1.1 Benford's law

Discovered in 1881 by Simon Newcomb, forgotten, and subsequently rediscovered and popularised by Frank Benford (Reference Benford1938), Benford's law asserts that digits of naturally occurring numbers conform to distributions based on logarithms. For example, the occurrence of the digit 1 as the first digit is expected to be [(LOG10(1 + (1/1)) ≈] 0.3010Footnote ¹. Figure 1 depicts the expected distribution of Benford's law. If a distribution of numbers is naturally occurring, then it is expected to follow the Benford distribution. Numbers such as city populations, levels of brightness in nature, and the size of lakes and ponds were all found to follow the expected logarithmic distribution (Benford (Reference Benford1938)).

Figure 1.

Panel A: Expected distribution of first digits. Panel B: Expected distribution of second digits. Panel A of Figure 1 is a histogram of the expected distribution of first digits for a given magnitude of 10 according to Benford's law. The vertical axis is the expected proportion of each of the possible first digits. The proportions sum to 1. Panel B is a histogram of the expected distribution of second digits according to Benford's law.

Benford's law has been used to identify non-random human behaviour to investigate tax evasion (Nigrini (Reference Nigrini1996)), crypto-currency manipulation (McInish and Miller (Reference McInish and Millerworking paper), and Covid-19 test results (Koch and Okamura (Reference Koch and Okamuraworking paper) and Lee et al. (Reference Lee, Han and Jeong2020)). The key aspect of our research that is in common with these papers is detecting non-random behaviour in an attempt to protect the public from being harmed by inefficient health care deliveryFootnote ². Specifically, deviations from Benford's law have indicated non-random, human intervention. In our study we evaluate total charges for health care consumption. If treatment is perfectly aligned with the medical ailment, and not influenced by non-random human behaviour, total charges for health care encounters are expected to follow Benford's law. While this figure is not the amount that is ultimately exchanged, it does represent the rawest indication of illnessFootnote ³. Avoiding public (and private) harm through efficient health care is an important endeavour for policy makers, academics, and the public. Using the Law of Anomalous Numbers, Benford's law, we analyse our distribution of encounter charges to determine if they are naturally occurring or if they are affected by non-clinical intervention. To our knowledge, we are the first to apply Benford's law to determine the non-constant effects of non-clinical factors on the cost of health care.

1.2 Hypotheses

Pointing to Winnie Langley, ‘Britain's oldest smoker’, Smith (Reference Smith2011) states that ‘in general, Epidemiologists do a rather poor job of predicting who is and who is not going to develop a disease.’ Smith argues that we should embrace the randomness of those with diseases. Yulmetyev et al. (Reference Yulmetyev, Yulmetyeva and Gafarov2005) model the chaos and randomness in human health as well as the effectiveness of treatment.

Observable factors, including a patients age, gender, and geographical location may influence the extent of health care needs. Provider and payer factors including hospital size, urban setting, and payer type may also influence the availability of treatment. Non-observable factors such as genetic markers, immunocompetence, and illness potency certainly contribute to an individual's health experience. The set of unobserved factors means that general population predictions can be dubious for an individual. In embracing the randomness of individual predictions as argued by Smith (Reference Smith2011), our first hypothesis becomes:

If the cost of treatment reflects the severity of an individual's illness, then health care treatment would reflect the random nature of disease. However, certain market factors incentivise non-random human behaviour to drive treatment costs. Bell (Reference Bell1984) discusses New York State medical malpractice reform laws that cap payments to injured patients. He argues that consumers should only care about the legislation if the legislation results in an increase in medically unsafe behaviour, but not based on reduced charges to patients. Kessler and McClellan (Reference Kessler and McClellan1996) identify the behaviour of defensive medicine, wherein physicians order or perform costly treatments with minimal beneficial effect to avoid the financial and non-financial consequences of a malpractice lawsuit. Interestingly, these additional procedures expose providers to more malpractice risk and increases the provider's implicit marginal cost per procedure, possibly reducing utilisation of ‘extra’ procedures (Chandra and Skinner (Reference Chandra and Skinner2012), Currie and MacLeod (Reference Currie and Bentley MacLeod2008), and Baicker et al. (Reference Baicker, Fisher and Chandra2007)).

