Chapter 6

All the real knowledge which we possess, depends on methods by which we distinguish the similar from the dissimilar.  The greater number of natural distinctions this method comprehends, the clearer becomes our idea of things.  The more numerous the objects which employ our attention the more difficult it becomes to form such a method and the more necessary.

For we must not join in the same genus the horse and the swine, tho' both species had been one hoof'd nor separate in different genera the goat, the reindeer and the elk, tho' they differ in the form of their horns.  We ought therefore by attentive and diligent observation to determine the limits of the genera, since they cannot be determined a priori.  This is the great work, the important labour, for should the Genera be confused, all would be confusion.  
[Carolus Linaeus, Swedish Botonist, Genera Plantarum, 1739]

General observations drawn from particulars are the jewels of knowledge, comprehending great store in a little room.  
[John Locke, 17th Century British Philosopher]

Science is built up with facts, as a house is with stones.  But a collection of facts is no more a science than a heap of stones is a house.  
[Jules Henri Poincare, French Mathematician, La Science et l'Hypothese, 1908]

Throughout the history of the development of scientific method the only lasting theories have been those that began with good observation, with noting peculiar relations among measurements, or with firm groundwork of classificatory, taxonomic, and clinical experience.  In those cases where theory appears to have preceded observation, it will often be found that the theory that preceded measurement is the same as the post-measurement theory in name only.  
[Raymond B. Cattell, Research Professor in Psychology at the University of Illinois (Urbana), in Personality and Motivation Structure and Measurement (New York: World Book Company, 1957, p. 3)]

Comparing mills is like comparing apples and oranges.  No two are identical and the local environmental problems and priorities are different.  
[J. L. McClintock, Weyerhaeuser Corporation, as quoted in Paper Profits: Pollution in the Pulp and Paper Industry (New York: Council on Economic Priorities, 1971)]

One picture is worth more than ten thousand words.  
[Anonymous Chinese Proverb]

  In thy face I see
The map of honor, truth, and loyalty.  
[Shakespeare, Henri VI]

His face is the worst thing about him.  
[Shakespeare, Measure for Measure]

When men are calling names and making faces,
    And all the world's ajangle and ajar,
I meditate on interstellar spaces
    And smoke a mild seegar.  
[Burt Leston Taylor, 19th Century Poet, Canopus]

6.1--Introduction

The purpose of this chapter is largely to consider a number of approaches in taxonomy and the quest for empirical types.  The approaches discussed later on in this chapter are those which either (i) result in sensory displays (confined here to visual displays) enabling human observers to search for "types" in a subjective manner, or (ii) result in mathematical partitionings of entities into "types" via numerical taxonomy techniques.  The analysis may consist of more than merely searching for types on the basis of multivariate corporate social impacts such as those illustrated in Appendix A.  A point made repeatedly in earlier chapters is that corporate social accountings will typically yield masses of data, some of which are qualitative and some of which are quantitative but measured in differing units (percentages, man-hours, tons, cubic yards, dollars, etc.).  In such situations some type of parsimony is needed for both reporting and analyzing such a hodgepodge of disconnected facts.  The accustomed accounting procedure of converting everything to monetary units and then aggregating by arithmetic methods (usually addition) to achieve parsimony in social accounting is fraught with difficulties.  The usual statistical multivariate data analysis techniques (e.g., multiple regression, discriminant, factor and variance analyses) are somewhat more flexible, but frequently suffer from overly restrictive assumptions and/or difficulties in interpretation.

The major purpose of Chapter 6 is to explore some more general techniques for condensing and evaluating multivariate quantitative data, although some of the techniques may also accommodate qualitative differences.  In an effort to avoid being too abstract, such techniques are applied to a number of social accounting variables observed on twelve electric utility companies.  Particular emphasis is placed upon graphic and other visual display techniques under varying circumstances.  Several important data transformations and numerical taxonomy are also examined.

6.2--Theory of Types

Raymond Cattell, authority of personality typology, once stated:
    ...The Experience of science is that a tidy taxonomy is never useless, but full of systematic profits for research.  For example, in many social psychological problems, in which one person is the stimulus situation for the behavior of another, perceptions depend on type affiliations.  Types are thus not just unnecessary intermediate concepts--not just another instance of academic punditry or compulsion--but, if properly conceived, necessary and economical operational concepts...¹

The term "type" has intuitive meaning to nearly everyone, although forming a precise definition (along with related concepts such as group, pattern, cluster, configuration, factor, genus, species, etc.) is difficult.²  Entities classified as a type supposedly are "more alike" in terms of certain properties than other entities not of that type.  Different properties (attributes, traits, etc.) may give rise to different groupings of entities into types.  In addition, what constitutes a "type" depends on the basis for defining similarity (association, distance, affinity, interaction, etc.) and precise constraints imposed by the definition of what constitutes or does not constitute a "type."  For example, "types" may be mutually exclusive versus intersecting, collectively exhaustive versus selective, discrete partitions versus having gradations of belongedness, and so on.

Ball lists seven uses of cluster analysis which apply to the quest for types in general:

1. Finding a true typology;

2. Model fitting;

3. Prediction based on groups;

4. Hypothesis testing;

5. Data exploration;

6. Hypothesis testing;

7. Data reduction.³

These are not necessarily mutually exclusive, and prediction seemingly may arise under any of the above purposes.  Cattell writes:
    Briefly to indicate what this second step may comprise, one should point out that Aristotelian classification permits one to make predictions of the kind: "This is a dog; therefore it may bite"; "This is a schizophrenic; therefore the prospect of remissions is not high."  In other words, a classification of objects by variables of one kind may permit prediction on others not at the time included.  Parenthetically, despite the illustrations, these predictions need not be categorical, but can be parametric.⁴

I do not pretend to be the first to suggest that business firms might be typed.  For many years business firms have been viewed according to industry types, size classifications, production or marketing regions, capital intensity, labor intensity, etc.  I am suggesting, however, that researchers devote more attention to classifying business firms into empirical types on the basis of social impacts.  In the next chapter (Chapter 7) some attention is devoted to classifying firms or persons on the basis of human perceptions.  In this chapter (Chapter 6) our concern will be more upon classifications based upon general statistics on businesses, e.g., earnings margins, product prices, pollution expenditures, etc.  Research along similar lines has taken place with respect to finding nation types.  Rummell, for example, writes:
    Students of comparative relations have always dealt with nation types.  One type that has played a dominat role in the theoretical and applied international relations is that of the powerful nation.  This type has become so widely recognized as implying set characteristics and international behavior that we readily employ the noun "powers" alone to refer to nations of this kind.  Such nation "types" as "modern," underdeveloped," "Constitutional," "status quo nations," "prismatic," "aggressive," "traditional," and "nationalistic," have only to be mentioned to evidence the prevalence of typal distinctions.

The problem with the prevailing types is that the rationale underlying the categorization is not explicit (and that it is not clear whether the type really divides different kinds of variance).  If we are to deal in types, a clear and empirical basis for the distinctions must be made.⁵

¹    R. B. Cattell, Personality and Motivation Structure and Measurement (Yonkers-on-Hudson, New York: World Book Company, 1957, p. 383).

²    Definition varieties for "type" are discussed by Cattell, Ibid, pp. 364-69.

³    G. H. Ball, Classification Analysis, Stanford Research Institute, Project 5533, Stanford, California, 1971.

⁴    R. B. Cattell, "Taxonomic Principles for Locating and Using Types (and the Derived Taxonome Computer Program)," in Formal Representation of Human Judgment, Edited by B. Kleinmuntz (New York: John Wiley & Sons, Inc., 1968, p. 104).

⁵    R. J. Rummell, The Dimensions of Nations (Beverly Hills, California: Sage Publications, 1972, p. 300).

6.3--Condensation of Data: The Need for Parsimony

In spite of the difficulties of detecting, recording, and attestation of corporate impact data, equally difficult problems arise in utilizing such data.  Decisions are made by humans (or decision rules set by humans) and, unfortunately, the human mind is easily boggled by relatively small amounts of data.  As facts and figures begin to pile up, the decision maker devises means of organizing, categorizing, and summarizing in an effort to achieve parsimony in what he or she must comprehend and evaluate.  At one end of the spectrum are masses of disconnected facts; at the other end are a few condensed statements or measures.

Within a firm, the degree of condensation of traditional accounting data varies with the manager's level in the organization and the use to which information is to be put.  In social accounting we are still at a stage where we have a basket of apples, oranges, rocks, carrots, thistles, roses, rabbits, turtles, monkeys and ad infinitum.  Methods of condensation of heterogeneous social accounting items are undeveloped.

In traditional accounting, condensation typically consists of additive aggregation, e.g., operating managers may only see labor cost aggregated over people and time.  Top management examines summary reports over multiple divisions, subsidiary companies, and longer intervals of time.  The investing public receives even more parsimonious aggregations.

Another means of data condensation is the filtering process.  For example, budget or standard items may automatically be compared (by computer) with actual out comes.  Operating managers may only act upon "exception" phenomena, e.g., aberrant phenomena which vary from standard by some predetermined amount.  The aberrant phenomena are "filtered" out and acted upon.  Similarly, public press releases are usually about aberrant events apart from routine day-to-day happenings.

Typically an analysis is conducted whenever hidden or obscure relationships are suspected which are not evident in either the basic or aggregated data.  Analysis may, in turn, facilitate further condensation and parsimony, especially if the analysis yields crucial "measurements" needed to achieve further condensation.  The term "analysis" has a connotation of breaking something down into component parts, whereas "condense" implies combining component parts into a denser whole.  However, in science the term "analysis" does not necessarily imply less parsimony, e.g., one of the objectives of factor "analysis," component "analysis," cluster "analysis," regression "analysis," and other statistical analysis tools may be that of achieving parsimony.  As such, some form of "analysis" may be part of a data condensation process.  Similarly, in accounting a cost analysis may entail decomposition of "total cost" into various "component costs."  However, this is not necessarily the same as moving a step backwards on the condensation spectrum.  For example, total cost may be analyzed to break it down into fixed and variable components.  The analysis may utilize detailed data from labor and materials records, but the analysis may identify a relationship (e.g., linear) which facilitates parsimony and condensation.

In corporate financial accounting, the higher-most levels of condensation (after much aggregation, filtering, and analysis) are financial statements items and various computed statistics (e.g., working capital ratios and earnings-per-share) derived from financial statement items.  For example, the total assets reported (in billions of dollars) at the bottom of a General Motors Corporation annual report is a condensed measure of the millions of heterogeneous items of value held by the company.  The condensation process which yielded such a figure for G. M. Assets entailed a myriad of accounting "rules" of measurement.

At nearly every point in the accounting condensation process, accountants disagree as to the proper "rule."  As the condensations become more parsimonious, the accounting disputes are more pronounced.  One of the constant sources of difficulty is the penchant (based on centuries of tradition) of condensing on the basis of monetary units (i.e., a numeraire).  For example, cash in bank accounts, inventories, land, buildings, and all other items termed "assets" in the General Motors balance sheet are measured in dollars, which in turn, makes the heterogeneous items additive in a common scale of measurement.

Since it is even more difficult to measure most corporate social impacts in monetary units, accountants are reluctant to extend financial boundaries into unexplored social accounting territory.  Attempts to do so (e.g., the Abt Associates Social Audits⁶) have been highly controversial both as to method and to purpose.  Social audits have primarily been confined to descriptive listings of corporate social endeavors, with little or no attempt to measure or aggregate over heterogeneous items.  The question is whether it is possible to do more than just hold forth a basket of social accounting apples, oranges, rocks, carrots, thistles, roses, rabbits, turtles, monkeys, and so on.

