Vince Kuraitis has posted a most excellent “Shake the Winter Blahs” Edition of Health Wonk Review at e-Care Management Blog.
How much would you pay to live longer? Most people would say an infinite amount. In practice, however, this is not the case. For instance, you can drive slower to reduce your risk of a car crash. In this case, you trade off your time with your probability of death. Or, to continue the care safety theme, one could buy the most expensive car seat with the highest safety track record. However, not everyone buys the most expensive car seat.
Economist measure preferences using a utility function and one can readily adapt a utility function to measure the value of a statistical life. Let one’s expected utility be:
where p is the probability of death in the current period, ua(w) and ud(w)are the utlity of weath conditional on suriving and not suriving. If we differentiate with respect to both w and p, we get:
How do we interpret this result? A paper by Hamitt et al. (2012) provides the answer:
…in the standard model expectancy conditional on surviving the current period increase the utility of survival ua(w) and may increase the marginal utility of wealth conditional on survival ua‘(w). Reductions in life expectancy and health clearly limit the opportunities for gaining utility from wealth…and there is some empirical evidence that impaired health reduces the marginal utility of wealth…Depending on the magnitudes of the effects on the total and the marginal utilities of wealth given survival, better health and increased longevity may increase, decrease, or not affect VSL.
Clearly, the value of wealth at dealth depends on the value individuals place on bequests. In individuals do not value bequests, then:
Many people are familiar with survival models. Survival models measure the probability of survival to a given time period. The “problem” addressed by these models is that some people are “censored”, in other words, the do not die in the sample time period. Although longer survival is good in practice, for statisticians it is problematic because one does not observe how long a person lives for.
Standard survival models can be written formally as:
where S(t) is the probability of surviving to time t, S*(t) is the baseline expected survival (e.g., survival without having a disease) and R(t) e.g., the effect of having a disease on survival probabilities.
The corresponding hazard functions are:
where h(t) is the all-cause hazard (mortality) rate, h*(t) is the expected hazard (mortality) rate, and λ(t) is the excess hazard (mortality) rate associated with the disease of interest.
However, what if there is some probability each period that you can be cured of a disease? A cure fraction model takes a standard survival model and adapts it for this case. As described in Lambert (2007):
Cure models are a special type of survival analysis model where it is assumed that there are a proportion of subjects who will never experience the event and thus the survival curve will eventually reach a plateau. In population based cancer studies, cure is said to occur when the mortality (hazard) rate in the diseased group of individuals returns to the same level as that expected in the general population.
Assume that in any time period, the likelihood of being cured is π. Then in this case, one can model survival as:
where Su(t)is the survival function for the uncured individuals. The corresponding hazard function is:
The mixture cure fraction model assumes that at diagnosis there is a group of individuals who experience no excess mortality compared to the general population.
In a series of papers, Coile, Milligan and Wise look at Social Security Programs and Retirement Around the World. In the sixth installment of the series, the authors look specifically at disability programs. Although stereotypically one would believe that most people exit the workforce due to choice and rely on Social Security, job pensions, and personal savings to finance their retirement, in many cases, workers become disabled and have significant difficulties working.
The paper finds that not only is there significant variation in the disability insurance programs across countries, but also the share of individuals who collect disability also varies across nations.
The authors key findings are described below:
First, the proportion of men 60 to 64 collecting disability benefits ranges widely across countries, ranging from 17 percent in Belgium to 16 percent in the UK to 14 percent in the US to 6 percent in Italy and France to 2 percent in Japan—including Belgium and Italy that use a DI proportion different from the other countries. Second, the data show that in all countries, with the exception of the United States, there was large variation over time in DI participation rates with substantial decline in participation beginning in the early to mid-1990s in many countries. For example, in Canada participation in the 60-64 age group declined 49.6 percent between 1995 and 2009. In the UK, DI participation declined 49.6 percent between 1996 and 2012. In the US on the other hand DI participation between 1990 and 2012 increased by over 30 percent. Third, variation in DI participation over time was unrelated to trends in health, which improved consistently over time based on declines in mortality. Fourth, and perhaps most striking, DI participation in all countries is very strongly related to education level, even controlling for health. Fifth, descriptive data show a noticeable inverse relationship between DI participation and employment over time.
According to a paper by Jena et al. (2014), the answer is no.
The paper examines 30-day mortality rates for Medicare patients admitted to the hospital with acute myocardial infarction (AMI) or heart failure and compares “…mortality and treatment differences…during dates of national cardiology meetings compared with nonmeeting dates.mortality rates “during dates of national cardiology meetings compared with nonmeeting dates.”
The authors compare mortality in the period just before and just after the conference to the meeting dates and find no differences in patient characteristics. However, they conclude that:
High-risk patients with heart failure and cardiac arrest hospitalized in teaching hospitals had lower 30-day mortality when admitted during dates of national cardiology meetings. High-risk patients with AMI admitted to teaching hospitals during meetings were less likely to receive PCI, without any mortality effect.
In layman’s terms, don’t worry too much if you are admitted to a teaching hospital and your doctor is out of town. Your chance of dying is likely the same or may even improve.
In discrete choice experiments (DCEs), respondents are asked to choose amoung different options which vary across different attributes. For instance, a DCE on mobile phone preferences could have processor speed, battery life, screen size and cost as attributes. A DCE looking at different treatments could have expected survival, anticipated side effects and cost as attributes.
DCEs assume that respondents are rational and have complete, monotonic, and continuous preferences. However, “Continuity of preferences implies that individuals use compensatory decision-making processes, meaning that they take into account all the available information to make their decisions.” In practice, however, this may not be the case. Instead, respondents may use simple strategies or heuristics to make their decisions. Under one common heuristic, known as attribute non-attendance (ANA), respondents ignore one or more attributes when deciding across DCE attributes.
Is ANA common? In short, yes.
In a recent study, Hole (2011) used a two-step process to account for endogenous ANA of respondents and found that ‘a substantial share of the respondents ignored one or more attributes when making their choices’.
So how do you fix this problem?
Today is a big day. Sure, it’s Christmas Eve Day. But it’s also the 224rd and final edition of the Cavalcade of Risk.
For those not familiar with CoR, “>it’s mission is summarized as follows:
The purpose of the C of R is to offer insights into the world of risk management; generally, this will be insurance-related, but that’s not a requirement. Our goal is to help folks understand what risk is, and how to manage it. It’s about business and finance, of course, but it’s also about risks in our everyday lives and personal relationships.
I would like to personally thank Hank Stern of InsureBlog of coordiniating the CoR over the past 10 years. These blog carnivals have provided great insight into insurance market (especially health insurance markets), financial matters, and forms of risk of all type. Hank has been a selfless and committed champion of CoR and I appreciate his leadership in the area of risk and risk management.
So, before the end of the year, be sure to check out the very last version of the Cavalcade of Risk over at InsureBlog.