McFarland et al. (working paper) tests the covariant relation between health care claim frequency and severity. They find that over the entire distribution of health care claims, the relation between frequency and severity is positive, but heterogeneous. As a patient access health care more frequently, the cost of each health care encounter increases. Patients with the most severe health ailments have more exposure defensive medicine, SID, and TDD, through both opportunity and costFootnote ⁴. These observations lead us to our second hypothesis.

Newhouse (Reference Newhouse1992) claims that a consequence of ‘too much’ health insurance is ‘too much’ technological change. He finds that having health insurance leads to patients receiving extra treatment with advanced medicine that they otherwise would not receive. This is because health insurance drives the marginal price of health care to near zero. This is especially pronounced for socially funded health care coverage that is costless, or nearly costless to the insured. This circumstance highlights the presence of moral hazard. Privately insured patients are typically responsible for co-pays and deductibles, possibly mitigating the wasteful nature of ‘too much’ insurance.

We further test insurance coverage by type of payer to determine the prevalence of non-clinical factors among privately insured patients compared to government insured patients. Two factors incentivise monitoring by private insurers but not government insurers. First, individuals and employers must cover the cost of their private health insurance and second, private insurers seek to earn a profit. Further, Pauly (Reference Pauly, Culyer and Newhouse2000) argues that spending effects increase when insurance shields the consumer from financial responsibility, as is the case with many government-funded insurance. These observations lead us to our third hypothesis.

Uninsured individuals face the unique non-clinical factor of health care consumption of complete risk acceptance. Being fully and personally financially responsible for health care consumption alters uninsured individual's consumption choices. Hadley et al. (Reference Hadley, Steinburg and Feder1991) find that uninsured individuals forego expensive treatment far more often than insured individuals. The increasing costs of health care coupled with the lack of risk sharing through insurance often prices uninsured individuals out of the market for health care services. Our fourth hypothesis is:

2.1 Data

Our primary data source is the electronic health records Health Facts EMR dataset, made available through the Center for Biomedical Informatics at the University of Tennessee Health Science Center, UTHSC. The Health Facts dataset includes over 49 million distinct patients with more than 290 million patient encounters from 2000 through 2015. Data in Health Facts are extracted directly from the EMR from hospitals in which Cerner has a data use agreement. Cerner Corporation has established Health Insurance Portability and Accountability Act-compliant operating policies to establish de-identification for Health Facts.

Each encounter begins upon admission and ends at discharge. The total charges for an encounter, our main variable of interest, are the summation of all provider-related charges for that encounter. Charges for outpatient prescriptions, when written by a primary care provider, but not filled during a clinical visit, are excluded from total charges. However, inpatient prescriptions administered by the provider during the encounter are included in total chargesFootnote ⁵.

It is preferable that the data set covers multiple magnitudes (1s, 10s, 100s), covers a full range of magnitudeFootnote ⁶, and that the data are not averaged. It is also critical that the numbers are not rounded or have minimumsFootnote ⁷ or maximums. Our data complies with these criteria.Footnote ⁸ Distributions that are expected to follow Benford's law include transactions-level data (e.g. sales, trade size), numbers that result from a combination of numbers – quantity × price. Data for which the mean is greater than the median are also more likely to follow Benford's law. McFarland et al. (working paper) find that the distribution of individual health care costs is positively skewed (Figure 2).

Figure 2.

Panel A: Price bucket 1, Panel B: Price bucket 2, Panel C: Price bucket 3, Panel D: Price bucket 4. Figure 2 is a set of histograms showing the distribution of total charges for each price bucket.

Using patient billing data, we evaluate the scope of total encounter expenditures by reviewing admission sources and discharge dispositions to identify patients who are expected to have additional encounters. This approach facilitates our developing parameters for estimating and capturing health care expenses. We apply our filters to ensure that we include only those patients for whom we have most claim dataFootnote ⁹. This leaves us with over 59 million encounters. For much of our study, we further limit our observations to encounter with a minimum charge of $100 and not exceeding $1,000,000. This limitation allows us to group encounters by severity while maintaining full magnitudes of 10 in the first digit. Given our filters, our sample covers the years 2000 through 2014 and includes over 47 million encounters. Table 1 reports descriptive statistics regarding patients, encounters, encounter charges, and length of stay.

Table 1.

Descriptive Statistics

We examine the cost of health care encounters for patients across the U.S. during the years 2000 through 2014 from the Health Facts EMR data. We report the distribution of encounter charges and encounter length of stay (LOS), measured in days, for all patients. We present our results for the entire sample (All) and classified by the charged amount (Charges), the patient gender (Female), patient age (Age), and payer type. For each category we report minimum, mean, and maximum charges and LOS.

