Time Dependency

Time Dependency

A model has time dependency when a quantity that affects its results changes with a relevant time scale. In health economic evaluation, risks, treatment effects, costs, utilities and the composition of a population may evolve as people age, progress through disease, spend time in a state, or encounter changes in calendar time. The essential task is to state which clock drives each change and how that clock is represented in the analysis.

Which clock is changing the model?

Elapsed follow-up time is only one possible clock. Two people observed in the same model cycle can have different ages, different times since treatment and different durations in a health state. Confusing those clocks can produce a plausible looking model with the wrong risk trajectory.

Time scales can operate together. For example, all-cause mortality may depend on attained age and calendar year, while post-event recurrence depends on time since remission. A time horizon is the total span analyzed; a cycle length is the simulation's time step; neither specifies how an individual input changes within that span.

How changing hazards become event probabilities

For a time-to-event process, let $h(t)$ be a hazard: the instantaneous event rate among people still at risk at time $t$. Its cumulative hazard over $[a,b]$ is $H(a,b)=\int_a^b h(s)\,ds$. If this is the only exit from the at-risk state, the conditional event probability in that interval is

$$p(a,b)=1-\exp\{-H(a,b)\}.$$

For a constant hazard $h$ over an interval of length $\Delta t$, this becomes $p=1-\exp(-h\Delta t)$. For a changing hazard, integrate it over the interval rather than inserting an arbitrary point hazard and assuming the same accuracy. Probabilities are dimensionless; hazards have units of inverse time. If several mutually exclusive events compete, the single-event formula cannot simply be applied to each cause-specific hazard as if it were alone: event probabilities must account for the combined exits and their timing.

This survival representation can supply cycle-specific transition probabilities in a state-transition model. It also clarifies why a constant transition probability over every cycle is a substantive assumption, even if cycle length is fixed.

Worked example: recurrence risk after remission

Consider an illustrative cohort entering remission at time zero with no competing exit. Suppose its recurrence hazard is $0.12$ per year during the first three years and $0.04$ per year for the following nine years. Under this piecewise-constant assumption, the first three-year cumulative hazard is $0.12(3)=0.36$ and the next nine-year contribution is $0.04(9)=0.36$.

The probability of recurrence by year 12 is $1-\exp(-0.72)=0.5132$. Within a one-year cycle entirely in the early period, the conditional recurrence probability is $1-\exp(-0.12)=0.1131$; entirely in the later period it is $1-\exp(-0.04)=0.0392$. For people who enter remission at different dates, the early or late hazard follows time since their own entry, not the cohort's common cycle number. The example is a teaching calculation, not an empirical recurrence estimate. It excludes death and other competing events so that these single-event expressions apply directly.

How to represent time dependency in models

The model structure must preserve enough information to evaluate each relevant clock. The simplest adequate representation depends on the decision question, data and required precision; adding a changing input without aligning it to the right people and intervals does not solve the problem.

• Cycle-specific inputs: A cohort model can apply $p_t$, $c_t$ or $u_t$ when all people in a given state and cycle share the same risk, cost or utility. Age-dependent mortality can be indexed by attained age for a cohort with a common starting age, with appropriate care for age mixtures.
• Tunnel states: Temporary substates can record successive periods since entry into a condition, such as the first year after an event. These can reproduce duration effects in a cohort state-transition model, subject to the chosen cycle length and number of tunnel stages.
• Individual simulation or event histories: A microsimulation or discrete-event simulation can track each person's age, treatment start, state-entry time and prior events. This is useful when the clocks differ across people or history affects subsequent risk.
• Time-to-event models: Piecewise, parametric or flexible survival models can describe changing baseline hazards; time-varying covariates or treatment effects may be needed when exposures or effects change. Model fitting and extrapolation need support beyond observed follow-up.

