Time-Dependent Utility and Delayed Harms

Why Timing Matters in Benefit-Risk Assessment

A treatment may improve symptoms soon after initiation while causing a harmful effect only after prolonged exposure. Comparing the size of these effects without representing their timing can obscure important trade-offs. A benefit that lasts several years is not equivalent to a brief improvement; nor is a short-lived adverse event equivalent to a persistent loss of health.

The introductory material on utility functions establishes how clinical outcomes can be represented on a preference-based scale. The additional task here is to represent how utility changes over time and, when appropriate, to give future utility a different weight from current utility. These are separate choices: time-varying utility describes the value of health states as they change over time; discounting describes how the decision analysis weights utility at different times.

Definition: Let $u_a(t)$ be the utility per unit time for regimen $a$ at time $t$, and let $u_c(t)$ be the corresponding utility under a comparator. The incremental utility trajectory is $\Delta u(t)=u_a(t)-u_c(t)$. For a time horizon $T$ and discount factor $D(t)$, the discounted incremental value is $$ V_a-V_c=\int_0^T D(t)\,E[\Delta u(t)]\,dt. $$ This expression applies when utility is accumulated continuously over time, such as utility per year. Its units are utility multiplied by time, for example utility-years.

The expectation allows outcomes to vary among patients or across possible clinical courses. It does not require that the utility trajectory be the same for every patient. For a deterministic hypothetical example, the expectation can be replaced by the specified trajectory itself. A comparison should use the same time origin, horizon, utility scale, and discount convention for both regimens.

Two Ways Time Enters the Calculation

A time-varying utility function may reflect a change in health state, a change in how a state is valued at different times, or both. For example, a patient may move from symptom relief to a later adverse-event state. Alternatively, an elicited utility assigned to the same health state may vary with age or clinical context. If the analysis assumes that preferences for a given state are stable over time, it should say so rather than treating stability as automatic.

Discounting is a separate weighting convention. Under continuous discounting at a constant rate $\rho$ per year, the discount factor is $D(t)=e^{-\rho t}$. Under discrete annual discounting at an effective annual rate $r$, a utility amount assigned to the end of year $k$ receives factor $D_k=(1+r)^{-k}$. The rate and time units must agree: a rate per year requires time measured in years. Discounting future utility is a modeling or decision convention; it is not a claim that a future health outcome is less real.

Important distinction: If $u(t)$ is utility per year while a health state persists, integrate it over the time spent in that state. If $v(t)$ is instead a one-time utility consequence of an event occurring at time $t$, count that event consequence once and discount it at its occurrence time. Integrating a one-time event consequence over every subsequent year would count it repeatedly.

The calculation also depends on the horizon. A finite horizon $T$ includes only utility through $T$; if a delayed harm begins after $T$, it contributes nothing to the calculation unless the horizon or model is extended. A horizon should therefore be long enough to capture clinically relevant delayed consequences, or the excluded consequences should be made explicit.

Deriving the Discount Factors

For continuous discounting, suppose the remaining weight on a unit of utility declines at a constant proportional rate $\rho$ as time advances. If $D(t)$ denotes that weight, the assumption is $dD(t)/dt=-\rho D(t)$, with $D(0)=1$. Separating variables and integrating gives the standard exponential factor:

\frac{dD(t)}{D(t)}=-\rho\,dt \quad\Longrightarrow\quad \log D(t)=-\rho t \quad\Longrightarrow\quad D(t)=e^{-\rho t}.

For a constant utility rate $q$ over an interval from $a$ to $b$, its discounted contribution follows by integrating that rate against the discount factor:

\int_a^b e^{-\rho t}q\,dt = \begin{cases} q\displaystyle\frac{e^{-\rho a}-e^{-\rho b}}{\rho}, & \rho>0,\\[6pt] q(b-a), & \rho=0. \end{cases}

The second case follows directly from $D(t)=1$ when $\rho=0$. For discrete annual calculations, the convention must specify when each period's utility is credited. The examples below assign the utility accumulated during year $k$ to the end of that year, so its factor is $(1+r)^{-k}$. Crediting utility at the beginning or midpoint of each year would produce different factors and should not be mixed with the end-of-year convention.

