Journal of Criminal and

Volume 66 | Issue 2 Article 8

1975 Scaling : An Evaluation of Magnitude and Category Scaling Techniques George S. Bridges

Nancy S. Lisagor

Follow this and additional works at: https://scholarlycommons.law.northwestern.edu/jclc Part of the Criminal Law Commons, Criminology Commons, and the Criminology and Criminal Justice Commons

Recommended Citation George S. Bridges, Nancy S. Lisagor, Scaling Seriousness: An Evaluation of Magnitude and Category Scaling Techniques, 66 J. Crim. L. & Criminology 215 (1975)

This Symposium is brought to you for free and open access by Northwestern University School of Law Scholarly Commons. It has been accepted for inclusion in Journal of Criminal Law and Criminology by an authorized editor of Northwestern University School of Law Scholarly Commons. TMI JoURNAL OFCRImINAL LW & CRIMINOLOGY Vol. 66, No. 2 Copyright g 1975 by Northwestern University School of Law Printed in U.S.A.

SCALING SERIOUSNESS: AN EVALUATION OF MAGNITUDE AND CATEGORY SCALING TECHNIQUES

GEORGE S. BRIDGES* AND NANCY S. LISAGOR**

Unidimensional attitude scaling in social re- by delinquency should be measured by scaling search encompasses a variety of measurement attitudes. It was argued: techniques. A relevant issue in the application The criteria for determining degrees of s- of any of these, however, is the extent to riousness must ultimately be determined by which different procedures yield similar re- someone's or some group's subjective interpre- sults. For example, one might expect that since tation. If weights were assigned by a few crim- magnitude and category scales represent two inologists engaged in the task of construct- distinct and different types, of scaling, each ing a mathematical model, we should regard would generate different sets of results. In cat- this as an arbitrary determination. But if egorical analysis, the subjects' judgements in- judgments were elicited from theoretically volve placing responses into intervals or cate- meaningful and large social groups, consensus gories; magnitude scaling involves judging might produce a series of weighted values that strength or salience and order on a more ex- meaningful and large social groups, consensus would have validity.... Although no external pansive and continuous scale. Research in objective criteria, beyond people's judgments, psychophysical scaling suggests, however, that exists for producing a continum of seriousness category and magnitude scales are logarithmi- of delinquent acts, there are objective methods cally related." The purpose of this present re- of measurement which have been developed port is to empirically display the relationship into psychological '' relating two different between these scale types within -the context of kinds of psychological scales, these methods the measurement of delinquency. It is thought can be applied to such nonphysical dimensions that insight on selecting scale types for delin- as the gradual seriousness of deviant behavior.3 quency research can be gained by examining To determine the scalability of their stimuli, whether similar seriousness scores result from Sellin and Wolfgang empirically compared cat- these methods. egory and magnitude scales. The latter tech- In 1964 Sellin and Wolfgang developed a nique was then selected for developing the se- seriousness index of delinquent events 2 where- riousness index. Their choice of method was in they asserted that the social harm caused based on the presumed theoretical strengths of * George S. Bridges is a Research Assistant at magnitude over category scaling, rather than the Center for Studies in Criminology and Criminal on any differences that were produced in the Law, University of Pennsylvania, Philadelphia, Pennsylvania. scale data. This decision is the focal point of ** Nancy S. Lisagor is an Assistant Professor analysis in this paper. First, however, a brief at the Department of at Stockton State review and discussion of scaling techniques is College, Pomona, New Jersey. The authors wish to express their gratitude to necessary. Professors Marvin E. Wolfgang and Phillip C. The unidimensional scaling methods that Sagi for their advice and support. Also, thanks go have been developed thus far serve at least two to Robert M. Figlio for the use of data from his dissertation. This paper grew out of a project un- important measurement functions. The most dertaken by Ms. Lisagor in a graduate seminar at common of these functions involves the con- the University of Pennsylvania. 1See A. Shinn, Relations Between Scales, in struction of empirical indices that describe the MEASUREMENT IN THE SOCIAL SCIENCES (H. Bla- strength and direction of individual and group lock ed. 1974). Shinn gives a comprehensive dis- attitudes. By these methods, data can be ana- cussion of this literature and the scaling principles related to this logarithmic transformation of the lyzed with a focus on the attitudinal composi- tion of either an entire sample or any of the scale2 scores. T. SELUN & M. WOLFGANG, THE MEAsURE- MENT OF DELINQUENCY (1964). 3 Id. at 237. BRIDGES AND LISAGOR [Vol. 66

