Arrest Rates and Crime Rates: When Does a Tipping Effect Occur?*

Arrest Rates and Crime Rates: When Does a Tipping Effect Occur?* D 0 N W. B R 0 W N, University of California, Riverside ABSTRACT The tipping effect of sanction certainty reported by Tittle and Rowe is of considerable theoretical import. This paper attempts to determine whether the tipping effect is peculiar to their data. An examination of sanction certainty indicators and the index crime rate in California cities and counties plus a further examination of Florida city and county data reveals that the tipping effect is an attribute of small populations: (1) it is present in smaller Florida cities but not larger ones and (2) the evidence of a deterrent effect of arrest certainty is stronger in smaller cities and counties than in larger ones. Three other explanations for this finding are considered and research directions specified. Tittle and Rowe's study of crime rates in Florida's cities and counties added a potentially significant (if stil amorphous) proposition to deterrence theory (but see Geerken and Gove; Silberman). They reported that a threshold of sanction certainty exists, above which official sanctions deter crime and below which they do not. Because of the importante of Tittle and Rowe's finding, we tried to determine whether it applies generally or is peculiar to certain conditions or data. Through an analysis of crime and arrest rates in California's counties and cities and a reanalysis of Tittle and Rowe's Florida data, this paper reports that (1) the sanction certainty tipping effect is not general but peculiar to smaller cities where (2) evidence supporting a deterrent effect of arrest certainty is stronger than in larger political units. Procedure Stated in its most general form, Tittle and Rowe's study generated the proposition that above a certain level the probability of official punishment for crimes will reveal a negative correlation with the crime rate while below that level the correlation will be absent. The operational shape that this *The Committee on Research of the University of California, Riverside provided funds for the analysis reported herein. Stephen McDougal's co-authorship of an earlier report of the California city data, which he collected, was important to the development of this paper. 671

672 / Social Forces / vol. 57:2, december 1978 proposition took in their analysis, however, was that beyond a 30 percent probability level the percentage of index crime cleared by arrest will show a higher negative correlation with the rate of reported index crime than it will below that level. The primary question of this study is this: To what extent is the general form of the proposition supported? DATA To answer this question, we analyzed four sets of data on crimes and arrests. The first set consisted of 1971 crime and arrest rates in California cities with populations over 25,000. 1 The second provided data on crime rates and arrest clearance rates for 1973 in California's counties. (Data from 1971 would have been used but clearance rates were reported only for 1973 in recent years, California Bureau of Criminal Statistics, b.) The third and fourth sets on crimes and arrests in Florida cities and counties (with populations over 2,500) were the same as those employed by Tittle and Rowe for reported crimes and arrest clearances (Florida Department of Law Enforcement). In each data set the crime rate variables are based not on all crimes reported but on the offenses classified as index crimes 2 under the FBI's Uniform Crime Reporting system (UCR) and reported to the appropriate state agency for the year indicated. Thus the crime data may not be parallel to the actual offense rate and is subject to the standard criticisms levelled at official statistics (Beattie; Seidman and Couzens; Wolfgang). But Skogan's comparison of the association between official and survey estimates of automobile thefts and robberies suggests that correlation statistics based on official data, as employed in this analysis, may not be severely biased. Nevertheless, readers should keep in mind the caveats implied by the criticisms. The arrest data are of two types. Tittle and Rowe used the percentage of index crimes reported that were cleared by arrest (crimes cleared by arrest/crimes reported). We used the same measure for California counties and for the extended analysis of Florida cities and counties. But sine the California Bureau of Criminal Statistics does not publish data on arrest clearance rates for cities we relied instead on alternative measures of sanction certainty: the ratio of index arrests (whether they cleared a crime) to reported crimes (arrests/crimes), and the ratio of persons charged with crimes (those not released after arrest) to reported crimes ([arrestsreleases]l crimes). 3 Justifications given by Logan and by Tittle and Rowe for the use of arrest measures as indicators of sanction certainty apply here as well. Thus the data sets include three different indicators of sanction certainty on two different states, with California county data on a different

