ESD

This post will review Han Bleichrodt’s lecture regarding the micro foundations of using QALY based utility functions.

QALYs

Many health economists use a QALY model to describe a person’s preference over health states. For instance let (q₁,…q_τ) be an individuals health profile from year 1 to year τ. The QALY model assumes that utility is additive so that U(q₁,…q_τ)=ΣV(q_t). For chronic health conditions where health states do not vary from year to year, one can further simplify the formulation so that U(Q,T)=V(Q)*T.

Standard Gamble

Calculating V(Q) is not as easy as it seems. One way to elicit the individuals preferences is to use the standard gamble and to assume that V(Death)=0 and V(Full Health)=1. Let us look at an example.

You currently have severe back pain. You are offered a surgery that with probability p will make you completely healthy for the next 30 years, but with probability 1-p you will die. If you do not preform the surgery, then you will have chronic back pain for the next 30 years.

One can ask the individual to choose whether or not they want to surgery and the experimenter can adjust the probabilities until the individual is indifferent between both states. In this case, we know that the QALY utility weight is equal to p:

• U(Back Pain, 30 yrs)=p*U(Health, 30 yrs) + (1-p)U(Death) = p*1+(1-p)0 = p

The total number of QALYs is p*T=p*30.

Time Trade-off

Another means of eliciting patient preference is to use the time trade-off. For instance, one is asked if they prefer to live 40 years with back pain, or 30 years in full health. The number of years in full health can be adjusted until the patient is indifferent between the two states. Thus, mathematically we can calculate the utility weight of the QALY as :

• (Q₁,T₁)~(Q₂,T₂)–>(Back pain, 40 yrs)~(Health, 30 yrs) –>
• U(Back Pain)*40=U(Health)*30 –>
• U(Back Pain)=1*30/40=0.75

Rating Scale

The final method to elicit QALY is to simply ask the person. One asks simply describes a disease and asks a person to rate it between 0 and 100, where 100 is perfect health and 0 is death. The QALY is the stated rating divided by 100.

Microeconomic Theoretical Foundations

What assumptions need to be satisfied in order for the QALY model to be an accurate depiction of reality? An article by Pliskin, Shepard and Weinstein (Operations Research 1980) derives 3 conditions in order for the QALY model to hold:

1. Mutual Utility Independence. This states that the utility function must be separable between quality of life and years of life. Mathematically U(Q,T)=V(Q)+W(T) or U(Q,T)=V(Q)*W(T).
2. Constant Proportional Tradeoff. This means that (Q₁,T₁)~(Q₂,T₂) if and only if (Q₁,αT₁)~(Q₂,αT₂), where α is non-negative.
3. Risk Neutrality with respect to life-years. This is probably the key assumption. One must assume that W(T)=T.

Empirical Findings

Does the QALY model hold empirically? Bleichrodt states that empirical evidence shows that people are not risk neutral with respect to life years. If a less restrictive model of the form U(Q,T)=V(Q)T^r were adopted, then empirically r ≈0.75.

Miyamoto and Eraker (1988) try to test utility independence. They find support for utility independence of life duration from health quality, but also found that for short life durations, about 25% of their subjects were not willing to give up any life years to improve their health status.

Bleichrodt, Pinto and Abellan-Perpiña (JHE 2003) test for constant proportional trade-offs using life durations of 13, 19, 24, 31 and 38 years. There was some support that constant proportional trade-offs holds, but the evidence was not overwhelming.

Bleichrodt claims that the Time trade-off QALY solicitation is the least biased while the ranting scale is the least accurate. Evidence for this comes from the Bleichrodt and Johannesson (JHE 1997) paper, but I believe that between the standard gamble and time trade-off there is no clear cut optimal method. In my opinion, the rating scale seems to be the worst method, with the least grounding in microeconomic traditions.

Bleichrodt also briefly discusses prospect theory, which claims that people do not weight probabilities accurately. For instance, people care much more about an increased probability of dying from 1% to 2% then an increase from 51% to 52%. To take this into account, prospect theory uses a weighting function. Let us give an example:

• {(Q₁,T₁),p; (Q₂,T₂),1-p}=w(p)U(Q₁,T₁) + (1-w(p))U(Q₂,T₂)

The function w(p) weights the probability to take into account the fact that individuals do not perceive probabilities accurately.