Worked Example: Early Benefit and Later Toxicity

Suppose regimen A, compared with a reference regimen, adds $0.20$ utility per year in years 1 and 2, then causes a utility decrement of $0.23$ per year in years 3 and 4. Assume these are utility rates sustained over each full year. Use end-of-year discrete discounting at $r=0.05$ per year and a four-year horizon.

The annual factors are $D_1=1/1.05=0.9524$, $D_2=1/1.05^2=0.9070$, $D_3=1/1.05^3=0.8638$, and $D_4=1/1.05^4=0.8227$. The discounted contributions are:

\begin{aligned} V_A-V_c &=0.20D_1+0.20D_2-0.23D_3-0.23D_4\\ &=(0.20)(0.9524)+(0.20)(0.9070) -(0.23)(0.8638)-(0.23)(0.8227)\\ &=0.1905+0.1814-0.1987-0.1892\\ &=-0.0160\ \text{utility-years}. \end{aligned}

Under these assumptions, regimen A has a small negative discounted incremental value relative to the reference over four years. Without discounting, the corresponding value is $0.20+0.20-0.23-0.23=-0.0600$ utility-years. Discounting makes the later harms count less relative to the early benefits, but it does not make the value positive in this example. This is a comparison on the specified utility scale, not a conclusion that the regimen is clinically unacceptable.

Worked Example: A Time-Varying Valuation of a Delayed Harm

Suppose a delayed adverse state occurs during years 3 and 4. Patients' elicited utility decrement for that state is assumed to be $-0.10$ per year in year 3 and $-0.30$ per year in year 4, reflecting a hypothetical change in the relevant clinical context. The regimen also provides $+0.16$ utility per year in years 1 and 2. Use the same end-of-year discount factors at $r=0.05$.

Here the harm has a time-varying utility rate; it would be inaccurate to use a single decrement at both times unless that simplification were justified. The discounted value is:

\begin{aligned} V-V_c &=0.16D_1+0.16D_2-0.10D_3-0.30D_4\\ &=(0.16)(0.9524)+(0.16)(0.9070) -(0.10)(0.8638)-(0.30)(0.8227)\\ &=0.1524+0.1451-0.0864-0.2468\\ &=-0.0357\ \text{utility-years}. \end{aligned}

The result is negative over this horizon. For comparison, replacing the two time-specific harm decrements with a constant $-0.20$ per year would give $0.2975-(0.20)(0.8638+0.8227)=-0.0398$ utility-years. The difference is modest here, but it arises from an assumption about utility over time, not from the discount rate. In an actual assessment, time-specific values need a defensible basis, such as appropriate preference data or a clearly identified scenario analysis.

Worked Example: Continuous Discounting With a Delayed Harm

Consider a hypothetical regimen that adds utility at a constant rate of $0.18$ per year from year 0 through year 2, then causes a decrement of $0.24$ per year from year 3 through year 5. Assume continuous discounting at $\rho=0.03$ per year and no additional utility changes. The horizon is five years.

The benefit occurs over $0\leq t\leq2$ and the harm over $3\leq t\leq5$. Applying the integrated-rate formula gives:

\begin{aligned} V-V_c &=0.18\frac{1-e^{-0.03(2)}}{0.03} -0.24\frac{e^{-0.03(3)}-e^{-0.03(5)}}{0.03}\\ &=0.18\frac{1-0.9418}{0.03} -0.24\frac{0.9139-0.8607}{0.03}\\ &=0.3494-0.4258\\ &=-0.0764\ \text{utility-years}. \end{aligned}

The discounted incremental value is negative. With no discounting, the same rates and durations give $(0.18)(2)-(0.24)(2)=-0.1200$ utility-years. The less negative discounted result reflects the later timing of the harm under the stated convention. The continuous calculation integrates utility through each interval; it is not equivalent to assigning the entire interval's utility to an arbitrary time point.