We report our descriptive statistics both as a single sample and segmented by charged amount. Encounters with charges between $100 and $999.99 (charges = 1), $1000 and $9999.99 (charges = 2), $10,000 and $99,999.99 (charges = 3), and $100,000 and $999,999.99 (charges = 4) are grouped together. We observe a negative relation between encounter severity (measured by encounter charges) and the number of encounters. Health care charges, like most insurable risks, are skewed distributions wherein relatively few people experience extremely high health care costs. Notwithstanding, we find a significant number of encounters at all severity levels, including nearly 200,000 encounters with charges in excess of $100,000. Our sample includes more female encounters (36 million) than male encounters (23 million). Most of our encounters are patients that are aged 18–65 (34 million) while the average encounter charge appears to increase with the age of the patient. With the notable exception of research-based encounters, our sample includes an even mix of payer types with over 19 million government payer encounters, 12 million commercial payers, and 3 million self-payers.

2.2 Methodology

Following Nigrini and Mittermaier (Reference Nigrini and Mittermaier1997) we use three tests: first-digits, second-digits, and first-two digits. We also follow Drake and Nigrini (Reference Drake and Nigrini2000) by calculating the mean of absolute deviations (MAD) to use as a way to assess conformity to the expected distribution. A naïve person choosing numbers at random would most likely guess a distribution would be uniform with 11% occurring for each digit 1–9. However, Benford's law states that for a group of natural occurring numbers the distribution of first digits occurs based on logarithmic properties as follows: 1s 0.3010, 2s 0.1761, 3s 0.1249, 4s 0.0969, 5s 0.0792, 6s 0.0669, 7s 0.0580, 8s 0.0512, and 9s 0.0458. In addition to the standard test of counts of first digits, we calculate the mean average deviations (MAD), which is called a reasonableness test by Drake and Nigrini (Reference Drake and Nigrini2000). To compute the MAD, we average the absolute value of deviations and divide by 9. For first digits, a MAD of 0.000 ± 0.004 indicates close conformity, 0.004 ± 0.008 acceptable conformity, 0.008 ± 0.012 marginally acceptable conformity, and a MAD greater than 0.012 nonconformity.

Smith (Reference Smith2011) acknowledges that epidemiologists do a poor job of determining who is going to get sick with what ailment and when. This is because of the readily acceptable fact that illness affects individuals randomly. Additionally, the causes and complications of diseases may be determined by variables not readily observable, either ax-ante or ex-post. We begin our analysis by identifying patient, facility, payer, or regional factors that contribute to the cost of a health care encounter. Master diagnostic codes (MDCs) are nationally recognised standard groupings that correspond to single organ system ailments or medical specialties e.g. respiratory. We expect that overall encounter cost levels vary by MDC group. However, each MDC group encompasses a wide range of encounter types, from low-cost preventative care to the severe emergent encounters. We begin by regressing encounter charges on MDC controls and estimate our first regression model as

(1)$$EC = \beta _1 + X_{{\rm MDC}} + \varepsilon $$

where EC is an abbreviation of encounter charges and X _MDC is a vector of MDC dummy variables. In model 2 we include patient and calendar year descriptive variables. Age is the patient's age in years at the time of admission. Year is the calendar year at admission to control for rising health care costs over time. We also include vectors of dummy variables for gender (G), race (R), marital status (M), and US census location (L) of the patient.

(2)$$EC = \beta _1 + \beta _2Age + \beta _3Year + X_{{\rm MDC}} + G_{{\rm Gender}} + R_{{\rm Race}} + M_{{\rm Marital}} + L_{{\rm location}} + \varepsilon $$

In model 3 we retain our previous control variables and include the treating facility variables urban, a dummy variable equal to 1 for all urban providers, size (the size of the facility based on licensed bed count), and teaching, a dummy variable equal to 1 for all teaching hospitals.

(3)$$\eqalign{EC = & \;\beta _1 + \beta _2Age + \beta _3Year + \beta _4Urban + \beta _5Size + \beta _6Teaching + X_{{\rm MDC}} \cr & + G_{{\rm Gender}} + R_{{\rm Race}} + M_{{\rm Marital}} + L_{{\rm Location}} + \varepsilon } $$

In model 4 we expand our control variables to include time-of-week and year. Weekday is a dummy variable equal to 1 for all admission that occur Monday through Friday. Holiday is a dummy variable equal to 1 for admissions that occur on a nationally recognised holiday. We also include a vector of monthly dummy variables (Mt) to account for seasonal effects.

