1 INTRODUCTION

Health and health care are surrounded by a lot of risk, implying that risk aversion plays a central role in health economics. Recently, several studies have convincingly shown that also some concepts beyond risk aversion, such as prudence (i.e., downside risk aversion or a preference for separating a mean-zero risk from a fixed loss, equivalent under expected utility to a positive sign of the third derivative of the utility function), are much more important than previously thought (Eeckhoudt & Schlesinger, 2006; Trautmann & van de Kuilen, 2018). These concepts relate to higher moments of a distribution than just variance, such as skewness and kurtosis, and are therefore coined higher-order risk attitudes. Hence, the necessity to look beyond second-order risk attitudes has become clear, also in the health care field. This knowledge is important for several reasons. First, it allows to test if the quality-adjusted life year (QALY) model represents individual health preferences, and hence if QALYs are a proper metric to value health improvements. Related to this, the current conduct of cost-effectiveness analysis (CEA) is usually to assume the QALY model without allowing for risk aversion for quality of life (QoL) or longevity, or third- and fourth-order risk attitudes. The same holds for the value of a statistical life (VSL) literature, where risk neutrality is typically assumed and the marginal value of a change in survival at a point in time is independent of the baseline survival level (Rosen, 1988). If individuals are instead risk averse in QoL, the cost-effectiveness threshold and the willingness to pay for marginal gains in QoL would vary with baseline health status (Lakdawalla & Phelps, 2020). Likewise, risk aversion for longevity increases the willingness to pay to avoid early death (Bommier & Villeneuve, 2012), while it can explain the sizable private healthcare expenditures at the end of life (Córdoba & Ripoll, 2017). Second, higher-order risk attitudes are relevant to many everyday health care decisions, such as risky treatment choices to combat a disease in the face of comorbidities. It is well known that many people suffer from two or more diseases at the same time (MacMahon, 2018), which may influence their preferences for treating their primary disease.

Courbage and Rey (2006) pointed out that the level of prudence is a main determinant of the optimal level of prevention for health risks, and Pauker (2014) advocated higher-order risk attitudes as a research topic that should receive priority on the research agenda in the domain of medical decision making. Moreover, Bleichrodt, Crainich, and Eeckhoudt (2003) have shown the importance of higher-order risk attitudes in treatment decisions in the presence of comorbidities influencing life expectancy. They demonstrated that economic evaluations and medical decision analyses that ignore comorbidities will lead to recommendations that are biased in the direction of too much treatment if aversion to health status risks increases with life expectancy. They also derived several predictions regarding treatment decisions under particular assumptions, but so far these predictions had not yet been tested empirically. In addition, Eeckhoudt et al. (2007) showed how investment in tertiary preventive care (i.e., the treatment of an established or chronic disease in order to minimize the negative health consequences of the disease) depends on cross-prudence of health and income, that is it depends on whether an individual prefers to disaggregate a zero-mean income risk and a fixed health reduction, or equivalently has a positive third cross-derivative of income with respect to health.

Krieger and Mayrhofer (2012) have explored higher-order risk attitudes in a health context empirically and observed both risk aversion and prudence. However, they only studied univariate risk attitudes and no multivariate risk attitudes, whereas in many settings a decision maker actually faces more than one attribute (Keeney & Raiffa, 1993). Eeckhoudt et al. (2007) and Ebert and van de Kuilen (2015) have stressed the importance of multi-attribute decision making, given the high prevalence of decisions where more than one attribute is involved. In the health domain, for instance, the widely used QALY model, which is the recommended metric to be used in health economic evaluations (Sanders et al., 2016), involves the attributes longevity and QoL.

In case of two attributes, correlation aversion means that an individual prefers a 50% chance of a loss in one attribute and a 50% chance of a loss in the other attribute over a 50–50 gamble offering a loss in neither attribute or a loss in both (Eeckhoudt et al., 2007). An example of correlation aversion in health is when a patient prefers a lottery where he will get either a lower QoL (50% chance) or a shorter life expectancy (50% chance) over a lottery where he has a 50% chance to get both a health deterioration and a lower life expectancy at the same time, and 50% chance to get no health losses at all. Bleichrodt, Crainich, and Eeckhoudt (2003) showed that various consequences of the QALY model can be tested by obtaining knowledge about higher-order (cross-) derivatives of the utility function for longevity and QoL. One of their predictions was that people are risk averse for both longevity and QoL, which are both established theoretical predictions (Lakdawalla & Phelps, 2020; Miyamoto & Eraker, 1988) that have been empirically confirmed in several studies (Attema et al., 2012, 2013, 2016; Bleichrodt & Pinto, 2005; Rouyard et al., 2018; Schosser et al., 2016; Wakker & Deneffe, 1996). Another prediction they made is correlation seeking for the combination of these two attributes. That is, people would prefer to combine a bad [good] health state with a short [long] life duration over mixing these two.1 The risk apportionment technique allows us to test these predictions.