⁶    See Chapter 3 of the book (cited at the top of this table).

6.4--Multivariate Data Analysis (MDA)

It is evident from preceding chapters (and Appendix A) that corporate social accounting entails multiple variates in areas of environmental impacts, consumer impacts, employee impacts, etc.  In this chapter I will turn to a number of multivariate data analysis (MDA) techniques employed in scientific research.  The objectives in most instances are to both achieve parsimony and to discover hidden unknown relationships.  It should be stressed, however, that rarely do MDA techniques disclose underlying casual mechanisms.  At best, the outcomes in MDA aid in prediction and possibly provide clues in the quest for discovery of causal relationships.

It should also be stressed that, in spite of intricate and complex mathematical formulations, the MDA outcomes are often not conducive to statistical inference testing.  Accordingly, MDA is usually a first exploratory step rather than a conclusive final stage in the analysis.

An extensive body of theory concerns MDA applied to continuous variates.⁷  Models used for such purposes include multiple regression, multiple discriminant analysis, canonical correlation, partial correlation, cluster analysis, factor analysis and related approaches.  Closely related are the classical experimental design models and analysis of variance (ANOVA) intended for analyzing a continuous criterion variate over discrete predictor variate cross-classifications.

Nominal variates may be analyzed in various ways.   Binary variates, for example, may often be included with continuous variates and treated as if they themselves are continuous, e.g., binary variates are commonly included as predictors in multiple regression equations.  Another means of nominal variate analysis is available in multivariate contingency table analysis.  For example, stepwise procedures utilizing maximum likelihood theory are availabe.⁸

Ordinal variates are usually the most difficult to analyze.  The usual procedure is either to (i) ignore the ordinal property and analyze ordinal variates in contingency tables, or (ii) ignore the discrete property and treat ordinal variates as continuous variates.  In recent years, however, multidimensional scaling (MDS) techniques have opened up a new line of approach.  In particular, MDS is useful in mapping preference or similarity orderings into metric space, and as such was a major breakthrough in analyzing subjective preferences.  This subject is taken up in greater detail later on in Chapter 7.

Few MDA techniques have been employed in corporate social accounting.  On occasion, social impact costs have been analyzed in some MDA models.  For example, studies utilizing regression techniques in air pollution impact measurement were reviewed in Chapter 4.  In the remainder of this chapter, potential applications of several other MDA tools will be explored, in particular general purpose multiple variate display and numerical taxonomy techniques.

⁷    References are legion.  I have compiled and abstracted thousands of MDA references on computer tape, R. E. Jensen, A Computerized Bibliography in Multivariate Data Analysis c/o South Stevens Hall, University of Main, Orono, Maine 04473.  Also see J. L. Dolby and J. W. Tuckey, The Statistics Cum Index (Los Altos, California: R&D Press, 1973).

⁸    See L. A. Goodman, "The Analysis of Multidimensional Contingency Tables: Stepwise Procedures and Direct Estimation Methods for Building Models for Multiple Classifications," Technometrics, Vol. 13, 1971, pp. 33-61.

6.5--An Illustration: Search for Types Among Twelve Electric Utility Companies

Throughout the remainder of this chapter, some electric utility company data will be analyzed for illustrative purposes using a variety of techniques.  It should be stressed that the intent is to illustrate the potential application of certain MDA techniques in comparing corporations in terms of multiple criteria.  In no way is this intended to be a thorough analysis of the companies involved.  It should also be noted at the onset that, although the data used in most of the illustrations in this chapter are continuous, many of the MDA approaches discussed are easily adapted to discrete data as well.

The electric utilities chosen for this section are the N=12 private power corporations listed in Table 6.1.  These were selected from the fifteen companies investigated in considerable depth by the Council on Economic Priorities.⁹  The three smallest companies are not included here, mainly for convenience in certain graphical displays presented later on.

The focal point for many of the illustrations which follow will be the Table 6.2 data on variates x₁,...x₁₀.  It might be noted that except for x₁ (megawattage), the other variates x₂,...x₁₀ are not necessarily directly associated with size of the companies involved.  For example, whereas pollutant volumes would normally be expected to increase with the size of an electric power company, percentage data such as that given for x₇,...x₁₀ pollution variates need not behave in such a manner.

The reader is cautioned about some of the conclusions which are either explicitly drawn or implicitly inferred in the illustrations which follow.  These conclusions follow only from the data as tabulated in the Council on Economic Priorities Study.  The write-up for the CEP study contains many footnotes and other explanations on the nature and limitations of this data.  Most of these explanations are not repeated here but should be carefully heeded before accepting my analysis of the published data as fact.

In some of the graphical displays it is difficult to handle more than a few variates at a time.  Therefore, from among the M=10 variates in Table 6.2, a select of subset four social impact criteria was extracted and is comprised of:

The above four variates cut across various interest groups, including shareholders, consumers, local communities, and the public-in-general (who might be especially interested in the R&D commitment.).

⁹    Charles Komanoff, Holly Miller, and Sandy Noyes, The Price of Power: Electric Utilities and the Environment, Edited by Joanna Underwood, (New York: The Council on Economic Priorities, 1972).

6.6--Graphic and Other Display Techniques

6.6.1--Purposes.  Numerical data are convenient to view in graphical form whenever possible.  For instance, continuous variates are often displayed in Cartesian scatter plots along one, two, and occasionally even three dimensions.  Discrete data are often represented in histograms, pie charts, etc.  Such display techniques are familiar and need not be elaborated upon here other than to mention that they might be effectively employed in corporate social accounting.  For example, wages might be displayed in relation to age, sex, race, plant location, etc.  Pollutant outputs might be plotted in relation to time, weather conditions, plant locations, etc.  Product performance and plant safety might similarly be displayed in various ways.  To date, however, graphic displays are sparingly employed in corporate social audit reports.  Conversely, in the public sector economic and social indicators are commonly displayed in graphic form.

Some of the more common purposes of graphical displays are mentioned below:

1. (COMMUNICATION).  Frequently the major intent is to communicate to other persons as concisely and efficiently as possible.  Graphical displays are advantageous first of all because they are more likely to capture attention than are long columns of numbers or paragraphs of text.  Secondly, graphical displays are frequently among the most parsimonious means of communicating data.

2. (DISCOVERY OF DISTRIBUTION PROPERTIES).  Sometimes the analyst constructs a graphical display of a single variate in order to discover its distributional properties, e.g., dispersion and skewness.  Following a mathematical analysis, outcomes or residuals are often plotted in order to identify violations of assumptions in the analysis.  For instance, regression residuals are frequently plotted in an effort to investigate conformance with normality, homoscedasticity, and independence assumptions.

3. (DETECTION OF ABERRANT PHENOMENA).  Often data are plotted in order to disclose phenomena deviating from norms.  Graphic displays are often a quick and simple means of detecting awry or extreme reactions.

4. (DETECTION OF LEVEL DIFFERENCES, SHAPES, AND CLUSTERS).  Graphical displays often disclose differences in levels of observations.  However, whereas level differences may often be discovered by merely scanning the data, hidden patterns, shapes, or clusters of phenomena may be disclosed (in graphical displays) which are almost impossible to discern by scanning the data itself.

5. (TRANSFORMATION AND CONCATENATION).  Graphics may assist the analyst in determining what, if any, transformations of the data (e.g., translation of axes, rotation, and scaling transformations) provide more useful results.  Often these become linked in a sequence and, through concatenation in interactive computer graphics, can be combined in one procedure.

6. (INVESTIGATION OF VARIATE AND ENTITY RELATIONSHIPS).  Another purpose of graphical displays may be to analyze the relationship between two or more variates.  For instance, scatter plots along two dimensions are frequently employed to study linear or nonlinear relations of two continuous variates.  Smooth functions may be fitted amongst data points.  If one of the variates is time, the purpose may be to identify trends, seasonal patterns, structural shifts, and drift of a variate of interest over time.

Patterns or clusters may also be detected among entities.  For instance, companies (or divisions within companies) might be first plotted according to pollutant discharges and then be partitioned into subsets according to visual scannings of plotted points.

An advantage of visual display is the tremendous ability and flexibility of humans for detecting spatially and temporally distributed features in data.  Mathematical models, though often an aid in discovering relationships, have much less flexibility and adaptive innovation ability.

6.6.2--Limitations.  Graphic displays are physical representations of properties.  One limitation is that qualitative properties are usually cumbersome to display relative to quantitative properties.  Quantitative properties, however, are also difficult to display in more than two dimensions, even though the analyst is frequently interested in detecting patterns in multivariate space.  Thirdly, in most graphical displays there is usually an upper bound on the number of entities that can be effectively plotted and compared.  Fourthly, it is a fallacy to assume that graphic displays are a substitute for mathematical analysis.  Often the detection or communication of phenomena depends upon making appropriate mathematical transformations of data to be plotted.  Developments in computer graphics have greatly facilitated the combining of mathematics and graphics.

Various approaches have been proposed for graphical display to overcome one or more of the above limitations, although usually trade-offs are encountered.  Several of these approaches are illustrated in the following discussion.  In many of these approaches an added difficulty arises in that how the variates (properties) are assigned to graphic pattern components either unintentionally or purposefully biases the outcomes.  Also, too many variates may obscure existent patterns in subsets of the variates.

6.6.3--Profile Line Plots and Shape Correlations.  Although quantitative variates are difficult to plot in more than two dimensions, various techniques may be employed.  One such technique is profile analysis in which entities are usually compared on the basis of their "profiles" on two or more variates under study.  Profile analysis is employed extensively in educational and psychological testing, i.e., persons are compared on the basis of graphical profiles of test scores.  If variates are not measured in the same scales, they are typically standardized to avoid scaling differences.

For illustrative purposes, four variates (x₃, x₄, x₅, and x₆) were selected from the Table 6.2 data presented previously.  Although the raw data could be plotted in profile charts, I elected to standardize (normalize) the variates using the customary transformation

The resultant standardized variate outcomes are shown in the STDVAR matrix in Table 6.3.  The electric utility company profiles derived from this data are shown in Exhibit 6.1.

It is immediately evident that no single company is consistently "best" or "worst" in terms of all four of these criteria.  For instances, Oklahoma Gas and electric (OGE) had the highest earnings margin (19.4%) and the lowest allocation to research and development (9% of revenues).   Similarly, The Southern Company (SOC) has a relatively poor performance on three criteria but generates the cheapest power (1.69¢ per kwh) for average residential users.  On two criteria (earnings margin and price per kwh) Consolidated Edison Company of N.Y. (CON) falls way below all the other companies in performance.

A careful inspection of Exhibit 6.1 reveals a number of profile similarities.  The Southern Company (SOC) and Florida Power and Light (FPL) have rather close profiles except for the x₅ (R&D) criterion.  Houston Lighting and Power (HLP), Oklahoma Gas and Electric (OGE), and Virginia Electric and Power (VEP) have similar profiles, especially in terms of the first three criteria.  Commonwealth Edison (COM) and Northern States Power (NSP) have somewhat close profiles on all three criteria.  Pacific Gas and Electric (PGE) and Southern California (SCE) are also similar except for the x₅ criterion (R&D allocation).

