Probabilistic choice models in health-state valuation research: background, theories, assumptions and applications

Abstract

Interest is rising in measuring subjective health outcomes, such as treatment outcomes that are not directly quantifiable (functional disability, symptoms, complaints, side effects and health-related quality of life). Health economists in particular have applied probabilistic choice models in the area of health evaluation. They increasingly use discrete choice models based on random utility theory to derive values for healthcare goods or services. Recent attempts have been made to use discrete choice models as an alternative method to derive values for health states. In this article, various probabilistic choice models are described according to their underlying theory. A historical overview traces their development and applications in diverse fields. The discussion highlights some theoretical and technical aspects of the choice models and their similarity and dissimilarity. The objective of the article is to elucidate the position of each model and their applications for health-state valuation.

Keywords::

• choice models
• health state
• quantification
• subjective measurement
• utilities
• values

Acknowledgement

We would like to thank four anonymous reviewers for their insightful and excellent suggestions on an earlier draft of this manuscript.

Financial and competing interests disclosure

The authors have no relevant affiliations or financial involvement with any organization or entity with a financial interest in or financial conflict with the subject matter or materials discussed in the manuscript apart from those disclosed.

No writing assistance was utilized in the production of this manuscript.

Appendix

A. Law of comparative judgment

Thurstone [1] proposed that perceived physical phenomena or subjective concepts (e.g., health states, treatment outcomes and process characteristics) can be expressed as follows:

(1)

where θ_i is the true weight of an object (e.g., item, stimulus, health state) i, α is the measurable component of that weight for the object i, and ϵ is a random error term. The assumption in the model proposed by Thurstone is that ϵ is normally distributed. This assumption yields the binomial probit model.

In Thurstone’s terminology, choices are mediated by a ‘discriminal process’. He defined this as the process by which an organism identifies, distinguishes or reacts to stimuli. Consider the theoretical distributions of the discriminal process for any two objects, like two different health states i and j. In the LCJ model, the standard deviation of the distribution associated with a given health state is called the discriminal dispersion of that health state. Discriminal dispersions may be different for different health states.

Let θ_i and θ_j correspond to the scale values of the two health states. The difference θ_i – θ_j is measured in units of discriminal differences. This difference process, θ_i – θ_j = (α_i − α_j) + (ϵ_i − ϵ_j), is normally distributed with mean θ_i – θ_j and variance corresponding to

(2)

Thurstone stated that the relation between the difference in the means of what he called the discriminable process, θ_i – θ_j, the z score of the probability of selecting the one object as larger (better) than the other, and the variance and correlations of the random variables θ_i and θ_j can be modeled. This is known as the law of comparative judgment:

(3)

where θ_i, θ_j denotes the standard deviations of the two stimuli (health states) i and j, ρ_ij denotes the correlation between the pairs of discriminal processes i and j, and z_ij is the unit normal deviate corresponding to the theoretical proportion of times health state j is judged greater than health state i. This basic form of the model can be represented as, θ_i – θ_j = z_ij, for which the probability that object j is judged to have more of an attribute than object i is

(4)

where ϕ is the cumulative normal distribution with mean zero and variance unity.

B. Bradley–Terry–Luce model

While the probit model (by Thurstone) has normally distributed error terms, a logit model is simply a log ratio of the probability of choosing a stimulus to the probability of not choosing a stimulus. If P is a probability, then P/(1 − P) is the corresponding odds, and the logit of the probability is the logarithm of the odds. The logit function is defined as the inverse of the logistic function. The logistic model is not linear, nor additive. Rather, it assumes an S-shaped response curve. One of the reasons the logit model was formulated was its’ ease of use. In comparison, probit models require the computation of integrals, which is why these models were less often used in the past. Modern computing however has made this computation fairly simple. The main difference between the logit and probit models lies on the distributional assumption of the error term. Consequently, the weighting of the cumulative probabilistic curve is different, as the logistic distribution tends to be a little flat tailed. The coefficients obtained with these two models are actually fairly close in most cases.

In the Bradley–Terry–Luce model [2,3], the probability that object i is judged to have more of an attribute than object j is:

(5)

where θ_i and θ_j are respectively the scale values or weights of the two objects.

C. Rasch model

In the Rasch model for dichotomous data [4], the probability that the outcome is correct (or better than another) is given by:

(6)

where η_n identifies a characteristic of the person n, as for instance his or her ability or the quality of his or her health status, and θ_i refers to the item i, as for instance the difficulty of an item (or seriousness of a health state). By an interactive conditional maximum likelihood estimation approach, an estimate θ_i − θ_j is obtained without involvement of η, which is specific to the Rasch model. This estimation approach leads to invariance: a fundamental aspect of measurement [5].

D. Multinomial logit models versus conditional multinomial logit models

There is much confusion in the literature about the differences and similarities between multinomial and conditional logit models. The authors have contacted several experts in this field of research and received almost as many different explanations as experts approached. Multinomial logistic and conditional (multinomial) logistic regression models are different but often the terminology to describe the model is used differently or incorrectly. In fact, the term multinomial logit is quite confusing because different fields and people use it to refer to different things. The term conditional logit unfortunately includes a wide array of sub-models that depend on whether certain effects of interest are generic or differ for at least one of the choice alternatives.

