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# Transitivity in health utility measurement: An experimental analysis

- Ulrich Schmidt
^{1, 2}and - Michael Stolpe
^{1}Email author

**1**:12

https://doi.org/10.1186/2191-1991-1-12

© Schmidt and Stolpe; licensee Springer. 2011

**Received:**28 February 2011**Accepted:**30 August 2011**Published:**30 August 2011

## Abstract

Several experimental studies have observed substantial violations of transitivity for decisions between risky lotteries over monetary outcomes. The goal of our experiment is to test whether these violations also affect the evaluation of health states. A particular feature of our experimental design is that it takes into account the possible role of decision errors for generating violations of transitivity. Since we find neither substantial nor systematic deviations from transitive choice behaviour, we can conclude that previously reported violations do not seem to bias health utility measurement.

## Keywords

- Transitivity
- health utility
- errors

## 1 Introduction

Health utility measurement plays an important role in medical decision making, in particular in cost-effectiveness analyses of alternative treatments. A central assumption in health utility measurement is that preferences satisfy transitivity. Transitivity demands that whenever option A is preferred to option B and B is preferred to C, then A has to be preferred to C. In the absence of transitivity, a well-defined utility function U (i.e. for all options A and B we have U(A) ≥ U(B) if and only if A is weakly preferred to B) does not exist. Consequently, standard methods in health utility measurement, such as the time tradeoff method or quality adjusted life years, cannot be meaningfully applied in the absence of transitivity. Several empirical studies observed substantial violations of transitivity, in particular for choice between risky options (e.g. [1–8]). The validity of health utility measurement would be seriously challenged if these violations carried over to the evaluation of health states: No consistent rankings of health states could be established and meaningful outcomes measures in many applications of cost-effectiveness analysis would simply become unavailable. However, since the evaluation of health states involves consequences composed of several attributes (i.e. at least health status and life duration), behaviour may be fundamentally different than in the above mentioned studies which are based on monetary consequences.

To the best of our knowledge, it has not been analyzed before whether transitivity empirically holds for the evaluation of health states. However, several studies observed preference reversals in health utility measurement (e.g. [9–11]) which are closely related to violations of transitivity. Preference reversals are an intensively discussed phenomenon in decision making under risk. Here, a preference reversal occurs if a risky option p is preferred to another risky option q (i.e. p ≻ q) in a straight choice, but a higher certainty equivalent (CE) is assigned to q (i.e. CE(q) > CE(p)). Assume such a reversal and consider an amount z with CE(q) > z > CE(p). Since the certainty equivalent of an option should be indifferent to that option, we get q ~ CE(q) ≻ z, z ≻ CE(p) ~ p, and p ≻ q from choice. Hence, we have an intransitive cycle p ≻ q ≻ z ≻ p, i.e. an intransitive preference cycle. Precisely this result has been utilized by many studies mentioned above-in particular by [2, 3, 5, 8]-in order to derive experimental designs from the preference reversal literature where substantial violation rates of transitivity have been observed. In one example from [3], 28 subjects exhibited the cycle p ≻ q, z ≻ p, and q ≻ z, whereas only one subject exhibited the opposite pattern-with lotteries given by the following state-dependent payoff-triples: p = (7.5, 7.5, 1), q = (10, 3, 3) and z = (5, 5, 5), where the first state has a probability of 0.4 and the second and third states a probability of 0.3 each. The motivation of our paper is that the existing evidence of preference reversals in health utility measurement suggests violations of transitivity might also occur in the evaluation of health states.

There is, however, some debate concerning the evidence of violations of transitivity [12–19]. In particular, it has been argued that the observed violations are not "true" violations since they may be simply caused by random errors. Consequently, we develop an experimental design which allows us to discriminate between true violations and violations caused by random error. Our design is presented in the next section where we also discuss the role of errors as potential causes of transitivity violations. Section 3 reports the results of our experiment and Section 4 offers some concluding observations.

## 2 Experimental Design and the Role of Errors

Experimental Design of Series I (left panel) and Series II (right panel)

Prob (Death) | Years | Migraine | Prob (Death) | Years | Migraine | ||
---|---|---|---|---|---|---|---|

A | 0 | 15 | 1 | A | 15 | 30 | 3 |

B | 33 (35) | 20 | 0 | B | 25 | 35 | 2 |

C | 13 (10) | 30 | 2 | B | 22 (20) | 24 (25) | 1 |

- (1)
A ≻ B ⇔ #{i: Ai ≻ Bi} > #{i: Bi ≻ Ai},

where A_{i} and B_{i} denote the attribute values of A and B. It is easy to see that for the values in Table 1 we have for Series I according to the majority rule B ≻ A (more years and less migraine), C ≻ B (more years and lower probability of death) and A ≻ C (lower probability of death and less migraine). Also for Series II the majority rule implies the cycle B ≻ A ≻ C ≻ B.