Assumptions, Diagnostics, and Alternatives

A time-aggregated comparison is interpretable only if its assumptions are made visible. At minimum, specify the utility scale and comparator, the time origin and horizon, whether utilities are rates or one-time event consequences, and the discount convention and rate. If utility varies over time, explain whether that variation represents changing health states, changing valuations, or both. If values are based on expectations, their calculation should represent the clinical outcome distribution relevant to the decision.

Several checks can expose a fragile analysis:

Plot or tabulate the trajectory. Show the incremental utility by interval alongside the cumulative discounted value. This reveals whether a conclusion is driven by one late interval or by an abrupt assumed utility change.
Check the horizon. Extend it when plausible delayed harms or persistent benefits fall beyond the original endpoint. If they cannot be modeled, state that limitation.
Vary the discount rate and utility assumptions. Recalculate at zero discount and at other defensible rates. Also examine plausible alternative durations and time-specific utility values. A conclusion that changes under small, reasonable changes is sensitive to those assumptions.
Audit units and timing. A utility rate per year multiplied by a duration in years yields utility-years. A one-time event utility should not be multiplied by its duration unless the quantity was defined as a rate.

Undiscounted aggregation is a useful alternative when the decision convention assigns equal weight to utility at all times, and it is an important comparison for showing the effect of discounting. A fixed utility value for a health state can be simpler when preferences are assumed stable and the state does not change; that simplification should be checked against the clinical trajectory. Discrete time steps can be practical when outcomes are observed by visit or year, while continuous integration can represent changes occurring between visits more naturally. These approaches are not interchangeable without assumptions about how utility changes within each interval.

No discount rate or utility trajectory is universally correct for every decision. Discounting may change rankings when regimens have different timing, but it cannot resolve uncertainty about whether a delayed harm occurs, how long it lasts, or how patients value it. Results should therefore be interpreted as conditional on the specified horizon, utility model, and discount convention, rather than as timeless properties of the treatments.

Common Pitfall

Common mistake: “The toxicity happens later, so discounting means it can be ignored.”

Correction: Discounting reduces the weight assigned to future utility under a stated convention; it does not remove the toxicity or establish that it is unimportant. Its contribution still depends on its probability, severity, duration, utility value, and timing. Report the timing assumptions and test whether the comparison changes under reasonable alternatives.

Check Your Understanding

Answer each question using the stated timing convention. Solutions are included below.

With $r=0.05$, what end-of-year factor applies to utility accumulated in year 2, and what factor applies in year 4?
A one-time event consequence has utility $-0.30$ and occurs at the end of year 3. What is its discounted value under $r=0.05$? Why should it not be integrated over years 3 and 4 as a utility rate?
A regimen adds $0.10$ utility per year in years 1 and 2 and causes a decrement of $0.10$ per year in years 3 and 4. Calculate its discounted incremental value at $r=0.05$, using end-of-year factors.
State two distinct reasons why utility might vary with time, and explain how each differs from discounting.

Solutions. (1) The factors are $D_2=(1.05)^{-2}=0.9070$ and $D_4=(1.05)^{-4}=0.8227$. (2) The event value is counted once: $(-0.30)(1.05)^{-3}=(-0.30)(0.8638)=-0.2592$ utility units. It is a one-time consequence, not a utility rate sustained across years. (3) The value is $0.10(D_1+D_2-D_3-D_4)=0.10(0.9524+0.9070-0.8638-0.8227)=0.0173$ utility-years. (4) Utility may vary because the patient's health state changes, or because the value assigned to a state changes with age or clinical context. Discounting instead changes the weight applied to utility because it occurs at a particular time; it does not itself change the underlying health state or its utility value.