well know paper by Altman (1968). Of course, in a study of this kind one would operate with more than five ratios, data reduction being one of the objectives of the exercise. We have chosen to operate with these five ratios only, as the data set is only illustrative.

A measure of dissimilarity between the firms has to created from the original data set. In this particular case, we operated with Euclidean distances. The ratios are measured in different units and distances cannot be calculated before this problem is solved. Our approach was to standardise the variables and to work with distances between standardised variables for each observation. In this way a matrix of dissimilarities was created from the data. If the ratio structure of two companies was similar, the dissimilarity distance was small, and if the ratio structure was very different, the dissimilarity matrix was a large one. Two observations are in order; first, MDS is robust to the presence of outliers, although the final maps look less cluttered if outliers are removed; second, here we are working with distances between cases, and this produces a large dissimilarity matrix, in practice we would work with distances between variables and we would use this information in order to derive information about companies on the basis of the ratios.

In this case we have simply produced a map in two dimensions. This map can be seen in Figure 4. 


Figure 4: MDS representation of company data

An indication of which firms are solvent and which ones have failed has been made by representing solvent firms with crosses and failed firms with dots. It can be clearly seen that solvent firms concentrate on the right hand side of the map while failed firms concentrate on the left hand side of the map. There almost no overlap between the area in which solvent firms operate and the area in which the failed firms operate. The MDS representation clearly shows that, given the sample of companies available, it is clearly possible to establish a difference between the companies that fail and the ones that succeed. It is no surprise that excellent results have been obtained when subjecting this data set to Linear Discriminant Analysis and to Neural Networks. Any method would work well with a data set that separates so neatly. The advantage of MDS is that we see the separation between the two sets of points, we do not have to infer it from the statistics.

To give a rationale for success and failure, arrows have been added to the map, which indicate the way in which four financial ratios change with the position of the firms. These arrows are the equivalent of North/South and East/West directions in geographical maps. For example, the arrow that corresponds to ratio 3 points towards the right hand side, indicating that this ratio takes high values for the companies that are situated towards the right of the map, and low values for the companies that are situated towards the left of the map. Ratio 3 is a profitability ratio, this means that failure is a characteristic of firms with low profitability. The technique that makes it possible to draw the arrows, Property Fitting, is based on Multiple Regression Analysis. It is customary not to represent the arrows that are associated with poor regression results, this is why Ratio 5 does not appear in the representation. This can be interpreted to mean that Ratio 5 is not related to failure or success in this data set.

Conclusion

We have argued here that highly accessible representations of multivariate data sets can be obtained by producing statistical maps that capture the structure of the data set. These maps are created by means of Multidimensional Scaling. MDS modules are now part of most standard statistical packages, and are easy to use. The maps shown above were created with SPSS, a very popular statistical package.

There are many issues that have not been touched here, and that make fascinating reading. One such issue is the analysis of three dimensional data sets. In the above example we could have had data for a series of years for each company and we might have been interested in exploring how the map changes from year to year. Another issue could have been the incorporation of extra information to a map: how would we assess if a company that has not been used in the analysis is healthy or is likely to fail? The techniques needed are not complex, but are outside the scope of this paper.

The example has illustrated the strength of a graphical representation. Try it yourself. Next time you have a problem that involves multivariate data do not try to show your boss sophisticated statistical methods. All you have to do is to produce a MDS map. Who knows, your success may be rewarded with a holiday to Tenerife. Personally, we prefer the North of the Island. It is green, it has a great variety of vegetation, there is less noise from the discos, and the south is only a short bus ride away.

For the interested reader

  • Altman EI (1968) Financial ratios, discriminant analysis and the prediction of company bankruptcy. Journal of Finance, 23, p589-609.
  • Bailey KE, Bylinsky JH and Shields MD (1983) Effects of audit report wording changes on the perceived message. Journal of Accounting Research, 21, p355-370.
  • Belkaoui A (1980) The interprofessional linguistic communication of accounting concepts: an experiment in Sociolinguistics. Journal of Accounting Research, 18, p362-374.
  • Kendall DG (1975) The recovery of structure from fragmentary information. Philosophical transactions of the Royal Society of London, 279, p547-582.
  • Kruskal JB and Wish M (1978) Multidimensional Scaling. Sage Publications. London.
  • Mantoura RFC, Gschwend FM, Zafiriou OC, and Clarke R (1982) Volatile organic compounds at a coastal site. Environmental Science Technology, 16, p38-45.
  • Mar Molinero C (1991) On the relationship between Multidimensional Scaling and other statistical techniques. Discussion Papers in Accounting and Management Science, number 91-16, Department of Management, University of Southampton, UK.
  • Mar Molinero C, Apellaniz Gomez P and Serrano Cinca C (1996) A multivariate study of Spanish bond ratings. Omega, 24, p451-462.
  • Mar Molinero C and Ezzamel M (1991) Multidimensional Scaling applied to corporate failure. Omega, 19, p259-274.
  • Mar Molinero C and Mao Qing (1990) Decision support systems for university undergraduate admissions. Journal of the Operational Research Society, 41, p219-228.
  • Mar Molinero C and Serrano Cinca C (2000). A multivariate analysis of bank failure in Spain. European Journal of Finance, forthcoming.
  • Odom MD and Sharda R (1993) A neural network model for bankruptcy prediction. In: Trippi and Turban (eds). Neural Networks in Finance and Investment. Probus Publishing Company, Chicago, USA.
  • Bankruptcy prediction by Neural Network. In: Trippi and Turban (eds). Neural Networks in Finance and Investment. Probus Publishing Company, Chicago, USA.
  • Rivett P (1980) Indifference mapping for multiple criteria decisions. Omega, 8, p81-93.
  • Rockness HO and Nikolai LA (1977) An assessment of APB voting patterns. Journal of Accounting Research, 15, p154-167.
  • Serrano Cinca C (1996) Self organizing neural networks for financial diagnosis. Decision Support Systems, 17, p227-238.

CARLOS SERRANO CINCA is a senior lecturer in Accounting and Finance at the University of Zaragoza (Spain) He is currently visiting scholar a the Department of Management of the University of Southampton (UK). His doctoral thesis on ANeural Networks in Financial Statement Analysis@ received the prize for the best PhD of the year 1994. His research interests include: the application of Artificial Intelligence in Accounting and Finance; multivariate statistics applied to financial information; and Information Technology in Accounting and Finance. He has published in journals such as The Journal of Forecasting, Decision Support Systems, Omega, Neural Computing and Applications, The European Journal of Finance etc. His personal web page is http://ciberconta.unizar.es/carlos.htm 

CECILIO MAR MOLINERO is a Reader in Operational Research at the University of Southampton (UK). His main interests are: Multidimensional Scaling, Data Envelopment Analysis, quantitative models in Accounting and Finance, OR in Education, and Community OR. He is a frequent contributor to the Journal of the Operational Research Society and other learned journals, but still does not know how to set up his own web page.

First published to members of the Operational Research Society in OR Insight January - March 2000