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 Readings in Statistical Analysis Labovitz: Treating Ordinal Data as Interval
 THE ASSIGNMENT OF NUMBERS TO RANK ORDER CATEGORIES * Sanford Labovitz The American Sociological Review, 35 (1970), 151-524. Edited by Kenneth Janda Abstract By both random and nionrandom, assignments of numbers to rank orders (which are consistent with the monotonic nature of the categories), it is shown that ordinal variables can be treated as if they conform to interval scales. The scoring systems, of which 18 were randomly generated by a computer, resulted in negligible error when comparing any assigned scoring system with any selected "true" scoring system. Errors are determined by the Pearsonian correlation coefficient (r) and r2. The advantages of treating ordinal variables as interval are demonstrated with regard to the relation between occupational prestige and suicide. These advantages include: (1) the use of more powerful, sensitive, better developed and interpretable statistics with known sampling error, (2) the retention of more knowledge about the characteristics of the data, and (3) greater versatility in statistical manipulation (e.g., partial and multiple correlation and regression, analysis of variance and covariance, and most pictorial presentation). The computer approach to this problem does not exhaust all possibilities for assigning numbers, which partially limits the generality of the findings.
 EMPIRICAL evidence supports the treatment of ordinal variables as if they conform to interval scales (Labovitz, 1967).[1] Although some small error may accompany the treatment of ordinal variables as interval.[2] this is offset by the use of more powerful, more sensitive, better developed, and more clearly interpretable statistics with known sampling error. For example, well-defined measures of dispersion (variance) require interval or ratio based measures. Furthermore, many more manipulations (which may be necessary to the problem in question) are possible with interval measurement, e.g., partial correlation, multivariate correlation and regression, analysis of variance and covariance, and most pictorial presentations. The arguments presented below are general enough to apply to any ordinal scale, and perhaps with even greater confidence they apply to variables that fall between ordinal and interval, e.g., I.Q. scores and formal education (Somers, 1962: 800). To determine the degree of error of results when treating ordinal variables as if they are interval, the relation between occupational prestige and suicide rates is analyzed. Prestige rankings obtained by NORC in its 1947 survey are related to suicides by occupation for males in the United States in 1950. The list of occupations, taken from Duncan's comparisons of occupational categories used in the survey, are matched to the detailed occupational classification in the U.S. Census of 1950 (Reiss et al., 1961). Because suicides are not reported for all of these occupations and sometimes the reported suicides are for two or three occupations grouped into one, 36 occupations were selected which contain the necessary data used in this study (see Table 1). Measurement of occupational prestige is based solely on the principle of ordinal ranking. In the survey, respondents were given occupations to rank by the method of paired comparisons; consequently, the resulting prestige scores indicate merely the rank of one occupation relative to the others (Reiss et al., 1961: 122 l23).[3] [Editor's note: Labovitz is using a well-known ranking of occupation prestige as a criterion variable which he will correlate with other scales having different values.]

 TABLE 1. Prestige, Income, Education, and Suicide Rates for 36 Occupations UNITED STATES, MALES, Circa, 1950 Occupation NORC Rating Scalea Male Suicide Rateb Median Incomec Median School Yrs . Completedd Accountants and auditors 82 23.8 3977 14.4 Architects 90 37.5 5509 16+ Authors, editors and reporters 76 37 4303 15.6 Chemists 90 20.7 4091 16+ Clergymen 87 10.6 2410 16+ College presidents, professors and instructors (n.e.c.) 93 14.2 4366 16+ Dentists 90 45.6 6448 16+ Engineers,civil 88 31.9 4590 16+ Lawyers and judges 89 24.3 6284 16+ Physiciansand surgeons 97 31.9 8302 16+ Social welfare, recreation and groupworkers 59 16 3176 15.8 Teachers (n.e.c.) 73 16.8 3465 16+ Managers, officials and proprietors (n.e.c.)