Encyclopedia of SociologyVol._3

in the form of monotonic propositions (‘‘The greater the X, the greater the Y’’) and hypotheses derived solely from concatenating (multiplying) the signs of the relationships. For example, the axioms ‘‘As A increases, B increases’’ and ‘‘As B increases, C decreases’’ lead to the hypothesis that ‘‘As A increases, C decreases.’’ Standard statistical tests of the deduced hypotheses were presumed to be proper tests of the theory that generated the axioms. The approach was quick, convenient, and readily understood, and did not require expertise in mathematics or statistics. Consequently, it quickly became popular.

Numerous inadequacies with such theories soon became apparent, however (see for example critiques in Hage 1994), and interest turned to path models based on earlier work by Wright (1934). Path models assume that empirical measurements coincide exactly with theoretical concepts; that all variables are continuous (or reasonably close to continuous); that all relationships (paths) are bivariate, linear, and causal; that there is no feedback in the system (the recursive assumption); and that all relevant variables have been included in the model (the ‘‘closed-system’’ assumption). The underlying theoretical model is therefore very simplistic but does allow for the introduction of assumptions about multiple causal paths, and for different types of causal relationships including intervening, spurious, modifying, and counteracting.

If the assumptions are reasonable, then the causal effects, or path coefficients, are equal to ordinary multiple regression coefficients. That is, the underlying mathematical model feeds directly into a well-known statistical model, and the tie between theory and research seems well established. Are the assumptions reasonable?

The closed-system assumption, for example, implies that there is no correlation between the prediction errors across the various equations. If error correlations appear, then path coefficient estimates based on the statistical model will be in error, or biased, so the theory will not be tested properly by the statistics. Only two solutions are possible: (1) add more variables or paths to the model or (2) develop a statistical model that can accommodate correlated errors.

The assumption that measurement equals concept poses a different problem whenever various

scales or multiple indicators are used to represent a theoretical concept not readily assessed in a simple measure. Traditionally, the statistical model called factor analysis has been used to handle this measurement-concept problem, but factor analysis was not traditionally linked to the analysis of theoretical systems.

Recent years have seen very extensive development and elaboration of statistical models for linear systems, and these models address both the correlated error and the measurement-con- cept problems while allowing departure from recursiveness. They are called linear structural models (Joreskog 1970; Hayduk 1987). The underlying mathematical model still said that all variables are continuous and all relationships bivariate, linear, and causal. This development focused entirely on technical statistical questions about bias in parameter estimates. Furthermore, although the theory generally supposed that one variable affects another over time, the data are normally from only one or a very few points of time. From a general theory point of view, the underlying linear model’s assumptions, which were imposed for tractability, are highly restrictive.

MOVING BEYOND LINEAR MODELS

Consider the assumption of linearity. If a dynamic process is being modeled, linear relationships will almost always prove faulty. How change in some causal factor induces change in some consequent system property is typically constrained by system limits and is likely to be altered over time through feedback from other system factors.

As a disease like acquired immune deficiency syndrome (AIDS) spreads, for example, the rate at which it spreads depends on how many people are already infected, how many have yet to be infected, and what conditions allow interaction between the not-yet-infecteds and the infecteds. With very few infected the rate of spread is very small because so few cannot quickly infect a very large number of others. As the number of infecteds grows, so does the rate of spread of the disease, because there are more to spread it. On the other hand, if nearly everyone had already been infected, the rate of spread would be small because there would be few left to spread it to. To complicate

theoretical matters further, as the disease has generated widespread concern, norms governing sexual contact have begun to change, influencing the probabilities of transmission of the virus. In mathematical terms, the implication is that the rate of change of the proportion infected (i.e., the rate of spread of the disease) is not constant over time, nor is it a constant proportion of change in any variable in the system. In short, the process in inherently nonlinear.

Nonlinear models in mathematics take many forms. If the variables are conceived as continuous over time, and the primary theoretical focus is on how variables change as a consequence of changes in other variables, then the most likely mathematical form is differential equations. The substantive theory is translated into statements about rates of change (Doreian and Hummon 1976). Most diffusion and epidemiology models, like the AIDS problem just noted, use differential equations. So do a number of demography models. Recently some of the numerous differential equations models dealing with topics such as conflict and arms control have been applied to theoretical questions of cooperation and competition in social interaction (e. g., Meeker and Leik 1997).

