Multiplicative Utility for Interacting Attributes

Why Model Interaction Between Clinical Attributes?

An additive benefit-risk model treats each attribute’s contribution as separate: the utility gain from improving one attribute does not depend on the levels of the others. That can be a useful and transparent starting point, but preferences may behave differently. A patient may value improved symptom control especially when treatment also avoids a serious adverse event. Alternatively, an added benefit may matter less when another outcome is already very favorable. A multiplicative utility model represents this kind of dependence in the way attribute preferences combine.

The interaction in this model is a feature of the preference aggregation, not evidence that one clinical outcome causes another. The model combines attribute utilities; it does not, by itself, describe clinical mechanisms or the statistical dependence of observed outcomes.

Definition: Let $u_i(x_i)$ be the single-attribute utility for clinical attribute $i$, scaled from 0 at its least-preferred level to 1 at its most-preferred level. A normalized multiplicative multi-attribute utility function is $$ U(x_1,\ldots,x_m)=\frac{\prod_{i=1}^{m}\left(1+K w_i u_i(x_i)\right)-1}{K},\qquad K\ne 0, $$ where $w_i$ is the utility of the outcome with attribute $i$ at its best level and every other attribute at its worst level. The parameter $K$ is constrained by $$ 1+K=\prod_{i=1}^{m}(1+K w_i). $$ This constraint makes the utility of the all-best outcome equal to 1. When $K=0$, the limiting form is additive: $U=\sum_{i=1}^{m}w_i u_i(x_i)$, with $\sum_{i=1}^{m}w_i=1$.

Meaning and Elicitation of the Interaction Parameter

For two attributes, write their normalized utilities as $u_1$ and $u_2$, and their single-attribute utilities as $a$ and $b$. Expanding the multiplicative form gives a particularly clear expression:

U=a u_1+b u_2+c u_1u_2,\qquad c=K ab.

Because $U(1,1)=1$, the coefficients obey $a+b+c=1$. The term $c u_1u_2$ is the interaction contribution. A positive $c$ increases utility when both attributes are favorable; a negative $c$ reduces the combined utility relative to the additive contribution. In this parameterization, $a$ and $b$ are corner utilities, not additive weights that must sum to 1 on their own.

The interaction parameter can be elicited from preference judgments. First establish the normalized utility scales and the utilities $a$ and $b$ of the two single-attribute best outcomes. Then elicit the utility $q$ of a mixed outcome with known $u_1$ and $u_2$. Under expected utility, for example, a person indifferent between that outcome and a lottery giving utility 1 with probability $q$ and utility 0 otherwise assigns it utility $q$. Rearranging the two-attribute model yields:

c=\frac{q-a u_1-b u_2}{u_1u_2},\qquad K=\frac{c}{ab},

provided $u_1u_2>0$ and $ab>0$. Elicited values should also be checked against the normalization constraint. A mismatch may reflect elicitation noise, inconsistent judgments, or a multiplicative form that does not adequately describe the preferences.

Worked Example: Estimating Interaction From an Indifference Judgment

Suppose a hypothetical patient’s single-attribute utilities are $a=0.40$ for symptom control and $b=0.35$ for avoiding a treatment-limiting adverse event. Each attribute utility ranges from 0 to 1. At the mixed profile $u_1=0.50,\ u_2=0.50$, the patient is indifferent between that profile and a lottery paying utility 1 with probability $q=0.4375$, and utility 0 otherwise. Estimate $c$ and $K$.

The elicited utility of the mixed profile is $q=0.4375$. Its additive contribution is $a u_1+b u_2=(0.40)(0.50)+(0.35)(0.50)=0.375$, and $u_1u_2=0.25$. Therefore,

c=\frac{0.4375-0.375}{0.25} =\frac{0.0625}{0.25}=0.2500.

Then $K=c/(ab)=0.2500/[(0.40)(0.35)]=1.786$ (to four significant figures). The normalization check gives $a+b+c=0.40+0.35+0.25=1.00$. Equivalently, $1+K=2.786$, while $(1+Ka)(1+Kb)$ is approximately $(1.714)(1.625)=2.786$. The elicited values are consistent with the normalized two-attribute multiplicative model.

Assumptions and a Short Derivation

A multiplicative form is not automatically appropriate just because there are several clinical outcomes. Its use assumes that the single-attribute utilities are meaningful across the levels of the other attributes, and that preferences satisfy the utility-independence conditions supporting a multiplicative representation. In practical terms, elicitation should not reveal that the patient’s utility scale for one attribute changes unpredictably depending on fixed levels of another. The utility scales, attribute ranges, and direction of preference must also be specified consistently.

For two attributes, the product expansion directly shows how the interaction term arises:

\frac{(1+K a u_1)(1+K b u_2)-1}{K} =\frac{1+K a u_1+K b u_2+K^2ab u_1u_2-1}{K} =a u_1+b u_2+K ab u_1u_2.

Setting both single-attribute utilities to 1 gives $U(1,1)=a+b+Kab$. Normalization requires $a+b+Kab=1$, so $c=Kab=1-a-b$. This also shows why the two-attribute interaction is determined once the two corner utilities are fixed and the all-best utility is fixed at 1. A mixed-profile elicitation is useful for checking whether observed judgments agree with that implied interaction.