Today I will review the insightful lecture of Willard Manning at European Science Days. Manning is most famous for his work with the RAND Health Insurance Experiment.

Problems with Healthcare Data

There are 4 major econometric problems one must consider when trying to analyze health care cost and utilization data:

1. There is a large mass of individuals with zero utilization (or expenditures) during a given time period,
2. Consumption among those with any care is very skewed (e.g.: visits, hospitalizations, expenditures),
3. The dependent variable often responds in a non-linear manner to many covariates,
4. demand response to covariates may change by the level of demand (e.g.: outpatient to inpatient, or low to high levels)

Log or Box-Cox Transformations

While using OLS is easy, it can often produce out-of-range predictions (i.e.: y_hat=xβ_hat<0). Since health care data is skewed, many researchers decide to log the dependent variable in order to have a more symetric distribution of errors. The tradeoff of using logs is that although one gains precision and robustness, no one is interested in log-scale results per se.

The Box-Cox transformation of y is as follows:

• [(y^λ-1)/λ]=xβ+ε, if λ≠0
• log(y)=xβ+ε, if λ=0

One estimates λ using MLE in order to minimize the skewness in the residuals.

Log Example

Using a log transformation implies that second moments often matter. For instance, let us assume log(y|g)~N(μ_g,σ_g), where treatment g=A, B. Then we know

• E(y|g=A) = exp[μ_a+ 0.5(σ_a)²].
• E(y|g=A)/E(y|g=B) = exp[(μ_a-μ_b)+ 0.5{(σ_a)²-(σ_b)²}]

We can see from the second equation above, that the second moment of the distributions matters if there is heteroskedasticity, but not if there is homoskedasticity (i.e.: σ_a=σ_b=σ)

Marginal Effects with log transformation

Calculating marginal effects with non-linear econometric formulations is often difficult.  For instance, we know that E(y)= exp(xβ)E{exp(ε)|x}. This implies that the marginal effect is equal to:

• dE(y)/d(x_k)=exp(xβ)[β_kE{exp(ε)|x}+ d E{exp(ε)|x}/d(x_k)]

This is much more complicated that the incorrect formulation that: dE(y)/d(x_k)=exp(xβ)β_k.

Generalized Linear Model Approach

In this method, one searches for the appropriate β’s to solve the following function:

• Σ dμ(xβ)/dβ*V(x)^-1*(y-μ(xβ))=0

In practice, one usually assumes that μ(xβ)=exp[xβ]. A variance structure is assumed so that Var(y|x)=α[E(y|x)]^γ. The γ’s correspond to some standard parametric distributions:

• Gaussian NLS: γ=0
• Poisson: γ=1
• Gamma: γ=2
• Wald or inverse Gamma: γ=3.

Two Part Models

To this point, we have been focusing on the skewness problem and been ignoring the fact that many of the observations also clump at zero. We can decompose the expected value as follows:

• E(y|x) = P(y>0)*E{y|y>0} + P(y=0)*0 = P(y>0)*E{y|y>0}

Now we must estimate P(y>0) and E(y|y>0) separately. The first part term we can estimate with a probit model [P(y>0)=Φ(xα). The second part one can log the y term to take into account skewness.

If the log-scale error term is normally distributed, then:

• y_hat= Φ(xα)*exp(xβ + .5σ²), where β, σ are estimated from the data.

If the log-scale error term is not normally distributed, than one can use the following formulation:

• y_hat= Φ(xα)*exp(xβ)*D
• D is Duan's (JASA 1983) smearing estimator:
• D=N^-1Σexp[ε]=N^-1Σexp[ln(y|y>0)-xβ_ols]

Count Data

Count data in health economics is very common. The number of doctor visits, hospitalizations and ER visits all are types of count data. Poisson and Negative Binomial regressions are frequently recommended for these types of data.

Below is a summary of Don Kenkel’s lecture regarding the Economics of Substance Abuse Use.