individuals within it. A second function, which scores. In this vein, Sellin and Wolfgang ini- is of specific interest to this research involves tially investigated whether this psychophysical estimating the magnitude of questionnaire or property held for their non-physical serious- interview stimuli. The focus here, as developed ness stimuli. They employed Helm's principle by Thurstone and later greatly extended by of equal precision or dispersion by standardiz- Stevens, is directed towards locating estimates ing the variances of their rater's catetory judg- of stimuli on an empirical range of strength or ments for each stimulus. This effectively trans- salience. Shinn, in his development of a scale formed the non-linear plot of the category and typology, suggested that these stimulus-cen- log-magnitude mean scores into a linear one. tered methods could be classified into two In so doing, the logarithmic relation between types: magnitude and category scales. The for- the scales was confirmed. mer amounts to a continuous ratio scale, a nu- This finding has at least two important im- mercial range with an absolute zero point. Sub- plications for measuring the seriousness of de- jects utilizing a magnitude scale evaluate the linquency. Perhaps most importantly, it sug- magnitude of stimuli by assigning them scores gests that seriousness can be measured psycho- that represent points on a psychological scale. physically, as a response to physical stimuli Category scaling, as primarily developed by would be. Secondly, the log-linear relation- Thurstone, involves constructing a rank- ship evidenced in this experiment between ordered continuum of stimuli. In the most com- scales confirms that the scale scores are a log- mon technique of successive categories, sub- linear function of one another. Thus, log-mag- jects are presented with a questionnaire item nitude and equal variance category scores with, or stimulus, and then are directed to locate it of course, some measurement error, are di- along a scale of successive categories (usually rectly related ways of estimating the ranging from low to high) that best describes seriousness of delinquent events. The proper- stimulus magnitude. Statistical estimates of the ties of psychophysical scaling developed by "true" stimuli magnitudes for both magnitude Stevens and others are, therefore applicable to and category techniques are then derived from this topic in criminological research. sample distributions of such judgments. In ef- A second aspect of Sellin and Wolfgang's fect, scales of the stimuli, as well as estimates scale development, however, is their final of their magnitude, are thus obtained. choice of scale design. Despite the direct loga- 4 It has been argued and shown by Helm, rithmic relation between the methods, it is 6 7 Ekman,5 Ekman and Kuennapas, and Stevens argued that the magnitude scales have added that the log-linear relationship connecting dimensions of validity that extend beyond this magnitude and category scales is dependent log-linear function. Since the scale values are upon the dispersions of the judgments made by determined by the subject rather than the ex- subjects. If judgments on the category scale perimenter, as is the case in the category de- are homoscedastic (have equal variance) for sign, Sellin and Wolfgang argue that magni- all stimuli, and the dispersions of judgments tude estimation better taps the "true" on the magnitude scales are heteroscedastic psychological effects of stimuli. Secondly, it is and directly function with scale value, then a implicitly argued that the breadth of range in logarithmic transformation of the magnitude magnitude scales, being so much greater than judgments will lead directly to the category that of category scales, enhances accuracy in measurement by providing "intrinsically more 4Helm, Messick & Tucker, Psychological Mod- information" about judgments. Due to these els for Relating Discriminationand Magnitude Es- advantages, Sellin and Wolfgang chose to se- timation Scales, 68 PSYCHOLOGICAL REVIEW 121 lect magnitude scaling over the category tech- (1961). 