“Doing the math” on the distribution of healthcare expenditures: a Pareto-like distribution is inevitable

Yesterday I explored one of the major challenges to affordable, universal health insurance, namely the high cost of providing care to the sickest patients. The extreme distribution of healthcare costs means that “Targeting the highest spenders represents the greatest opportunity to have a significant impact on overall spending”, an opportunity for insurance carriers  to reduce costs by risk selection, as well as for public policy. Here is a deeper look into the math behind the distribution of healthcare expenditures, using 2012 US data as a model.

One can fit a Pareto (power law, 80/20) distribution with scale coefficient \alpha – that is, prob(expenditure)\sim 1/expenditure^{\alpha+1} – to the data in several ways. For a Pareto distribution with scale coefficient \alpha, the per-capita expenditure at a given percentile from the top scales as 1/\%ile^{1/\alpha}. The first two of these approaches yield a scale coefficient 1/\%ile^{0.893}, with expenditures scaling as :

  1. Use the 80/20 rule modified to fit the data: the top 25% ranked by healthcare expenditures account for 86.7% of costs; thus \alpha=1.115.
  2. Use the ratio of mean to median expenditure, 5.05:1; thus \alpha=1.119.
    However, a graphical analysis finds that the data does not follow such a Pareto distribution, shown as a black dashed line in the following figure (representing a Pareto distribution with \alpha=1.117 and median expenditure $854, the actual median expenditure).
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  3. Use data for the most expensive patients (10% through 30% percentiles from the top), for these patients, per-capita expenditure scales as 1/\%ile^{1.24}, (R^{2}=0.994), shown as a dashed red line in the figure above; thus \alpha=0.806.
  4. Use the fraction of total expenses paid by the most expensive patients. A comparison of the fraction of expenses paid by the most expensive 1%, 5% and 10% finds that this scales as x^{0.4228}, (R^{2}=0.987), shown as a dashed black line in the figure below. This scaling exponent is 1-1/\alpha; thus \alpha=1.733. (Scaling added to figure modified from Cohen, 2014)

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Thus, there really is no typical patient. For discussion and implications, see Feyman, who called the empirical distribution of healthcare costs “worse than Pareto”. The Pareto-like (hyper-Pareto?) empirical distribution of expenditures presents a severe challenge to risk pooling through insurance without limiting the highest expenditures through risk selection (illegal!).

Pareto distributions differ sharply from normal distributions, with important consequences for payment models. For a Pareto-like distribution with \alpha\leq2 at large expenditures, the variance is not defined, and sample variance approaches infinity with increasing sample size. Therefore, unlike the case of distributions with finite variance, variability in the mean of a sample of size N does not decrease with N. This violates a standard requirement for insurance; that risk pooling over a large sample reduces variability in the mean expenditure, and thus, standard insurance models cannot effectively price health insurance when the highest per capita expenditures follow Pareto distributions.

Moreover, a Pareto-like distribution may be a natural consequence of advances in healthcare: our growing ability to manage multiple simultaneous chronic conditions, with consequent exponential growth in costs, while extending life expectancy, so that the probability of dying is not only not reduced, but may actually increase. In a mathematically limiting case, with no bound on healthcare costs, these dynamics yield a Pareto distribution.

In fact, if one extrapolates the power law for a broad range of the sickest patients (the 10th through 30th percentiles of expenditures from the top), obtaining a Pareto distribution with \alpha\leq1, even the mean is not defined and the sample mean approaches infinity with increasing sample size. The actual distribution of healthcare cost for the very sickest patients clearly falls below the empirical Pareto distribution with \alpha=0.806, such a distribution predicts a cost at the 1st percentile of $178,194, well above the average for the top 1% of $97,956. Deviations from this distribution for the very sickest patients may reflect current limits on healthcare and thus healthcare expenses. These limits may be relaxed with advances in healthcare, causing further growth in costs.

A Pareto-like distribution of healthcare costs is here to stay, and must be reflected in how we share the burden of healthcare and provide care to our sickest patients.