For relatively simple differential equations models, once the model is developed it is possible to determine the trajectory over time of any of the properties of the system to ascertain under what conditions the covariation of system properties will shift or remain stable, and to ask whether that system will tend toward equilibrium or some other theoretical limit, oscillate in regular patterns, or even ‘‘explode.’’

None of these questions could be asked of a linear model because the mathematics of the linear model leaves nothing about the model itself to be deduced. Only statistical questions can be asked: estimates of the regression coefficients that fit the model to the data and the closeness of that fit. To the extent that sociology addresses questions of process, appropriate theory requires nonlinear models.

What about the assumption of continuous variables? For cross-sectional data, or data from only two or three time points, new techniques of statistical analysis suitable for categorical data have been developed. However, these focus once again

on technical statistical problems of bias in estimation of parameters and generally rely on assumptions of linearity. Within mathematical sociology, other approaches exist. If time or time-related variables were to be treated in discrete units, there are at least three different approaches available. For handling dynamic systems without the calculus of differential equations, difference equations are the appropriate form. Huckfeldt and colleagues (1982) provide a convenient overview of this approach.

For extensive time series with relatively few variables, there are Box-Jenkins and related types of models, although these have seen relatively little use in sociology. Because they are closer to statistical models than general theoretical models, they are only mentioned in passing.

Many theories treat systems as represented by discrete states. For example, over a lifetime, an individual is likely to move into and out of several different occupational statuses. Are certain status transitions more likely than others? In simplest form, the implied theoretical model involves a matrix algebra formulation that specifies the probability of moving from each of the states (occupational statuses for this example) to each of the other states. Then the mathematics of matrix algebra allows deduction of a number of system consequences from this ‘‘transition matrix.’’ Such a treatment is called a Markov chain. An early application by Blumen and colleagues (1955) demonstrated that certain modifications of a Markov chain were needed for the theory to fit the data they had available on occupational transitions. Their work was the initial inspiration for a distinguished string of mathematical models of social mobility. Another classic is by White (1970), who conceptualized mobility of vacant positions as well as of individuals into and out of positions.

Another type of substantive problem that deals with discrete data is the analysis of social networks such as friendship structures. These can be modeled using the mathematics of graph theory, the basic concepts of which include nodes (or points) and relationships between pairs of nodes (or lines). One of the most vigorous modeling areas in sociology, network analysis has produced a rich and elaborate literature addressing a wealth of substantive issues. Typically, network data consists of

whether or not any two cases (nodes in the network) are linked in one or more ways. The resulting data set, then, usually consists of presence or absence of a link of a given type over all pairs of nodes.

Early network analyses concerned friendships, cliques, and rudimentary concepts of structurally based social power. With the introduction of directed graph theory (Harary et al. 1965), random or probabilistic net theory, and block modeling, powerful tools have been developed to approach social structure and its consequences from a network point of view. As those tools emerged, the range of questions addressed via network analysis has greatly expanded. Over the past twenty years, numerous articles in the Journal of Mathematical Sociology have dealt with networks. The journal Social Networks also publishes work in this area. Overviews and examples can be found in Burt and Minor (1983), and in Wellman and Berkowitz (1988).

Graph theory may be applied to other theoretical issues; for example, a graph-theoretic model developed by Berger and colleagues (1977) helps explain the processes by which people combine information about the various characteristics of themselves and others to form expectations for task performance.

Small group processes have also generated a variety of mathematical formulations. Because observations of groups often generate counts of various types of acts (for example amount of talking by members of a discussion group) that display remarkable empirical regularity, these processes have intrigued model builders since the early 1950’s. Recent developments, combining network analyses with Markov chains, include Robinson and Balkwell (1995) and Skvoretz and Fararo (1996).

At the most micro level of sociology, the analysis of individual behavior in social contexts has a long tradition of mathematical models of individual decision making. Recent developments include the satisfaction-balance decision-making models of Gray and Tallman (Gray et al. 1998).

An exciting aspect of these different levels of development is that, increasingly, inquiries into microdynamics based on social exchange theory are working toward formulations compatible with the more general network structural analyses. These

joint developments, therefore, promise a much more powerful linking of micro-system dynamics with macro-system structural modeling (Cook 1987; Willer 1987). Recent work on power as a function of the linkages that define the exchange system is an example. The use of mathematics to express formal definitions has enabled researchers to pinpoint where there are theoretical differences, leading to productive debate (Markovsky et al. 1988; Cook and Yamagishi 1992; Friedkin 1993; Bonacich 1998).

