Trusting Math

Sometimes caring about numbers is how we care for the vulnerable. Sometimes it’s a way of effacing the image of God
Appears in Fall 2015 Issue: Health Beyond The Hospital
September 1 st 2015

For better or ill, my academic meanderings have brought me to a career where I spend the majority of my time building mathematical models to aid health-care managers in solving complex scheduling and capacityplanning problems. In other words, I try to convince health-care managers, on the strength of my word, to adopt often counterintuitive policies based on complex mathematical models they cannot hope to understand—and that doing so will provide better care for those who need it. Think of it as bringing Walmart's supply-chain sophistication to the world of health care. But what makes my work most difficult is not solving equations, or even explaining them. Rather, those I seek to convince are largely driven by a utilitarian ethic that uses mathematics to justify ends that, in my mind, contradict the proper goals of medicine.

One of the symptoms of this utilitarian ethic is the ubiquitous use of what the industry calls "quality-adjusted life years" (QALYs) as one of the primary justifications for deciding how resources should be allocated. QALYs are a measure designed to determine the benefit of a given intervention. Suppose your doctor presents you with two potential treatments for a given condition: one provides you with twenty additional years of life, but throughout those twenty years you will suffer from severe headaches; the other provides you with nineteen additional years of life but with no discernible side effects. A strict assessment of the two interventions based solely on life years gained would prioritize the first over the second. But many, if not most, would prefer the second. QALYs attempt to deal with this added dimension to the decision-making process by assigning a reduced weight to years spent in less than complete health.

The cold calculations of QALYs look at the work of the Sisters of Mercy in Calcutta and pronounce over it the damning word "inefficient."

Suppose it was somehow determined that a year with severe headaches was worth only half a year in perfect health. The first intervention would thus only provide 20 x 0.5 = 10 QALYs, while the second would produce 1 x 19 = 19, thus providing a measure that coincides more readily with our intuition. In one measure, health-care managers can assess both quantity (years of life saved by an intervention) as well as quality. The primary use of QALYs is in a cost-benefit analysis that compares interventions based on the cost per QALY gained. The lower the cost per QALY, the more efficient or beneficial the intervention.

Confined to comparing two potential treatments for the same condition, as in the example above, the use of QALYs seems fairly innocuous. However, health-care managers also use QALYs to compare different interventions for completely different conditions in order to determine which intervention is more worthy of funding. When administrators determine a cost-per-QALY threshold, they deem interventions with costs below the threshold fundable while those above they do not. Thus what ought to be a complex moral question (how should scarce health resources be allocated?) is reduced to a simple mathematical formula that allows administrators to avoid responsibility for human judgment by appealing to an apparently impartial and objective method.

Who counts?

This use of QALYs as the sole means of allocating health resources is troublesome for a number of reasons.

First of all, it places the disabled and the aged at a clear disadvantage. Peter Singer, a proponent of the use of QALYs, is refreshingly candid about these biases inherent in QALY assessments. In a Journal of Medical Ethics article, he presents two examples. In the first, two women are injured in a car accident. The first escapes with a limp as the only long-term consequence, while the second is confined to a wheelchair. Though both are expected to live another forty years, under any quality adjustment, the second woman is undoubtedly going to have fewer QALYs. A couple years later they are both afflicted with a heart condition that requires immediate transplant. Clearly in this case a QALY-maximization approach would favour treating the first woman over the second. The second woman's prior misfortune thus plays against her.

Or take another example: two men face the same heart condition. Without treatment they will die, but with treatment they will return to full health. But the second man has a separate medical condition, latent at present but that will inevitably flare up at some point and end his life prematurely. In this case, any assessment based on expected life years gained will favour the first. In both cases, Singer and his co-authors state that they "do not find it selfevident that it would be unfair to give lower priority to the disabled person." And he is right, of course, if justice consists solely in allocating resources to those who will benefit the greatest. But we can accept that QALYs are an appropriate and just means of allocating health resources only if we assume issues of equity and compassion play no part. As with so many things, we cannot avoid the question of what is just even in what most would consider banal, mathematical, administrative decisions. Sometimes, math isn't simply math.

You might say that I (or really Singer) have misrepresented the use of QALYs, which are not (yet) used to determine resource allocations between individuals but rather to judge the benefits of interventions at a population level. But bias doesn't simply disappear when we move it from the level of the individual to the level of subpopulations. It is hard to believe, for instance, that a system measuring utility based on QALYs would treat palliative care favourably. Even if quality of life were vastly improved under palliative care, the number of life years gained would likely be minimal.