These profile similarities seem to suggest certain geographic "types" since the above-mentioned likenesses are mostly between companies operating in somewhat contiguous regions.  This is interesting since some of the paired companies along these criteria have major differences as well, e.g., whereas SOC is a large holding company across various southern states and in 1970 generated electric power with 79.1% coal, 20.6% gas, and 0.3% oil, FPL is a much smaller southern company using 56% oil and 44% gas.¹⁰

When examining profiles, analysts are sometimes interested in comparing profile shapes (configurations) irrespective of differences in profile levels and/or scatter.  A transformation which facilitates such comparisons is the profile scatter transformation

This transformation eliminates both profile level (elevation) and profile scatter (standard deviation) differences.  The effect of profile elevation removal, in particular, is to bring profiles with similar configurations (at different levels) closer together.¹¹  The profile scatter transformation yields what are called "pure shape" proviles.¹²  Profile charts derived after such a transformation of the data conform to the profile shape correlation coefficients computed from the formula

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This correlation coefficient (sometimes call a Q-technique correlation) is used when the analyst is interested in comparing profile shapes aside from elevation and scatter considerations.  In other words, the profile shape correlation coefficients are invariant under profile elevation and scatter transformations.  Other pairwise coefficients (such as Euclidean distances) are not necessarily invariant under such transformations, i.e., Euclidean distances reflect differences in profile levels whereas profile shape correlations measure differences in profile shapes (configurations).¹³

¹⁰    The Council on Economic Priorities, The Price of Power: Electric Utilities and the Environment, Op. Cit., p. 144.

¹¹    From a mathematical standpoint, the profile elevation transformation (i.e., the subtraction of entity means) projects the entity scores from N space to a hyperplane of N-1 dimensions.

¹²    In mathematical terms, the profile scatter transformation projects N entity scores to a hypershpere of N - 2 dimensions of constant radius lying in a hyperplane.

¹³    The profile shape correlation coefficients can, however, be shown to be related to Euclidean distance by the formula

CORENT(I,H) = 1 - DISENT(I,H))²
_____________________________
2(M - 1)

where DISENT(I,H) is the Euclidean distance between Entity I and Entity H using STDENT data.

The profile scatter transformation was performed on the STDVAR data in Table 6.3, yielding the STDENT standardized entity matrix also shown in Table 6.3.  The STDENT profiles are plotted in Exhibit 6.2.  One surprising and quite unexpected outcome is the near congruence of the Pacific Gas and Electric (PGE) and Baltimore Gas and Electric (BGE) profiles in Exhibit 6.2.  This indicates almost identical profile shapes for these two companies on the four criteria being analyzed, i.e., the two companies have almost identical profile "shapes" in Exhibit 6.1.  Similarly, the Oklahoma Gas and Electric (OGE) profile is closely related in shape to both the PGE and BGE profiles.  This indicates that these three companies must also have high profile shape correlations coefficients.  Another surprising likeness in profile shapes, as revealed in Exhibit 6.2., arises between Commonwealth Edison (COM) and Southern California Edison (SCE).  In this case, the two companies have similar profile shapes but differ in terms of profile elevation (in Exhibit 6.1).

The above visual conclusions from Exhibit 6.2 are borne out by the profile shape correlation coefficients shown in Table 6.4.  The five highest correlations are as follows:

What is a little less obvious in Exhibit 6.2 are the profile shapes least congruent.  In Table 6.4, however, the most negative profile shape correlation coefficients are revealed as:

These differences are not especially surprising except for the Northern States Power (NSP) and Virginia Electric Power (VEP) profiles.  These two companies are somewhat similar in size and in fuel usage.¹⁴  However, whereas the NSP profile in Exhibit 6.1 is relatively flat, the VEP profile moves from a high on earnings margin and cost per kwh to lows on R&D and pollution control inadequacy.

¹⁴    In 1970, the fuel use for NSP was 66% coal, 33% gas, and 1% oil.  For VEP the percentages were 53.8% coal, 46% oil, and 0.2% gas.

6.6.4--Principal Component (Factor Score) Profiles.  Profile analysis becomes clumsy when more than five or six variates (criteria) are under study, e.g., imagine trying to compare profile patterns over twenty or thirty social criteria.  Often, however, multicollinearities exist such that one, two, or several principal components or factors account for much or most of the variation in an entire system of variates.

One approach is to transform the original variates into factors and then plot entity factor scores.  For one or two principal factors, entities can be plotted in scatter plots.  For more than two factors, entity profile configurations can be examined using underlying factors in lieu of original variates.

Suppose there are M variates under study.  There are two major reasons why factor scores may be more of interest than original data:

(1) Whereas the M variates under study may be systematically intercorrelated with one another, the factors (principal components) are linearly independent (orthogonal).  This is helpful in data analysis techniques which are linearly independent (orthogonal).  This is helpful in data analysis techniques which assume linear independence.

(2) The factors (principal components) are extracted in such a manner that they successively account for smaller portions of the total variation among the M original variates.  If the first few factors account for a large share of this variation, and if they can be meaningfully interpreted, it may be possible to describe the system more parsimoniously (i.e., in fewer than M variates).

The major difficulty in principal component or factor analysis often lies in interpreting the importance and meaning of the factors extracted from the original variates.  The relative importance of successive factors can be estimated by comparing their latent roots (eigenvalues).  Finding descriptive interpretations is more difficult.  The usual approach is to examine the factor loadings (eigenvectors), which are correlations between factors and original variates.  Frequently, subsets of the original variates having highest correlations with a given factor have something in common which is suggestive of what the factor depicts.¹⁵

¹⁵    This approach was illustrated in the Chapter 4 principal component analysis of air pollution and human mortality data.  An excellent elementary example is also provided in W. W. Cooley and P. R. Lohnes, Multivariate Data Analysis (New York: John Wiley & Sons, Inc., Second Edition, 1971, pp. 133-36).

For illustrative purposes, the pairwise correlations between variates x₁,...,x₁₀ given in Table 6.2 are given in the CORVAR matrix in Table 6.5.  An underlying factor structure is not easily determinable from merely scanning this correlation matrix.  A

I.    PAIRWISE CORRELATIONS (CORVAIR) BETWEEN TEN VARIATES IN TABLE 6.2

II.    FACTOR LOADINGS

III.    FACTOR INTERPRETATIONS

(1) Factor 1 (Air Pollution Control Inadequacy): This factor loads highly on overall pollution inadequacy (x₆) and sulphur dioxide control inadequacy (x₈), both of which reflect air pollution under-investment in state-of-the-art controls available.  This factor also loads relatively high on coal usage (x₂), suggesting that heavy coal burning companies have a more serious under-investment in such controls, although there is considerable dispute over what constitutes "state-of-the-art" control, e.g., the wet scrubber dispute is discussed later on.

(2) Factor 1 (Technology): This appears to be largely an R&D (x₅) and nitrogen oxides control inadequacy (x₉) factor, the two variates being highly correlated at -.818.  Size of company in terms of megawattage (x₁) also loads highly on Factor 2, partly reflecting the fact that there is a tendency for larger companies to have a higher R&D proportion and lower nitrogen oxides control inadequacy.

(3) Factor 3 (Financial): This appears to be a combination of the company's earnings margin (x₃) and average customer price per kwh (x₄), the two being negatively correlated at -.5145.

(4) Factor 4 thru 10 (Junk): These factors have latent roots less than one, and hence, are not viewed as relevant underlying factors.

IV.    LATENT ROOTS (EIGENVALUES)

principal component analysis on the variates x₁,...,x₁₀ in Table 6.2 yielded the outcomes in Table 6.5.  Three factors emerged with latent roots exceeding one.  These three factors account for 79.1% of the variance in the ten-variate system.  Interpretations of these factors are not at all obvious or concise.  Based upon the rotated factor loadings shown in Table 6.5, the best interpretations I could come up with are also given in Table 6.5.

The illustration points out one of the potential frustrations with principal component or factor analysis in general, i.e., a frequently encountered situation arises in which there is no concise and all-embracing concept for two or more rather heterogeneous variates closely correlated with a factor.  This is particularly evident in Factor 2 in Table 6.5, which loads highly on research and development (x₅), nitrogen oxide control inadequacy (x₉), and megawattage (x₁).  It is also evident in Factor 3, which loads highly on earnings margin (x₃) and cost (price) per kwh to an average residential electricity consumer (x₄).

The outcomes in Table 6.5 were utilized in transforming the M=10 variates (in Table 6.2) into the major factor scores (on each entity) sown in Table 6.6.  The company (entity) profiles derived from the standardized factor scores (SFSCENT) are shown in Exhibit 6.3.  No company consistently performs highest on all criteria, although SCE performs relatively well on all three major underlying factors, brief interpretations for which were given in Table 6.5.  The inconsistent performance of CON is manifested in its somewhat reasonable performance on Factor 1 (pollution control) relative to falling way below other companies on Factor 3 (financial performance) due to a combination of having both the lowest earnings margin and the highest kwh rates.  The inconsistent performance of AEP is also evident in its poor showing on Factor 1 (pollution control) relative to the highest showing on Factor 2 (technology) due to a combination of having a relatively high R&D commitment (x₅) and a low nitrogen oxides state-of-the-art underinvestment (x₉).  As indicated previously, however, the AEP performance on x₉ is misleading since it is the lack of technology for "state-of-the-art" pollution control rather than investment in pollution controls which gives the coal-fired AEP such a good score on x₉.

Similarity in both level and shape on the three principal underlying factor profiles in Exhibit 6.3 are also evident.  For example, the large coal burning companies (AEP, COM, and SOC) have very similar profiles, with AEP pulling ahead on Factor 2 due to a higher R&D commitment.  In contrast, the smaller natural gas-fired OGE and HLP companies have almost congruent profiles with shapes nearly opposite those of the large coal-fired companies.  The larger SCE, however, does not succumb to the OGE and HLP drop along Factor 2 because of the exceptional performance of SCE on both R&D (x₅) and nitrogen oxides (x₉) criteria.

One of the most important outcomes in the factor score profiles in Exhibit 6.3 arises in the amazing similarity between the Florida Power and Light (FPL) and Northern States Power (NSP) profiles.  In contrast, the M=10 variate raw scores for these companies (see Table 6.2) are much more divergent.¹⁶  This phenomenon provides an important illustration of how principal components or other types of factor analyses can be used to reduce a large number of variates into a more parsimonious subset of underlying principal factors.  At the same time it also illustrates "overkill" in the sense that the outcome may be too parsimonious.  For example, the primary determinants of Factor 3 appear to be quite different social impact criteria which, at least in this data, are negatively correlated.  Company scores on Factor 3 are caught between opposing forces.  For example, the FPL "poor" showing on earnings margin (x₃) pulls against the FPL "good" score on electricity pricing (x₄).  Similar negative correlations in performance criteria are present in other factors.  Hence, this is the case where, because of opposing interests in given factors, less parsimony in terms of keeping opposing criteria separated is probably more meaningful.

¹⁶    Also note the divergent FPL and NSP profiles in Exhibit 6.1.

6.6.5--Fourier Series Profiles.  In the preceding section, a principal component analysis was reported in which M=10 variates were parsimoniously reduced to M'=3 factors (principal components).  The resultant factor scores were plotted in the Exhibit 6.3.  Suppose, however, that such an analysis yielded a substantially larger number of underlying factors, e.g., suppose M=50 variates produced M'=15 factors of interest.  Profile charts are difficult to construct and evaluate for more than a few factors.