There are other examples of work in mathematical sociology involving attempts to develop appropriate mathematical functions to describe a theoretically important concept. One is Jasso’s innovative work on models of distributive justice (e.g., 1999). Similarly, affect control theory (e.g., Smith-Lovin and Heise 1988) represents mathematical formulation in an area (symbolic interaction and sociology of emotions) typically considered not subject to such treatment.

One other area of vigorous development deserves attention; the treatment of strings of events that constitute the history of a particular case, process, or situation (Allison 1984; Tuma and Hannan 1984; Heise 1989). Event history analysis has some of its origins in traditional demographers’ life tables, but methods and models have experienced a great deal of attention and growth in recent years. If one had lifetime data on job placements, advancements, demotions, and firings (i.e., employment event histories) for a sample of individuals, then event history methods could be used for examining what contributes to differential risks of one of those events occurring, how long someone is likely to be in a given situation (‘‘waiting time’’ between events), and so forth.

An important recent development is the use of complex computer simulations for developing and exploring mathematically expressed theories. We have noted above the tension between simple models that are mathematically tractable and more complex models that may be more realistic as for example including feedback loops and random processes. Computer simulation is a way of showing what can be derived from the assumptions of a model without an analytic mathematical solution; examples include Macy and Skvoretz (1998), Carley (1997), and Hanneman (1995).

Like many other areas of social and behavioral science, mathematical sociology has been influenced by developments in game theory, from early work by Rapoport (1960) to the more recent idea of evolutionary games introduced by Axelrod (1984) and extensive interest in problems of collective action (Marwell and Oliver 1993). These consider how actions of individuals may produce unintended outcomes because of the logic of their interdependence with actions of others. The related theoretical area of rational choice theory also has a strong mathematical component (Coleman 1990); see also recent issues of the journal Rationality and Society.

A notable feature of current work in mathematical sociology is that the development, testing, and refinement of mathematical models is located within substantive research programs. Mathematical formulations appear in mainstream sociology journals and are becoming accepted as one component of continuing programs of research along with development and refinement of theory and collection and analysis of empirical data (several examples can be found in Berger and Zelditch 1993).

One indication of continuing interest in mathematical sociology is the recent formation of the Mathematical Sociology Section of the American Sociological Association. There is also a large amount of work internationally, including in Japan (Kosaka 1995) and in England and Europe (Hegselmann et al. 1996). Mathematical work in sociology is alive and vigorous. It truly does promise a higher level of theoretical precision and integration across the discipline.

———, and Morris Zelditch, Jr., eds. 1993 Theoretical Research Programs: Studies in the Growth of Theory. Stanford, Calif: Stanford University Press.

Blumen, Isadore, Marvin Kogan, and Philip H. McCarthy 1955 The Industrial Mobility of Labor as a Probability Process. Ithaca, N.Y.: Cornell University Press.

Bonacich, Phillip 1998 ‘‘A Behavioral Foundation for a Structural Theory of Power in Exchange Networks.’’

Social Psychology Quarterly 61:185–198.

Burt, Ronald S., and Michael J. Minor 1983 Applied Network Analysis. Beverly Hills, Calif.: Sage.

Carley, Kathleen, ed. 1997 Computational Organizational Theory. New York: Gordon and Breach (special issue of the Journal of Mathematical Sociology).

Coleman, James S. 1964 Introduction to Mathematical Sociology. New York: Free Press.

——— 1990 Foundations of Social Theory. Cambridge, Mass.: Harvard University Press.

Cook, Karen S., ed. 1987 Social Exchange Theory. Newbury Park, Calif.: Sage.

———, and Toshio Yamagishi 1992 ‘‘Power in Exchange Networks: a Power-Dependence Formulation.’’ Social Networks 14:245–266.

Doreian, Patrick, and Norman P. Hummon 1976 Modeling Social Processes. New York: Elsevier.

Fararo, Thomas J. 1973 Mathematical Sociology. New

York: Wiley.

———, ed. 1984 Mathematical Ideas and Sociological Theory. New York: Gordon and Breach (special issue of the Journal of Mathematical Sociology).

Friedkin, Noah E. 1993 ‘‘An Expected Value Model of Social Exchange Outcomes.’’ Pp. 163–193 in E. J. Lawler, B. Markovsky, K. Heimer, and J. O’Brien, eds., Advances in Group Processes, vol. 10. Greenwich, Conn.: JAI Press.

