Testing Benford's Law on CDCR uses of force

Benford's Law theorizes that the numerals occupying the first digit in a number—including counts of CDCR uses of force—are distributed $P( d) = \log _{10} ( 1 + {1\over d})\; \rm { for\; all }\; {\it d} \in {\{ 1,\; ...,\; 9\} }$.Footnote ¹ A wide array of phenomena follow the distribution—stock market prices (Ley, Reference Ley1996), the winning bids in auctions (Giles, Reference Giles2007), and the population of US counties (Hill, Reference Hill1995, p. 355 ).Footnote ² Given widespread adherence to Benford's distribution, deviations thereof are often understood as evidence of irregular or fraudulent data (Varian, Reference Varian1972). The intuition behind forensics investigations is that when humans manipulate or falsify data, they are typically unable to do so in a way that adheres to the Benford distribution because of ignorance of the law's existence, psychological biases, strategic considerations, and inconsistent record keeping. Thus, violation from Benford's Law is understood as indicating human interference in the collection of data. Although legitimate data generating processes can naturally produce some deviations from Benford's Law,Footnote ³ deviations indicate—at a minimum—that data should be inspected closely before being used for hypothesis testing.

Since 2008, the 35 institutions overseen by the California Department of Corrections have been mandated to collect and publish use-of-force data. Pursuant to California Code of Regulations Title 15, §3268.1 (California Codes of Regulations, Title 15, 3268.1, 1999), “any employee who uses force or observes a staff use of force” is mandated to report details of the incident. Published monthly by the CDCR's Office of Research, each report includes counts of the use of force in each institution, disaggregated by type of force across ten categories. All uses of force can be applied against a single inmate or a group of inmates, and more than one type of force can be used per incident.Footnote ⁴

We test statistical hypotheses about the extent to which CDCR data conform to Benford's Law and probe subsamples of our results to shed light on practical hypotheses about the extent to which CDCR data are fraudulent.Footnote ⁵ We use four statistical tests to evaluate the statistical conformity of CDCR use-of-force data with Benford's Law. The first is the $\chi ^2$ goodness-of-fit test, the most commonly-used test in existing scholarship. The test statistic is calculated

(1)$$\chi^2 = \sum_{i = 1}^{k} = {( O_{i} - E_{i}) ^{2}\over E_{i}},\; $$

where $O_{i}$ is the observed frequency of digit i, and $E_{i}$ is the frequency expected by the Benford distribution. The statistic is compared to critical values in standard $\chi ^{2}$ tables at 8 degrees of freedom with a 95 percent critical value of 15.5 and a 99 percent critical value of 20.1.

A drawback of the $\chi ^{2}$ test is that its considerable power in large samples means that even slight deviations or just a handful of outliers can result in large test statistics and correspondingly small p-values (Cho and Gaines, Reference Cho and Gaines2007). Accordingly, our second test draws from Cho and Gaines (Reference Cho and Gaines2007) and calculates the Euclidean distance of the observed data from the Benford distribution:

(2)$$d = \sqrt{\sum_{i = 1}^{k} ( p_i - b_i) ^2}.$$

In Equation 2, $p_{i}$ is the proportion of observations with numeral i in the first digit, and $b_{i}$ is the proportion expected by Benford's Law. As Cho and Gaines (Reference Cho and Gaines2007) note, d is not derived from the hypothesis testing framework, so there are no critical values to compare it against. Instead, they recommend calculating $d^{\ast } = {d\over 1.036}$, where 1.036 is the maximum possible Euclidean distance between observed data and the Benford distribution—that is, the value that occurs when 9 is the only numeral in the first digit. Thus, $d^{\ast }$ ranges between 0 and 1, where larger values indicate more significant deviation. For the sake of comparison, Cho and Gaines (Reference Cho and Gaines2007, Table 1) show that 0.02 is a realistically small value of $d^{\ast }$ that would indicate conformity and values approaching 0.1 indicate more extensive deviation. Given this, we use $d^{\ast } = 0.06$ as a useful heuristic; values below it indicate data fairly consistent with Benford's Law; values above it indicate nonconformity.