The term multinomial logit (MNL) model refers to a regression logit model that generalizes logistic regression by allowing more than two discrete outcomes. This model assumes that data are case specific; that is, each independent variable has a single value for each case. Consider an individual choosing among K alternatives (e.g., health states) in a choice set. Let X_j represent the characteristics of individual j and β_k the regression parameters (each of which is different, even though X_j is constant across alternatives):

(7)

Let P_jk denote the probability that individual j chooses alternative k. The probability that individual j chooses alternative k is

(8)

It is important to clarify some terminology. The model mentioned and used for behavioral modeling of polytomous choice situations, developed by McFadden [6], is generally called MNL. Yet some important distinctions have to be made between the (conventional) MNL model and the conditional MNL model. Although the McFadden model is often simply referred to as the MNL model, this refers to the conditional model. In conditional logistic regression, none, some or all of the observations in a choice set may be labeled. Thus, McFadden’s choice model (discrete choice) is a special case of conditional logistic regression (conditional logistic analysis is also applied in epidemiology when analyzing matched case control data). In the conditional logit model, θ is a single vector of regression coefficients; the explanatory variables Z assume different values for each alternative; and the impact of a unit of Z is assumed to be constant across alternatives:

(9)

The probability that the individual j chooses alternative k is

(10)

Both models can be used to analyze the choice of an individual among a set of K alternatives. The central difference between the two is that the conventional MNL model focuses on the individual as the unit of analysis and uses the individual’s characteristics (e.g., gender, age, religion) as explanatory variables. In contrast, the conditional MNL model focuses on the set of alternatives for each individual and the explanatory variable comprises characteristics of those alternatives. This is the typical mechanism (see Figure 1) that seems required in the case of measurement (health-state valuation), whereas the conventional MNL model is used for the prediction of choice behavior. It is possible to combine the two models to simultaneously take into account both the alternatives’ and the individual’s characteristics as explanatory variables. This is called a mixed-logit model:

(11)

Where U_jk is the utility of the alternative k assigned by the individual j. U_jk depends on both the alternatives’ characteristics X and on the individuals’ characteristics Z, plus a nonestimable part represented by ϵ.

In addition to the mixed-logit model (where ‘mixed’ refers to characteristics), where both respondents’ and stimuli characteristics are being taken into account, an even more general model is the logit-mixture model (where ‘mixture’ refers to the distributions of error terms). This model also takes individual taste variation into account, by partitioning the error term in a random part (or any other type of distribution) and an extreme value part. The model has the following form:

(12)

where βx_jk is the systematic component of the utility (which can include both respondent and attribute characteristics) and µ_jz_jk and ϵ_jk are error terms; µ is a vector of random terms with a mean of zero (or of any other distribution than the normal distribution) and ϵ_jk is IID and has an extreme value type 1 distribution. The component µ_jz_jk allows for the induction of heteroscedasticity and correlation across the random part of the utility of the different alternatives in the choice set. It is this model that in the literature is most often referred to as the mixed-logit model. As stated before, these types of models, with a component directed on the prediction of respondent characteristics, are less valuable in the case of the measurement (valuation) of health states, but of course, may be very relevant for evaluation research in general.

Interestingly, the conditional multinomial logistic model could be extended to analyze ordinal preferences. Accordingly, it is conceivable that rank orderings can be generated by a process in which an individual first chooses his most preferred alternative from all available alternatives. From the remaining alternatives he again chooses his most preferred one – thereby stating his second preference – and so on, until there is only one remaining alternative, which is, of course, his last preference. Thus, an observed preference order can be understood as being generated by a repeated selection process in which the best alternative is always chosen and subsequently deleted from the choice set. The later decisions are assumed to be independent of the previous ones, which is to say that IIA holds. This model is also called ‘conditional logit’, ‘exploded logit’ or ‘rank-ordered logit’, as the ranking of K states is exploded into K − 1 decision stages (see Figure 3). The contribution of using a rank-ordered logit model is that more information is incorporated in the estimation of the representative function compared with the standard logit models.

E. Independence of irrelevant alternatives

The IIA property, which arises from the assumption of independent random errors and equal variances for the choice alternatives (IID assumption), implies that the odds of choosing one alternative over another must be constant regardless of whatever other alternatives are present [3]. To give an example put forward by Debreu [7] where IIA does not hold: suppose an individual wants to buy a CD, and she is equally likely to choose a Beethoven or a Debussy recording (Pr{B|B, D} = Pr{D|B, D} = 0.5). Now suppose that she encounters a second Beethoven recording that she likes just as much as the first (Pr{B₁|B₁, B₂} = Pr{B₂ |B₁, B₂} = 0.5). If she were rational, how would she choose among all three recordings {B₁, B₂, D}? We would expect Pr{B₁|B₁, B₂, D} = 0.25, Pr{B₂|B₁, B₂, D} = 0.25 and Pr{D|B₁, B₂, D} = 0.5. However, IIA implies that Pr{B₁|B₁, B₂, D} = 1/3, Pr{B₂|B₁, B₂, D} = 1/3, and Pr{D|B₁, B₂, D} = 1/3 (in this context this makes perfect sense, as a second Beethoven recording is unlikely to be irrelevant from the first). This IIA assumption may be too restrictive in practical situations can be unrealistic in many settings. The outcomes that could theoretically violate IIA (such as the outcome of multicandidate elections, or according to Arrow [8] any choice made by humans) may make conditional MNL an invalid estimator. Nonetheless, when IIA reflects reality, it offers many advantages, but whether IIA holds in a particular setting is an empirical question amenable to statistical investigation. There seems to be ample scope for research aimed at developing models that allow for managing contexts where IIA may not hold. Some models such as the logit-mixture relaxed the assumption of IIA. This means that these models can allow for random taste variation, correlations in unobserved factors over time and unrestricted substitution patterns. McFadden and Train showed that given an appropriate specification of variables and distribution of coefficients, a logit-mixture can approximate to any degree of accuracy any true random utility model of discrete choice [9].

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