(SEE ALSO: Paradigms and Models; Scientific Explanation)

REFERENCES

Allison, Paul D. 1984 Event History Analysis: Regression for Longitudinal Event Data. Beverly Hills, Calif: Sage.

Axelrod, Robert M. 1984 The Evolution of Cooperation. New York: Basic Books.

Berger, Joseph, Bernard P. Cohen, J. Laurie Snell, and Morris Zelditch, Jr. 1962 Types of Formalization in Small Group Research. Boston: Houghton Mifflin.

———, M. Hamit Fisek, Robert Z. Norman, and Morris Zelditch, Jr. 1977 Status Characteristics and Social Interaction: An Expectation-States Approach. New York: Elsevier.

Gray, Louis N., Irving Tallman, Dean H. Judson, and Candan Duran-Aydintug 1998 ‘‘Cost-Equalization Applications to Asymmetric Influence Processes.’’ Social Psychology Quarterly 61:259–269.

Hage, Jerald, ed. 1994 Formal Theory in Sociology : Opportunity or Pitfall? Albany: State University of New York Press.

Hanneman, Robert A., ed. 1995 Computer Simulations and Sociological Theory. Greenwich Conn.: JAI (special issue of Sociological Perspectives).

Harary, Frank, Robert Z. Norman, and Dorwin Cartwright 1965 Structural Models. New York: Wiley.

Hayduk, Leslie A. 1987 Structural Equation Modeling with LISREL. Baltimore, Md.: Johns Hopkins University Press.

Hegselmann, Rainer, Ulrich Meuller, and Klaus G. Troitzsch 1996 Modelling and Simulation in the Social Sciences from the Philosophy of Science Point of View

Dordrecht, The Netherlands: Kluwer Academic.

Heise, David R. 1989 ‘‘Modeling Event Structures.’’

Journal of Mathematical Sociology 14:139–169.

Huckfeldt, Robert R., C. W. Kohfeld, and Thomas W. Likens 1982 Dynamic Modeling: An Introduction. Beverly Hills, Calif.: Sage.

Jasso, Guillermina 1999 ‘‘How Much Injustice Is There in the World? Two New Justice Indexes.’’ American Sociological Review 64:133–167.

Joreskog, Karl G. 1970 ‘‘A General Method for Analysis of Covariance Structures.’’ Biometrika 57:239–232.

Kosaka, Kenji, ed. 1995 Mathematical Sociology in Japan. New York: Gordon and Breach (special issue of the

Journal of Mathematical Sociology).

Leik, Robert K., and Barbara F. Meeker 1975 Mathematical Sociology. Englewood Cliffs, N.J.: Prentice-Hall.

Macy, Michael W., and John Skvoretz 1998 ‘‘The Evolution of Trust and Cooperation between Strangers: A Computational Model.’’ American Sociological Review

63:638–660.

Markovsky, Barry, David Willer, and Travis Patton 1988 ‘‘Power Relations in Exchange Networks.’’ American Sociological Review 53:220–236.

Marwell, Gerald, and Pamela Oliver 1993 The Critical Mass in Collective Action: A Micro-Social Theory. Cambridge, England: Cambridge University Press.

Meeker, Barbara F., and Robert K. Leik 1997 ‘‘Computer Simulation: an Evolving Component of Theoretical Research Programs.’’ Pp. 47–70 in J. Szmatka, J. Skvoretz, and J. Berger, eds., Status, Network and Structure: Theory Construction and Theory Development. Palo Alto, Calif.: Stanford University Press.

Rapoport, Anatol 1960 Fights, Games, and Debates. Ann Arbor: University of Michigan Press.

Robinson, Dawn T., and James W. Balkwell 1995 ‘‘Density, Transitivity, and Diffuse Status in Task-Oriented Groups.’’ Social Psychology Quarterly 58:241–254.

Simon, Herbert A. 1957 Models of Man: Social and Rational. New York: Wiley.

Skvoretz, John, and Thomas J. Fararo 1996 ‘‘Status and Participation in Task Groups: a Dynamic Network Model.’’ American Journal of Sociology 101:1355–1414.

Smith-Lovin, Lynn, and David R. Heise 1988 Affect Control Theory: Research Advances. New York: Academic.