An alternate approach which is especially interesting when there are more than a handful of underlying major factors is to use a Fourier series method originally proposed by Andrews.¹⁷  The procedure for plotting multivariate observations on each entity is to compute the following Fourier series transform on each entity (e.g., each company):

The f(t) function is then plotted (best results are obtained from a computer plotter) for values of t over the range ±3.1416, such that each entity receives a plotted curve over this range of t.  Profiles of entities may then be compared both as to level and to configuration.  The number of variates is not a limiting constraint, i.e., the f(t) function is plotted against t rather than the x_J variates.  When the x_J variates are linearly independent and certain other assumptions are met, the f(t) outcomes have a number of interesting properties and are conducive to statistical inference testing of differences between entity profiles.

Proceeding by way of illustration, consider the factor scores shown previously in Table 6.6.  These outcomes were transformed into Fourier series curves plotted in Exhibit 6.4.  Most plotted f(t) profiles yield conclusions similar to those derived previously from the profiles in Exhibit 6.3.  For example, in Exhibit 6.4 the FPL(E) and NSP(G) curves are nearly congruent, indicating that these two companies have almost identical scores on the three major underlying factors.  The similarity among the three largest coal-fired companies (AEP(A), COM(C), and SOC(K)) are also evident in their bell-shaped curves which differ markedly from the curves of the other companies.  The natural gas burning companies HLP(F) and OGE(H) also have similar profiles.  The widely differing performances of CON and SCE are also evident.

When there are only a few factors (e.g., the three factors in Exhibit 6.3) there seems to be little advantage in resorting to the more complex Fourier series profiles such as those in Exhibit 6.4.  The Fourier series approach becomes more interesting when the number of factors becomes too unwieldy for a profile analysis on all factors simultaneously.  However, both approaches (e.g., those in Exhibits 6.3 and 6.4) are cumbersome when there are very many entities, e.g., the N=12 profiles plotted in the preceding profile exhibits approach the limit of human ability to visually compare profiles.

¹⁷    D. F. Andrews, "Plots of High Dimensional Data," Biometrics, Vol. 28, March 1973, pp. 125-36.

6.6.6--Geometric Patterns and Plotted Caricatures.  Instead of plotting multivariate data as scatter plots or profile line plots, it is sometimes better to consider other geometric patterns (e.g., triangles, rectangles, etc.) or caricatures (e.g., facial sketches).  It may be particularly advantageous to do so when:

(1) The number of entities (N) is such that profile lines overlap and crisscross so much that entity comparisons are difficult, e.g., previous profile plots of N=12 electric utility companies were difficult to evaluate because of numerous intersecting line segments.

(2) There are discrete qualitative variates under study which can be depicted as varying geometric shapes or caricature components.

There is a limit to how many entities (N) can be depicted or how many variates (M) can be incorporated as features in geometric patterns or caricatures.  In recent years, however, a number of interesting innovations in these areas have arisen, some of which will be illustrated here.

For example, Edgar Anderson proposed the drawing of geometric patterns which he termed "glyphs."¹⁸  These were intended primarily for the graphical display of multiattribute discrete variates in biology.  A glyph has a base (or core) with rays pointed upward, where each ray depicts a different attribute.  For example, an attribute having three categories is depicted by Anderson as a ray having three lengths, i.e., zero, medium, and long.

A slightly modified glyph approach is illustrated in Exhibit 6.5.  In this case the standardized variates on x₃, x₄, x₅, and x₆ social impact criteria in Table 6.3 are depicted as separate rays (in clockwise order).  Each glyph corresponds to a different electric utility company.  The ray lengths are marked into unit gradations where:

The origin on a standardized variate (which is also the mean of a standardized variate) is marked with a "o" on those rays for which companies scored at or above the mean on the criterion in question.

In Exhibit 6.5 each glyph is plotted in a two-dimensional Euclidean space, where the horizontal axis corresponds to x₁ (megawattage) and the verticle axis corresponds to x₂ (coal usage) raw data scores from Table 6.2.  Note that the largest coal burning companies (AEP, COM, and SOC) are isolated by themselves in x₁ and x₂ space.  Smaller companies which also rely heavily on coal (NSP, BGE, and VEP) also cluster by themselves.  Companies which use little or no coal are also clustered on the x₁ axis as large (SCE, PGE, and CON), medium (HLP and FPL) and small (OGE).

The net result is that in Exhibit 6.5 multivariate data in six dimensions are plotted in two-dimensional space.  The company glyphs resemble frontal views of wounded biplanes returning from battle.  Performances on the x₃, x₄, x₅, and x₆ standardized criteria appear as wings (rays) of varying lengths.  If the origin, "o," is shown on the wing (ray), the company performed at or above the mean on the criterion in question.  The "o" origins resemble engines beneath a wing.  In this context, a company has an "engine" on a wing if it performed at or above the mean performance on that criterion.

In this sense, the "best" performing companies in Exhibit 6.5 are those with the longest wings.  The only company performing above the standardized mean (zero) on all four social impact criteria (and therefore having all four "engines" intact under its glyph wings) is Pacific Gas and Electric (PGE).  Both HLP and OGE are natural gas burning companies which perform at or near the best on three criteria (x₃, x₄, and x₆) but have little or no wing (ray) length on the x₅ (R&D) criterion.  Similarly, SCE performs quite well on three criteria but falls slightly below the mean on the x₄ (kwh price) criterion.  AEP and NSP are also "three-engine" glyph biplanes, where AEP falls short on x₆ (pollution control inadequacy) and NSP falls short on x₃ (earnings margin).

In contrast, CON barely flies along on its single x₆ (pollution control inadequacy) engine whereas FPL limps on its x₄ (price per kwh) performer.  Other single-engine glyphs (BGE and COM) have better balance in terms of wing (ray) length on all four criteria in Exhibit 6.5.

Among all the graphic display approaches illustrated thus far, I find the glyph approach quite appealing.  Anderson's glyph rays are plotted according to discrete ordinal scales, although nominal or continuous (as illustrated in Exhibit 6.5) variates may be plotted as glyph rays.  Glyphs may also be used as geometric pattern representations without having to be plotted in Euclidean space.  Anderson recommends no more than seven rays and that rays do no extend in all directions.  He also recommends having no more than three discrete levels for ray length (a recommendation which was not followed in Exhibit 6.5).  Continuous variates may also be transformed into these three discrete ordinal categories.  Multiple rays may be used for more than three categories or complexes of related variates.  Anderson writes:

In attempting to work out complexes of related qualities, the analysis is facilitated if the ray lengths are coded in such a way that all the extreme values characteristic of one complex are assigned long rays and those characteristic of the other are assigned no rays.  For example, in studying hybridization between two subspecies of Campsis, one of the subspecies had a short tube, a wide limb, and much red in the flower; the other had a long tube, a small limb, and little red.  Redness and limb width were coded with long rays for much red and for wide limbs, tube length was coded in reverse with a long ray for short tubes.  This meant that those hybrids closely resembling the other parent as (sic.) a rayless dot.¹⁹

For purposes of graphic plotting, the symbols drawn may be triangles, line segments, polygons, or most any caricature imaginable.  One of the most unique caricature plotting ideas is described by Tversky and Krantz.²⁰  They depict alternate sketches of face shape (long versus wide), eyes (empty  versus filled-in), and mouth (straight versus curved) to represent three binary variates in two-dimensional plots.  The facial sketches were then used in a visual perception test of interdimensional additivity, i.e., that overall dissimilarity between faces could be decomposed into additive components represented by varying facial features.

A more extensive and general facial plotting program was apparently developed independently by Chernoff,²¹ although both Tversky-Krantz and Chernoff utilize elliptical components.  Each variate (initially the computer program developed by Chernoff can handle up to 18 variates, but the program can be modified to accommodate more variates) is represented as a feature (eye shape, eye size, mouth shape, mouth size, etc.) in a computer-sketched face.  Differing values of the variate are distinguished by different sizes and/or shapes of the feature in question.  Each entity is depicted by a particular face whose features are determined by observed values of variates on that entity.  An advantage of facial caricatures over glyph plots is that numerous features can be depicted in faces whereas Anderson found that glyphs with more than seven rays were too cumbersome.

The facial features in Chernoff's original program are listed in Table 6.7.  If there are fewer than M=18 variates under study, a given variate may (i) be assigned to more than one feature or (ii) certain features may remain fixed.

For instance, the N=12 entities (electric utility companies) measured on M=4 social impact criteria in Table 6.3 are plotted as faces in Exhibit 6.6.  In this case the M=4 variates were randomly assigned to four different facial features, giving rise to 16 features which vary among the N=12 faces plotted in Exhibit 6.6.  The faces have been arranged in two-dimensional Euclidean space on x₁ (megawattage) and x₂ (coal usage) from Table 6.2, i.e., the exhibit depicts two Cartesian variates and sixteen facial variations determined by x₃ (earnings margin), x₄ (kwh pricing), x₅ (R&D), and x₆ (pollution control inadequacy).  Recall that the latter four criteria were also displayed in Exhibits 6.1 and 6.2 in profile charts and Exhibit 6.5 as glyph rays.

After plotting the faces, I had a number of students, businessmen (e.g., those who attended my N.A.A. courses on accounting for corporate social responsibility²²), and other friends try to match up the faces.  For this purpose the faces were not plotted in Euclidean space on x₁ and x₂ as they are in Exhibit 6.6 nor was there any indication as to what the faces depicted.  Interestingly, rather consistent partitionings of these N=12 faces into G=5 clusters (groups) emerged from those subjective evaluations.

  The most consistent clustering were as follows:

Variations in the above clusterings tended to arise mainly in differing partitionings among the Cluster 1 and 2 companies, all of which tend to be the "good guys" in terms of Table 6.3 data relative to the companies in Clusters 3, 4, and 5.²⁴  In any case, the subjective clusterings differed greatly in terms of x₁ size and x₂ coal usage variates (see Exhibit 6.6).  For example, BGE is a small and relatively heavy coal user whereas SCE is a much larger power company with only light usage of coal.  Similarly, AEP, NSP, and FPL vary widely in terms of size and/or coal usage.  It might also be noted that I tended to get fairly consistent outcomes when human subjects clustered faces obtained under two other random assignments of particular facial features to the M=4 social impact criteria in Table 6.3.

In a second effort, I used the standardized (SFSCENT) factor scores in Table 6.6 (which in turn were derived from the M=10 social impact criteria in Table 6.2) to obtain the electric utility company faces shown in Exhibit 6.7.  The most consistent subjective clusterings (among the human subjects I persuaded to match up the faces) correspond to companies allocated to G=4 clusters (groups) as follows:

Faces are grouped in Exhibit 6.7 to reflect these clusters.  Variations arose mainly when a few subjects matched CON, SOC, and SCE, apparently on the basis of head shape but ignoring major differences in length of nose, length of mouth, height of centers of eyes, separation of centers of eyes, half-length of eyes, position of pupils, eccentricities of eyes, and eyebrow features.²⁶  The fact that some persons matched CON, SOC, and SCE faces highlights the need to make several plottings of faces with different random assignments of variates (in this case factors) to facial features.  The Exhibit 6.7 faces are the result of only one such random assignment.

The more frequent clusterings of faces into G=4 clusters (groups) shown in Exhibit 6.7 conform fairly well with the Exhibit 6.3 profiles.  Both FPL and NSP faces are closely matched in Cluster 2, whereas CON by itself in Cluster 1 stands apart from the rest of the faces in feature combinations.  The Cluster 3 companies AEP, COM, and SOC have similar profiles in Exhibit 6.3, whereas the VEP difference in profile shape is not reflected in the Exhibit 6.7 faces.  In order to capture profile shape comparisons it would be better to first remove entity elevation and scatter (to arrive at STDENT values in the manner described previously) before plotting the faces.