&emdash;self-employed&emdash;manufacturing 81 64.8 4700 12.2 Managers, officials and proprietors (n.e.c.) -- selfemployed -- wholesale and retail trade 45 47.3 33806 11.6 Bookkeepers 39 21.9 2828 12.7 Mail-carriers 34 16.5 3480 12.2 Insurance agents and brokers 41 32.4 3771 12.7 Salesmen and sales clerks (n.e.c.), retail trade 16 24.1 2543 12.1 Carpenters 33 32.7 2450 8.7 Electricians 53 30.8 3447 11.1 Locomotive engineers 67 34.2 4648 8.8 Machinists and jobsetters, metal 57 34.5 3303 9.6 Mechanics and repairmen, automobile 26 24.4 2693 9.4 Plumbers and pipe fitters 29 29.4 3353 9.3 Attendents,autoservice and parking 10 14.4 1898 10.3 Mine operatives and laborers (n.e.c.) 15 41.7 2410 8.2 Motormen, street, subway, and elevated railway 19 19.2 3424 9.2 Taxscab-drivers and chauffeurs 10 24.9 2213 8.9 Truck and tractor drivers, deliverymen and routemen 13 17.9 2590 9.6 Operatives and kindred workers, (n.e.c.), machinery, except electrical 24 15.7 2915 9.6 Barbers, beauticians and manicurists 20 36 2357 8.8 Waiters, bartenders and counter and fountain workers 7 24.4 1942 9.8 Cooks, except private household 16 42.2 2249 8.7 Guards and watchmen 11 38.2 2551 8.5 Janitors, sextons and porters 8 20.3 1866 8.2 Policemen, detectives, sheriffs, bailiffs, marshals and constables 41 47.6 2866 10.6 a. Albert J. Reiss, Jr., et al., 1961:122&emdash;123. The scale ls based on a 1947 survey. b. Males, aged 20-64. National Office of Vital Statistics, Vital Statistics&emdash;Special Report, Vol. 53, No. 3 c. 1949 Median Income. United States Census of Population, 1950. Occupational Characteristics. d. 1950 Median School Years Completed, Ibid.
 The rank correlation (rho) between occupational prestige and suicide is .07. The scatter diagram of the NORC prestige ratings and suicide rates suggests that the relation is roughly linear, although the plotted points are widely scattered. The Pearsonian correlation coefficient (r) on the same data is slightly larger (.11). The .04 discrepancy between the two measures is due to the magnitude of the differences between adjacent scores which are not considered in rho, but do influence the value of r. ASSIGNMENT OF SCORING SYSTEMS TO ORDINAL CATEGORIES Twenty [different] scoring systems are used on NORC's occupational prestige values. One scoring system is the actual prestige ratings resulting from the study (the NORC Prestige Rating Scale in Table 1). A second scoring system is the assignment of equidistant numbers (i.e., an equal distance between assigned numbers) to the occupational categories (Table 2). The remaining scoring systems in Table 2 were generated from a computer according to the following conditions: (1) the assigned numbers lie between the range of 1 and 10,000, (2) the assignment of numbers is consistent with the monotonic function of the ordinal rankings, (3) any ties in the ordinal rankings are assigned identical numbers, and (4) the selection of a number is made on the basis of a random generator in the computer program. To be consistent with the monotonic function, any subsequent randomly selected numbers must be higher than previous ones (except for ties). The resulting largely random scoring systems vary among themselves (sometimes to a large extent) on the actual values assigned to each rank, the range of values, and the size of the differences between adjacent values. Although all are necessarily consistent with the monotonicity of the ordinal rankings, they vary widely among themselves. In fact, some of the scoring systems show definite curvilinear patterns--logarithmic, exponential or higher order curves (two or more inflection points).
 TABLE 2. NORC PRESTIGE RATINGS, LINEAR SCORES, AND FIVE MONOTONIC RANDOM GENERATED SCORING SYSTEMS Linear NORC Monotonic Random Generated Scoring Systems (1) (2) (3) (5) (9) (13) (18) 1 7 13 79 52 849 418 2 8 34 105 109 909 585 3.5 10 99 233 380 923 648 3.5 10 99 233 380 923 648 5 11 248 389 518 1152 820 6 13 407 580 557 1167 869 7 15 727 605 799 2300 1271 8.5 16 1824 771 2167 2343 1478 8.5 16 1824 771 2167 2343 1478 10 19 1897 1042 2790 2845 1647 11 20 2021 1287 2796 2876 1789 12 24 2470 1374 3209 3107 2112 13 26 2978 1713 3558 3159 2627 14 29 2995 2083 3598 3231 2628 15 33 3330 2595 3808 3409 2777 16 34 3412 2715 3945 3760 2921 17 39 3535 2751 4087 4238 3077 18.5 41 3952 2861 4094 4898 3156 18.5 41 3952 2861 4094 4898 3156 20 45 4082 3003 4745 5336 3209 21 53 4485 3266 4885 5903 3600 22 57 4865 4013 4892 6016 4304 23 59 5091 4267 5044 6106 4323 24 67 5146 4449 5300 6242 4762 25 73 5349 5318 5819 6270 5020 26 76 5775 6330 5876 6681 5528 27 81 5995 6547 5923 6787 5797 28 82 6304 6810 5932 6915 6027 29 87 6356 6974 5976 7118 6388 30 88 6644 7660 5995 7229 6471 31 89 6742 8145 6160 7652 6560 33 90 7657 9085 6231 7926 6911 33 90 7657 9085 6231 7926 6911 33 90 7657 9085 6231 7926 6911 35 93 7841 9108 6458 8283 6972 36 97 8164 9461 7094 8472 7588 See text for an explanation of the scoring systems. The five random scoring systems are indicative of the 18 used in the study.