Tuma, Nancy B., and Michael T. Hannan 1984 Social Dynamics: Models and Methods. New York: Academic.

Wellman, Barry, and S. D. Berkowitz, eds. 1988 Social Structures: A Network Approach. New York: Gordon and Breach.

White, Harrison C. 1970 Chains of Opportunity; System Models of Mobility in Organizations. Cambridge, Mass.: Harvard University Press.

Willer, David 1987 Theory and the Experimental Investigation of Social Structures. New York: Gordon and Breach.

Wright, Sewell 1934 ‘‘The Method of Path Coefficients.’’

Annals of Mathematical Statistics 5:161–215.

Zetterberg, Hans 1965 On Theory and Verification in Sociology. Totowa, N.J.: Bedminster.

BARBARA F. MEEKER

ROBERT K. LEIK

MEASUREMENT

There are many standards that can be used to evaluate the status of a science, and one of the most important is how well variables are measured. The idea of measurement is relatively simple. It is associating numbers with aspects of objects, events, or other entities according to rules, and so measurement has existed for as long as there have been numbers, counting, and concepts of magnitude. In daily living, measurement is encountered in myriad ways. For example, measurement is used in considering time, temperature, distance, and weight. It happens that these concepts and quite a few others are basic to many sciences. The notion of measurement as expressing magnitudes is fundamental, and the observation that if something exists, it must exist in some quantity is probably too old to attribute to the proper authority. This notion of quantification is associated with a common dictionary definition of measurement: ‘‘The extent, capacity, or amount ascertained by measuring.’’

A concept such as distance may be considered to explore the meaning of measurement. To measure distance, one may turn to a simple example of a straight line drawn between two points on a sheet of paper. There is an origin or beginning point and an end point, and an infinite number of points between the beginning and the end. To measure in a standard way, a unit of distance has to be arbitrarily defined, such as an inch. Then the

distance of any straight line can be observed in inches or fractions of inches. For convenience, arbitrary rules can be established for designating number of inches, such as feet, yards, and miles. If another standard is used—say, meters—the relationship between inches and meters is one in which no information is lost in going from one to the other. So, in summary, in the concept of measurement as considered thus far, there are several properties, two of which should be noted particularly: the use of arbitrary standardized units and the assumption of continuous possible points between any two given points. The case would be similar if time, temperature, or weight were used as an example.

There is another property mentioned above, a beginning point, and the notion of the beginning point has to be examined more carefully. In distance, if one measures 1 inch from a beginning point on a straight line, and then measures to a second point 2 inches, one may say that the distance from the beginning point of the second point is twice that of the first point. With temperature, however, there is a problem. If one measures temperature from the point of freezing using the Celsius scale, which sets 0 degrees at the freezing point of water under specified conditions, then one can observe temperatures of 10 degrees and of 20 degrees. It is now proper to say that the second measurement is twice as many degrees from the origin as the first measure, but one cannot say that it is twice the temperature. The reason for this is that the origin that has been chosen is not the origin that is required to make that kind of mathematical statement. For temperature, the origin is a value known as absolute zero, the absence of any heat, a value that is known only theoretically but has been approximated.

This problem is usually understood easily, but it can be made more simple to understand by illustrating how it operates in measuring distance. Suppose a surveyor is measuring distance along a road from A to B to C to D. A is a long distance from B. Arriving at B, the surveyor measures the distance from B to C and finds it is 10 miles, and then the distance from B to D is found to be 20 miles. The surveyor can say that the distance is twice as many miles from B to D as from B to C, but he cannot say that the distance from A to D is twice the distance from A to C, which is the error one

would make if one used the Celsius temperature scale improperly. Measuring from the absolute origin for the purpose of carrying out mathematical operations has become known as ratio level measurement.

The idea of ‘‘levels of measurement’’ has been popularized following the formulation by the psychologist S. S. Stevens (1966). Stevens first identifies scales of measurement much as measurement is defined above, and then notes that the type of scale achieved depends upon the basic empirical operations performed. The operations performed are limited by the concrete procedures and by the ‘‘peculiarities of the thing being scaled.’’ This leads to the types of scales—nominal, ordinal, interval, and ratio—which are characterized ‘‘by the kinds of transformations that leave the ‘structure’ of the scale undistorted.’’ This ‘‘sets limits to the kinds of statistical manipulation that can legitimately be applied to the scaled data.’’

