Evaluation of Medicine

• Two types:

– Societal level• Economic evaluation

– Individual level• medical decision making

Economic Evaluation

• Comparison of costs and benefits

• Central question: How can we express the benefits of health care numerically?

Approaches

• Ignore health benefits - Cost minimization

• Express benefits as life-years gained

• Contingent valuation - CBA

• Express benefits as utilities - CUA

Medical Decision Making

• Optimal treatment selection

• Common approach: perform decision analysis expressing benefits in utility terms

QALYs

• Question: which utility model?

• Most common model:

– Quality-Adjusted Life-Years (QALYs)

QALYs

• Additive model:

• Let (q1,..., qT) be health profile

• QALY model: U (q1,..., qT) = V(qi )

Example

• Living for 10 years with asthma

• Suppose V(asthma) = 0.5

• QALYs (10 years asthma) = 5

QALYs

• Advantages– intuitively appealing– easy to use in practice

• Disadvantages– may be too restrictive

Two questions

• QALY model: U (q1,..., qT) = V(qi )

– How do we determine the utilities V(qi )?

– Which are the assumptions underlying the QALY model?

Basic Ingredients

• Set of health states H

• Set of lotteries P over H

• Preference relation R over P

• Representing function V:PIR such that

V(P) V(Q) iff x R y

Assumptions

• Health states are chronic: H = TQ

• Health states are positive, i.e., preferred to death

• Expected utility holds

Expected Utility

• V((Q1,T1), p, (Q2,T2)) =

pU(Q1,T1) + (1p)U(Q2,T2)

• U = a + bU, a real, b > 0

Chronic Health States

• Two attribute utility function

U(Q,T) = V(Q)*T

Q set of positive health states, T set of durations

First characterization

• Pliskin, Shepard & Weinstein (1980)

• 3 conditions– (mutual) utility independence– constant proportional trade-off– risk neutrality wrt life-years

Utility independence

• Quality of life is utility independent of duration if preferences over lotteries on quality of life holding duration fixed do not depend on the level at which duration is held fixed

• Duration is utility independent of quality of life if preferences over lotteries on duration holding quality of life fixed do not depend on the level at which quality of life is held fixed

Formally

• If (p1,(Q1,T);….;pm,(Qm,T)) R (r1,(Q1,T);….;rm,

(Qm,T))

• then

• (p1,(Q1,T);….;pm,(Qm,T)) R (r1,(Q1,T);….;rm,

(Qm,T))

And

• If (p1,(Q,T1);….;pm,(Q,Tm)) R (r1,(Q,T1);….;rm,

(Q,Tm))

• then

• (p1,(Q,T1);….;pm,(Q,Tm)) R (r1,(Q,T1);….;rm,

(Q,Tm))

Standard gamble

• (20y, Asthma) ((20y, FH), 2/3, (20y, Death))

• Then also

• (40y, Asthma) ((40y, FH), 2/3, (40y, Death))

Intermediate result

• The following statements are equivalent:

– utility independence holds

– U is either additive, U(Q,T) = V(Q) + W(T), or multiplicative U(Q,T) = V(Q)*W(T)

Hence

• To arrive at the QALY model must

(i) exclude the additive model

and

(ii) ensure linearity of W(T).

Constant proportional tradeoffs

• The preference relation satisfies constant proportional tradeoffs if

(Q1,T1) (Q2,T2) iff (Q1,T1) (Q2, T2)

for all Q1, Q2 in Q, T1, T2, T1, T2 in T and nonnegative

Time Trade-off

• If (Q1,T1) (Q2,T2) then

• U(Q1) = T2/T1

Example

• (10 years asthma) ~ (8 years FH)

• U(asthma) = 0.8

• (20 years asthma) ~ (16 years FH)

Exercise

• Show that CPT excludes the additive model

• Hence– U(Q,T) = V(Q)*W(T)

Risk neutrality

• Risk neutrality wrt life-years

• Risk neutrality for duration holds if for a fixed health status level all treatments with equal expected life duration are equivalent.

• (20y., FH) ~ ((40y., FH), 0.5; (0y, FH))

Implications

• W(T) linear

• Hence, have derived the QALY model

Utility Independence

10 30 50 70

Uti

lity

Duration

CPT

10 30 50 70

Uti

lity

Duration

Risk Neutrality

10 30 50 70

Uti

lity

Duration

Uniqueness

10 30 50 70

Uti

lity

Duration

Theorem

• Under EU the following two statements are equivalent

– The QALY model represents preferences for health

– The preference relation satisfies utility independence, constant proportional tradeoffs and risk neutrality wrt life-years

Less restrictive result

• Take the opposite route

• Start with risk neutrality wrt life-years

• U(Q,T) is linear in life-years

Hence

• U(Q,T) = A(Q) + V(Q)*T

• Note: negative health states are allowed

Hence

• Have to get rid of term A(Q) to obtain QALY model.

• Assume zero condition: for duration zero all health states are equivalent

Exercise

• Show that the zero condition implies that A(Q) = 0 for all Q.

Risk Neutrality

10 30 50 70

Uti

lity

Duration

Zero Condition

10 30 50 70

Uti

lity

Duration

Uniqueness

10 30 50 70

Uti

lity

Duration

Theorem

• Under EU the following two statements are equivalent

– The QALY model is representing– The preference relation satisfies risk neutrality

wrt life-years and the zero condition

Hence

• In PSW representation can drop

• utility independence

• and can weaken CPT to zero condition

Empirical evidence

• Zero condition unobjectionable

• People are risk averse wrt life-years

• Hence, QALY model not descriptively valid

• Normative status risk neutrality?

More general model

• U(Q,T) = V(Q)*W(T)

• Miyamoto, Wakker, Bleichrodt & Peters (1998)

Standard Gamble Invariance

• For Q and Q unequal to death:

(Q,T) ((Q,Y), p, (Q,Z))

iff

(Q,T) ((Q,Y), p, (Q,Z))

Then

• U(Q,T) = V(Q)*W(T) + A(Q)

• Zero condition: A(Q) = 0

Theorem

• Under EU the following two statements are equivalent

– The nonlinear QALY model is representing– The preference relation satisfies standard

gamble invariance and the zero condition

One more result

• Under EU the following two statements are equivalent

– U(Q,T) = V(Q)*T

– The preference relation satisfies standard gamble invariance and constant proportional trade-offs

Empirical evidence

There is support for

utility independence Miyamoto&ErakerBleichrodt&Johannesson Guerrero Bleichrodt&Pinto

and constant proportional tradeoffs

Bleichrodt&Johannesson

However

• Maximal endurable time– violates utility independence

• Lexicographic preferences for low durations– violates constant proportional tradeoffs– more in line with increasing proportional

tradeoffs

Maximal Endurable Time

10 20

Uti

lity

Duration

Lexicographic Preferences

10 20

Uti

lity

Duration

Health state A

Health state B

Alternative measures

• Mehrez & Gafni (1989): Healthy-years equivalent (HYEs)

• (q1,..., qT) (Full health, T´)

• HYEs (q1,..., qT) = T´

Claim

• ´´HYEs impose no assumptions on the utility function and are therefore entirely general´´

• Q: Is this claim true?

2-stage measurement procedure

• First stage: determine p such that

(q1,..., qT) ((FH,T), p, death)

• Second stage: determine T´ such that ((FH,T), p, death) (FH,T´)

Argument

• Two-stage gamble leads to same result as directly determining T´ from(q1,..., qT) (Full health, T´)

Questions:

– Is this argument correct?

– If so, is it true that HYEs are exactly as restrictive as QALYs?