Interventions designed for those who are already severely disabled would be similarly disadvantaged. To put it bluntly, if the utility and funding of an intervention are solely based on the "ability to benefit," then interventions targeted at, say, the aged or disabled will receive scant funding. This cuts right to the roots of the Western medical tradition, which was originally created precisely to care for the most disadvantaged.

Capturing Quality?

We ought to greet with scepticism the contention that we can truly capture "quality of life" in a quantitative measure. Even proponents of the use of QALYs admit that it is fraught with difficulties; yet they still contend that it is better than any alternative. QALYs acknowledge the importance of "quality of life" as a component of resource-allocation decisions, and surely this is better than a simple assessment of the quantity of years gained. But are they a sufficient or even appropriate acknowledgement of the complexity of the question? No, they are not. Rather they represent a clumsy attempt to quantify the unquantifiable, which has the dangerous tendency to make us believe that we have adequately accounted for the qualitative aspects of the decision-making process. In truth we have simply hidden human lives behind the numbers (through subjective weighting of conditions) and reduced a complex moral decision to a simple calculation.

Moreover, QALYs are based primarily on physical disability, with the occasional nod to "mental anguish" or "anxiety." But this is a gross simplification of the rich and complicated relation of health and life. Our physical well-being is not the only thing that makes life worth living, and it takes a certain type of unthinking arrogance to believe otherwise. There are simply too many examples of those who have flourished despite and perhaps even because of suffering. Beethoven, Terry Fox, Stephen Hawking—these are just three of a myriad of examples that populate human history. In fact, it is often because of their ailments that they accomplished what they did. (Hawking has even said as much.) My mother had polio at the age of eight and lost the use of her left arm. Any QALY measure would have viewed her life years since as diminished in quality, but the perseverance and strength of will she developed as a result of that ailment cannot be measured. Helen Keller was blind, deaf, and dumb, but no one would argue that her quality of life was less than that of a perfect physical specimen like Mike Tyson.

But what alternative is there?

Humanist Math

The alternative is to admit that we cannot measure the quality of life in numbers and to reject the idea that questions of resource allocation can be answered by a mathematical formula. Health-care professionals must recognize that other aspects of resource allocation deserve attention, including the question: Does the allocation of these health resources offer an adequate response to what

We ought to greet with skepticism the contention that we can truly capture "quality of life" in a quantitative measure.

it means to be human? QALYs are not simply a "neutral" means of determining resource allocation. They are a mathematical tool that incorporates, and serves, a particular view of health and humanity. Accepting QALYs as the means of determining resource-allocation decisions means accepting that the primary purpose of health care is to maximize future benefit. This is the case regardless of whether that benefit marginalizes certain segments of the population and regardless of the claims those in significant hardship may have on the larger society. In other words, accepting the use of QALYs ignores the voices of need, equity, and compassion that have traditionally played a role in these decisions, and it ignores them precisely because they cannot be easily captured in a quantifiable manner. The philosopher Michael Banner makes this same point.

The question which the advocate of QALY maximisation will ask, namely, "why should we invest so much in what promises so little return?," is one which evidences a failure to recognize the demand made upon us by the existence of those whose needs are so patent partly because human endeavour can do so little to meet them. Thus we should see in the attempt to meet those needs as best we can, not an inefficient or unproductive use of resources, but rather something which is at the heart of humane medicine as it is at the heart of a humane, or we might just say human, society: that is the expression of a sympathy which asserts a desire to maintain, and does indeed maintain, our kinship and community with those who, through the approach of death or through physical or mental deprivation, stand in different ways on the edges of the most limited and perhaps most obvious of communities, namely the community of those who are fit, strong and productive.

The origins of the Western medical tradition— unarguably entwined with the Christian tradition—suggest that the purpose of medicine is not simply to maximize future benefits (though that plays a role) but also to alleviate current suffering. In fact the greater the suffering, the greater the claim for care regardless of whether that care results in dramatic improvements in health. The cold calculations of QALYs, on the other hand, look at the work of the Sisters of Mercy in rescuing the dying on the streets of Calcutta and pronounce over it the damning word "inefficient." It is a frightening thought that, with the looming threat of euthanasia, it might easily be seen as more "efficient" to end the suffering of an individual with an inexpensive drug than to devote significant resources to his or her care. When that day comes, one can guarantee that the argument will be based on a quantitative analysis demonstrating the immense benefit to efficiently and humanely "ending the suffering" of a portion of the population in order to divert more funding to those who can benefit most.