In the pretest, all subjects received an identical booklet with all six choices, each choice on a separate sheet. In the main test there were two booklets, one with three choices (two from one series and one from the other series) at the beginning of class and a second booklet with the remaining three choices at the end of class. This procedure ruled out that subjects could make consistency checks. The main test also divided subjects into two groups, to control for ordering effects. Compared to the first group, the order of the two booklets and the order of alternatives in the booklets were reversed in the second group. Since there are no significant differences in the results for these two groups we do not distinguish them in the following. Finally, in order to motivate subjects for careful consideration of the questions, 12 subjects drawn randomly received a flat payment of 20 Euros in the main test.

Results of Starmer and Sugden (1998) [5]

ABA | ACA | BBA | BBC | ACC | BCC | ABC | BCA | |
---|---|---|---|---|---|---|---|---|

Case 2 | 5 | 2 | 12 | 9 | 4 | 45 | 3 (3.3%) | 10 (11.1%) |

To control for the possible role of errors, [18, 19] estimate an explicit error model using data from an identically repeated experiment. However, this procedure requires additional assumptions on the structure of the error model. Moreover, it is unclear whether the error model does accommodate real behaviour sufficiently well, and if not, it is hard to interpret the results. We therefore follow a different route in this article: We simply adjust the attributes of the alternatives such that the number of subjects choosing patterns ABA, BBC, or ACC roughly equals the number of subjects choosing the other three transitive patterns ACA, BBA, or BCC. In this case random errors would imply an approximately equal frequency of the two intransitive patterns so that we can use the cycling asymmetry to test for intransitivities. Given the categorical nature of our observations, we shall use a one-tailed binomial test to check if the null of equal frequency is rejected in favour of a greater frequency of one intransitive pattern in line with the cycling asymmetry.

## 3 Results

### (i) the pretest

Results of the Pretest

ABA | ACA | BBA | BBC | ACC | BCC | ABC | BCA | |
---|---|---|---|---|---|---|---|---|

Series I | 0 | 11 | 5 | 2 | 14 | 4 | 2 (5.0%) | 2 (5.0%) |

Series II | 2 | 8 | 2 | 5 | 2 | 21 | 0 | 0 |

### (ii) the main test

Results of the Main Test

ABA | ACA | BBA | BBC | ACC | BCC | ABC | BCA | |
---|---|---|---|---|---|---|---|---|

Series I | 9 | 15 | 12 | 14 | 29 | 13 | 5 (5.1%) | 1 (1.0%) |

Series II | 20 | 6 | 7 | 20 | 7 | 30 | 6 (6.1%) | 2 (2.0%) |

In both series, the cycle ABC occurs more frequently than the cycle BCA. However, according to a one-tailed binominal test (which was also used by [5] and is appropriate for observations falling into two categories, such as ours, where only a *greater* sample frequency of observations in one category than expected under the null hypothesis justifies rejection), this difference is not statistically significant, so we can not conclude that a cycling asymmetry exists. Since the rarely observed cycle BCA is the one implied by the majority rule for our design, this rule appears to perform rather poorly in our experiment.

## 4 Conclusions

This paper has presented an experiment aimed at testing transitivity in the valuation of health states, as required in health utility assessments. A particular feature of our design is the use of a balanced set of alternatives such that an asymmetric frequency of the two intransitive cycles is unlikely to result from random errors. In contrast to many previous experimental studies, we find neither a substantial frequency of subjects violating transitivity nor a significant cycling asymmetry. Further research may be required to understand whether the absence of the intransitivity problem is mainly due to the control for errors in our design or due to the fact that our alternatives are not lotteries over monetary amounts, as in previous studies, but composed of health states. Moreover, additional studies of transitivity in the health domain would be useful trying to overcome some of the obvious limitations of our study which include a non-representative sample of students and the small number of stimuli. In particular, the fact that many of our subjects had no experience with the presented health states (i.e. days per week with migraine) may have contributed to our results.

Altogether, we do not find evidence that people's evaluation of discrete health states is substantially biased by violations of transitivity. This finding is in line with other recent experimental studies [18–20] that test transitivity while controlling for the possible role of errors. If transitivity holds, the evaluation of any one alternative does not depend on the other alternative with which it is compared. This is a necessary prerequisite for many tools and concepts in the health domain like quality-adjusted life years or cost-effectiveness analyses. It is therefore encouraging that there really does not seem to be much evidence for intransitive choice being a reality.

## Declarations

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.