Bleichrodt, Crainich, and Eeckhoudt (2003) also showed that, according to the QALY model, risk aversion for QoL should not depend on having a comorbidity that only affects longevity. In addition, they predicted that decreases in the riskiness of longevity caused by this comorbidity will generally lead to more treatment-prone behavior (i.e., people get less risk averse for QoL). Finally, Bleichrodt, Crainich, and Eeckhoudt (2003) derived how risk aversion, and hence treatment intensity, depend on higher-order multivariate risk preferences (i.e., risk aversion, correlation aversion, cross-prudence, and cross-temperance – a preference for disaggregating a zero-mean longevity risk and a zero-mean QoL risk).

Attema et al. (2019) recently applied the risk apportionment technique to the health field, when they measured multivariate risk preferences, up to the fourth order, for longevity and wealth. They reported substantial risk aversion and correlation aversion for gains, but the opposite was found for losses. Furthermore, they observed less substantial amounts of prudence and temperance, but still significantly more than 50%. However, that study only investigated the duration component of the QALY model and hence could not test all the propositions from Bleichrodt, Crainich, and Eeckhoudt (2003). In fact, to the best of our knowledge, no assessments of (cross-)prudence and (cross-)temperance are available yet for QoL.

In this paper we are the first to empirically study several higher-order properties of the QALY model. This design enables us to test the theoretical predictions put forward by Bleichrodt, Crainich, and Eeckhoudt (2003). In a nutshell, we combine an implementation of the risk apportionment technique with a treatment intensity task in a lab experiment, in which we measure risk aversion for QoL for different life durations. First, we obtain evidence on individuals' correlation attitude between longevity and QoL. Second, we elicit their third- and fourth-order multivariate risk attitudes, that is, cross-prudence and cross-temperance. Finally, we measure preferred treatment intensity for treating a disease affecting only QoL for patients also suffering from a comorbidity which affects longevity. Here, a higher treatment intensity increases the spread in the potential QoL outcomes. The latter measure enables us to test several theoretical predictions based on the QALY model as suggested by Bleichrodt, Crainich, and Eeckhoudt (2003).

Our results show that subjects have marked risk preferences for longevity and QoL. First, we find a lot of risk aversion for both attributes, confirming most theoretical models. Second, we confirm Bleichrodt, Crainich, and Eeckhoudt's (2003) prediction of correlation seeking, with an overwhelming majority of subjects showing this preference. Furthermore, in contrast to most studies using monetary outcomes, we also find highly significant evidence for cross-imprudence and cross-intemperance. However, we observe no systematic correlation between treatment intensity and duration. Finally, we observe a marginally significant relation between treatment intensity and riskiness of life duration, in agreement with the intuition of Bleichrodt, Crainich, and Eeckhoudt (2003).

2 METHOD We assume preferences satisfy a weak-order, that is they are complete and transitive. Individuals care about QoL (q) and longevity (t). According to the QALY model, preferences for chronic health states are evaluated by: (1)

If expected utility holds, a subject is risk averse for QoL if and risk averse for longevity if . Prudence for QoL holds if , prudence for longevity implies and temperance holds if for QoL and for longevity. Concerning multivariate risk preferences, a subject is correlation averse if , cross-prudent for longevity if , cross-prudent for QoL if , and cross-temperate if . Opposite signs define correlation seeking, cross-imprudence and cross-intemperance, respectively. Throughout this paper, we only consider health states better than dead, that is, we assume utility is increasing in life duration: .

The general QALY model of Eq. (1) does not give any prediction about univariate (, . , , ) or multivariate () risk preferences For instance, in addition to (i.e., correlation seeking), we have (i.e., cross-imprudence for longevity) in case of risk aversion for QoL, and (i.e., cross-imprudence for QoL) in case of risk aversion for longevity and finally, if the decision maker is risk averse in both longevity and QoL. The linear QALY model, , which is often applied in economic evaluations, provides more specific predictions. The linear QALY model implies that  = 0, that is, people are risk neutral with regard to longevity. From this it follows that , , , and are also 0 for the linear QALY model, while if , and in case of risk aversion for QoL.