General

The World Health Organization and the United Nations International Drug Control Programme (UNDCP) have statistics on the number of drug users around the world. They claim that in 2002 there were 2 billion alcohol users, 1.3 billion smokers, and 185 million illicit drug users.  These number represent 32%, 21% and 3% of the world’s population.  Of illegal drug users, 69% use cannabis, 16% amphetamines, 6% cocaine, 4% heroin, 3% ecstasy and the rest other opiates.

Defining what the term ’substance’ is also difficult.  Alcohol and marijuana have been legal and illegal in different times and places.  Is caffeine a substance?  What about prescription drugs?  Are betel or khat substances?

Rational Addiction Model

Kenkel briefly discusses a model of a rational addict.  The rational addict is forward looking and takes into account that choice of consumption today will affect the marginal utility of consuming the substance in the next period.  A rational addict may have the following utility function:

• Σβ^t-1 U(C_t,C_t-1, Y_t, e_t)
• s.t.: I_t=Y_t+pC_t
• C: number of cigarettes, Y: other consumption goods, I: Income

Assuming a quadratic utility function, one can show that the structural demand equation is:

• C_t=θC_t-1+βθC_t+1+θ₁P_t+θ₂e_t+θ₃e_t+1

This can be estimated using 2SLS with P_t-1, P_t+1 as instruments for C_t-1, C_t+1.

The rational addiction model is appealing for a number of reasons.  It explains many features of addictive consumption and features two unstable steady states: one with low and one with high consumption of cigarettes.  Life cycles shocks can move consumers from the low to high cigarette consumption states.  The model also explains why quitting cold turkey can be optimal.

Quasi Hyperbolic Discounting

This was developed by Laibson (1997).  The utility function is time-inconsistent in that the future is discounted by more than the present.  For instance:

• U_t=u_t + δΣβ^τ-1 u_τ

The term δ represents the taste for immediate gratification. Psychological experiments tend to confirm quasi-hyperbolic discounting. Undergraduates given choices between a delayed reward of $1000 and immediate rewards ranging from $1 to $1000 revealed a year 1 discount rate of 60%, but a 16% discount rate for years 2-5. Also, this model would explain the existence of commitment devices (e.g.: former smokers supporting restaurant smoking bans as a commitment device).

Cue triggered addiction

The seminal work here is Berheim and Rangel (AER 2004).  The paper claims that individuals operate in either a ‘cold’ mode where decision processes are made rationally, and a ‘hot’ mode where decisions and preferences diverge and this results in substance use.  Addicts know they make bad decisions while in hot mode and can choose their lifestyle to alter the probability of being in the hot mode.  The Neuroeconomics blog from George Mason University has some commentary on this paper.

Quotation.
Entertaining quotation regarding the 1998 Master Settlement Agreement.

Q: Could you please explain the recent historic tobacco settlement?
A: Sure. Basically, the tobacco industry has admitted that it is killing people by the millions, and has agreed that from now on it will do this under the strict supervision of the federal government.
-Dave Barry

One session of the European Science Days summer school involved a presentation on long-term care (LTC) from Volker Meier. The main question is why is there so little demand for LTC insurance in the United States as well as in other countries? Below are some explanations as to why the LTC insurance market is so small and why–when people do buy LTC insurance–do they make the purchase so late in life:

• Loading factors. If loading factors are too high, this may possibly justify government intervention. However a paper by Brown and Finkelstein (JPubE 2007) finds that loading factors are comparable to those of life annuities. Meier claims that if there are fixed loading costs, purchasing later in life can save on these costs. However, I believe that these most insurance contracts likely have a fixed and variable loading factor even if economists do not typically model it this way, and this fixed loading cost explanation is likely a poor one.
• Adverse Selection. It seems likely that individuals have some information regarding the probability they will need LTC in the near future. Most individuals purchase health insurance because there is a non-trivial probability that they will become sick each year. Most younger individuals in good health, however, will not need LTC in the next year. Even if one falls sick, it may be years before LTC is needed. Thus, if one can predict the need for LTC with near certainty two years in advance, no one would purchase LTC insurance.
• Medicaid. In the U.S., Medicaid pays for about 40% of nursing home stays (Forbes). Thus, poor people have no incentive to purchase LTC insurance since Medicaid will pay. The middle class can run down their assets in the case where they need LTC and have Medicaid foot the bill. Thus, only for the rich will purchase LTC insurance in the United States in order to protect their accumulated assets.
• Proposal: Matching life insurance with LTC insurance. Peter Zweifel proposed this idea which seems logical. Individuals who live past 65 can use the proceeds from their life insurance policy to pay for LTC insurance. However, there would seem to be significant problems with insurance companies pricing LTC years in advance. Further, healthy individuals who reach age 65 may simply prefer to use the life insurance proceeds to spend on other things. This is especially true if LTC needs are predictable a few years in advance.