Arrest & Crime Rates 1 673 year than the others. They provide material appropriate to explore the generality of the tipping effect proposition that Tittle and Rowe developed. ANALYSIS We first examined the California data sets to ask (1) is there a negative correlation between sanction certainty and crime rate, and (2) does a tipping effect for sanction certainty exist and if so, at what point? To answer these questions, scatter diagrams were constructed and Pearson correlation coefficients computed for the relationship between sanction probabilities and crime rates at various levels of sanction certainty. Among the 58 California counties, the anticipated negative association between the arrest clearance rate and the reported rate for index crimes was found (r =.39). The correlation coefficient is smaller than that reported by Tittle and Rowe for Florida counties (r =.65) but it is negative and supports a deterrence interpretation. What of the tipping effect? The correlation between the arrest clearance rate and the crime rate is.32 for arrest probabilities above 25 percent and.24 for counties with clearance rates below 25 percent. The correlation between clearance rate and crime rate is indeed slightly larger among counties where the likelihood of a crime being cleared by arrest is greater. But the difference between the two correlations does not approach that which Tittle and Rowe found. They reported a.58 coefficient for the association between sanction certainty and crime rate among Florida counties with clearance rates above the threshold but only.13 for those below it. The difference in the explanatory power of clearance rate between the counties above and below the critical level was 31 percent compared to a difference of only 4 percent between the two sets of California counties. The difference in the California data is small enough to raise doubt that it supports the interpretation that "there is a critical level that certainty of punishment must reach before there is a noticeable change in volume of crime" (Tittle and Rowe, 458). Neither does a tipping effect for sanction certainty seem to obtain in the California city data. As explained earlier, both the straight probability of arrest and the probability of being charged with a crime were employed as indices of sanction certainty. The relationship between arrest probability and the crime rate is negative although low (r =.17) and beyond one level of certainty a tipping effect does seem to exist. The 67 cities with arrest probabilities below 25 percent showed a correlation coefficient of.01, while the 47 cities with higher arrest probabilities revealed a correlation of.32, not large but considerably higher in relative terras. But another discovery shakes our confidence that a tipping. effect operates. Presumably, were there a critical level above which sanctions exerted a deterrent effect

674 1 Social Forces / vol. 57:2, december 1978 there would be a consistent negative relationship between the probability of arrest and the crime rate at each level of arrest probability above the critical level. But the tipping effect that seemed to begin at an arrest probability of 30 percent starts to disappear when we test for a threshold at 33 percent and vanishes beyond the 35 percent level (Table 1). At that point the number of higher probability cities drops to 27, approaching a size where we would be suspicious of correlation coefficient reliability since the elimination of only a case or two would sharply alter its magnitude. However, the same concern would have to apply to Tittle and Rowe's Florida city data in which there were but 29 cities and 26 counties above the cirtical level. We concluded that the California city arrest data did not support a tipping effect interpretation. As an alternative indicator of sanction certainty in California cities the probability of being charged with a crime was used. All of the reasons for using probability of arrest as a sanction indicator apply to the probability of being charged. Furthermore, its use should avoid the errors in the arrest measure that arise when a police department employs arrest not to bring criminals into the sanctioning system but to handle otherwise unmanageable situations. The probability of being charged with a crime is correlated negatively (r =.23) with the crime rate. But once again, it shows no evidence of a tipping effect. While we find differences in the correlations above and below any given point of the probability axis, in all instances the correlations are (1) both negative, (2) smaller than the overall correlation, and (3) small in difference (Table 2). In sum, for the two sets of California data, we consistently find a small negative correlation between our indicators of sanction certainty and the overall index crime rate, but we do not find evidence of a tipping effect. We are forced to conclude that the more general form of the tipping effect proposition is not supported and must be rejected. Yet the Florida data are sharply at odds; Tittle and Rowe did find a tipping effect for sanction certainty. Our question then becomes "Under what conditions does a tipping effect exist, or alternatively, what proposition about a tipping effect is supportable?" Unless we can locate an explanation for the disparities, Tittle and Rowe's interesting and potentially important contribution to deterrence theory is forced into the shadowy realm occupied by other partially supported hypotheses of uncertain theoretical status. With data from two states we have a rich opportunity to locate the source of the differences. What dissimilarities in the data might account for the difference between our findings and those of Tittle and Rowe? A different standard for reporting and classifying crimes and arrests is a possible explanation, but both Florida and California follow the same FBI standards in building their statistics. Still, our first guess was that some unknown peculiarity of the Florida arrest clearance data led to the tipping