“Time-dependent transition probability” describes a probability indexed by a specified clock. “Time-dependent covariate” describes a predictor whose value changes during follow-up. A time-varying treatment effect means its relative or absolute effect changes over time. These are related possibilities, but one does not establish the others. A model can have time-varying baseline risk while treatment's hazard ratio remains constant, or a changing treatment effect while another input remains fixed.

Consequences for costs, outcomes and decisions

Timing affects both the number of events and the time spent accruing costs and health. A year of benefit early in the horizon can contribute differently from a year late in the horizon because survival, quality of life, treatment use and discounting may differ. Thus two models with the same average annual hazard can produce different QALYs and costs if events occur at different times.

For a continuous illustration, expected QALYs over horizon $T$ may be represented as $Q=\int_0^T S(t)u(t)\,dt$ under a simple alive/dead setup, where $S(t)$ is survival and $u(t)$ is utility while alive. If discounting is required at annual continuous rate $r$, the expression becomes $Q_{\mathrm{disc}}=\int_0^T S(t)u(t)\exp(-rt)\,dt$. These are illustrative constructions: richer state structures require state-specific occupancy and utility, and the selected discount convention must match the analysis. Discounting values occurring at different times is distinct from making the underlying risk or utility time dependent.

In technology assessment, assuming that a treatment effect lasts indefinitely or wanes after a chosen interval can substantially change long-term incremental outcomes. Alternative shapes and timing should be justified with trial follow-up, plausible mechanisms and external evidence where available, then examined in scenario and uncertainty analyses. A time-varying pattern should not be selected simply to make a preferred result emerge.

Data, assumptions and common errors

Observed follow-up may be too short to identify long-term hazards or treatment-effect waning. Fit within the trial period does not alone validate an extrapolation, and an age effect estimated cross-sectionally need not equal change within an individual over time. State clearly which input is estimated from data, calibrated, borrowed from an external source or imposed as a scenario.

• Clock mismatch: Applying a recurrence curve by model cycle to people who entered remission in different cycles gives some people the wrong duration-specific risk.
• Hazard–probability confusion: A hazard of $0.12$ per year is not exactly a one-year event probability of $0.12$; under a constant sole hazard the probability is about $0.1131$.
• History loss: An ordinary memoryless state-transition specification conditions on the current state; a duration effect requires extra states, an indexed history, or a different model structure.
• Inconsistent time steps: Annual hazards converted to monthly probabilities need months expressed in years, for example $\Delta t=1/12$, and costs and utilities must be accrued on compatible periods.
• Competing exits and selection: Recurrence, death and treatment switching can alter who remains at risk. A naive independent per-event conversion or comparison of survivors at later times can misstate event probabilities or effects.
• Extrapolation certainty: The date on which an effect wanes and the shape of the change may be uncertain. Report and test those assumptions rather than hiding them in a single transition table.

What to report and check

An auditable analysis identifies the clocks, time origin, interval boundaries, cycle length, horizon and which inputs depend on each clock. It explains how the relevant data or assumptions define hazard, probability, treatment effect, cost and utility trajectories. The model should be checked against observed event curves, plausible age patterns and external constraints where relevant.

For implementation, test probability bounds and row sums, correct conversion from rates to interval probabilities, continuity at piecewise boundaries, treatment and state-entry offsets, survival monotonicity where applicable, and sensitivity to cycle length and uncertain long-term patterns. Where an approximation such as tunnel states is used, show that it is adequate for the decision. Report illustrative examples separately from empirical parameter estimates and flag unresolved extrapolation judgments for expert review.

Sources and further reading

The ISPOR-SMDM state-transition modeling task force report discusses time-dependent parameters and the history limitations of simple state-transition models. The NICE economic evaluation manual addresses time horizon and uncertainty around extrapolating costs and outcomes. The NICE Decision Support Unit survival-analysis technical support document treats patient-level survival extrapolation for economic evaluation. These sources guide modeling choices; the remission example and its figures are original illustrations.