Care in a nursing home or care at home

Much debate in Europe and the U.S. has wondered whether paying family members to care for the elderly can save taxpayers money and increase the quality of care. Dr. Meier creates a model with the following characteristics:

• Individuals fall sick and the cost to care for them is variable. Individuals who go to the nursing home have a cost of K_n, while individuals can also receive care at home for a cost of β_iK_h, where β_minK_h < K_n < β_maxK_n.
• β_i can depend on the dependent’s opportunity cost or the seriousness of the disease. Thus, we would predict that individuals with better paying jobs and individuals who’s parents have more serious diseases will prefer to put their parents in nursing homes compared to at-home long term care.
• Another problem with paying individuals to care for their parents at home is the issue of fraud. Individuals can claim to be helping their parents cope with some diseases in order to receive extra cash, even though the work they do may be minimal or non-existent.
• Meier finds that it is optimal to offer two types of LTC insurance contracts. The first is for full insurance for nursing home care. Another insurance contract will give partial insurance for care at home. The premium for the first contract will be more expensive than for the second. The partial insurance will help to discourage fraud. Meier claims that this is how the German public long-term care insurance operates.

Reading Recommendations:

• Brown and Finkelstein (2007) “Why is the Market for Long-Term Care Insurance So Small?” J Pub Econ forthcoming
• Pauly, M.V. (1990) “The rational nonpurchase of long-term-care insurance.” J Pub Econ 98, 153-168.

Below is a summary of some of the interesting points in the lecture of Mathias Kifman.

Gouveia model

This is a topping up political economy model. First, we have individuals who get utility from consumption and health care. There are two types: high risk π_h and low risk π_l. There are also two incomes, y_i: rich y_r and poor y_p. Individuals have the following utility function:

• max u(c) + π_iv(h)
• c=y_i-Πt(y)g
• h=g

Π is equal to the average risk of the population. The first order condition is:

• u’Πt(y_i)+π_iv’

The comparative statics show that dg/dπ_i is positive (sicker people prefer more government health care), but the sign of dg/dy is unknown. Rich sick people prefer to have some government provided insurance since they can purchase the insurance at a lower rate, but since progressive or proportional taxation finances the system, they dislike the redistribution aspects of this program.

Topping up
We now allow individuals to buy additional medical care, m, even after paying for the public insurance program. For instance, if the government covers 5 doctors visits per year, and individual can buy additional insurance to pay for additional visits. Here, our model is:

• max u(c) + &pi_i;v(h)
• c=y_i-Πt(y)g-πm_i
• h=g+m

The first order condition for the choice of m is:

• -u’(y_i-Πt(y_i)g-πm)+v’(g+m)=0

The comparative statics reveal the following:

• dm/dg is negative. More public health insurance will reduce the quantity of private health insurance purchased. This makes perfect sense since in the model public and private health insurance are perfect substitutes.
• dm/dy is positive. Richer people have higher demand for all goods so it is sensible that they wish to purchase more private health insurance.
• dm/dπ_i is negative. Sicker people purchase less private health insurance since the price of private health insurance is based on their risk level. Thus, high risks pay much more for private health insurance than low risks, while in public health insurance the price of health insurance is based on income rather than risk level.