Cluster 4 in Exhibit 6.7 contains the least homogeneous profiles (from Exhibit 6.3).  In particular, BGE, HLP, OGE, and PGE are joined together, whereas both the BGE and PGE profiles differ rather markedly from the HLP and OGE profiles in Exhibit 6.3.  Once again this demonstrates that, if profile shape (rather than level) is of primary interest, a profile scatter transformation should be made prior to forming the faces.  Cluster 4 does tend to contain the "clean-guys" with higher proportions of natural gas-generated electric power.  The noteworthy exception in Exhibit 6.7 cluster 4 is Baltimore Gas and Electric (BGE) which in 1970 utilized 59.1% coal as opposed to 0.1% gas.  In terms of size and coal usage, BGE is much more like NSP and VEP, but its face (and its standardized principal factor scores) differs markedly from the NSP and VEP faces in Exhibit 6.7.

An apropos question is (among the infinite patterns or caricatures which might be used)--"Why faces?".  Probably the best argument which might be raised in favor of faces is that all people with sight are used to seeing faces.  At an early age humans learn to distinguish, on the basis of manifest facial features, hundreds or even thousands of faces (both real and cartoon).  A second argument is that numerous variates can be depicted by facial features (jaw line, cheeks,  nose, eyes, ears, hair, dimples, wrinkles, etc.) in terms of shape, size, and orientation.  If additional body features (neck, chest, abdomen, etc.) are added in, thousands of variates can, in theory, be included.  Prior to computer-aided plotting, however, slight variations in continuous variates would have been difficult to precisely portray.

It is usually possible to compare more entities in caricature plotting than in profile analysis.  Chernoff, for example, provides two empirical illustrations comprised of 88 and 53 entities (faces) respectively.²⁷  A visual cluster analysis was attempted by various persons in both instances, with consistent agreement on clusterings of Chernoff's many facial caricatures.

There is a limit, however, to how many faces can be visually compared and clustered by human analysts.  I cannot imagine, for example, comparing N=729 caricatures in the Pickett and White study to be mentioned later on, i.e., if faces were drawn smaller and condensed for "texture' comparisons, features in each face would be obscured.  Thus, the facial caricature approach would probably be used for a fewer number of individual comparisons, although the maximum upper bound of faces that can be compared depends upon many circumstances.

Another drawback of the facial caricature approach, it seems to me, is that in a given facial feature only extreme variations are easily discerned.  This can be partly overcome by assigning a variate to two or more features which, in combination, serve to bring out lesser variations.

Still another drawback is that some facial features may have more importance than others in distinguishing faces.  This implies that clustering outcomes may may be biased when assigning variates to facial features.  This can be partly overcome by repeating the analysis several times under alternative assignments of variates to features.  This approach, of course, increases the time, effort, and cost of the study in terms of computers, plotters, and persons examining facial caricatures.

An especially bothersome phenomenon in both profile and pattern display approaches (including facial caricatures) is that the addition of too many variates may tend to obscure patterns in smaller subsets of the variates under study.  The solution seems to fall back on repeated attempts under judicious selections of subsets of variates.  In this regard, statistical analysis and graphic analysis might work hand-in-hand.  For instance, a multiple regression might be performed to "take out the effects" of certain variates (as in covariance analysis) prior to plotting regression residuals.  Similarly, a principal component analysis might be performed in order to extract interpretable orthogonal factors to be used in lieu of intercorrelated variates.  This latter approach was illustrated previously in Exhibit 6.7.

¹⁸    Edgar Anderson, "A Semigraphical Method for the Analysis of Complex Problems," Technometrics, Vol. 2, August 1960, pp. 387-91.

¹⁹    Ibid, p. 391.

²⁰    Amos Tversky and David H. Krantz, "Similarity in Schematic Faces: A Test of Interdimensional Additivity," Perception and Psychophysics, Vol. 5, 1969, pp. 124-28.

²¹    Herman Chernoff, "The Use of Faces to Represent Points in n-Dimensional Space Graphically," Technical Report No. 71, Department of Statistics, Stanford University, December 27, 1971.  Portions of this paper are also published in the Journal of the American Statistical Association, June 1973, pp. 361-68.

²²     These N.A.A. courses were mentioned in greater detail in Chapter 3.

²³    Using Exhibit 6.6 faces, which in turn were derived using STDVAR data from Table 6.3 on x₃, x₄, x₅, and x₆.  I hesitated to conduct a formal analysis of the subjective clusterings for a number of reasons, one of which is that time constraints under which subjects were asked to compare faces varied greatly due to circumstances outside of my control.  Only 33 persons submitted completed subjective clusterings according to my instructions, which allowed them to choose both the number of clusters and the assignment of faces to clusters.  The mode clustering outcome (12 cases) was that shown above.  Variations tended to not differ greatly from this mode.

²⁴    There are exceptions noted previously, however, such as the low R&D commitments (x₅) of HLP and OGE relative to AEP and COM.  The ultimate judgment of "good versus "bad" entails consideration of other criteria and operating constraints.

²⁵    The three factors (components are the Table 6.6 standardized factor scores underlying the M=10 variates in Table 6.2.  I hesitated to conduct a formal analysis of subjective clustering variations for reasons noted previously.

²⁶    Each of the three standardized factors (from Table 6.7) was randomly assigned to six facial features giving rise to eighteen facial feature variations in Exhibit 6.7.

²⁷    The first of these involved eight variates observed on each of 88 specimens from the Encene Limestone Formation in northwestern Jamaica.  The second involved twelve variates observed on each of 53 mineral core specimen from a core drilled in a Colorado mountainside.  In both instances the variates were all quantitative in nature.

6.6.7--Texture Analysis in Large Sample Graphs.  If geometric patterns or caricatures are to be compared for a large number of entities, comparisons of individual entities may become futile (unless the intent is to discover one or a few aberrant entities which stand out from the crowd).  In such instances, however, it may be possible to identify patterns among dense groupings of entities.  In information display terminology this is sometimes called analyzing the "texture" patterns.  For example, Pickett and White²⁸ use computer-graphic triangles to represent N=729 college students.  The triangles are drawn quite small in order to fit on a single page.  Each triangle depicts five variates in the manner described below:

Each triangle presents five measures.  Two of the measures control the position of the triangle in its unmarked 30x30 raster unit cell.  Another measure controls the altitude of the triangle, another its orientation and another the width of its base.²⁹

Whereas in preceding Exhibits 6.1 thru 6.7, individual entity (company) profiles could be compared with one another, it is difficult to imagine such comparisons among the mass of N=729 triangles (depicting college students) drawn by Pickett and White.  Many of their triangles are so small that their plot is hardly more than small, faint lines.  Instead of individual comparisons, the Pickett and White approach is normally used to compare predefined groups or classes of entities.  For this reason, entities are arranged in the Pickett and White illustration as described below:

The data are arranged into three groups, forming verticle bands of equal width.  The left band contains profiles of dropouts, the middle band profiles of regular graduates, the right band profiles of honor graduates.³⁰

From these outcomes, Pickett and White concluded the following:

Again the hope would be that some new hints of differences among these three groups might be derived by looking at such a display.  One intriguing thing that has been suggested by brief perusals so far is that honor students may be more similar to dropouts than they are to regular graduates.  It is insights of this rather unexpected sort which, if they prove to be valid, would make such a regular display technique very much worth while.³¹

In graphic displays with densities such as that illustrated by Pickett and White, the images resemble something analogous to the texture of interwoven or interwined threads.  Human perception of visual "texture" has been the subject of behavioral study.³²  The objective might be to perform either:

(i)    Cluster Analysis--to identify similar clusters or areas having common "texture" in visual image;

(ii)    Discrimination--to compare "textures" of different groupings of entities in order to determine whether variates under study differentiate the (known) groups.

It is important to note that in discrimination efforts the groupings are predefined for graphic display purposes.  In their college student illustration, for instance, Pickett and White determined in advance the student dropout, regular student, and honor student groupings.  The students were plotted in three contiguous verticle "bands" of triangles according to which group they belonged.  In contrast, for cluster analysis purposes entities would not be plotted according to such predefined structure.  Analysts would instead plot the entities at random and then attempt to determine "if" and "how many" clusters seemed to emerge on the basis of visual texture similarities.  Attempts would be made subsequently to identify and interpret the groupings.  Cluster analysis is discussed in greater detail later on.

²⁸    Ronald M. Pickett and Benjamin W. White, "Constructing Data Pictures," Seventh National Symposium on Information Display, Society for Information Display, 1966, pp. 75-81.

²⁹    Ibid, p. 80.  Pickett and White note that in a stero (three-dimensional) display two additional variates could be represented by the depth and tilt of each triangle.

³⁰    Ibid, pp. 79-80.

³¹    Ibid, p. 80.

³²    See R. M. Pickett, "The Perception of Visual Texture," Journal of Experimental Psychology, Vol. 68, 1964, pp. 13-20.

6.6.8--A Crystal-Ball Look Into the Future.  Tremendous strides have been made in graphics in recent years, particularly computer graphics.  There have been significant advances in plotting accuracy, shading interactive graphics, luminescence, cathode ray tube techniques, film recording, and large screen projection, not to mention related advances in color television, photography, and picture transmission.  The future holds forth laser displays, light modulation techniques, and improved use of color, e.g., multicolor phospher.  There are also harbingers of total sensual experience systems using visual, sound, touch, and odor stimuli.  The idea of combing of such inputs (not merely for entertainment but for serious analysis of multivariate properties) is fascinating to conjecture about in armchair speculation.  Information display is in fact a bright spot amidst the gloom of being swamped in the spate of a data floodtide in corporate social accounting.

From the standpoint of visual display, effective three-dimensional plotting would be a tremendous help in analyzing data.  There have been some advances in line perspective displays and shading.³³  Stereoscopic displays hold forth some potential,³⁴ along with holographic display techniques.³⁵  However, nothing seems as effective as three-dimensional physical models capable of being viewed from varying perspectives.  Efficient ways of constructing three-dimensional displays have yet to be developed.

Also of special interest in data analysis is interactive computer graphics, which allows the computer and the analyst to "interact" in determining the nature of graphic displays.³⁶  The computer is utilized for various purposes, the major ones being data transformation and concatenation.  Translation, rotation, and scaling changes are commonly performed in interactive sequences as analyst and machine interact.³⁷  In addition, more complex data analysis routines (e.g., principal component analysis, multidimensional scaling, etc.) may be called up from the computer library to produce outcomes which the analyst becomes interested in seeing displayed.  Although most interactive computer graphic systems are still exploratory in nature, it does appear that such capabilities are in the horizon.  This newer technology may revolutionize both corporate social accounting and traditional financial and managerial accounting as well.

³³    An excellent discussion can be found in Part 4 of William M. Newman and Robert F. Sproull, Principles of Interactive Computer Graphics (New York: McGraw-Hill Book Company, 1973).

³⁴    See, for example, Richard Stover, "Autostereoscopic Three Dimensional Display," Information Display, Vol. 9, January/February 1972.

³⁵    See A. D. Jacobson, "Requirements for Holographic Display," Information Display, Vol. 7, Nov./Dec. 1970.