5 Ekman, Measurement of Moral Judgment: A nique for the index construction of seriousness. Comparison of Scaling Methods, 15 PERCEPTUAL The focus of this present research is to dem- AND MOTOR SKILLS 3 (1962). and 6 Ekman & Kuennapas, Scales of Aesthetic onstrate the similarity between magnitude Value, 14 PERCEPTUAL AND MOTOR SKILLS 19 category scales of seriousness. The following (1962). analysis is directed initially to the log-trans- 7 S. STEVENS, PSYCHOPHYSICS AND SOCIAL SCALING (1972). form property that ties category and magnitude 1975] SYMPOSIUM scores, and itwill compare the distributions of "most students enroll for the course in intro- magnitude and category scale data for a range ductory sociology without any special factor of of offense stimuli. It is thought that this ap- student enrollment that might cause them to proach, differing from -the mean score plot cluster in a particular way." 10 Thus, it was as- that is commonly evidenced in the psychophys- sumed that the students represented samples ical literature, permits a closer examination of from a larger student population that would be the relationship between the scales. Rather free of any unusual attitudinal or intellectual than focusing simply on mean stimulus scores, qualities." It was further assumed that the it was intended .that the research examine also multitude of factors that contribute to students' the shapes of the distributions for each stimu- decisions about which course offering to take, lus type. The extent to which these distribu- as well as which class times and the teachers tions "fit" one another will then be tested sta- preferred in each course, necessarily added to tistically between the scaling methods. this representativeness and also introduced in- A second consideration given in the analysis dependence between the sample groups. At a involves the implications for measuring de- large university with many course offerings, linquency that directly arise from the results this assumption seems reasonable. obtained in the study. If the expected logarith- Since the purpose of our analysis was mic relationship between category and magni- merely to demonstrate scale differences rather tude scales is demonstrated, then consideration than make generalizations about the perceived in measuring delinquency's seriousness should seriousness of delinquency, we proceeded with be given to any differences that extend beyond comparisons and tests of the scaling tech- the log transform. Sellin and Wolfgang's con- niques. Any other biases that may have been cern about intrinsic differences in validity operant in the data due to sampling and meas- would, therefore, be justified. Otherwise, any urement error were beyond the control of the empirical differences that arise in the analysis investigators. should be examined more closely and Sellin MEASURES and Wolfgang's considerations re-evaluated. Figlio provided a concise description of the METHOD category and magnitude scales that best suited Data were drawn from an analysis of of- the discussion to follow. He noted: fense seriousness and respondent background In the category scale each subject was made by Figlio.8 In one part of Figlio's re- asked to circle the number from one to eleven search, two samples of subjects judged the se- (least to most serious) which best represented riousness of twenty delinquent events; one how serious he thought that particular offense group employed a magnitude estimation scale was. In the magnitude scale, the subject was asked to choose any number which adequately (N = 158) and the other a category scale represented the seriousness of that particular (N = 58). Subjects were given test booklets 9 offense description. The category scale has the similar to those used by Sellin and Wolfgang. advantages of being easy to visualize and to Each booklet contained a thorough set of in- understand, it is also numerically constraining. structions as well as a list of the twenty delin- The magnitude scale, while having no such quent events to be judged. Data generated constraint, requires greater abstraction in the from these judgments served as the basis for thought process.' 2 comparing the scaling techniques. :0 Id. " As noted previously, the assumption of hom- SAMPLES oscedastic or equal dispersion category scores un- The samples were drawn from undergradu- derlies the log-linear transform linking category and magnitude scale scores. An examination of the ate sociology classes at the University of sample category scores indicated that the disper- Pennsylvania. As Sellin and Wolfgang noted, sions around the twenty offense stimuli were ap- proximately equal. Also, regression analyses of the 8 Figlio, Tie Seriousness of Offenses: An Eval- mean category scores (raw) on the mean log- nation by Offendmers and Nonoffenders, 66 J. CRIM. magnitude scores gave a very tight linear fit. By L. & C. 189 (1975). these findings, it was assumed that the property of 9 T. SELLIN & M. WOLFGANG, supra note 2, at homoscedasticity was operant in the category scale. 237. 12 Figlio, supra note 8. BRIDGES AND LISAGOR [Vol. 66