Nominal scales can be of a type like numbering individuals for identification, which creates a class for each individual. Or there can be classes for placement on the basis of equality within each class with regard to some characteristic of the object. Ordinal scales arise from the operation of rank ordering. Stevens expressed the opinion that most of the scales used by psychologists are ordinal, which means that there is a determination of whether objects are greater than or less than each other on characteristics of the object, and thus there is an ordering from smallest to largest. This is a crucial point that is examined below. Interval scales (equal-interval scales) are of the type discussed above, like temperature and these are subject to linear transformation with invariance. There are some limitations on the mathematical operations that can be carried out, but in general these limitations do not impede use of most statistical and other operations carried out in science. As noted, when the equal-interval scales have an absolute zero, they are called ratio scales. A lucid presentation of the issue of invariance of transformations and the limitations of use of mathematical operations (such as addition, subtraction, multiplication, and division) on interval scales is readily available in Nunnally (1978).

What is important to emphasize is that how the scales are constructed, as well as how the scales

are used, determines the level of measurement. With regard to ordinal scales, Nunnally makes a concise and precise statement that should be read carefully: ‘‘With ordinal scales, none of the fundamental operations of algebra may be applied. In the use of descriptive statistics, it makes no sense to add, subtract, divide, or multiply ranks. Since an ordinal scale is defined entirely in terms of inequalities, only the algebra of inequalities can be used to analyze measures made on such scales’’ (1978, p. 22). What this means is that if one carries out a set of operations that are described as making an ordinal scale, the moment one adds, subtracts, divides, or multiplies the ranks, one has treated the scale as a particular type of interval scale. Most commonly, the type of scale that is de facto created when ordinal data are subject to ordinary procedures like addition, subtraction, division, and/or multiplication is through the assumption that the difference between ranks are equal, leading to sets like 1, 2, 3, 4, 5, 6, and thus to the treatment for ties such as 1, 2, 4, 4, 4, 6. This is sometimes called a flat distribution with an interval of one unit between each pair of ordered cases. Effectively, this is the same kind of distribution in principle as the use of ordered categories, such as quartiles, deciles, or percentiles, but it is a more restrictively defined distribution of one case per category. To repeat, for emphasis: The use of addition, subtraction, division, and/or multiplication with ordinal data automatically requires assumptions of intervals, and one is no longer at the level of ordinal analysis. Thus, virtually all statistical procedures based on collected ordered or rank data actually assume a special form of interval data.

ISSUES ON LEVEL OF MEASUREMENT

A number of issues are associated with the notion of levels of measurement. For example, are all types of measurement included in the concepts of nominal, ordinal, interval, and ratio? What is the impact of using particular statistical procedures when data are not in the form of well-measured interval scales? What kind of measurement appears (epistemologically) appropriate for the social and behavioral sciences?

The last question should probably be examined first. For example, are measures made about attributes of persons nominal, ordinal, or interval?

In general, we cannot think of meaningful variables unless they at least imply order, but is order all that one thinks of when one thinks about characteristics of persons? For example, if one thinks of heights, say of all males of a given age, such as 25, does measurement imply ordering them on an interval scale? We know that height is a measure of distance, so we assume the way one should measure this is by using a standard. For purposes of the example here, an interval scale is proposed, and the construction is as follows. The shortest, 25 year-old male (the category defined as 25 years and 0 days to 25 years and 365 days of age) and the tallest are identified. The two persons are placed back to back, front to front, and every other possible way, and the distance between the height of the shortest and the tallest is estimated on a stick. Many estimates are made on the stick, until the spots where agreement begins to show discretely are evident; and so, with whatever error occurs in the process, the locations of beginning and end are indicated on the stick. Now the distance between the beginning and the end is divided into equal intervals, and thus an interval scale has been created. On this scale it is possible to measure every other male who is 25 years old, and the measure can be stated in terms of the number of intervals taller than the shortest person. Note that all possible values can be anticipated, and this is a continuous distribution.

Now if a million persons were so measured, how would they be distributed? Here the answer is on the basis of naive experience, as follows. First, there would be very few people who would be nearly as short as the shortest or as tall as the tallest. Where would one expect to find most persons? In the middle of the distance, or at some place not too far from it. Where would the next greatest number of persons be found? Close to the biggest. With questions of this sort one ends up describing a well-distributed curve, possibly a normal curve or something near it.