³⁶    See D. J. Hall, G. H. Ball, and J. W. Eusebio, "Promenade--An Interactive Graphics Pattern-Recognition System," Information Display, Vol. 5, Nov/Dec 1968.  Also see S. A. Watson, "Dataplot: A System for On-Line Graphical Display of Statistical Data," Information Display, Vol. 4, July/August 1967.

³⁷    An excellent discussion is given in Newman and Sproll, Op Cit.

6.7--Numerical Taxonomy

6.7.1--Definition of Terms.  Natural scientists have long been faced with situations in which they attempt to compare entities (organisms, subjects, specimens, or "organizational taxonomic units" called OTU's) on the basis of multiple variates (characteristics, properties, attributes).  In taxonomy such comparions are made for purposes of both defining taxa (groups, classifications, or subsets) and assigning entities to taxa.

Taxonomic procedures also take place in economics and business (e.g., the definitions of industries and assignment of companies to industry classes) although the terminology is quite different.  Natural scientists (with the help of scholars from various other disciplines) have, however, developed certain numerical taxonomy procedures which have only rarely been applied in business and economics.³⁸  The purpose of this section will be to illustrate how some of these numerical procedures might be useful in corporate social accounting.  First, however, some of the taxonomy terminology will be more precisely defines as presented in Sneath and Sokal:³⁹

(1) SYSTEMATICS.  Sneath and Sokal borrow Simpson's definition of "systematics" as "the scientific study of the kinds and diversity of organisms and any and all relationships among them."⁴⁰  In corporate social accounting such "organisms" might be companies in general, companies in a given industry, factories, mines, mills, or some other subdivision of corporations.  However, they might also be interest groups among employees, customers, investors, etc.

(2) CLASSIFICATION.  Again borrowing from Simpson, classification is defined as "the ordering of organisms into groups (or sets) on the basis of their relationships."⁴¹  Classification is common in defining industry groupings of companies.  However, other types of classification may also be of interest, e.g., classification of social impacts or interest groups.

(3) IDENTIFICATION.  Sneath and Sokal define identification as the "allocation of additional unidentified objects to the correct class once its classification has been established."⁴²  Relating this to social accounting, suppose firms in a given region are classified as either meeting or not meeting a set of norms or standards (e.g., pollution levels, minority employment, etc.).  Variables of interest would be observed and then identification of class membership could be established.

(4) TAXONOMY.  Simpson defined taxonomy as "the theoretical study of classification, including its bases, principles, procedures and rules."⁴³

(5) NUMERICAL TAXONOMY.  Sneath and Sokal define numerical taxonomy as "the grouping by numerical methods of taxonomic units into taxa on the basis of their character states."⁴⁴  Various related terms (e.g., mathematical taxonomy, quantitative taxonomy, numerical systematics, taxometrics, and multivariate morphometrics) have also been used, but numerical taxonomy seems to be the most widely employed, even in fields outside the natural sciences.

³⁸    There are some applications in business and economics, a few of which are as follows: W. D. Fisher, Clustering and Aggregation in Economics (Baltimore: John Hopkins Press, 1969); R. G. Fisher, W. T. Williams, and G. N. Lance, "An Application of Techniques of Numerical Taxonomy to Company Information," Econ. Rec., Vol. 43 pp. 566-87; F. Goronzy, "A Numerical Taxonomy on Business Enterprises," in A. J. Cole (Ed.), Numerical Taxonomy (London: Academic Press, 1969, pp. 42-52; T. Joyce and C. Channon, "Classifying Market Survey Respondents," Applied Statistics, Vol. 15, 1966, pp. 191-215; R. E. Frank and P. E. Green, "Numerical Taxonomy in Marketing Analysis.  A Review Article," Journal of Marketing Research, Vol. 5, 1968, pp. 83-94; P. E. Green, R. E. Frank, and P. J. Robinson, "Cluster Analysis in Test Market Selection," Management Science, Vol. 13, 1967, pp. B387-B400; R. E. Jensen, "A Cluster Analysis Study of Financial Performance of Selected Business Firms," The Accounting Review, Vol. XLVI, January 1971, pp. 36-56; B. King, "Market and Industry Factors in Stock Price Behavior," The Journal of Business, Supplement 1966; F. M. Bass, "A Taxonomy of Magazine Readership," The Journal of Business, Vol. 42, 1969, pp. 337-63; A.S.C. Ehrenberg, "Factor Analytic Search for Program Types," Journal of Advertising Research, 1968, pp. 55-63; J. G. Myers, "On Some Applications of Cluster Analysis for the Study of Consumer Typologies and Attitudinal Behavior Change" In Johan Ardnt (Editor), Insights Into Consumer Behavior (New York: Allyn and Bacon, 1968); J. G. Myers and F. M. Nicosia, "On the Study of Consumer Typologies," Journal of Marketing Research, Vol. 5, 1968, pp. 182-93; J. N. Sheth, "The Multivariate Revolution in Marketing Research," Journal of Marketing, Vol. 35, 1971, pp.3-19.

³⁹    Peter H. A. Sneath and Robert R. Sokal, Numerical Taxonomy: The Principles and Practice of Numerical Classification (San Francisco: W. H. Freeman and  Company, 1973).

⁴⁰    G. G. Simpson, Principles of Animal Taxonomy (New York: Columbia University Press, 1961, p. 7).

⁴¹    Ibid, p. 9.

⁴²    Sneath and  Sokal, Op. Cit., p. 3.

⁴³    Simpson, Op. Cit., p. 11.

⁴⁴    Sneath and  Sokal, Op. Cit., p. 4.

6.7.2--Purpose.  The fundamentals of numerical taxonomy in the natural sciences are grounded in the early works of an 18th century French botanist named Adanson.  These fundamentals, sometimes called neo-Adansonian, are summarized by Sneath and Sokal as follows:

1.    The greater the content of information in the taxa of a classification and the more characters on which it is based, the better a given classification will be.

2.    A priori, every character is of equal weight in creating natural taxa.

3.    Overall similarity between any two entities is a function of their individual similarities in each of the many characters in which they are being compared.

4.    Distinct taxa can be recognized because correlations of characters differ in the groups under study.

5.    Phylogenetic inferences can be made from the taxonomic structures of a group and from character correlations, given certain assumptions about evolutionary pathways and mechanisms.

6.    Taxonomy is viewed and practiced as an empirical science.

7.    Classifications are based on phenetic similarity.⁴⁵

In practice such fundamentals are not always adhered to literally.  For instance, rather than include all possible variates (characters), subsets of variates are sometimes selectively chosen.  Similarly, equal weighting is not always employed, nor is classification limited solely to phenetic similarity.

Obviously, certain of these fundamentals grounded in the natural sciences are not directly applicable to corporate social accounting.  However, the point to be stressed is that in the process of information condensation (in the context discussed in the early parts of this chapter) some of the fundamentals are inherent, particularly similarity based upon character states, discovery of taxa, and the sorting or classification of entities into taxa.

Information display, clustering, and discrimination techniques illustrated previously for corporate social accounting might be used in subjective taxonomy efforts.  The purpose of this section will be to explore numerical techniques commonly used in numerical taxonomy for such purposes.  After variates (e.g., corporate social impact criteria) have been observed the next step is usually to define some measure of association (resemblance, distance, correlation, similarity, likeness, etc.) between entities being compared.  For instance, rather than human visual scanning of the data or data displays, numerical indices of association (similarity or dissimilarity) between entities or groups of entities are explicitly defined.

Subsequent steps depend upon the purpose of the investigation.  If classes (groups, clusters, taxa, subsets, etc.) have been predefined, a purpose may be merely to assign entities to classes (identification).  More often, however, the analyst does not know "if" or "how many" groupings exist among entities.  Instead, a cluster analysis may be performed to discover the existence of "natural" clusters.  The usual procedure is to utilize some clustering algorithm which partitions entities into subsets (clusters) based upon their pairwise measures of association.  Sneath and Sokal write:

These numerical methods are collectively called cluster analysis.  They are methods for establishing and defining clusters of mutually similar entities from the t x t resemblance matrix.  These clusters may be likened to hills and peaks on a topographic chart, and the criteria for establishing the clusters are analogous to the contour lines of such a map.  Rigid criteria correspond to high elevation lines that surround isolated high peaks--for example, species groups in a matrix of resemblances between species.  As the criteria become more relaxed the clusters grow and become interrelated in the same way that isolated peaks acquire broader bases and become connected to form mountain complexes and eventually chains, with progress from higher to lower-level contour lines...Differences in methods of clustering refer mainly to rules for forming clusters and for partitioning the organisms in taxonomic (character) space.

The important common aspect of all these methods is that they permit the delimitation of taxonomic groups in an objective manner, given a matrix of coefficients of relationship.  Boundaries for taxonomic groups can be visualized as the contour lines already discussed or they can be represented as the intersections of horizontal transects with the branches of the tree-like diagrams of relationship commonly employed in numerical taxonomy.  Comparable limits can be drawn for all taxonomic groups within a particular study.  Boundaries or transects at progressively lower levels of resemblance would create taxa of increasingly higher taxonomic rank.⁴⁶

The numerical taxonomy methods referred to above may serve any or all of the purposes of condensation of data discussed at the beginning of this chapter--aggregation, filtering, and analysis.  Entities (or variates) are aggregated when being clustered into groups or sets.  Filtering takes place in the sense that unique items are isolated by remaining apart and not joining in multiple-entity clusters,⁴⁷ or if forced to merge with others in a cluster, the "compactness" of the cluster explodes.  Data analysis may be facilitated in a number of ways, a major one being the discovery of "natural' or "unsuspected" groupings which provide clues for further investigation.  Sometimes these numerical methods are utilized in the search for underlying structure in a mass of data.

⁴⁵    Sneath and Sokal, Op. Cit., p. 5.

⁴⁶    Sneath and Sokal, Op. Cit., p. 7.

⁴⁷    For example, in an earlier cluster analysis of major corporations on the basis of financial performance and stock trading data, I found that top performers (in terms of ex-post price appreciation) tended to remain isolated apart from companies that merged more readily into clusters.  See R. E. Jensen, "A Cluster Analysis Study of Financial Performance of Selected Business Firms," The Accounting Review, Vol. XLVI, January 1971, pp. 36-56.

6.7.3--Factor Analysis.  The many items in Appendix A illustrate that possible variates on corporate social actions and impacts abound.  Factor analysis (or related multidimensional scaling) may be used to condense such a multitude of variates into a more parsimonious set of underlying factors.  Use of principal component factor analysis for this purpose was illustrated both in this chapter (see Table 6.5) and in Chapter 4.

Use of factor analysis in comparing companies might follow along similar lines of several political science studies in comparing nations.  For example, Rummel collected observations on 236 variates on over 80 nations of each of several studies in the Dimensionality of Nations (DON) project.⁴⁸  This is one of many studies⁴⁹ in which factor analysis was used to achieve both parsimony and identification of underlying orthogonal factors.  Since the listing of corporate social impact variates in Appendix A is  so overwhelming, it may be useful to search in a similar manner for underlying factors among various subsets of these variates.

⁴⁸    See Rudolph J. Rummell, "The Dimensionality of Nations Project," in Comparing Nations: The Use of Quantitative Data in Cross-National Research, Edited by Richard L. Merritt and Stein Rokkan (New Haven: Yale University Press, 1966, pp. 109-30).  Also see R. J. Rummell, The Dimensions of Nations (Beverly Hills: Sage Publications, 1972).

⁴⁹    For example, see Jack E. Vincent, Factor Analysis in International Relations (Gainesville: University of Florida Press, 1971).