FIGURE 1 RAW CATEGORY AND LOG-MAGNITUDE SCALE

Rape (forcible) Runaway

01 3 4 5 6 7 10 11 6 7 8 9 10 11

category scale ...... log magnitude scale * Frequency distributions in percentages (abscissa is scale value, ordinate is percentage value)

Particular consideration was also given to data were calculated,' 4 2) a transformation biases introduced by stimulus and response ef- was made on the category scale data so that fects on the judging process. As Sellin and the discriminatory dispersions of each subject Wolfgang assert: "A larcency following a disor- were made equal, 3) the log-magnitude scores derly conduct might be judged more serious per subject were standardized so that compari- than a larceny following a murder; a charge of sons between the distributions could be made intoxication might be judged more serious if it on a -single numeric continuum and, 4) the followed a runaway offense than if it were pre- strength of the relationship between the scale ceeded by a larceny; and so forth." 13 In order data for each stimulus was examined statisti- to reduce these potential distortions, the same cally. items were used for each scaling technique and were randomly assigned positions in the test RESULTS booklet. Thus, the final measurement tool was The frequency distributions of the log-mag- two identical sets of randomized offense de- nitude and raw category scores for two of the scriptions, one that had category response twenty offense stimuli, forcible rape and run- scales and the other with boxes for magnitude ning away from home are found in Figure 1.15 scoring. From simple inspections of the graphs, the similarity between the scale curves is evident ANALYSIS in their general shapes. For most offenses, the Our primary concern was to establish the ef- 14 Naperian or "natural" logarithms were taken fects of the logarithmic transform property on in the transformation of the magnitude scores. category and magnitude scale distributions. 15 The presentation of the entire set of scale Four steps were taken to meet this goal: distributions was saved for the sake of brevity in the report. Distributions for forcible rape and run- 1) the logarithms of the magnitude scale ning away from home were selected because they are representative of other serious and non-serious 3 T. SELLIN & M. WOLFGANG, sutpra note 2, at offense stimuli. Copies of the entire set of distribu- 253. tions are available from the authors upon request. 19751 SYMPOSIUM

FIGURE 2 1 , RAW CATEGORY AND LOG-MAGNITUDE SCALE

Rape (forcible) Runaway 100 90 80 70 60 50 40 30. 20 20

-3 -2 0 1 2 3 4 5-3 -2 -1 0 1 2 3 4 5

category scale ...... log magnitude scale * Frequency distributions in percentages (abscissa is scale value, ordinate is percentage value) modal levels of the distributions were found to tion around each stimulus would differ in a be approximately equal. It seemed, however, misleading manner. In order' to: standardize that one distinguishable difference between the this subject effect, location and scale transfor- distributions was location. The category scale mations were made on each subject's evalua- scores were consistently located at points tions. This amounted to transforming the data higher on the seriousness scale than the mag- such that all 'had mean sdore 'of -zero and nitude scores. It was also evident that the cate- equal variances. These standardized judgments, gory distributions had slightly higher disper- plotted for forcible rape and running away, are sions than the magnitude distributions. These found in Figure 2. The marked similarity be- differences indicated that the log-magnitude tween the curves then becomes even more ap- and category scores, although obviously corre- parent; it is evident that, by performing a sim- lated by their similarity in shape, differed by ple standardization of the log-maguiitude and location and scale factors. It should be noted, category scale data, the expected logarithmic however, that this was not inconsistent with relationship between the scale types - is re- Sellin and Wolfgang's findings. The relation- vealed. The scales produced'extremely similar ship they demonstrated between the scales was results across all of the stimuli under measure. log-linear; category scales were related to In order to describe this relationship gtatisfi- magnitude scales by multiplicative and additive cally, chi-square goodness-of-fit tests were per- constants. This suggests that the scale distribu- formed on 'each of the offense stimuli. This tions should have differed only as a function of statistical method involves measuring the' ex- these. constant terms. Thus, a simple transfor- tent to which two frequency distributions dif- mation of the data to eliminate these constant fer at points along a' common numerical effects was sought in order to render the dis- scale.16 As is evident from Table I, two of the tributions comparable and equivalent. To determine the nature of this transforma- 36 One important consideration that must be tion, the study was next directed toward the made when using the chi-square' goodness-of-fit test relates to the number. of intervals into which variation in the judgments of individual sub- data are grouped. Differences betweenr distributions jects. It was thought that unless the variation may vary directly with this number. Thus, between individual subjects in the way each whether an hypothesis is accepted or rejected is often dependent upon the number of degrees of judge stimuli is taken into account, the varia- freedom associated with the test. In order to free- BRIDGES AND LISAGOR [Vol. 66

