Unbiased Analysis of Today's Healthcare Issues

What are cure fraction models?

Written By: Jason Shafrin - Jan• 12•15

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:

  • S(t)=S*(t)R(t)

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:

  • h(t)=h*(t) + λ(t)

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:

  • S(t)=S*(t){π+(1-π)Su(t)}

where Su(t)is the survival function for the uncured individuals. The corresponding hazard function is:

h(t)=h*(t) +

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.

Paul C. Lambert derives additional fractional cure survival models in his 2007 Stata Journal article.

Disability Benefits Around the World

Written By: Jason Shafrin - Jan• 11•15

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.

disability insurance around the world

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.


Start of Year Links

Written By: Jason Shafrin - Jan• 01•15

I’m headed off on a week of vacation starting on Saturday.  Blog posting will resume Jan 12.  Before I leave, here are some more interesting reads to start off the new year.


End of Year Links

Written By: Jason Shafrin - Dec• 31•14

Does your mortality rate increase when your doctor is out of town?

Written By: Jason Shafrin - Dec• 30•14

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.

What is attribute non-attendance?

Written By: Jason Shafrin - Dec• 28•14

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?


Quotation of the Day

Written By: Jason Shafrin - Dec• 25•14

…in the next 10 years, data science and software will do more for medicine than all of the biological sciences together.

The Final Cavalcade of Risk

Written By: Jason Shafrin - Dec• 24•14

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.

The Standard Gamble

Written By: Jason Shafrin - Dec• 22•14

One concept often used in healthcare is the quality-adjusted life years (QALY).  The concept is fairly simple.  It assumes that people value one year of life in perfect health at 1; people who die have a value of a life year of 0.  One year of life where you have 50% health is then valued at 0.5.

The question is, how do you determine what share of a life year different health states are.  What share of a life year do I value if I break my arm?  What if I am paralyzed?

One approach to quantifying the value of different health states is through a standard gamble.  Hammitt et al. (2012) describes how this is calculated. The standard gamble gives people two option. One option is living for t years in health state h.  For instance, one could live for 5 years with a broken arm. In the second option, the patient has a probability p of living t’ years in perfect health (i.e., h=1) or immediate death (i.e., t”=0).

The goal of the standard gamble is to determine the value of p.  If a patient prefers to live for 10 years with a broken arm compared to 5 years in perfect health, but would prefer to live 5 years in perfect health to 9 years and 11 months with a broken arm, then, the value of is 0.5.  In essence, 0.5 years of perfect health is equivalent to 1 year with a broken arm.  Thus, the value of a QALY with a broken arm (in my simple and very unrealistic example) is 0.5.


How did Obamacare affect Medicaid enrollment?

Written By: Jason Shafrin - Dec• 21•14

The Affordable Care Act (ACA) contained the provision requiring States to significantly expand Medicaid eligibility. The Supreme Court, however, ruled that states had the option of implementing this provision.  Thus, the change in Medicaid eligibility and subsequent enrollment varied significantly across states based largely on whether or not they were an expansion state.  A recent PwC study finds that Nevada, Oregon and Colorado had the largest percentage increase (all >50%), but three states Alaska, Missouri, and Georgia actually experienced decreases in Medicaid enrollment.


Overall, 3/50 states experienced a decrease in enrollemnt, 4/50 states experienced no change, and 43/50 experienced an increase in Medicaid enrollment since 2013. In fact, 27/50 states experienced an increase in Medicaid enrollment of 10% of more.