 Because this computer approach to assigning numbers to rank order data partially is based on a random selection of numbers, the generality of the findings is somewhat limited. It is possible that some systematic selection of numbers will not yield such consistent results as those reported herein. The similarity among the scoring systems can be assessed by their matrix of intercorrelations (Table 3). By assuming, in turn, that each scoring system is the "true" one, the intercorrelations (Pearson product-moment coefficients) indicate the extent of "error" of using one of the other 19 scoring systems. For example, if (4) is the "true" system and (7) has been used in its place, then .97 (the correlation between the two scoring systems) indicates the degree to which the two systems vary together. On the other hand, r2 (the values below the diagonal in Table 3) indicates "error" in terms of the amount of variance in the assigned scoring system accounted for by the variation in the "true" scoring system (Abelson and Tukey, 1959).[4] In this instance, between scoring systems (4) and (7), 94% of the variance in (7) is accounted for by the variation in (4).
 TABLE 3. INTERCORRELATIONS (r) AMONG TWENTY SCORING SYSTEMS Scoring Systems Twenty Different Scoring Systems (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (1)b ... .98 1 .99 .97 .99 .98 .99 .97 .98 .99 .99 .99 .99 1 .98 .99 .99 .99 1 (2) .98 ... .97 .96 .97 .98 .95 .97 .96 .94 .97 .97 .96 .98 .93 .97 .94 .99 .98 .96 (3) 1 .94 ... .99 .97 .99 .98 .98 .98 .98 .99 .99 .99 .99 .99 .98 .99 .99 .99 .99 ( 4) .98 .92 .98 ... .98 .99 .97 .99 .95 .99 .98 .98 .98 .99 .99 .99 .99 1 .98 .99 ( 5) .94 .94 .94 .96 ... .99 .93 .99 .91 .99 .96 .95 .96 .96 .97 .99 .98 .99 .94 .96 ( 6) .98 .96 .98 .98 .98 ... .97 .99 .94 .99 .97 .98 .98 .98 .99 .99 .99 .99 .97 .98 ( 7) .96 .9 .96 .94 .86 .94 ... .96 .98 .95 .96 .98 .98 .98 .98 .94 .96 .96 .99 .98 ( 8) .98 .94 .96 .98 .98 .98 .92 ... .93 .99 .97 .97 .98 .98 .99 .99 .98 .99 .96 .98 ( 9) .94 .92 .96 .9 .83 .88 .96 .86 ... .93 .97 .98 .97 .97 .96 .92 .96 .9S .99 .98 ( 10) .96 .88 .96 .98 .98 .98 .9 .98 .86 ... .97 .97 .97 .98 .98 .99 .99 1 .96 .98 ( 11) .98 .94 .98 .96 .92 .94 .96 .94 .94 .94 ... .99 .98 .98 .99 .98 .99 .98 .98 .99 ( 12) .98 .94 .98 .96 .9 .96 .96 .94 .96 .94 .98 ... .99 .99 .98 .97 .98 .98 .99 .99 ( 13) .98 .92 .98 .96 .92 .96 .96 .96 .94 .94 .96 .98 ... .99 .99 .97 .98 .98 .99 .99 ( 14) .98 .96 .98 .98 .92 .96 .96 .96 .94 .96 .96 .98 .98 ... .99 .97 .98 .99 .98 .99 ( 15) 1 .86 .98 .98 .94 .98 .96 .98 .92 .96 .98 .96 .98 .98 ... .98 .99 .99 .98 .99 (16) .96 .94 .96 .98 .98 .98 .88 .98 .85 .98 .96 .94 .94 .94 .96 ... .99 .99 .96 .97 ( 17) .98 .88 .98 .98 .96 .98 .92 .96 .92 .98 .98 .96 .96 .96 .98 .98 ... .99 .98 .99 ( 18) .98 .98 .98 1 .98 .98 .92 .98 .90 1 .96 .96 .96 .98 .98 .98 .98 ... .97 .98 ( 19) .98 .96 .98 .96 .88 .94 .98 .92 .98 .92 .96 .98 .98 .96 .96 .92 .96 .94 ... .99 ( 20) 1 .92 .98 .98 .92 .96 .96 .96 .96 .96 .98 .98 .98 .98 .98 .94 .98 .96 .98 ... a. r above the diagonal; r2 below b. linear scoring system. c. NORC prestige ratings.