TABLE I slightly lower dispersions than the category TESTS OF SCALE DIFFERENCES FOR TWENTY distributions for serious offenses. This "peak- OFFENSE TYPES ing" effect, however, contributed to only two significant differences between the scales in the 2 Offense Stimulus X , degrees Probability twenty comparisons. Thus, it is impossible to of freedom be confident statistically that these differences represent differential stimulus effects. Larceny, $5 ...... 3.67 6df p > .20 Although these differences are unexpected in Larceny, $20 ...... 5.78 5df p > .20 light of the scaling principles that have been $50 ...... Larceny, 13.25 7df p < .10 presented thus far, the two scale distributions Larceny, $1000 ...... 9.50 6df p < .20 Larceny, $5000 ...... 7.35 6df p > .20 closely approximate one another across most of Burglary, $5 ...... 13.95 7df p < .10 the offense stimuli. This similarity in the dis- Robbery (no weapon) $5 7.52 7df p > .20 tributions supplements our understanding of Robbery (weapon) 5.. 5.73 6df p > .20 the differences between the techniques. It is Assault (death) 14.38 7df p < .05* shown, through simple transformations of the Assault (hospitalization) 11.80 6df p < .10 scale data, that comparable distributions of se- Assault (theft & damage) 6.33 6df p > .20 riousness scores are produced. It is important Assault (minor) ...... 6.70 6df p > .20 to note that these transformations in no way Rape (forcible) ...... 8.85 7df p > .20 violate the study's earlier assumptions about Auto Theft ...... 27.70 7df p < .05* Rifle (no permit) ...... 5.52 7df p > .20 the discriminatory dispersions around stimuli. Trespassing ...... 14.04 7df p < .10 This analysis has involved no more than a shift Illegal Liquor ...... 6.28 5df p > .20 of the scale axes so that their distributions could Disorderly Conduct ..... 8.85 5df p < .20 be more easily examined. Runaway ...... 10.10 6df p < .20 ...... Hookey 10.70 5df p < .10 CONCLUSION This study has illustrated that magnitude * Significant beyond conventional levels of proba- and category scaling techniques produce quite bility. similar distributions and estimates of serious- ness magnitude. Although the analysis repre- sents empirically no more than a further vali- twenty distributions differed beyond the con- dation of a fundamental principle in attitude ventional levels of probability used in hypothe- scaling, the implications of its findings for de- sis testing. Since it was decided to reject the linquency research are important. Initially, null hypothesis of no significant differences be- these results indicate that rigor spent on tween the distributions at the .05 level of sig- choosing a "best" scale type for measuring se- nificance, one would expect that one of the riousness does not significantly affect the final (.05) would be rejected solely due twenty tests research product. Regardless of the technique to sampling error. The data, however, indicate employed (category or magnitude), quite simi- that the scales differ for two offense stimuli: lar findings result. It should be noted, how- assault resulting in the death of the victim and ever, that there are some intrinsic differences auto theft. In explanation, one might argue between magnitude and category scales that that magnitude and category scales differ as a warrant consideration. These differences be- function of stimulus magnitude or offense seri- yond the log transform cannot be directly re- ousness. This appears consistent with the scale solved by simple comparisons of data. The data because the magnitude distributions had properties of each method must be more closely examined. If the properties of one technique the analysis from this source of bias, the scale dis- tributions were divided, wherever possible, into are more directly suited than another to a spe- similar numbers of groups. It was thought that cific measurement problem in delinquency re- this procedure, while introducing an element of search, then this technique should be em- consistency in our method, would best reveal any differences existing between the scale distributions. ployed. Inherent differences in validity as well 1975] SYMPOSIUM as in the efficiency of administration and anal- better established. The above finding that the ysis between the scale types should be consid- different scaling methods produce similar re- ered in this choice of method. sults provides a validation of psychometric re- A second area to which these findings relate search; the log-linear relation between the is the research process itself. A common prob- scales had been demonstrated by simple al- lem in many areas of social research involves gebraic transformations of the scale data. More employing measurement techniques that are importantly, it assures the criminologist that best suited to research needs. As research magnitude and category scales provide similar grows in analytic complexity, it becomes in- estimations of delinquency's seriousness. Both creasingly more difficult to ensure this suitabil- methods are suited to their research problems; ity. By incorporating comparative analyses of each, however, has properties that are appeal- different techniques into specific substantive ing to specific research needs and must be con- areas, the suitability of these techniques can be sidered in future analyses.