Arrest & Crime Rates / 675 Table 1. CORRELATION OF PROBABILITY OF ARREST WITH CRIME RATE, CALIFORNIA CITIES Correlation Coefficients Arrest Probability Below Threshold Above Threshold Threshold Point (N) (N) 25 percent -.01 -.32* (47) (67) 30 percent.09 -.16 (71) (43) 33 percent.01 -.14 (85) (29) 35 percent.01 -.04 (87) (27) *P<.01. Tablet. CORRELATION OF PROBABILITY OF BEING CHARGED WITH CRIME RATE, CALIFORNIA CITIES Correlation Coefficients Charge Probability Below Threshold Above Threshold Threshold Point (N) (N) 18 percent -.28-26* (24) (90) 22 percent -.21-33^ (52) (62) 26 percent -.09 -.11 (73) (41) 30 percent -.l9 -.02 (89) (25) *P <.01. t P <.05. effect that Tittle and Rowe found. To explore this possibility, we reanalyzed the Florida evidence using as an independent variable not the clearance rate but the straight probability of arrest the same measure used in the California city data. Were it not to show a tipping effect, our hypothesis would stand. On the contrary, the arrest rate manifests a clear tipping effect among both cities and counties in Florida, just as does the arrest clearance rate. For the 79 cities with arrest probabilities above 30 percent the correlation of arrest certainty with crime rate is.43, whereas the correlation is but.17 for cities with lower arrest vrobabilities (Table 3). Among Florida

676 1 Social Forces 1 vol. 57:2, december 1978 counties, the tipping effect is even more striking. Counties with arrest certainties higher than 30 percent show a.70 correlation of arrest with crime, while lower certainty counties show an arrest-crime correlation of.02. Our first alternative hypothesis must be rejected. What other differences in the data sets might account for the disparities in the Florida and California findings? Perhaps the most obvious distinction lies in the size of the two groups of cities and counties. All the California cities in the data set we analyzed had more than 25,000 inhabitants in the 1970 census; only 26 Florida municipalities exceeded that figure, 110 had fewer than 10,000 residents and several approached the minimum population limit of 2,500. Almost two-thirds of the California counties, but just one-third of those in Florida had populations that surpassed 50,000. Consequently, our second alternative hypothesis was that the tipping effect appears only among smaller units of analysis, a possibility supported by Logan's study which found no tipping effect in statelevel data on arrest rates. To test the second alternative hypothesis the most desirable course would have been to determine whether a tipping effect for sanction certainty existed in Florida cities with populations at least as large as those in the California city data set. But the small number of Florida municipalities with populations over 25,000 was insufficient for reliable analysis. Instead, we separated the 180 Florida cities into those with less than 10,000 people (N = 110) and those with more (N = 70) and then examined the two subsets for signs of a sanction certainty tipping effect. The two subsets of city data illuminate an interesting distinction that seems to support the second hypothesis. There are but 5 larger cities with arrest clearance rates higher than the 30 percent critical level reported by Tittle and Rowe; 23 of their 28 higher certainty cities were small. Their inference that a tipping effect existed arose from a comparison of two sets of cities that differed not only in sanction certainty but in population size. This suggests that the tipping effect may be an attribute of small cities rather than medium-sized or large ones. To explore further, we calculated the sanction-crime correlation coefficients above and below a 20 percent arrest clearance rate. Two characteristics distinguish the larger cities from the smaller ones in addition to the smaller cities being overrepresented among those with higher arrest clearance rates. First, the statistical evidence for an arrest clearance rate deterrence effect is absent among the subset of larger cities (r =.16) and still present among the subset of smaller cities (r =.23). Second, the tipping effect is either reduced or missing in the larger cities, depending on whether one aberrant city with an extreme value 4 for its arrest clearance rate is excluded, relative to its appearance in the smaller cities. Among the bigger cities, with the extreme case inciuded, the arrest clearance rate for the high probability cities exhibits a.25 correlation with the crime rate,