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Sharing the burden of healthcare: providing care to our sickest patients

One of the major challenges to affordable, universal health insurance is the high cost of providing care to the sickest patients. According to Roy Vaughn, senior vice president at BlueCross BlueShield of Tennessee, “just 5 percent of the company’s marketplace customers had accounted for nearly 75 percent of its claims costs.” What is the cost of healthcare for the typical person in the United States?Distribution of per capita US health expenditures 2012

Data from 2012, the last year for which a full analysis is available, presents a complex and confusing picture. The graph above shows per capita expenditures by percentile starting with the highest per capita expenditure. 10% face expenditures of at least $10,250. The median per capita expenditure was $854. The mean average per capita expenditure was $4309 – five times the median – and “the top 1 percent ranked by their healthcare expenses accounted for 22.7 percent of total healthcare expenditures with an annual mean expenditure of $97,956″. In brief, there is no typical person: since the bottom 50% accounted for 2.7% of total expenditures, the average per capita expenditure of the top 1% was 420 times that of the bottom 50%. There really is no typical person in terms of healthcare expenditures.

Pareto/ power law distribution of healthcare costs

This extreme distribution of healthcare costs (approximately an “80/20”, Pareto/ power law distribution) poses a major challenge to providing universal healthcare through traditional insurance models based upon risk pooling. Prior to the Affordable Care Act (ACA), the US health insurance industry addressed these challenges with risk selection – adjusting premiums or denying insurance to patients with high predicted risks, such as those with pre-existing conditions, and imposing caps on annual and/or lifetime benefits, much like the way the auto insurance industry sets premiums and limits benefits to address extreme differences in projected driver risks. Come back tomorrow for another blog post with more technical details about the Pareto distribution and healthcare costs.

Risk selection is illegal but prevalent

The ACA makes both caps on benefits and risk selection based upon pre-existing conditions illegal. In particular, US insurance carriers are required to provide coverage to all, at rates independent of pre-existing conditions, a requirement which President-Elect Donald Trump would like to keep.

However, the extreme distribution of healthcare costs means that “Targeting the highest spenders represents the greatest opportunity to have a significant impact on overall spending”; an opportunity for insurance carriers as well as for public policy. Moreover, there are good predictors for high spending: age and end of life, chronic conditions, and high spending in a previous year. For example 44.8% of the top decile in 2008 healthcare expenditures “retained this top decile ranking with respect to their 2009 healthcare expenditures”; a fact cited in an extensive Forbes report. Swiss and Dutch experience found risk selection prevalent and persistent. However, with every adult paying the same premium – within a given fund for the same type of contract – but expected healthcare expenditure (HCE) varying widely, strong incentives for risk selection are created in the absence of an adequate risk adjustment scheme. Although risk selection is illegal, it is prevalent. Swiss conglomerates of insurance carriers have been reported to achieve risk selection by assigning applicants to “specific carriers based on their risk profiles.”

Removing the economic incentives for risk selection

There is one clear way to avoid built-in economic incentives for risk selection (incentives which seem to drive insurance company behavior); that is, a single payer system, universally or as excess coverage for significant, predictable expenses. The United States now has several parallel single payer systems, namely Medicare for the elderly, Medicaid for the very poor and CHIP for children; thus, in effect, a public/private partnership in healthcare. These pre-existing single-payer systems might serve as models for a more inclusive US single payer system. Alternatively, the United States might act as an insurer of last resort, providing umbrella insurance covering individual expenses above some relatively high limit, or for costly but treatable conditions using the End Stage Renal Disease (ESRD) Program, passed in 1972 as a model. This approach would also remove extreme costs from the health insurance risk pool, as both Medicare and the ESRD Program do now, by providing near-universal coverage for our sickest patients outside the private insurance system (elderly US citizens and those with severe chronic kidney disease, respectively).

Tomorrow I will return to the Pareto-like distribution of healthcare expenditures and its consequences for any competitive insurance program. But for now, a few conclusions. Medicare and the ESRD program provide models for a smooth transition from health insurance pre-ACA with its caps and limitations to a more universal system. Medicare can be expanded to a broader public alternative. Universal coverage for additional treatable but high-risk conditions can be modeled on the ESRD program. These steps should provide the basis for further evolution of the present public/private partnership into a more universal, more cost-effective system.

In my opinion, the extreme distribution of healthcare expenditures and the ability to perform risk selection, even though illegal, present a strong, essentially irrefutable argument for a single payer system; either overall, or for chronic conditions and expenditures predictable through risk selection. In the US, Medicare and the ESRD program provide illustrative, successful and useful models.

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