Opting out vs. Topping up
While the topping up model shown above gives a compelling arguement for allowing private health insurance to be purchased in addition to some basic level of public health insurance, one can not always ‘top up’ with medical care. For instance, individuals may wish to avoid public health services altogether in order to avoid waiting list or in order to meet with higher quality doctors. Dr. Kifmann notes the following:

• If the quality of public services is low, the typical individual prefers private services. Increasing public services (and raising taxes) decreases utility as long as private services are consumed.
• Once quality is sufficiently high, however, individuals prefer public services. Increasing the quality of public health further can make individuals better off until their preferred qualitz is reached

Of course, it is difficult to ensure quality in a government monopolized system where there is no competition.

Political Economy
The political economy analysis reveals an ‘ends against the middle’ phenomenom. Individuals poorer than the median voter and the wealthiest voters prefer lower services. The poor have a high marginal utility of income and would rather spend their money on themselves rather than a public health care system, while the rich dislike the redistributionary aspect of social health insurance. The middle class, however, generally supports public health insurance. Thus, if the middle class makes up a large proportion of your society, it is more likely that government provided health insurance will be approved by the median voter.

In the forthcoming days, I will be summarizing some of the lectures given at the European Science Days summer school in Steyr, Austria. On the first day, there was an interesting lecture by Louis Eeckhoudt about risk and pain disaggregation.

Most individuals are familiar with the concept of risk aversion. However, the lecture spoke extensively regarding the issue of prudence, presented in a paper by Kimball (1990).

Example

Which lottery would you prefer?

• In lottery A you have a 1/4 chance of getting 0 € and a 3/4 chance of getting 2000 €.
• In lottery B you have a 3/4 chance of getting 1000 € and a 1/4 chance of getting 3000 €.

What did you choose? Be honest…

Most people prefer lottery B. Why? The expected value and variance of the two lotteries are identical. However, the skewness of lottery B is positive, but is negative for lottery A. If people are prudent, they choose lottery B. Mathematically, prudence occurs if the third derivative of the utility function is positive (U”’) is positive. Intuitively, people would rather have an upside risk with small probability than a downside risk with small probability even if the mean and variance of the two lotteries are equal.

Dr. Eeckhoudt also introduced the concept of temperance. An individual is temperate if an an exogenous increase in one risk leads them to reduce risk in other areas. Applying this to alcohol, a temparate person would reduce their wine consumption as their beer consumption increase in order to moderate their aggregate risk of getting drunk. Mathematically, temperate individuals have a utility funciton where the fourth derivative (U””) is negative.

Applications to Health

Dr. Eeckhoudt spoke about the well known phenomenon that risk averse people would like to purchase some sort of insurance. If possible, they will self insure. This differs from self-prevention. Let us look at two examples:

• Self insurance: With probability p, you will become sick and have a utility of x-L(e)-e. With probability 1-p, you will be healthy utility x-e. In this example, by exerting effort, e, you can decrease your health loss L. Thus, the more effort you put forth, the closer will be the two utility levels in each state and thus risk will be dimished.
• Self-prevention: With probability p(e), you will become sick and have a utility of x-L-e. With probability 1-p(e), you will be healthy and have utility x-e. In this example, by exerting effort, e, you can decrease your the probability of becoming sick, p(e). The key insight from Dr. Eeckhoudt was that more prudent people have a lower level of self prevention!

This is certainly an important societal issue since prudence may decrease healthy behaviors such as self-prevention measures against sickness (e.g.: excercising, quitting smoking, immunizations).

Multi-dimensional prudence
Dr. Eeckhoudt underlying message was that if people prefer to disagregate their pain/risk level, rather than combining losses or risk into a single time period or payoff, this has interesting implications. Let us look at prudence in the mutli-dimensional case.

Each individual recieves utility from income, x, and health, h. The variable s represents sickness and e is a zero mean stochastic term which one could interperet as income risk. Which of the following lotteries should people prefer?

• In lottery A you have a 1/2 chance of being healthy and having a risky job (x+e,h) and 1/2 chance of having having a safe job but getting sick (x,h-s).
• In lottery B you have a 1/2 chance being sick and having a risky job (x+e,h-s) and a 1/2 chance of being healthy and having a safe job (x,h).

A prudent person will prefer lottery A to lottery B. In lottery B, sickness and risk are concentrated into one state, where as the risk/losses are disaggregated or spread between the two states.