Arrest & Crime Rates 1 677 Tabla 3. CORRELATION OF PROBABILITY OF ARREST WITH CRIME RATE, FLORIDA CITIES AND COUNTIES Correlation Coefficient Arrest Probability Below Threshold Above Threshold Threshold Point (N) (N) Counties: 30 percent -.02-70* (22) (45) 35 percent.17 -.65* (30) (37) Cities: 25 percent.32 -.35* (81) (99) 30 percent.17-43 (101) (76) yp <.01. compared to.39 for the lower probability cities; removal of the single extreme case drops the high probability correlation coefficient to.06. In the cities with fewer than 10,000 people, the tipping effect is prominent. The clearance rate correlation with the crime rate is.40 for those with higher clearance rates and.35 for those where sanction certainty is lower. Parallel results are produced when the straight arrest rate is employed as the measure of sanction certainty, for which it is again possible to treat 30 percent as a possible critical level since there are 31 large cities with arrest rates at least that high. There is essentially no relationship between the arrest rate and crime rate in the subset of larger cities (r =.13) but a slight one in the subset of smaller cities (r =.24) and the tipping effect is much more pronounced in the smaller cities (Table 4). The arrest crime correlation for the larger, above-threshold cities is negative but not statistically significant beyond the.05 level. While it would be desirable to perform the same analysis with the Florida county data, their number (67) is too few. Even dividing the smaller and larger counties into two approximately equal subsets of 33 and 34 respectively would create Ns in the higher and lower probability cells that would be unacceptably low for us to place confidence in correlation analysis. Nevertheless, the differences in the magnitude of the arrest crime rate correlation coefficients between the more and less populous counties run in the same direction as for the cities. Counties with populations over 40,000 show lower correlation coefficients (.41 and.45, respectively) for the association of arrest clearance rate and arrest rate with crime rate

678 1 Social Forces I vol. 57:2, december 1978 Tabla 4. CORRELATION OF SANCTION CERTAINTY INDICATORS WITH CRIME RATES IN FLORIDA CITIES, BY POPULATION Correlation Coefficient Independent Variable Cities over 10,000 Below Threshold Above Threshold (N) (N) Overall (N) Arrest clearance rate.39.06*.16 (20 percent threshold) (50) (19) (70) Arrest rate.00 -.20 -.13 (30 percent threshold) (39) (31) (70) Cities 2,500-9,999 Arrest clearance rate.35 -.40fi -.23t (20 percent threshold) (53) (57) (110) Arrest rate.10 -.40t -.24t (30 percent threshold) (62) (48) (110) *-.25 with extreme case not excluded. tp <.05. than do smaller counties (.61 and.71). Once again evidence suggestive of a deterrent effect for sanction certainty is greater in the smaller political units. Our examination of the differences in sanction crime correlations across the larger and smaller Florida cities leads to two conclusions. First, the second alternative hypothesis that the sanction certainty tipping effect is found among smaller but not larger cities cannot be rejected; furthermore, 24 of Tittle and Rowe's set of 29 cities with arrest clearance rates greater than 30 percent were small. Second, the statistical evidence for a deterrent effect of arrest certainty is negatively associated with the population of cities and counties: the negative correlation of both arrest clearance rate and arrest rate with crime rate is greater among Florida's less populous cities and counties. While the analysis points to a conclusion that the absence of a tipping effect in the California data sets and its presence in the Florida data are produced by the predominance of many smaller units in the laffer, an important question remains. Why should the tipping effect and deterrent effect of sanction certainty be associated with smallness? There are several possible explanations, none satisfactorily testable with the present data and all suggesting research directions. First, the apparent tipping effect of sanction certainty may be spurious, a creature of theoretically irrelevant characteristics of crime and sanctioning rates in smaller political units. After all, even though it must be used with caution, psychological láboratory research provides no evidence of a tipping effect of sanction certainty.