The success of QALYs despite their obvious deficiencies lies largely, I believe, in the fact that they transform a difficult moral decision into an easy-to-implement guideline. The administrator can avoid responsibility by hiding behind an apparently objective and impartial method for decision-making. Hopefully I have made it clear that QALYs are far from objective or impartial. They incorporate subjective evaluations of the "quality of life" under a given condition, and they are inherently biased against both the elderly and the disabled. Moreover, they by their very nature instantiate a utilitarian ethic whose only criterion for intervention is the extension of healthy life. This ethic stands in direct contradiction to the Judeo-Christian ethic, upon which Western medicine was founded. On Judeo-Christian view, all life is equally valuable, and the healthy have a moral responsibility to care for the sick.

Again, Banner puts it nicely.

The desire for administrative clarity is a legitimate one, but such clarity ought not to be achieved by discounting values, such as justice, which make the task of devising guidelines more difficult. . . . It is not that our values must be set aside if they prove recalcitrant in the hands of tidy-minded administrators, but rather that a "rational" distribution of resources has no claim on our attentions unless it expresses our values, unless it has struggled, for example, to relate the claim that health care should be efficient to the claims of those in need. . . . Administrators who are asked to distribute resources with an eye on fairness and need, as well as on benefit, have no ground of complaint just because they are being asked to exercise a certain amount of judgment rather than merely to do sums.

What, then, to do with morally significant public decisions?

Healing Trust

As I have been writing this essay, I have also been enjoying Chaim Potok's novel The Promise. One of the characters, Abraham Gordon, is forced to make a tough decision based on evidence he can't really understand. He says to his friend Reuven, "But how does one make a decision in such a situation? How does a person decide, for example, to accept a doctor's judgment that he requires surgery?" Reuven responds by saying that the most important thing is to feel you trust the doctor. "Yes," he replies. "I cannot think of anything more important than that."

But what if, through betrayals in the past and fractures in the moral consensus in the present, that trust is now often hard to find, or create?

My research, for instance, focuses largely on capacity planning and scheduling. Given the resources that are going to be made available, how do we determine the optimal capacity and how do we schedule in order to make the most efficient use of that capacity? Those are questions that mathematics can readily answer, and yet I find administrators reluctant to entrust the decision to the model. They face a dilemma: the capacity-planning and scheduling problems they deal with are complex enough that they require sophisticated

mathematical models well outside the expertise of most managers. But how do they trust a model they cannot understand? In such a situation they must do one of three things: develop the expertise to understand the complexity (often not a viable option), trust those who claim to have the expertise (despite plenty of instances of past failures by the experts), or simplify the analysis to fit their current knowledge. Thus they hire consultants who provide them with simple (dare I say simplistic?) solutions to complex problems. These administrators' motivation is largely the same as those who rely on QALYs—they reject a complex solution for a simple, seemingly "objective" measure that suggests a clear way forward.

We likely don't recognize how much of what we do is based on trust, or how many of the decisions we claim are based on evidence are based instead on the trust we have chosen to place in another. We trust the doctor knows what is best for our well-being. Imagine, though, that you belonged to a culture in which life after age eighty was not considered worth living. Would you trust a doctor raised in such a culture to care for you once you reached that age? So much of our trust is founded on the assumption that we are surrounded by people who share a common understanding of the good and the true. But what if what we value as good and true no longer coincides with the understanding of our neighbour? Would that trust begin to erode? In such a culture, how are morally loaded decisions such as those related to health-resource allocation to be made?

There is no easy answer, but we must emphatically reject the answer that these decisions can be decided by simple mathematical calculations masquerading as objective measures. For these "objective measures" are really the triumph of a specific, utilitarian understanding of the purpose of medicine—an understanding that discards such cherished virtues as compassion for the most vulnerable; that sees suffering and disability as conditions with no redemptive possibilities; that imposes a concept of justice antithetical to that which founded and inspired the Western medical tradition.

The answer must lie not in an unjustifiable resort to mathematics, but in the kinds of conversations that recognize the fractured state of our culture and the competing and often contradictory understandings of the nature of our world and humanity's place in it. These fractures can be seen in troublesome issues such as euthanasia, but they also show up in more seemingly banal health issues like where hospitals should be located, how much doctors should be paid, and how long their hours should be. I confess that I am not overly sanguine about the success of these conversations in healing the growing rifts in our culture. But of this I am sure, there is no path forward without them.


Jonathan Patrick is an associate professor at the Telfer School of Management at the University of Ottawa. His research centres on the application of mathematical models to health care management decision problems and in particular scheduling and capacity planning problems. The goal is to provide mathematical tools that can help determine the necessary capacity and the optimal schedule so that patients receive service in a timely fashion. He did his Ph.D. at the University of British Columbia in Operations and Logistics. He is married to Jennifer Patrick and they are blessed with four sons and a daughter.