Eeckhoudt and Schlesinger (2006) were the first to operationalize (higher-order) risk preferences in terms of choices between two binary lotteries with equally likely outcomes that distribute harms and benefits differently, as illustrated below. An example of an item revealing risk aversion for QoL is the following (Table 1):

TABLE 1. Question to test for risk aversion for QoL What is your most preferred alternative? Option A Option B 50%: Live with 40% of full health for 40 years 50%: Live with 30% of full health for 40 years 50%: Live with 50% of full health for 40 years 50%: Live with 60% of full health for 40 years Note: Bold text shows the answer revealing risk aversion for QoL. Abbreviation: QoL, quality of life.

Here, the risk averse individual would choose Option A, because it offers the same expected QoL as Option B (i.e., 45%), but with a lower spread. In fact, Option B is a mean-preserving spread of Option A. the general idea of the risk apportionment method is to have these kinds of choices between two-outcome gambles, with one resulting from the other from a mean-preserving spread. Similarly, risk aversion for longevity could be determined by gambles such as the following (Table 2):

TABLE 2. Question to test for risk aversion for longevity What is your most preferred alternative? Option A Option B 50%: Live with 60% of full health for 40 years 50%: Live with 60% of full health for 30 years 50%: Live with 60% of full health for 40 years 50%: Live with 60% of full health for 50 years Note: Bold text shows the answer revealing risk aversion for longevity.

In this example, Option A is riskless and Option B involves a mean-preserving spread of the same longevity. The risk apportionment method also allows for eliciting higher-order risk attitudes by adding different sources of uncertainty. For example, prudence for longevity can be elicited by the following choice (Table 3):

TABLE 3. Question to test for prudence for longevity What is your most preferred alternative? Option A Option B 50%: Live with 60% of full health for 40 years 50%: Live with 60% of full health for 30 years OR 50 years 50%: Live with 60% of full health for 10 OR 30 years 50%: Live with 60% of full health for 20 years Note: Bold text shows the answer revealing prudence for longevity.

In this case, QoL is always 60% and longevity is either 40 or 20 years. The choice involves distributing a zero-mean longevity risk of ±10 years to the bad longevity outcome (20 years, Option A) or the good longevity outcome (40 years, Option B). The former choice reflects imprudence and the latter choice reflects prudence. Similarly, temperance can be elicited by including two independent longevity or QoL risks and determining if the respondent prefers to aggregate (intemperance) or disaggregate (temperance) these risks. An example is shown in the Appendix.

Eeckhoudt et al. (2007) have demonstrated that the risk apportionment method can also be extended to elicit (higher-order) cross-risk attitudes when risk in both attributes is involved. For example, consider the following gamble (Table 4):

TABLE 4. Question to test for correlation aversion What is your most preferred alternative? Option A Option B 50%: Live with 60% of full health for 40 years 50%: Live with 60% of full health for 20 years 50%: Live with 30% of full health for 20 years 50%: Live with 30% of full health for 40 years Note: Bold text shows the answer revealing correlation aversion.

This gamble involves risk in both QoL (30% or 60%) and longevity (20 or 40 years). The essential choice is if one prefers to combine the good outcome for QoL with the good outcome for longevity, while at the same time combining the bad outcomes for both (Option A), or if one prefers to spread the risks and combine the good outcome for the one attribute with the bad outcome for the other attribute (Option B). The former is deemed correlation seeking and the latter correlation aversion. Tests of cross-prudence and cross-temperance can be conducted in a similar fashion. The below question could for instance be used for cross-prudence for longevity (Table 5).

TABLE 5. Question to test for cross-prudence for longevity What is your most preferred alternative? Option A Option B 50%: Live with 60% of full health for 30 years 50%: Live with 40% OR 80% of full health for 30 years 50%: Live with 40% OR 80% of full health for 40 years 50%: Live with 60% of full health for 40 years Note: Bold text shows the answer revealing cross-prudence for longevity.

Looking closely, we can see that one lives either 30 or 40 more years in both gambles. Furthermore, QoL may be 60% or it may be another gamble, resulting in either 40% or 80%. In effect, a zero-mean risk on QoL ( ∼ ±20%) has to be apportioned to either the good outcome of the gamble (i.e., t = 40 years, Option A) or the bad outcome of the gamble (i.e., t = 30 years, Option B). Someone who prefers to combine the zero-mean risk with the good longevity outcome is said to be cross-prudent for longevity, whilst someone who prefers combining the zero-mean risk with the bad longevity outcome is called cross-imprudent for longevity. Tests for cross-prudence for QoL and (cross-)temperance can be done similarly, as shown in the Appendix.

Embedded in our study is the assumption that, generally, individuals prefer both higher levels of longevity and higher levels of QoL. While this method relies on the assumption that individuals aim to maximize their utility, it does not require assumptions about the functional form of the utility function (Attema et al., 2019). The risk apportionment technique can also be applied to elicit the other traits mentioned above.