Arrest & Crime Rates 1 679 For example, Horai and Tedeschi found that compliance was a positive straightline function of threat credibility. Among smaller Florida cities the actual number of index crimes committed is low; the cities under 10,000 average less than one such crime per day, and several reported fewer than ten per year, with correspondingly small arrest totals. In cities with few crimes and arrests, a random difference of only a small number of arrests can mean the difference between a very high and a very low arrest clearance rate (or arrest rate). The variance of the arrest clearance rate is four times greater in the smaller cities as compared to the larger (320 versus 70) while the variance of the crime rate in the two city subsets is small (326 versus 284). Hence this situation produces scatter diagrams with smaller cities arrayed over the clearance rate axis from zero to almost 100 percent, but distributed relatively narrowly on the crime rate axis. Many smaller cities have both low clearance rates and low crime rates, but enough have high clearance rates and low crime rates that the relationship between the two variables is negative, particularly among the cities in which arrest clearances are very high. Of course, the pattern just described could be compatible with a deterrence and tipping effect explanation. Only if over time we found that small cities with low crime rates revealed high variation in their arrest or arrest clearance rates with no corresponding change in their crime rates could we conclude that the small town tipping effect arises not from a true deterrence effect of arrest but from chance variation in the sanction rate due in turn to the low number of crimes and arrests common to smaller jurisdictions. Our data, being on a single year, do not permit a test of this possibility. A second possible explanation is that the larger cities would manifest a tipping effect of sanction certainty if enough of them had arrest rates that were above the critical level. The tipping effect may be real, in other words, but perhaps it is only in smaller cities that the likelihood of arrest reaches a point where it tends to exert a deterrent influence on crime. As was reported above, only 5 larger Florida cities had clearance rates that exceeded the 30 percent critical level identified by Tittle and Rowe. Obviously, we cannot experimentally raise the sanction certainty in the more populous cities in order to determine whether a tipping effect appears. However some indirect evidence is relevant and provides no support for this explanation. There are 30 Florida cities with populations over 10,000 and 41 California cities that have straight arrest rates over 30 percent; they show no consistent evidence of the smaller cities' tipping effect, even though their arrest rates are above the hypothesized critical level. The third possible explanation is the most intriguing. Perhaps official legal sanctions exert an increased deterrent effect on crime in the smaller cities because in them social cohesion, face-to-face contacts among most citizens, and accompanying information links are greater. Their in-

680 1 Social Forces 1 vol. 57:2, december 1978 habitants may have a more accurate recognition of the probability of being arrested upon commission of a crime both when the probability is low (no deterrent effect) and high (deterrent tipping effect). Arrest may be perceived as more severe than in larger cities, where as a result even high probabilities of arrest may have little impact on individuals' decisions about whether to engage in criminal behavior. In order to judge the fit of this explanation, we need research on the association between city size and inhabitants' perceptions of the probability and severity of official sanctions. There are some data on public perceptions of sanctions (California State Assembly) but none on their links with size of the relevant political unit. Summary This paper first asked whether the tipping effect for sanction certainty identified by Tittle and Rowe is peculiar to their data. By means of an examination of sanction certainty indicators and the index crime rate in California cities and counties plus a further examination of Florida city and county data it was found that the tipping effect is an attribute of small populations: (1) a sanction certainty tipping effect is present in smaller cities 'but not larger ones and (2) the statistical evidence supporting a deterrent effect of arrest certainty is stronger in smaller cities and counties than in larger ones. The analysis thus specifies conditions under which both a tipping and a deterrence effect may and may not be expected. The statement of conditions raises a further question, the possible answers to which identify relevant routes for additional inquiry. Why should the tipping and deterrence effect of arrest certainty be more pronounced in smaller political units? Three possible explanations, each of which would require additional data, were considered. The tipping effect itself could be spurious, the arrest certainties in larger cities may not exceed the critical level necessary for a tipping effect to operate, or public perceptions of sanction certainty may be more accurate in smaller cities. Finally, however, while this analysis reports an apparent deterrent influence of official sanctions on crime, at least under certain conditions, two points must be recognized. The question of causal direction between sanction certainty and crime rates has yet to be satisfactorily answered (Brown and McDougal) despite the use of simultaneous equations techniques by some researchers in an attempt to handle the problem of reciprocity. While higher arrest rates could lower crime rates, so lower crime rates could raise arrest rates. Logan's study, although based on perhaps tenuous assumptions, suggests that the primary causal influence is from crime rates to arrest rates. Second, while we must not neglect the possible deterrent effect of official sanctions on crime, as a host of studies have rightly emphasized,