Monotonicity is another required check: improving an attribute should not lower utility if higher values of its utility scale mean more-preferred outcomes. In the two-attribute form, the marginal utility contributions are $a+c u_2$ for attribute 1 and $b+c u_1$ for attribute 2. Both must be nonnegative throughout $0\leq u_1,u_2\leq1$. Thus, it is enough to check $a\geq0,\ a+c\geq0,\ b\geq0,$ and $b+c\geq0$.

Worked Example: Checking Monotonicity and Calculating Utility

Use the hypothetical model $U=0.40u_1+0.35u_2+0.25u_1u_2$. Check that increasing either attribute cannot decrease utility, then calculate utility for $u_1=0.80,\ u_2=0.60$.

For attribute 1, the marginal contribution is $0.40+0.25u_2$, which ranges from 0.40 to 0.65 as $u_2$ ranges from 0 to 1. For attribute 2, it is $0.35+0.25u_1$, which ranges from 0.35 to 0.60. Both are positive throughout the utility domain, so the model is monotone.

Substituting the specified profile gives $U=(0.40)(0.80)+(0.35)(0.60)+(0.25)(0.80)(0.60)=0.320+0.210+0.120=0.650$. The profile has normalized utility 0.650 on the elicited scale.

Comparing Treatment Rankings

An additive model is the main alternative. It gives each attribute a separate weighted contribution and has no cross-product term. When comparing rankings, state clearly how its weights were obtained: the corner utilities $a$ and $b$ in a multiplicative model do not themselves sum to 1 when $c\ne0$. One transparent comparison is to retain their relative weighting and normalize them for an additive score, using $U_{\mathrm{add}}=(a u_1+b u_2)/(a+b)$. The scores from the two models are on different scales, so compare their ordering, not their numerical values.

Worked Example: An Interaction Changes the Ranking

Consider two hypothetical treatments evaluated with $U=0.40u_1+0.35u_2+0.25u_1u_2$. Treatment A has $(u_1,u_2)=(0.80,0.80)$; treatment B has $(u_1,u_2)=(1.00,0.58)$. Compare their ranking under the multiplicative model and under the additive comparator with weights proportional to 0.40 and 0.35.

Under the multiplicative model, treatment A has utility $(0.40)(0.80)+(0.35)(0.80)+(0.25)(0.80)(0.80)=0.320+0.280+0.160=0.760$. Treatment B has utility $(0.40)(1.00)+(0.35)(0.58)+(0.25)(1.00)(0.58)=0.400+0.203+0.145=0.748$. The model ranks A above B.

For the additive comparator, the denominator is $0.40+0.35=0.75$. Treatment A scores $[0.320+0.280]/0.75=0.800$, while treatment B scores $[0.400+0.203]/0.75=0.804$. The additive model ranks B slightly above A. Here, the positive interaction favors A’s more balanced profile enough to change the ranking. This illustrates model sensitivity; it does not establish which ranking is correct without preference evidence supporting the interaction.

Diagnostics, Alternatives, and Limitations

A practical assessment should check whether the fitted form reproduces elicited judgments, not merely whether its formula can be calculated. Useful checks include comparing predicted and elicited utility at several mixed profiles, verifying the all-best and all-worst endpoints, and checking that no clinically preferred improvement decreases utility. If implied utilities fall outside the intended 0-to-1 range or violate monotonicity, revisit the elicited values and model specification.

The additive alternative is simpler, easier to interpret, and appropriate when the evidence supports separable contributions. The multiplicative alternative adds a structured interaction while retaining relatively few parameters. It can still be too restrictive: one global interaction structure may not capture more complex preference patterns, and an elicited interaction may vary across patients. The function also depends on chosen attribute ranges and single-attribute utility scales; changing these can change the resulting combined utility and ranking.

Do not interpret a positive interaction as proof that clinical benefits reinforce one another biologically. It means only that, on the specified utility scales, the combined preference exceeds the additive contribution. Likewise, do not infer that a model is justified simply because its treatment ranking seems plausible. Elicitation consistency and assumptions about preference structure are essential.

Common pitfall: Treating $a$ and $b$ as additive weights that must sum to 1, then adding a positive interaction on top, can make $U(1,1)>1$. In the two-attribute multiplicative model, the correct normalization is $a+b+c=1$, where $c=Kab$. Check the all-best outcome before interpreting any treatment score.

Check Your Understanding

Answer each question, then compare with the solutions.

In a two-attribute model, $a=0.30$, $b=0.50$, and $c=0.20$. What is the utility of the all-best profile, and is the model normalized?
For $a=0.40$, $b=0.35$, and $c=0.25$, calculate utility at $u_1=0.20,\ u_2=0.80$.
For the model in question 2, can improving attribute 1 reduce utility? Justify your answer over the full utility range.
Explain what a positive interaction means and name one thing it does not establish.

Solutions. (1) $U(1,1)=a+b+c=0.30+0.50+0.20=1.00$, so the model is normalized. (2) $U=(0.40)(0.20)+(0.35)(0.80)+(0.25)(0.20)(0.80)=0.080+0.280+0.040=0.400$. (3) No. The marginal contribution from improving attribute 1 is $a+c u_2=0.40+0.25u_2$, which ranges from 0.40 to 0.65 and is always positive for $0\leq u_2\leq1$. (4) A positive interaction raises combined utility through the cross-product term when both attributes have favorable utility. It does not establish a biological or causal interaction between the clinical outcomes.