Table 1.

Tests of conformity to Benford's Law by year

Note: Cells display test statistics in the $\chi ^2$ and $d^{\ast }_N$ columns, proportions in the $d^\ast$ column, and the first digit mean and bootstrapped confidence interval in the Digit Mean Test columns.

Our third test is also based on Euclidean distance. Following Morrow (Reference Morrow2014), we calculate

(3)$$d^\ast _N = \sqrt{N} \ast \sqrt{\sum_{i = 1}^k ( p_i - b_i) ^2}.$$

The advantage of $d^{\ast }_{N}$ over $d^{\ast }$ is that it has critical values that allow for hypothesis testing. Specifically, Morrow (Reference Morrow2014, pp. 4–5) shows that 1.33 is the critical value to reject the null hypothesis that the data are distributed per Benford's Law with 95 percent confidence, and 1.57 is sufficient to do so at the 99 percent confidence level.

Our fourth test is the simplest. If a set of numbers follow Benford's Law, then the numerals occupying the first digits will have a mean value of 3.441. Hicken and Mebane (Reference Hicken and Mebane2015) recommend calculating the mean from the observed data and using nonparametric bootstrapping to assess whether the 95 percent confidence interval contains that value.

Empirical results

In testing whether the CDCR data conform to Benford's Law, we limit our analyses to evaluating the aggregated total uses of force, because several of the force types have counts that are too small for reliable inferences about their digit distributions. The dashed line in Figure 1 plots the proportions with which each numeral appears in the first digit of the total data, across all 35 institutions for the period from 2008 to 2017. The solid line plots the proportions expected by Benford's Law. It is evident that the observed data exhibit some inconsistencies with the expected distribution.

Figure 1.

Observed and expected first digit distributions.

Aggregating CDCR data across all institutions over the entire decade conceals important nuance; further interrogation of the data is warranted to determine potential explanations for these anomalies. We rely on theoretical expectations about subsets of the data to further interrogate our results (Cleary and Thibodeau, Reference Cleary and Thibodeau2015, p. 212; Rauch et al., Reference Rauch, Göttsche, Brähler, Engel and Miller2015, p. 262). In addition to the aggregated data being inconsistent with Benford's Law, we might also expect the data to exhibit particular temporal and cross-sectional patterns if the CDCR consistently manipulated its use-of-force data.

Consider the effect of changes in California's legal or political context on the behavior of prison guards. During the time period covered by our data, two major policy interventions were implemented that reformed when corrections officers could use force and how uses-of-force were to be reported. In August 2010, the CDCR filed notice that it had adopted and implemented statewide required use-of-force reforms associated with a 1995 US District Court ruling that correctional staff at Pelican Bay State Prison routinely used unusual and excessive force against inmates and were negligent in reporting it (Madrid v. Gomez, 889 F. Supp. 1146 (N.D. Cal. 1995)). A second set of policy reforms went into effect in 2016, when the state further clarified staff reporting responsibilities, updated official reporting procedures, and revised the definitions of acceptable use of force options in the CDCR Department of Operations Manual.

If these reforms had the intended effects of clarifying the conditions under which force should be used and of improving how uses of force are reported, we might expect the data to conform more closely to Benford's Law at two discrete moments. The first is in 2010, when the CDCR implemented use-of-force reforms. If these reforms worked, data from 2010 onward might adhere more closely than the data from 2008 and 2009. The second moment is in 2016, when the second set of reforms came into effect. These additional reforms might further improve the data's conformity to the Benford Distribution compared even to the 2010 reforms. Figure 2 graphs the observed and expected distributions by year and shows significant temporal variation. Compared to 2008 and 2009, data from 2010 onward appear to conform more closely with the Benford distribution; results from 2016 and 2017 appear to conform still more closely to Benford's Law.

Figure 2.