(4)$$\eqalign{EC = \, & \;\beta _1 + \beta _2Age + \beta _3Year + \beta _4Urban + \beta _5Size + \beta _6Teaching + \beta _7Weekday \cr & + \beta _8Holiday + X_{{\rm MDC}} + G_{{\rm Gender}} + R_{{\rm Race}} + M_{{\rm Marital}} + L_{{\rm Location}} \cr & + Mt_{{\rm Month}} + \varepsilon } $$

Finally, our last model includes the aforementioned control variables and price bucket.

(5)$$\eqalign{EC & = \beta _1 + \beta _2Age + \beta _3Year + \beta _4Urban + \beta _5Size + \beta _6Teaching + \beta _7Weekday \cr & + \beta _8Holiday + \beta _9Price\;Bucket + X_{{\rm MDC}} + G_{{\rm Gender}} + R_{{\rm Race}} \cr & + M_{{\rm Marital}} + L_{{\rm Location}} + Mt_{{\rm month}} + \varepsilon } $$

2.3 Empirical results

2.3.1 Does encounter severity reflect the random nature of illness severity?

In Table 2 we subdivide our sample into four price buckets, the first for all encounters that incur costs of at least $100Footnote ¹⁰ and not more than $999.99. The second price bucket includes encounters with charges ranging from $1000 to $9999.99. Price bucket 3 includes all encounters with charges between $10,000 and $99,999.99, and the final price bucket includes charges of at least $100,000 but not exceeding $999,999.99. We find that while many of our control variables are statistically significant, and in some cases economically significant as well, the best any of these models can do is return an r-squared of less than 0.03. In all cases except for model 3 the largest coefficient in terms of magnitude is the intercept term. These results support our first hypothesis that the severity of illness, and the associated costs are random at the individual level.

Table 2.

OLS estimated encounter expense.

We estimate OLS regressions to determine factors that predict individual encounter charges from the Health Facts EMR data. Age is the patient's age in years, Year is the calendar year, Urban is a dummy variable equal to 1 for patients that access an urban provider. Size is the bed size of the providing hospital. Teaching is a dummy variable equal to 1 if the hospital is a teaching hospital. Weekend and Holiday are dummy variables indicating the day of admission. We include fixed effects for MDC code, gender, race, marital status, US census location, and month. In model one we include only controls for MDC. In model two we also include age and year controls. Model three includes provider variables and in model four we also include weekday and holiday control variables. Final, in model five we include a control for price bucket. *, **, and *** indicate significance at the 10%, 5%, and 1% significance levels, respectively.

2.3.2 Non-clinical factors

In a frictionless environment the cost of a health care encounter should reflect the severity of the health care ailment. However, it is well understood that the real world is not frictionless. Within the setting of health care, there are some readily identifiable frictions, non-clinical factors that directly affect the cost of health care. Defensive medicine, SID, and TID are some of frictions. The magnitude and presence of these factors are difficult, if not impossible to identify at the encounter level. Additional unnecessary procedures, examinations, or diagnostic tests are justified by individual provider judgements or out of an abundance of diagnostic scepticism as opposed to nefarious motives. Many studies identify the presence of non-clinical cost factors by observing changes in expenditure before and after legislative developments (Kessler and McClellan (Reference Kessler and McClellan1996), Sloan and Shadle (Reference Sloan and Shadle2009), adoption of technologically advanced treatments (Chandra and Skinner (Reference Chandra and Skinner2012), and R&D spending. We execute an alternative research design to identify the presence of non-clinical factors in health care charges by applying Benford's law to encounter charges. This methodology does not distinguish between different non-clinical factors but does provide insight into the magnitude of the non-clinical factors at varying levels of encounter severity. This identification strategy is important to policy makers, medical practitioners and providers because it allows them to focus cost efficiency efforts on the relatively few encounters with the most inefficiencies. Benford's law applies nicely to number distributions that are naturally occurring, cover multiple magnitudes, and are positively skewed, as is the case with our data. Encounter level health care costs should meet these three criteria well if non-clinical charges are not present. We can therefore attribute most non-conformity to non-clinical factors.