It is proper now to make a small diversion before going on with answering the questions about the issues associated with level of measurement. In particular, it should be noted that there are many sources of error in the measurement that has just been described. First, of course, the age variable is specified with limited accuracy. At the limits, it may be difficult to determine exact age

because of the way data are recorded. There are differences implied by the fact that where one is born makes a difference in time, and so on. This may seem facetious, but it illustrates how easily sources of error are bypassed without examination. Then it was noted that there were different estimates of the right location for the point of the shortest and the tallest person as marked on the stick. This is an error of observation and recording, and clearly the points selected are taken as mean values. Who are the persons doing the measuring? Does it make a difference if the person measuring is short or tall? These kinds of errors will exist for all persons measured. Further, it was not specified under what conditions the measurements taken or were to be taken. Are the persons barefoot? How are they asked to stand? Are they asked to relax to a normal position or to try to stretch upward? What time of day is used, since the amount of time after getting up from sleep may have an influence? Is the measurement before or after a meal? And so forth.

The point is that there are many sources of error in taking measures, even direct measures of this sort, and one must be alert to the consequences of these errors on what one does with the data collected. Errors of observation are common, and one aspect of this is the limit of the discriminations an observer can make. One type of error that is usually built into the measurement is rounding error, which is based on the estimated need for accuracy. So, for example, heights are rarely measured more accurately than to the half-inch or centimeter, depending on the standard used. There is still the error of classification up or down, by whatever rule is used for rounding, at the decision point between the intervals used for rounding. Rounding usually follows a consistent arbitrary rule, such as ‘‘half adjusting,’’ which means keeping the digit value if the next value in the number is 0 to 4 (e.g., 24.456 = 24) or increasing the value of a digit by 1 if the next value is 5 to 9 (e.g., 24.789 = 25). Another common rounding rule is simply to drop numbers (e.g., 24.456 = 24 and 24.789 = 24). It is important to be aware of which rounding rule is being used and what impact it may have on conclusions drawn when the data collected are used.

The use of distribution-free statistics (often called nonparametric statistics) was popularized beginning in the mid-1950s, and quickly came to

be erroneously associated with the notion that most of the measurement in the social and behavioral sciences is of an ordinal nature. Actually, the use of the distribution-free statistics was given impetus because some tests, such as the sign test, did not require use of all the information available to do a statistical test of significance of differences. Thus, instead of using a test of differences of means, one could quickly convert the data to plus and minus scores, using some arbitrary rule, and do a ‘‘quick-and-dirty’’ sign test. Then, if one found significant differences, the more refined test could be carried out at one’s leisure. Some of the orientation was related to computing time available, which meant time at a mechanical calculator. Similarly, it was well known that if one used a Spearman rank correlation with larger samples, and if one were interested in measuring statistical significance, one would have to make the same assumptions as for the Pearson product moment correlation, but with less efficiency.

However, this early observation about distri- bution-free statistics suggests that measures can be thought of in another way. Namely, one can think of measures in terms of how much they are degraded (or imperfect) interval measures. This leads to two questions that are proper to consider. First, what kind of measure is implied as appropriate by the concept? And second, how much error is there in how the measure is constructed if one wants to use procedures that imply interval measurement, including addition, subtraction, multiplication, and division?

What kind of measure is implied by the concept? One way of answering this is to go through the following procedure. As an example, consider a personal attribute, such as aggressiveness. Is it possible to conceive of the existence of a least aggressive person and a most aggressive person? Obviously, whether or not such persons can be located, they can be conceived of. Then, is there any reason to think that persons cannot have any and all possible quantities of aggressiveness between the least and the most aggressive persons? Of course not. Thus, what has been described is a continuous distribution, and with the application of a standard unit, it is appropriately an interval scale. It is improper to think of this variable as intrinsically one that is ordinal because it is continuous. In fact, it is difficult to think of even plausible examples of

variables that are intrinsically ordinal. As Kendall puts it, ‘‘the essence of ranking is that the objects shall be orderable, and the totality of values of a continuous variate cannot be ordered in this sense. They can be regarded as constituting a range of values, but between any two different values there is always another value, so that we cannot number them as would be required for ranking purposes’’ (1948, p. 105). While a few comments were published that attempted to clarify these issues of measurement in the 1960s (Borgatta 1968), most methodologists accepted the mystique of ordinal measurement uncritically.