Digit distributions by year

Table 1 shows tests of conformity to Benford's Law by year. The first row in the second column reports the $\chi ^{2}$ test statistic when the data are aggregated across all years and all institutions types. The test statistic is very large, allowing us to reject the null hypothesis that the observed data follow Benford's distribution. The Cho and Gaines (Reference Cho and Gaines2007) Euclidean Distance proportion ($d^{\ast }$) is more conservative than the $\chi ^{2}$ test, but it too indicates nonconformity. Like the $\chi ^{2}$, Morrow's ($d^{\ast }_{N}$) Euclidean Distance test yields a large test statistic that rejects the null at the 99 percent level of confidence. The last two columns show similar results. If the observed data are consistent with Benford's Law, the mean of the first digit would equal 3.441. In these data, the observed mean is 2.96 with 95 percent confidence intervals that do not include the Benford value.

Table 1 also allows assessment of how conformity changes over time. First, though the $\chi ^{2}$ test statistics are very large early in the period, they decline sharply toward the end of it. According to the $\chi ^{2}$ test, in 2014, 2016, and 2017, we cannot reject the null hypothesis that the data appear to conform to Benford's Law. The two Euclidean distance tests and the digit mean tests also identify 2016 and 2017 as years where the data deviate little from Benford's expectations. A second feature of Table 2 is the notable improvement in conformity that occurred in 2010. Compared to 2009, the $\chi ^{2}$ statistic in 2010 is about half the size, while the two distance tests also yield considerably smaller values. These results suggest that the 2010 reforms may have had some effect in improving how force is reported. Conformity improves even more in 2016 and 2017, which coincides with the second set of reforms implemented in 2016.

In addition to temporal variance, we investigate nonconformance with Benford's Law across institution type as indicative of purposeful misreporting of CDCR use-of-force data.Footnote ⁶ The CDCR divides its institutions into four types. New inmates arrive in the CDCR system at reception centers, where their placements needs are recorded and they are assigned to one of 34 institutions. Inmates can remain in reception centers for up to 120 days. While in reception, inmates are assigned classification scores that facilitate placement at an appropriate institution. High security institutions house the CDCR's violent male offenders; general population facilities house minimum to medium custody male inmates while providing them with opportunities for participation in vocational and academic programs. Female institutions house women offenders across classification scores. Across institution types, we might expect heterogeneity in guard incentives to misreport (or fail to report) violence.

Figure 3 graphs the digit proportions for each institution type and plots them against the Benford distribution. The most glaring result is the poor fit of high security institutions, where digits 1–4 are almost uniformly distributed. General population and female institutions also fail to conform to Benford's Law; reception institutions adhere more closely to expectations. Table 2 performs the four statistical tests described above to determine whether there is heterogeneity with regard to institution type. In every case—across every test and every institution type—our tests show nonconformity with the expectations of Benford's Law. In the second column, which shows the results of the $\chi ^{2}$ test, and the fourth column, which shows the results of Morrow's ($d^{\ast }_{N}$) Euclidean Distance test, we are universally able to reject the null hypothesis that the observed data follow Benford's distribution. The Cho and Gaines (Reference Cho and Gaines2007) Euclidean Distance proportion ($d^{\ast }$) and the test of the mean of the first digit similarly indicate reason to be concerned with potential irregularities in CDCR-reported data across institution type. Although the data do not conform to expectations for any institution type, we see the lowest conformance with Benford's Law in high security male prisons, presumably locations where guards face high incentives to engage in violence and misreport their behavior. We see the highest conformance with Benford's Law in reception centers—locations where inmates are held for 120 days before being assigned to more permanent locations.Footnote ⁷

Figure 3.

Digit distribution by institution type

Table 2.

Tests of conformity to Benford's first digit distribution

Note: Cells display test statistics in the $\chi ^2$ and $d^{\ast }_N$ columns, proportions in the $d^\ast$ column, and the first digit mean and bootstrapped confidence interval in the Digit Mean Test columns.