We segment our sample of encounters into subsamples based on the charges for each encounter. We identify four subsamples of encounters consistent with the price bucket variable defined in Table 2. Segmenting the encounters this way provides at least two important benefits to our study. The first is that each price bucket contains a range that begins with a lowest possible first-digit number equal to 1 and ends with a highest possible first-digit number equal to 9. Furthermore, the distribution spans only a single magnitude of 10 for each distribution, meaning each total-charges bucket provides the same a priori probability to all digits 1 through 9 of being the first digit. The second important benefit of partitioning our sample is that it allows us to evaluate the conformity of Benford's law across different encounter severitiesFootnote ¹¹. We expect that the non-clinical factors are more prevalent for more severe encounters. For example, defensive medicine occurs as a deterrent to malpractice lawsuits. However, the risk of a malpractice lawsuit is less when the severity of an illness is small. Therefore, physicians will be more likely to practice defensive medicine, and in greater quantities, as the encounter severity increasesFootnote ¹². Because of this we expect that the distribution of total charges deviates in greater degree as total-charges increase. The same can be said for SID and TDD factors.

We employ the count test to examine first-digits (leading-digits). Table 3 presents our results. We find that at the MAD for the first bucket of charges (0.010) shows a marginally acceptable conformity to Benford's law. However, for the second (0.023), third (0.049), and fourth (0.092) buckets the MAD is greater than 0.012 indicating nonconformity. As expected, the MAD increases with the level of total-charges. Interestingly, we find consistent deviations from the predicted probabilities at the price bucket boundaries. 1-as-the-leading-digit is consistently over-represented while 9 is consistently under-represented. An additional possible explanation for this finding is hospital pricing strategies (Krishnan (Reference Krishnan2001), Sutherland (Reference Sutherland2015)). If providers are strategically pricing their services, then either prices are being strategically raised or lowered. If prices are being raised, our boundary observations show that services near the high end of a price range are being raised sufficiently to move those services into the next price bucket. This would cause an increase in 1-as-the-leading-digit occurrences and a decrease in 9-as-the-leading-digit occurrences, consistent with our observation. Alternatively, if prices are strategically lowered, then we would find the opposite result. In this case, strategic pricing strategies are muting the effect of other non-clinical factors affecting health care costs, and actual inefficiencies are more severe than we can identify. Hospital pricing strategies are beyond the scope of our research but present a compelling case for further research. This finding supports our second hypothesis that encounter charges deviate from Benford's law as the total-charges increase.

Table 3.

Distribution of first-digits

Using the Health Facts EMR data, we partition the observed encounters into four buckets based on total charges. The first bucket includes all observations with total charges of at least $100 and less than $1000. The second bucket includes total charges of at least $1000 and less than$10,000. The third bucket includes all charges of at least $10,000 and less than $100,000. The fourth bucket includes all charges of at least $100,000 and less than $1,000,000. We report the frequency of each digit (1 through 9) occurring as the first digit as a frequency as well as a per cent of the total in the bucket. Mean absolute deviations (MAD) are presented with 1 for indicating close conformity, 2 acceptable conformity, 3 marginally acceptable conformity, and 4 non-conformity.

According to Benford's law the second-digits expected probabilities for each number are as follows: 0s 0.1197, 1s 0.1139, 2s, 0.1088, 3s 0.1043, 4s 0.1003, 5s 0.0967, 6s 0.0934, 7s 0.0904, 8s 0.0876, 9s 0.085. This is much closer to uniform, but a dataset of uniform second-digits is statistically different from the expected distribution. We repeat our previous procedure on the second-digits, but the MAD thresholds are slightly different. For second digits, a MAD of 0.000 ± 0.008 indicates close conformity, 0.008 ± 0.012 acceptable conformity, 0.016 ± 0.016 marginally acceptable conformity, and a MAD greater than 0.016 nonconformity (Table 4).

Table 4.

Distribution of second-digits

Using the Health Facts EMR data, we partition the observed encounters into four buckets based on total charges. The first bucket includes all observations with total charges of at least $100 and less than $1000. The second bucket includes total charges of at least $1000 and less than $10,000. The third bucket includes all charges of at least $10,000 and less than $100,000. The fourth bucket includes all charges of at least $100,000 and less than $1,000,000. We report the frequency of each digit (0 through 9) occurring as the second digit as a frequency as well as a per cent of the total in the bucket. Mean absolute deviations (MAD) are presented with 1 for indicating close conformity, 2 acceptable conformity, 3 marginally acceptable conformity, and 4 non-conformity.