Often measures of a concept tend to be simple questions with ordered response categories. These do not correspond to ordinal measures in the sense of ordering persons or objects into ranks, but the responses to such questions have been asserted to be ordinal level measurement because of the lack of information about the intervals. So, for example, suppose one is attempting to measure aggressiveness using a question such as ‘‘When you are in a group, how much of the time do you try to get your way about what kinds of activities the group should do next?’’ Answer categories are ‘‘never,’’ ‘‘rarely,’’ ‘‘sometimes,’’ ‘‘often,’’ ‘‘very often,’’ and ‘‘always.’’ Why don’t these categories form an interval scale? The incorrect answer usually given is ‘‘because if one assumes an interval scale, one doesn’t know where the answer categories intersect the interval scale.’’ However, this does not create an ordinal scale. It creates an interval scale with unknown error with regard to the spacing of the intervals created by the categories.

Thus, attention is now focused on the second question: How much error is involved in creating interval scales? This question can be answered in several ways. A positive way of answering is by asking how much difference it makes to distort an interval scale. For example, if normally distributed variables (which are assumed as the basis for statistical inference) are transformed to flat distributions, such as percentiles, how much impact does this have on statistical operations that are carried out? The answer is ‘‘very little.’’ This property of not affecting results of statistical operations has been called robustness. Suppose a more gross set of transformations is carried out, such as deciles. How much impact does this have on statistical operations? The answer is ‘‘not much.’’ However,

when the transformations are to even grosser categories, such as quintiles, quartiles, thirds, or halves, the answer is that because one is throwing away even more information by grouping into fewer categories, the impact is progressively greater. What has been suggested in this example has involved two aspects: the transformation of the shape of the distribution, and the loss of discrimination (or information) by use of progressively fewer categories.

CONSEQUENCES OF USING LESS THAN NORMALLY DISTRIBUTED VARIABLES

If one has normally distributed variables, the distribution can be divided into categories. The interval units usually of interest with normally distributed variables are technically identified as standard deviation units, but other units can be used. When normally distributed variables are reduced to a small number of (gross) categories, substantial loss of discrimination or information occurs. This can be illustrated by doing a systematic exercise, the results of which are reported in Table 1. The data that are used for the exercise are generated from theoretical distributions of random normal variables with a mean of 0 and a standard deviation of 1. In the exercise, one aspect is examining the relationship among normally distributed variables, but the major part of the exercise involves the data in ordered categorical form, much as it is encountered in ‘‘real’’ data. The exercise permits reviewing several aspects associated with knowledge about one’s measurement.

As noted, the exercise is based on unit normal variables (mean = 0, standard deviation = 1) that are sampled and thus are subject to the errors of random sampling that one encounters with ‘‘real’’ data. The theoretical underlying model is one that is recommended in practice, specified as follows:

(1) A criterion variable is to be predicted, that is, the relationships of some independent (predictor) variables are to be assessed with regard to the criterion variable.

(2.) For each sample in which the relationship of the independent variables is assessed, the theoretical underlying relationship of the independent variables is specified, and thus for the purposes of the exercise is known. For the exercise, four levels of underlying theoretical relationship are product

moment correlation coefficients of .8, .6, .4, and

.2, between the predictor variables and the criterion variable. This represents different levels of relationship corresponding to magnitudes commonly encountered with ‘‘real’’ social and behavioral data.

(3.) For each sample that is used in the exercise, totally independent distributions are drawn from the theoretical distributions. That is, each sample in the exercise is created independently of all other samples, and in each case the criterion and independent predictor variables are drawn independently within the definition of the theoretical distributions.

(4.) Corresponding to a common model of prediction, assume that the independent variables are of a type that can be used to create scores. Here we will follow a common rule of thumb and use four independent variables to create a simple additive score; that is, the values of the four independent variables are simply added together. There are many ways to create scores, but this procedure has

many virtues, including simplicity, and it permits examining some of the consequences of using scores to measure a concept.

(5.) The independent variables are modified to correspond to grouped or categorical data. This is done before adding the variables together to create scores, as this would be the way variables encountered in research with ‘‘real’’ data are usually defined and used. The grouped or categorical independent variables are modified in the following ways:

A. Four groups or categories are created by using three dividing points in the theoretical distribution, −1 standard deviation, the mean of 0, and +1 standard deviation. The values for four variables, now grouped or categorical data, are added to give the score used in the correlation with the criterion variable. Thus, looking at Table 1, the row SumA.8 involves samples in which four independent variables based on data having four categories as defined above, and