Arrest & Crime Rates 1 681 neither must we forget that decisions about whether to commit criminal acts may be shaped not only by perceived costs of crime, but also its perceived benefits and the perceived benefits and costs of compliant behavior. Hence, students of deterrence might fruitfully examine the detergent effect of sanctions relative to these other sets of variables (Brown and McDougal; Stover and Brown). Notes 1. The data used were collected as part of a larger study of crime rates in Califomia's cities with populations greater than 25,000 in 1970 and with local police departments (Brown and McDougal); 114 cities met these criteria. Crime and arrest data for cities were obtained from the California Bureau of Criminal Statistics (a). 2. The index crimes are homicide, rape, assault, robbery, larceny of fifty dollars and over, automobile theft, and burglary. 3. Whenever the term "clearance rate" is used in this paper, it refers to the "arrest clearance rate." Both California and Florida are supposed to adhere to FBI criteria in computing this statistic. The Uniform Crime Reporting Handbook (Federal Bureau of Investigation) states that an offense is cleared by arrest when at least one person is : (1) arrested; (2) charged with the commission of the offense; and (3) turned over to the court for prosecution. Since offense is the unit of analysis, the arrest of several persons may clear only one offense, and the arrest of an individual may clear several offenses. Given the FBI criteria, the variable, "probability of being charged with a crime" in the California city data would be equivalent to the arrest clearance rate, except that its unit of analysis is the person rather than an offense. 4. One larger city had an arrest clearance rate of 56 percent, two standard deviations removed from the next city, which had a rate of 40 percent. References Beattie, R. H. 1960. "Criminal Statistics in the United States." Journal of Criminal Law, Criminology, and Police Science 51(May-June):49-65. Brown, D. W., and S. L. McDougal. 1977. "Noncompliance With Law: A Utility Analysis of City Crime Rates." Social Science Quarterly. California Bureau of Criminal Statistics. a:1972. "Crimes and Arrests, 1971." Crime and Delinquency in California, 1971. Sacramento: Bureau of Criminal Statistics.. b:1975. California Comprehensive Data Systems Criminal Justice Profile: Counties. Sacramento: Bureau of Criminal Statistics. Federal Bureau of Investigation. 1974. Uniform Crime Reporting Handbook. Washington: Government Printing Office. Florida Uniform Crime Reports Burea. 1972. Uniform Crime Reports, State of Florida, 1971. Tallahassee: Uniform Crime Reports Bureau. Geerken, M. R., and W. R. Gove. 1975. "Deterrence: Some Theoretical Considerations." Law and Society Review 9(Spring):497-513. Horai, J., and J. T. Tedeschi. 1969. "Effects of Credibility and Magnitude of Punishment on Compliance to Threats." Journal of Personality and Social Psychology 12:164-69. Logan. C. H. 1975. "Arrest Rates and Deterrence." Social Science Quarterly 56 (December):376-89. Seidman, D., and M. Couzens. 1974. "Getting the Crime Rate Down: Politica! Pressure and Crime Reporting." Law and Society Review 8(Spring):457-593. Silberman, Matthew. 1976. "Toward a Theory of Criminal Deterrence." American Sociological Review 41(June):442-61. Skogan, W. 1974. "The Validity of Official Crime Statistics: An Empirical Investigation." Social Science Quarterltj 55(June):25-38.

682 1 Social Forces 1 vol. 57:2, december 1978 Stover, R. V., and D. W. Brown. 1975. "Understanding Compliance and Noncompliance with Law: The Contributions of Utility Theory." Social Science Quarterly 56(December):363-75. Tittle, C. R., and A. R. Rowe. 1974. "Certainty or Arrest and Crime Rates: A Further Test of the Deterrence Hypothesis. " Social Forces 52(June):455-62. Wolfgang, Marvin E. 1963. "Uniform Crime Reports: A Critical Appraisal." University of Pennsylvania Law Review 111(April): 708-38.