Examining the distribution of second digits we find that total charges in our first two buckets, $100–$999.99 and $1000–$9999.99 closely conform to the Benford distribution. For our third bucket, encounters with total charges between $10,000 and $99,999.99 we see acceptable conformity. For our bucket with our most expensive encounters, like our test of first digits we find non-conformity indicating some type of thoughtful intervention. The consistent decline in conformity to the expected logarithmic distribution supports our second hypothesis and indicates that more non-clinical factors such as defensive medicine, SID, and TDD are present as the total charges of a patient's encounter increase.

2.3.3 Insurance and payer type

Health care in the US is predominantly financed through insurance. The risk sharing characteristics of insurance creates some separation between the consumer of health care and the payer. When the government, at any level, is the payer, this separation is amplified. A relatively small fraction of patients, however, are self-insured or otherwise pay for health care services directly. These important differences in the degree of separation between consumer and payer may lead to different applications or magnitudes of non-clinical cost factors. For example, Newhouse (Reference Newhouse1992) argues that ‘too much’ insurance may lead to ‘too much’ technological change.

Government insured patients are the consumers of health care that are furthest removed from the costs of health care. This is due to government-funded health insurance is generally made available at little or no cost to those who cannot otherwise afford private health insurance. Being so far removed from the cost of health care means that however minimal the benefits of additional treatment, the patient will likely accept treatment since they are not responsible for the costs. The patient's financial incentives for partaking in defensive medicine, SID, or TDD are aligned with the provider's and suppliers' non-clinical incentives to provide such care. On the other hand, those who are privately insured are financially responsible for co-pays, deductibles, or a portion of treatment costs. These financial responsibilities detach the patient's clinical incentives from provider's non-clinical incentives. Self-insured patients represent the extreme disconnect between clinical and non-clinical incentives to the point that self-insured patients may forego clinically prescribed treatments if the costs are prohibitively expensive or if the costs outweigh the perceived benefits (Hadley et al. (Reference Hadley, Steinburg and Feder1991)).

Patient and provider incentives are not the only means by which payer types may result in different results to our study. Private insurers operate to earn a profit, whereas government insurance programmes do not. The profit motive creates the incentive for private insurers to monitor and prevent non-clinical cost factors. We test these observations by segmenting our sample into three subsamples based on payer type. The first sub-sample comprises those patients with government-funded health insurance. Second, private health insurance and lastly self-insured patients. We remove from these subsamples of payers' patient encounters whose payment source is unidentified or research based. We calculate the distribution of first- and second-digit numbers and calculate the corresponding MAD values. Table 5 reports our results. Both government (0.012) and privately insured (0.009) patients show marginally acceptable conformity for the lowest severity bucket and nonconformity for all other severity buckets. While the MAD value for the lowest severity privately insured patients (0.009) is less than the MAD value for the lowest severity government insured patients (0.0112), in buckets 2 and 3 the rank order reverses. This finding is contrary to our third hypothesis that charges become less random when health care is government funded. The inability to effectively monitor providers, either due to cost-based reimbursement or loss-estimation difficulties (Pauly (Reference Pauly, Culyer and Newhouse2000)) are likely contributors to this finding.

Table 5.

Distribution of first- and second-digits by payer

Using the Health Facts EMR data, we segment the observed encounters by payer type and partition each sub-sample into four buckets based on total charges. The first bucket includes all observations with total charges of at least $100 and less than $1000. The second bucket includes total charges of at least $1000 and less than $10,000. The third bucket includes all charges of at least $10,000 and less than $100,000. The fourth bucket includes all charges of at least $100,000 and less than $1,000,000. We report the mean absolute deviations (MAD) and the corresponding level of conformity. 1 indicating close conformity, 2 acceptable conformity, 3 marginally acceptable conformity, and 4 non-conformity.

When examining the count of first digits we also observe that in all cases self-insured patients do not conform to Benford's law. MAD values for self-insured patients are 0.013, 0.034, 0.055, and 0.104 for buckets 1 through 4, respectively. This suggests that non-clinical factors, both cost increasing (defensive medicine, SID, and TDD) and cost reducing (refusal of treatment), affect self-insured patients more than insured patients. This finding is in support of our fourth hypothesis that charges for self-insured patients deviate from Benford's law across the entire distribution of claim severityFootnote ¹³.