In order to test the other predictions of Bleichrodt, Crainich, and Eeckhoudt (2003), as described in the introduction, we elicit the sign of several (higher-order) risk traits. Table 6 gives an overview of all traits we elicited and the associated implications for the utility function in case of EU.

TABLE 6. Overview of elicited traits and their implied EU condition Trait if prospect 1 is chosen Prospect 1 Prospect 2 EU condition prospect 1 is chosen Risk aversion for QoL () (0.5,) Risk aversion for longevity (t ) (0.5, ) Correlation aversion () Cross-prudence for QoL () Cross-prudence for longevity (t) () Cross-temperance (,) Abbreviation: QoL, quality of life.

In Table 6, Prospect 1 of the first row denotes a prospect where the subject has 50% probability to live with a QoL of for T years, and 50% to live in QoL of for T years. The other prospect of this first row is riskier, since it involves a lower minimum () and a higher maximum (q). The other prospects can be interpreted similarly. For cross-prudence and cross-temperance, and , denote zero-mean risks on longevity and QoL, respectively.

In the model of Bleichrodt, Crainich, and Eeckhoudt (2003), patients can choose the intensity n of a treatment combatting a disease. This only affects their QoL q and is risky, since it can either be effective, improving the patient's health by b*n, or it can be detrimental due to side effects, in which case the patient's health will deteriorate by c*n.2 Hence, the amount of upside and downside potential depends on the treatment intensity chosen by the patient; the higher the intensity, the more extreme the outcomes will be. In this study we test the predictions of the (linear) QALY model, as shown by Bleichrodt, Crainich, and Eeckhoudt (2003), by asking subjects to choose the amount n in this decision context, for different life durations t. For instance, in one of the questions the subject had to choose n such that they would live 20 more years with q , with n measured in percentages, and b = 0.4, c = −0.1; for example, n = 50% would correspond to . Repeating this for several durations t, we could test the correlation with the risk traits from Table 6.

3 EXPERIMENT 3.1 Subjects

Participants were recruited randomly through a faculty internal recruitment system available to all undergraduate business students at the Rotterdam School of Management. As an incentive for taking part, participants were awarded with course credits. On arrival at the laboratory, a maximum of four students completed the procedure in the same room. A total of 124 students took part in the study. For two subjects, a program failure occurred during data collection. One student re-contacted us, asking to be excluded from the study because he had not answered faithfully. Therefore, a total of three cases were excluded from the study. The final sample size was N = 121 (51.2% female). The average age of participants was 20.1 years (SD = 1.44). n = 19 participants reported a physical health condition (16.0%), and n = 7 a mental health condition (5.8%), and the average self-reported QoL on the visual analog scale ranging from 0 (death) to 100 (best possible health) was 83.48 (SD = 9.57). The average BMI was 21.52 (SD = 2.26), and n = 13 participants were considered underweight (10.7%), while n = 9 were considered overweight (7.4%).

3.2 Procedure

Subjects were first asked to provide their informed consent and signed a form of solemn commitment. Signing such a solemn commitment has been shown to increase diligent responding (Jacquemet et al., 2018, 2019). Subsequently, subjects received instructions to complete a part eliciting their risk attitudes and treatment proneness and completed 5 practice questions (1 for risk aversion with respect to QoL, 1 for correlation attitude, 1 for cross-prudence, 1 for cross-temperance, and 1 for treatment intensity). The order of the tasks was randomized. Within each trait, questions were not interspersed to avoid subjects having to switch between tasks continuously. Within each part, the questions were randomized. At the end of this part, four questions were repeated in order to test consistency (one for question on correlation attitude, one one cross-prudence for longevity, one on risk aversion for longevity and one for treatment intensity). The experiment was programmed in Matlab. A researcher was in the room with the participants during all sessions.

3.3 Stimuli

For all tasks, we took a QoL level of q = 60% of full health to be the base QoL. For longevity, this base was t = 40 life years. As a result, risk aversion for QoL was elicited by fixing longevity at 40 years while varying the variance of QoL. Likewise, risk aversion for longevity was assessed by fixing QoL at 60% while varying the variance of longevity between the options. A similar procedure was used for the other traits. Table 7 shows the stimuli for all traits.

TABLE 7. Stimuli for the risk apportionment tasks Taska Trait Prospect A Prospect B 1 Risk aversion for QoL [] [] 2 [] [] 3 [] [] 4 Risk aversion for longevity [] [] 5 [] [] 6* [] [] 7 Correlation attitude [] [] 8 [] [