6.7.4--Cluster Analysis.  The term cluster analysis⁵⁰ is commonly used to refer to a wide assortment of techniques for partitioning N entities into G clusters (groups, clumps, categories, classes, subsets, types, etc.).  Usually neither the number (G) of clusters nor their meanings are predefined.  Instead, clustering methods seek to find natural (heuristic, hidden, latent, etc.) groupings.

Some clustering methods are subjective, often employing visual comparisons.  For example, if profiles or caricatures are compared and then sorted into subsets according to which ones seem to be "more alike," this is a type of cluster analysis.  Such visual clusterings were illustrated previously when comparing the N=12 electric utility companies, e.g., subjective clustering can be attempted on Exhibits 6.1 thru 6.7.

In contrast, there are also wide assortments of numerical techniques available, most of which employ computer algorithms for sorting and assigning entities into clusters.  Such numerical techniques are "objective" in the sense that, once the variates and entities are determined and the clustering approach is specified, the clustering outcomes are not affected by human judgment.  Human judgment, of course, must enter into the selection of entities, choice of variates to observe, and the interpretation of clustering outcomes.

Proceeding by way of illustration, consider the M=3 standardized factor scores on each of the N=12 electric utility companies in Table 6.6.  In general, the number of ways in which N entities may be allocated among G nonempty and mutually exclusive groups is given by the formula

The above S(N,G) formula is known as the closed-form formula for Stirlings' Numbers of the Second Kind.  A serious problem in both deriving and evaluating clusters is that S(N,G) explodes into astronomical values for even small N values.  An added complication in cluster analysis is that the nature or number of groups (clusters) to be formed is not usually specified in advance.  Hence, the total number of clustering outcomes in such circumstances may include all possible numbers of groups (clusters) from G=1,...,N.  As a result, the total number of possible clustering outcomes becomes an even more astronomical TOTS(N,N) value given by

For example, the TOTS(12,12)=4,213,597 number of feasible ways of partitioning the N=12 electric utility companies into nonempty and mutually exclusive clusters is derived in Table 6.8.  Even in this "small" clustering problem, total enumeration of all clustering alternatives is computationally very expensive.  Given some type of clustering homogeneity criterion, however, the cluster analyst would like to know the "best" clustering alternative for each G value of interest.

Some years back I developed a dynamic programming algorithm designed to yield the optimal clustering solutions without having to enumerate all feasible clustering alternatives.⁵¹  A number of other researchers have also formulated integer programming approaches.⁵²  But neither dynamic programming nor integer programming are sufficiently efficient for most clustering problems (other than very small or large problems having special structure).  In most instances, a heuristic (hierarchical, linkage) algorithm must be resorted to, some of which are extremely efficient and popular.⁵³  A review of various techniques is given by Anderberg, Everitt and Duran and Odell.⁵⁴

There are many variations in cluster analysis, some which are discussed below:

(1) Whether or not entities are directly traceable to one or more groups.  For example, factor analysis, principal component analysis, and multidimensional scaling can sometimes be viewed as types of cluster analysis in which items (entities or variates) are represented by a smaller  number of factor, component, or axis groupings which are composites of the items.  However, individual items (entities) are  not sorted and allocated to groupings in such a manner that any group contains an identifiable subset of specific items (entities).⁵⁵  Unless noted otherwise, I will usually assume that items being grouped are directly traceable to one or more groups, as is usually the case in clustering and classification models.

(2) Whether or not entities are partitioned into mutually exclusive groups.  In most instances, cluster analysis has been applied where N entities are to be partitioned into both mutually exclusive and collectively exhaustive groups.  However, some applications exist where groups (clusters) overlap.⁵⁶  Overlapping groups greatly complicate the number of clustering alternatives and the analysis and interpretation of clustering outcomes.  Unless noted otherwise, it is generally assumed implicitly that groups do  not overlap, i.e., that groups are mutually exclusive.

(3) Whether or not pairwise association indices of similarity (e.g., correlation coefficients) or dissimilarity (e.g., Euclidean distances) are utilized in the clustering algorithm.  In most instances such indices between pairs of items are utilized.  A number of clustering algorithms do not, however, require pairwise comparisons using such indices, e.g., direct clustering methods of Hartigan⁵⁷ and normix (or mixture) analysis by Wolfe.⁵⁸

(4) Whether or not clustering algorithms are objective or subjective.  In most instances, clusters are generated by "objective" numerical algorithms on a computer.  In other instances, however, display techniques may be used, requiring human observers subjectively to form the clusters, e.g., clustering by visual comparisons of profiles and caricatures illustrated previously in this chapter.

(5) Whether or not clusters are "optimal" in terms of a stated objective function.  Many clustering algorithms are hierarchical in nature such that the number of groups (clusters) varies from stage to stage of the algorithm.  At a given stage, the partitionings of the entities are not necessarily "optimal" in terms of the clustering objective function.  Often, however, it is not practical to further attempt to find the optimal clusters at each stage of the hierarchy.

Hierarchical clustering techniques may be further subdivided into those which are divisive versus agglomerative.  A divisive hierarchical technique beings with all entities lumped together and then splits them into smaller and smaller subsets until each entity comprises a cluster by itself.  An agglomerative hierarchical approach begins with each entity in a separate cluster and then successively combines (merges) clusters until at the final stage all entities fall into a single cluster.  The analyst then chooses one or more intervening stages to analyze in greater detail.

(6) Whether entities or variates are to be clustered.  In most instances, entities are clustered on the basis of variates observed on each of the entities.  It is possible, however, to cluster the variates rather than entities.  It is also possible to cluster both variates and entities at the same time.

(7) Whether or not all variates are measured in mixed scales.  In most instances, variates under study are all continuous, all ordinal, or all nominal.  In a mixed-scale problem some of the variates may be a combination of continuous, ordinal, and nominal.  Mixed scale problems are difficult to deal with in numerical approaches, and hence, the analyst may prefer to resort to subjective approaches.

(8) Whether clusters are to be predictive rather than merely descriptive.  Descriptive clusters are evaluated only in terms of the internal input variates used in the clustering process.  Predictive clusters are normally evaluated in terms of some external criterion variate not used in the clustering process but subsequently of interest after the clusters are identified.

⁵⁰    Terms other than cluster analysis which arise in the literature include clumping, partitioning, grouping, or classifying theory.

⁵¹    Robert E. Jensen, "A Dynamic Programming Algorithm for Cluster Analysis," The Journal of Operations Research, Vol. 17, 1969, pp. 1034-57.  Also reproduced in B. S. Duran and P. L. Odell, Cluster Analysis: A Survey (New York: Springer-Verlag, Chapter 3, 1974).

⁵²    See, for example, H. D. Vinod, "Integer Programming and the Theory of Grouping," Journal of the American Statistical Association, June 1969, pp. 506-19.  One of the best formulations to date was presented by George Diehr, "Minimum Variance Partitions and Mathematical Programming," Paper Presented at the National Meetings of The Classification Society, Atlanta, Georgia, April 1973.

⁵³    One exceedingly popular formulation is the hierarchical algorithm described by J. H. Ward, "Hierarchical Grouping to Optimize an Objective Function," Journal of the American Statistical Association, Vol. 58, 1963, pp. 236-44.  An even more computationally efficient approach for large N is the k-means algorithm derived by J. B. MacQueen, "Some Methods of Classification and Analysis of Multivariate Observations," Proceedings of the Fifth Berkeley Symposium On Mathematical Statistics and Probability (Berkeley: University of California Press, 1967, pp. 280-98).

⁵⁴    M. R. Anderberg, Cluster Analysis for Applications (New York: Academic Press, 1973).  B. Everitt, Cluster Analysis (New York: Halsted Press, 1974).  B. S. Duran and P. L. Odell, Cluster Analysis: A Survey (New York: Springer-Verlag, 1974.)

⁵⁵    The term item is used since either entities or variates may be grouped, and in some cases both entities and variates are grouped.  It will be convenient to assume, however, that items to be grouped are entities (rather than variates) unless explicitly stated otherwise.

⁵⁶    Examples of cluster analysis with overlapping groups include: R. M. Needham, "A Method for Using Computers in Information Classification," Proceedings of I.F.I.P. Congress, 1962, p. 284-298; R. M. Needham and K. S. Jones, "Keyword and Clumps," Journal of Documentation, Vol. 20, 1964, pp. 5-15; A. G. Dale, N. Dale, and E. D. Pendergraft, "A Programming System for Automatic Classification With Applications in Linguistic and Information Retrieval Research," Paper No. LRC64, WTM-5, Linguistics Research Center, 1964; M. G. Kendall, "Discrimination and Classification," in P. R. Krishnaiah (Editor), Multivariate Analysis (New York: Academic Press, 1966, 165-84); L. L. McQuitty, "Agreement Analysis: Classifying Persons by Predominant Patterns of Responses," The British J. of Statistical Psychology, Vol. 9, 1956, pp. 5-16.

⁵⁷    J. A. Hartigan, "Direct Clustering of a Data Matrix," Journal of the American Statistical Association, Vol. 67, 1972, pp. 123-29.

⁵⁸    J. H. Wolfe, "NORMIX: Computational Methods for Estimating the Parameters of Multivariate Mixtures of Distributions," Research Memorandum, SRM 68-2, U.S. Naval Personnel Research Activity, San Diego, 1967; also see "Pattern Clustering by Multivariate Mixture Analysis," Multivariate Behavioral Research, Vol. 5, 1970, pp. 329-50.

6.7.5--Euclidean Distances Between Companies.  In earlier sections clusterings of N=12 electric utility companies on the basis of profile or caricature visual displays were illustrated.  A numerical clustering approach will now be illustrated.  The following elements are utilized:

DATA--Standardized factor scores in Table 6.6:

MEASURES OF ASSOCIATION--Euclidean distances between pairs of companies;

CLUSTERING ALGORITHM--The JENCLS General Classification Program at the University of Maine (a clustering routine comparable to Ward's hierarchical grouping routine mentioned previously.

DENDOGRAPH--A tree-like diagram depicting the entity (company) mergings into clusters at sequential stages in the hierarchical clustering algorithm.

The data and Euclidean distances are shown in Table 6.9.  The pair of companies most alike (with Euclidean distance of 0.38 in terms of the three major factors from Table 6.6) underlying the ten criteria (from Table 6.2) are Florida Power and Light (FPL) and Northern States Power (NSP).  Their factor score similarities (see Exhibit 6.3) are somewhat surprising since these two companies exhibit rather large differences among the M=10 original criteria in Table 6.2.  The next closest pair of companies consists of Commonwealth Edison (COM) and The Southern Company (COM) with a Euclidean distance of 0.61 in Table 6.10.

In contrast, American Electric Power (AEP) and Oklahoma Gas and Electric (OGE) are least alike with a Euclidean distance of 4.48.  This is not surprising since AEP is an immense coal-fired conglomerate with severe pollution problems but a relatively high R&D commitment.  The OGE company, on the other hand, is a much smaller natural gas-fired company with almost no particulate and sulphur dioxide emission problems but higher electricity prices and underinvestment in nitrogen oxides emission control.  The OGE company also has a much lower R&D expenditure as a proportion of revenues.

