**Editors’ Note:** *The mythology surrounding Florence Nightingale has often ignored or glossed over her role as an innovative applied statistician. Nightingale was doing sophisticated polar graph charts and thought experiments before think tanks and blogs existed. As we wrap up this year’s National Nurses Week and celebrate Nightingale’s 195th birthday, we thought it would be a good idea to look at how Nightingale would approach our modern health care issues. What follows is a fascinating scenario from Nurse and Mathematician Thomas Cox which positions Nightingale in the 21 ^{st} century to make sense of our current healthcare reforms in the US.*

**By Thomas Cox, PhD**

How would Florence Nightingale feel about Medicaid for the poor, sick and disabled? Would she embrace rugged individualism or equal care for rich and poor? Would she support a multi-tiered health care system that delivers care according to ability to pay instead? I believe given her advocacy of social reform based on detailed statistical analysis, she would see that our health care (finance) reform debates serve consumers poorly and she would criticize insurance executives, physicians and health facility executives who obtain bloated compensation while health care costs rise, quality falls and millions go without health insurance. She would certainly be dumbfounded that politicians think we could reduce premiums, and increase benefits, by augmenting our more than 1,200 health insurers.

Nightingale would likely proceed, as I have, through quantitative analysis that would review insurance, actuarial and risk theoretic mathematics, financial analysis, accounting, and economics. She would embrace health insurance but would be bewildered by our “third party payer mechanisms,” particularly capitation (Arrow et al, 2009), which is the fixed payment paid per patient to health care providers such as nurse practitioners or physicians for the services they provide.

How would Nightingale compare “fee for service” vs capitation-like payment mechanisms? The core principle of capitation is that we can transfer insurance risks from large, inefficient entities (Insurers, managed care organizations, union benefit plans…) to small entities, insurance risk-assuming health care providers who will manage these financial risks more efficiently. So Nightingale would ask: Are small insurers more efficient risk managers than large insurers? Nightingale would also recognize that the answer hinges on the relationship between insurer portfolio size and financial risk as specified by the Central Limit Theorem (CLT) (De Moivre, 1733).

**WWFND: NIGHTINGALE’S RESEARCH ON HOW INSURANCE WORKS**

Let’s say Florence Nightingale does a “thought experiment”: Three random insurers select policyholders, at random, from sample population P. They know how many claims they will get and their average loss ratio (Claims Costs/Premiums). (Note: For more detailed analysis, you can download this paper , this presentation and these spreadsheets: Loss and Profit as what follows is only a sketch of what would be Nightingale’s full analysis.)

In this scenario, Nightingale knows that the CLT suggests that all three insurers’ future loss ratios will be normally distributed “estimates” of population P’s loss ratio (PLR). She knows that large insurers’ PLR estimates cluster closer to the PLR than smaller insurers’ PLR. Nightingale uses the CLT to calculate the variability of any insurer’s PLR estimate, given its portfolio size, and the portfolio size and standard error of a second insurer’s PLR estimate. All insurers’ PLR estimates would vary by the square root of their relative portfolio sizes (Hogg, 1978).

At this point Nightingale thinks: Suppose Paradigm Insurer’s (PI) expected loss ratio = PLR = 0.7500 (Claims Costs/Premiums = 0.7500) and PI’s PLR estimate varies between 0.6500 and 0.8500 about 95 years in 100, when PI issues 1,000,000 policies. PI’s PLR estimate (PILR) will be normally distributed as N(0.7500, 0.0500). Nightingale knows that insurers’ non-loss costs tend to run about 15% of premiums, and she assumes that insurers seek 5% profit margins.

The remaining 5% of revenues are a “risk load” that increases each insurer’s probability of meeting their profit goals. In bad years, insurers cover PLR estimates as high as 0.8000 without sacrificing profitability.

Nightingale consults a Normal distribution table (Hogg, 1978; Cox, 2011), finding PI’s operating result probabilities:

Nightingale knows all insurer’s PLR estimates are Normal (0.7500, S * 0.0500), where S = square root(1,000,000/M) is a second insurer’s adjustment to PI’s standard error, for portfolio size M. If Little Insurer (LI) issues 50,000 policies, its PLR estimate (LILR) varies more (standard error = 0.2236 = 0.0500 * sqrt(1,000,000/50,000) than Pis. LI’s operating result probabilities are:

Nightingale realizes that more competition means smaller portfolios, lower probabilities of modest, sustainable profits and higher probabilities of operating losses. Nightingale wonders what happens if we have a national health insurer (NHI) issuing M = 318,000,000 policies? NHI’s PLR estimate, NHILR ~ N(0.7500, 0.0028), where 0.0028 = 0.0500 * sqrt(1,000,000/318,000,000). Nightingale’s calculations show that NHI’s operating result probabilities are far better than PI’s and LI’s:

NHI’s large portfolio accurately estimates the PLR and it earns profits greater than 8% every year and never incurs operating losses. **She concludes that NHI is the most mathematically efficient insurer possible for population P.**

Nightingale rejects all “capitation-like”, insurance risk transferring, health care finance mechanisms because they could never steer our health care (finance) systems toward greater efficiency, only further fracturing and wasteful spending. When clinically efficient providers accept capitation-like payment mechanisms, they become inefficient insurers, not more efficient clinicians.