 The r and r2 values in Table 3 are consistently and substantially high, indicating a high degree of interchangeability among the 20 scoring systems. Out of 190 correlation coefficients, all are above .90 (a few even reach unity), and 157 are .97 and above. Therefore, even without a rationale concerning the differences between ranks, by using a nearly random method of assigning scoring systems (consistent with the monotonic function), it is possible that under specific conditions the selected scoring system will deviate from the "true" system by a near zero or negligible amount. The r2 values are slightly lower than the r values, but still exceedingly high. For example, only nine of the 190 are below .90, and none are below .83. (Since r2 is the square of a decimal fraction, it is necessarily smaller than r.) Note that if the equidistant (linear) scoring system is always selected (no matter what the "true" scoring system may be), the expected error is smaller than the larger errors cited above. Almost all the r's and r2's for the linear system (1) are near unity, with the lowest r being .97 and the lowest r2 being .94. The linear scoring system lies midway between the other scoring systems (in correlational terms), which by definition excludes the most extreme scoring systems in each direction. The correlations between the extremes are lowest, and, therefore, selecting the linear scoring system eliminates the lowest r's and the highest potential "errors" in selecting a scoring system different from the "true" one. Possessing some knowledge about the amount of differences between ranks can reduce the small error even further, if the linear scoring system has been assigned to the ordinal categories. Perhaps, the best strategy, if there is some knowledge of the differences between ranks, is to modify the linear scoring system accordingly. For example, in the relation X1 > X2> X3, X2 is assumed to be closer to X3 than to X,. Consequently, the linear scoring system of 10, 20 and 30 (as values for Xs, X2 and X3) can be modified to 10, 25 and 30 to account for this additional knowledge. It should be stressed that without prior knowledge or theory such score assignments are not likely to prove useful for analysis. Table 4 offers further evidence that ordinal data can be treated as if they are interval by assigning scoring systems to the ordered categories. In this instance, the predictive ability of each scoring system is assessed in terms of its relation to suicide rates. As indicated previously, the rho value between the NORC prestige scale and 1950 suicide rates for males in 36 occupations is .07; for the same data, r is .11. Table 4 reports the r and r2 values between the 20 scoring systems and the suicide rate. (The last two columns in Table 4 are r values for 20 and 10 occupations respectively and will be discussed later in the paper.) The similarity in predicting an outside variable is extremely high. The r's vary between .09 and .15, and the 3 values are either .01 or .02.[5] Given some degree of unreliability in occupational prestige and suicide data, and the rather crude measurement procedures, these results substantiate the point that different systems yield interchangeable variables. Each indicates a quite low positive (statistically nonsignificant) relation between occupational prestige and suicide. These results are consistent with a previous study (Labovitz, 1967), which also found the relations to be very similar; however, in the previous study, the relations are somewhat higher and statistically significant.
 TABLE 4: CORRELATION COEFFICIENTS (r) BETWEEN SUICIDE RATES AND TWENTY SCORING SYSTEMS OF OCCUPATIONAL PRESTIGE Scoring System r(N=36) r2 r(N=20) r(N=10) (linear) (1) 0.13 0.02 0.35 0.28 (prestige rankings) ( 2) 0.11 0.01 0.35 0.25 ( 3) 0.13 0.02 0.31 0.24 ( 4) 0.11 0.01 0.32 0.3 ( 5) 0.1 0.01 0.3 0.21 ( 6) 0.11 0.01 0.35 0.18 ( 7) 0.14 0.02 0.28 0.33 ( 8) 0.12 0.01 0.38 0.34 ( 9) 0.14 0.02 0.26 0.15 ( 0) 0.09 0.01 0.29 0.24 ( 11) 0.13 0.02 0.3 0.24 ( 12) 0.11 0.01 0.28 0.22 ( 13) 0.15 0.02 0.41 0.35 ( 14) 0.14 0.02 0.37 0.25 ( 15) 0.13 0.02 0.35 0.32 ( 16) 0.09 0.01 0.33 0.18 ( 17) 0.12 0.01 0.37 0.34 ( 18) 0.11 0.01 0.3 0.33 ( 19) 0.15 0.02 0.33 0.25 ( 20) 0.14 0.02 0.38 0.41 Partially based on the data in Tables 1 and 2. Scoring systems 3 to 18 are randomly generated.