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 (q_{1},…q_{τ}) be an individuals health profile from year 1 to year τ. The QALY model assumes that utility is additive so that U(q_{1},…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 :

- (Q
_{1},T_{1})~(Q_{2},T_{2})–>(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:

**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).**Constant Proportional Tradeoff**. This means that (Q_{1},T_{1})~(Q_{2},T_{2}) if and only if (Q_{1},αT_{1})~(Q_{2},αT_{2}), where α is non-negative.**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:

- {(Q
_{1},T_{1}),p; (Q_{2},T_{2}),1-p}=w(p)U(Q_{1},T_{1}) + (1-w(p))U(Q_{2},T_{2})

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

From

Douglas Foerster:I want use the occasion and add a small coment: you state that using the TTO and SG method yield to the QALY.

“SG…In this case, we know that the QALY is equal to p”

“TTO…Thus, mathematically we can calculate the QALY as…U(Back Pain)=1*30/40=0.75”

IMO, SG and TTO deliver the utility of a health state in question. This utility is the weight of a QALY that means to derive the quality adjusted life years you have to multiply the duration of the health state with its utility.

Living 40 yrs. with back pain – and U(back pain)=0.75 – yields to 40yrs * 0.75 = 30 QALYs