Of course, one could argue that this thought experiment was a waste of time because if Nightingale was transported to the 21^{st} century she would be unable to perform the calculations presented. However, there is a twist and a fruitful avenue for Nightingale historians. She could have done this analysis in the mid-19th century because probabilists and astronomers had worked out the normal distribution probabilities decades before her birth. As an astronomer and mathematician, her statistical mentor, Quetelet (Mawhin, 2010, Quetelet, 1842) was well-versed in the “Theory of Errors”. That, and Quetelet’s relationship to Gauss, is a far more interesting tale.

**REFERENCES AND FURTHER READING**

Arrow K, et al. (2009). “Toward a 21st-century health care system: Recommendations for health care reform”. Ann Intern Med. 150(7), 493-5.

Beard, R. E., Pentikainen, T. and Pesonen, E. (1984). “Risk theory: The stochastic basis of insurance”. New York: Chapman and Hall.

Borch, K., (1974). “The Mathematical Theory of Insurance”. Lexington, MA: D. C. Heath and Company.

Bourdon, T.W, Passwater, K. and Priven, M., (1997), “An Introduction to

Capitation and Health Care Provider Excess Insurance”, Proceedings of the

Casualty Actuarial Society, Arlington, VA: Casualty Actuarial Society.

Bourdon, T.W., (1998), “Reinsurance Implications of Managed Care: Capitation

and Healthcare Provider Excess Insurance”, Proceedings of the Casualty

Actuarial Society, Arlington, VA: Casualty Actuarial Society.

Bowers, N, Gerber, H, Hickman, J, Jones, D and Nesbitt, C., (1997). “Actuarial Mathematics”. 2nd ed., Schaumburg, IL: Society of Actuaries.

Cox, T. (2011). “Standard Errors: Statistical Consequences of Health Care Provider Insurance Risk Assumption”. In JSM Proceedings, Section on Government Statistics. Alexandria, VA: American Statistical Association. 5180-5194.

Cox, T. (2011). “The Impact of Size on Success of Health Insurance Companies”. Nurse Leader, 9(5):38-41.

Cox, T. (2011). “Exposing the true risks of capitation financed healthcare”. Journal of Healthcare Risk Management, 30: 34–41.

Cox, T. (2010). “Legal and Ethical Implications of Health Care Provider Insurance Risk Assumption”. JONA’S Healthcare Law, Ethics, and Regulation, 12(4): 106-116.

Cox, T. (2006). “Professional caregiver insurance risk: A brief primer for nurse executives and decisionmakers”. Nurse Leader, 4(2): 48-51.

Doctoral Dissertation in Nursing: Cox, T. “Risk induced professional caregiver despair: A unitary appreciative inquiry.” Defended April 7, 2004.

Cox, T. (2001). “Risk theory, reinsurance, and capitation”. Issues in Interdisciplinary Care, 3(3): 213-218.

Hogg, R.V. and Craig, A.T., (1978). “Introduction to Mathematical Statistics”. 4th ed., New York, NY: Macmillan.

Mawhin, J (2010) “Some Direct and Remote Relations of Gauss with Belgian Mathematicians”. Jahresbericht Der Deutschen Mathematiker-vereinigung}, 112(2): 99–116.

McDonald, L. (Ed) (2004). “Florence Nightingale on Public Health Care: Collected Works of Florence Nightingale”. Volume 6. Wilfrid Laurier Univ. Press.

McDonald, L. (Ed) (2003). Florence Nightingale on Society and Politics, Philosophy, Science, Education and Literature. Volume 5. Wilfrid Laurier Univ. Press.

Quetelet, A.J (1842). “A treatise on man and the development of his faculties”. First translation into English. Edinburgh: W. and R. Chambers.

**Thomas Cox**,** PhD** like Nightingale, was fascinated first by physics, mathematics and statistics; earning a bachelor’s degree in pure mathematics, and a master’s degree in applied mathematics and statistics, 3 decades before completing his BSN, MSN and PhD in nursing. He also spent a decade doing insurance and reinsurance rate-making, reserving and expense reporting, which led to his analysis of insurance risk transferring health care finance mechanisms and his nursing-centered work on ‘Professional Caregiver Insurance Risk” and this post.

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I actually wondered questions like this and ‘interviewed’ Ms. Nightingale ten years ago. I have just reposted that interview on my Facebook page Cathy Crowe.

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For those who may not find it… https://www.facebook.com/cathy.crowe.18?fref=ts

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