Use of standardized scores has an advantage of not differentially weighting individual variates because of scaling differences.  Use of factor scores (from Table 6.6) in lieu of the M=10 original variates has an added advantage of linear independence (orthogonality) between inputs in the Euclidean distance calculations.  For example, if Euclidean distances were calculated from the M=10 variates in Table 6.2 (after variate standardization), correlated pollution or other variates would be "double counted" and, thereby, tend to overwhelm individual variates by themselves.  Factor analysis recasts the entire system of variates into fewer "factors" which are not correlated, and hence, are not double counted in Euclidean distance calculations.

6.7.6--Hierarchical Clustering Outcomes.  Enumeration of all TOTS(12,12)=4,213,597 possible clustering outcomes (see Table 6.9) seemed impractical for this study.  Instead an agglomerative hierarchical approach⁵⁹ was used in which all N=12 companies are first viewed as G=12 single-entity clusters at Stage 1.  At Stage 2, the closest pair of companies (in terms of Table 6.9 Euclidean distances) are merged into one cluster, thereby leaving G=11 clusters at Stage 2.  Two clusters are merged in each succeeding stage until at Stage 12 all N=12 companies are forced into G=1 cluster.

The hierarchical mergings of these electric utility companies into clusters are pictured in a dendograph-type diagram in Exhibit 6.8.  Unfortunately, there are no statistical tests or generally accepted mathematical criteria as to what stage (i.e., as to the number, G, of cluster groupings) should be considered "best."  Parsimony increases as there are fewer and fewer clusters, i.e., as G becomes smaller.  Usually, however, this parsimony is offset by decreasing within-group homogeneity as entities (in this case electric utility companies) are forced into larger and larger clusters.

One clustering homogeneity criterion is the pooled within-groups sums of squares, otherwise known as "Trace W" criterion,⁶⁰ where W is the dispersion matrix on all the variates (in this case the three-factors in Table 6.10).  Although the use of Trace W as a "stopping" criterion is somewhat controversial,⁶¹ the Trace W values at each clustering stage are shown in Exhibit 6.8.⁶²  The large jump in Trace W between Stages 8 and 9 suggests that Stage 8 yields relatively homogeneous clusters and parsimonious groupings of the companies into G=5 groups (clusters), which might be viewed here as empirical "types" in terms of the original M=10 social impact criteria in Table 6.2.

The Stage 8 clusterings into "types are as follows:

Cluster 1 is comprised of the largest coal-fired companies.  Clusters 4 and 5 contain the least-polluting companies with much higher natural gas usage.  Conversely, Clusters 4 versus 5 differ primarily in terms of size and R&D expenditures as a proportion of revenues (the three underlying major factors on which the Exhibit 6.8 clusterings are based were briefly interpreted in Table 6.5), i.e., Cluster 5 companies have a much higher commitment to R&D than Cluster 4 companies.  Consolidated Edison of New York (CON) stands apart (Cluster 3) from all other companies, in large measure due to its exceptionally poor performance on Factor 3, i.e., due to having the lowest earnings margin and the highest kwh prices on electricity of all the companies in the study.

⁵⁹    My JENCLS General Classification Program at the University of Maine was utilized.  The program contains a hierarchical clustering algorithm which, for this data, yields clusterings in a similar manner to Ward's hierarchical grouping program, i.e., See J. H. Ward, Op. Cit.

⁶⁰    Other criteria such as |W| and G|W[ are discussed elsewhere.  See, for example, F. H. C. Marriott, "Practical Problems in a Method of Cluster Analysis," Biometrics, Vol. 27, 1971 pp. 501-14.

⁶¹    See Robert L. Thorndike, "Who Belongs in the Family?", Psychometrika, Vol. 18, 1953, pp. 267-76.

⁶²    When Euclidean distances are available it is easier to compute Trace W from averaged sums of all pairwise Euclidean distances (squared) in the manner described in Robert E. Jensen, "A Dynamic Programming Algorithm for Cluster Analysis," Journal of Operations Research, Vol. 17, 1969, pp. 1034-57.

⁶³    These clustering outcomes are given at Stage 8 in Exhibit 6.8.

6.8--Summary

The major intent of this chapter was to explore means by which multivariate social criteria can be simultaneously compared on companies without having to convert everything into monetary units (as is the case in traditional financial accounting).  Graphic and other display techniques were considered.  An important advantage of display techniques lies in the ability to exploit human mental powers in sorting and making comparisons between entities and/or variates.  Another advantage is the ability to combine both quantitative and qualitative variations in a single display.

Drawbacks of visual displays lie mainly in the subjectivity and obvious cumbersomeness of making comparisons if many entities and/or considerable detail are included in the display.  Profile charts, for example, are highly satisfactory where there are small numbers (e.g., less than twelve) of entities and a few (e.g., less than six) quantitative variates.  Certain mathematical transformations (e.g., principal component analysis) may help to reduce the number of variates to be treated in the display, although interpretations may be somewhat complex.  Fourier series plots appear to have few advantages over profile plots except where there are too many variates (or factors) for profile plots.  Also statistical inference testing becomes possible when a number of restrictive assumptions are satisfied in the Fourier series model.

Geometric pattern and/or caricature displays may accommodate more entities and variates than do profile charts.  In addition, qualitative variations may be accommodated in many types of such displays.  Two such approaches illustrated in this chapter were glyph plots and facial caricatures.  Facial caricatures can be utilized for a greater number of variates than can glyphs, although the facial comparisons become quite dependent upon how human viewers subjectively weight different features when comparing faces.  There will also be skeptics who view comparisons of abstract representations (such as faces) as being nonsense or silly fun-and-games.

Cluster analysis and other tools in numerical taxonomy were designed primarily to overcome some of the difficulties caused by subjectivity in taxonomy classifications in the natural sciences.  Numerical techniques have the important advantage of yielding "objective" groupings (provided the variates and appropriate mathematical approaches can be agreed upon) in the sense that allocations of entities (or variates) to groups is accomplished by mathematical techniques (usually on a computer) rather than human observers.  This advantage, however, is offset by computational difficulties and the fact that different approaches work better than others for certain types of clusterings.  In contrast, the human mind is much more flexible (e.g., when presented with visual displays) in detecting clusters and abberrations.

In practice it is probably best to compare "subjective" visual display clusterings with "objective" numerical clusterings.  Both approaches were illustrated in this chapter.  For example, Table 6.2 listed performance data for N=12 private electric utility companies on M=10 social impact criteria.  A principal component analysis was performed, reducing these criteria to three underlying independent factors (interpreted in Table 6.5).  Rotated factor scores were then analyzed in visual displays (profile charts and facial caricatures) and in a hierarchical clustering algorithm.  Although the number of clusters which emerge is open to debate, it seemed to me that Stage 8 was a reasonable stopping point (where G=5 clusters) in Exhibit 6.8.  The human observers I presented with the Exhibit 6.7 faces tended to choose G=4 clusters.  The groupings in both cases were similar but not identical, as is indicated in Exhibit 6.9.  In particular, the BGE classification seems to be the least consistent.  The "objective" numerical clusterings in Exhibit 6.8 include BGE with CEP, FPL, and NSP.  This is consistent with their Factor 2 (Technology) and Factor 3 (Financial) similarities evidenced in Exhibit 6.3.  However, Exhibit 6.3 also reveals how BGE (and CON) pull ahead of the pack with respect to Factor 1 (State-of-the-Art Pollution Control).  This is especially surprising since BGE is a relatively heavy coal user (59.1% under x₂ in Table 6.2).  The exceptional performance (relative to other coal and/or oil burning companies) on particulate and sulphur dioxide control seems to be the reason for BGE's inclusion with the "clean guys" in Cluster 4 in Exhibit 6.7.  But the falling down of BGE on Factors 2 and 3 (see Exhibit 6.3), however, partly explains the inconsistencies in BGE classifications in Exhibit 6.7 versus Exhibit 6.8.  Similarly, the exceptional Factor 1 performance of CON also gives it certain facial features resembling Cluster 4 "clean guys" in Exhibit 6.7.

As expected, the "objective" hierarchical clusterings in Exhibit 6.8 (based on Euclidean distances) agree more closely with profile similarities in Exhibit 6.3 than do the "subjective" clusterings in Exhibit 6.7.  The "subjective" facial clusterings differ largely because of apparent unequal weightings given to different facial features by human observers.  For example, eyebrow size and shape variations seem to be much less important than eye size and shape variations.  One means of overcoming this problem is to make a number of facial plottings under different assignments of variates (or factors) to facial features and attempt to discover if human observers tend to detect consistent clusters under such variations.

I stress that no significance whatever should be placed upon which companies have the most "agreeable," "appealing," or "happy" faces.  In both Exhibits 6.6 and 6.7, the social impact variates (or factors) were randomly assigned to facial features.  The purpose is merely to compare faces with one another in an effort to discover subsets which seem to be most (or least) alike.  One advantage of the caricature (e.g., glyphs or faces) comparisons (e.g., see Exhibit 6.7) relative to numerical clustering outcomes (see Exhibit 6.8) is that alternative clusterings are a little more evident.  For example, in Exhibit 6.7 it is evident that, although BGE has certain things in common with most other Cluster 4 companies (e.g., head size, head shape, nose length, mouth length, position of center of mouth, separation between centers of eyes, and position of pupils), BGE also has features in common with CON in Cluster 1 (e.g., eyebrow length, height of centers of eyes, half-length of eyes, and angle of brows).  This similarity is also evident in the profiles in Exhibit 6.3 but is not shown as clearly in the cluster-merging (dendograph) diagram in Exhibit 6.8.

Both "objective" and "subjective" cluster analysis approaches illustrated in this chapter are means by which entities (e.g., companies) may be sorted into "types" on the basis of multivariate criteria (e.g., the M=10 social impact criteria in Table 6.2).  The interpretation of these "types," and more particularly the ranking of the "types" along a "good versus bad" or "high versus low" composite of all criteria simultaneously, is a much more difficult and controversial undertaking.  Material in Chapters 7 and 8 have some relevance to such endeavors.

6.9--Suggestions for Further Research

The number of criteria (i.e., the M=10 variates in Table 6.2) is too small for a thorough taxonomy study of corporate social criteria.  Many additional criteria (e.g., see Appendix A ) must be considered.  However, relevant data on which corporations can be compared along a much wider spectrum are lacking.  It seems that future corporate comparisons such as those illustrated in this chapter will await better data.  Such data might either be generated in large-scale studies of companies or from required (and uniform) reporting practices imposed upon corporations.  Internal studies by the companies themselves are of less use due to likely inconsistencies in definitions, measurement techniques, accuracy, and scope of investigation.

Much further study is obviously needed to determine what attributes (criteria) are most important to study.  The added Chapter 5 considerations must be better resolved.  If data become available, however, multivariate analyses such as those mentioned in this chapter are especially interesting to pursue, e.g., in both seeking underlying factors amidst criteria and empirical "types" of companies.  Dimensions of social conflict and interaction are also of interest in future research.  Chapter 7, in particular, bears upon such issues.

Clusterings in this chapter concerned companies at a point in time.  Another area of interest might be the study of evolutionary patterns over time with respect to economic, social, and environmental criteria.  For example, cladistic taxonomy⁶⁴ reconstructs the branching patterns over different time planes.  This might be further extended to considerations of evolutionary rates, parallelism, and convergence.

⁶⁴    Classification by clades is discussed in J. S. Huxley, "Evolutionary Processes and Taxonomy With Special Reference to Grades," Uppsala University Arssks, 1958, pp. 21-39.  For other references, see Chapter 6 in P. H. A. Sneath and R. R. Sokal, Numerical Taxonomy (San Francisco: W. H. Freeman and Company, 1973).