Delayed Treatment Effects in Piecewise Hazard Models

Why Specify a Treatment Lag?

A treatment may have little or no effect immediately after randomization, with its effect emerging only after a biologically or clinically motivated delay. A single hazard ratio applied over all follow-up does not describe that pattern: it assumes the same relative hazard at every time. A piecewise hazard model can represent the delay directly and show how the treatment contrast accumulates as follow-up continues.

The lag model is a description of a time pattern, not a claim that all treatment effects truly switch on at one exact instant. Its value is that it makes the assumed onset explicit. The resulting survival difference depends not just on the post-lag hazard ratio, but also on the lag length, the baseline hazard over time, and the time at which the contrast is evaluated.

Definition: Let \(t\) be time since randomization, \(\tau \geq 0\) a prespecified lag, \(\lambda_0(t)\) the control-arm hazard, and \(\theta\) the treatment-to-control hazard ratio after the lag. A delayed-effect model specifies \(\lambda_1(t)=\lambda_0(t)\) for \(0 \leq t \leq \tau\), and \(\lambda_1(t)=\theta\lambda_0(t)\) for \(t>\tau\). Here \(\lambda_1(t)\) is the treatment-arm hazard. The parameter \(\theta\) is positive; \(\theta<1\) represents a lower treatment hazard after the lag, while \(\theta>1\) represents a higher treatment hazard after the lag.

From Hazards to Survival Contrasts

A hazard is an instantaneous event rate among individuals who have survived up to time \(t\); it is not the probability of an event over an interval. For a hazard function \(\lambda_j(t)\), where \(j=0\) denotes control and \(j=1\) denotes treatment, define the cumulative hazard as \(H_j(t)=\int_0^t \lambda_j(u)\,du\). The corresponding survival function is \(S_j(t)=\exp[-H_j(t)]\). These relationships require nonnegative, integrable hazards over the time interval being considered.

The baseline hazard need not be constant. It may be piecewise constant, or otherwise vary with time. Under the delayed-effect model, treatment and control have identical hazards up to \(\tau\). After \(\tau\), the treatment hazard is a constant multiple \(\theta\) of the contemporaneous control hazard. Integrating the two parts gives the central result:

H_1(t)= \begin{cases} H_0(t), & 0 \leq t \leq \tau,\\ H_0(\tau)+\theta\{H_0(t)-H_0(\tau)\}, & t>\tau. \end{cases}

The second line says that the treatment cumulative hazard consists of the common cumulative hazard accumulated during the lag, plus a fraction \(\theta\) of the control cumulative hazard accumulated afterward. Survival follows by exponentiating the negative cumulative hazard:

S_1(t)= \begin{cases} S_0(t), & 0 \leq t \leq \tau,\\ \exp[-H_0(\tau)-\theta\{H_0(t)-H_0(\tau)\}], & t>\tau. \end{cases}

This expression also gives a useful interpretation. Before the lag ends, the model predicts no survival contrast. Afterward, a contrast develops as the post-lag cumulative hazard difference grows. If \(\theta<1\), then \(H_1(t)\leq H_0(t)\), so \(S_1(t)\geq S_0(t)\). If the control hazard is positive over some portion of follow-up after \(\tau\), the inequality in survival is strict after that portion. This is a consequence of the specified model, not evidence that the model is appropriate for a particular trial.

Derivation: Since \(H_1(t)=\int_0^t\lambda_1(u)\,du\), for \(t>\tau\) split the integral at the lag: \(\int_0^\tau\lambda_0(u)\,du+\int_\tau^t\theta\lambda_0(u)\,du\). The first integral is \(H_0(\tau)\), and the second is \(\theta\{H_0(t)-H_0(\tau)\}\). Thus the stated expression for \(H_1(t)\) follows. Applying \(S_1(t)=\exp[-H_1(t)]\) gives the treatment survival function. At \(t=\tau\), the post-lag contribution is zero, so both cumulative hazards and both survival probabilities agree at the boundary.

The hazard ratio is time-specific: it equals 1 during the lag and \(\theta\) afterward. The cumulative hazard ratio \(H_1(t)/H_0(t)\), when \(H_0(t)>0\), generally is neither of those values after the lag. It averages the common pre-lag contribution and the reduced or increased post-lag contribution. The absolute survival difference \(S_1(t)-S_0(t)\) is another quantity again; it changes with time and is measured in probability units, or percentage points.

Worked Calculations

Worked Example: Constant Baseline Hazard and a Prespecified Lag

Suppose, hypothetically, that the control hazard is constant at \(0.050\) events per person-year. A prespecified model has a lag of \(\tau=0.75\) years, followed by a treatment-to-control hazard ratio \(\theta=0.60\). Calculate the survival probabilities and their difference at \(t=1.5\) years.

The model assumes a constant control hazard and no treatment effect during the first 0.75 years. The treatment hazard afterward is \(0.60(0.050)=0.030\) events per person-year. Both hazards are nonnegative, and the specified post-lag ratio is applied over the entire interval from 0.75 to 1.5 years.

The control cumulative hazard is \(H_0(1.5)=0.050(1.5)=0.075\). The treatment cumulative hazard includes 0.75 years at the common hazard and 0.75 years at the reduced hazard:

$$ H_1(1.5)=0.050(0.75)+0.60(0.050)(0.75) =0.0375+0.0225=0.0600. $$

Therefore, \(S_0(1.5)=\exp(-0.075)=0.9277\), and \(S_1(1.5)=\exp(-0.0600)=0.9418\). Using the unrounded exponential values for the subtraction gives \(S_1(1.5)-S_0(1.5)=\exp(-0.0600)-\exp(-0.075)=0.01402\), or about 1.40 percentage points. Under this hypothetical model, treatment survival at 1.5 years is about 1.40 percentage points higher than control survival. The difference is not obtained by subtracting rounded survival values.

Worked Example: A Piecewise Baseline Hazard

Suppose, hypothetically, that the control hazard is \(0.040\) events per person-year during year 0 to year 1 and \(0.080\) events per person-year after year 1. The lag is \(\tau=0.50\) years, and \(\theta=0.70\) thereafter. Calculate the treatment and control survival probabilities at \(t=0.75\) and \(t=2\) years.

This model allows the baseline hazard to change at year 1 while retaining the same treatment-to-control ratio after the prespecified lag. The intervals do not conflict: the treatment ratio applies from year 0.50 onward, including the change in baseline hazard at year 1.

At \(t=0.75\), the control cumulative hazard is \(0.040(0.75)=0.030\). Treatment accumulates \(0.040(0.50)=0.020\) during the lag and \(0.70(0.040)(0.25)=0.007\) after it. Thus \(H_1(0.75)=0.027\). The survivals are \(S_0(0.75)=\exp(-0.030)=0.9704\) and \(S_1(0.75)=\exp(-0.027)=0.9734\). The difference, using unrounded values, is \(\exp(-0.027)-\exp(-0.030)=0.002915\), about 0.292 percentage points.

At \(t=2\), the control cumulative hazard is \(H_0(2)=0.040(1)+0.080(1)=0.120\). Treatment accumulates \(0.040(0.50)=0.020\) during the lag, \(0.70(0.040)(0.50)=0.014\) from year 0.50 to year 1, and \(0.70(0.080)(1)=0.056\) from year 1 to year 2. Therefore \(H_1(2)=0.020+0.014+0.056=0.090\). The survival estimates are \(S_0(2)=\exp(-0.120)=0.8869\) and \(S_1(2)=\exp(-0.090)=0.9139\). Their difference is \(\exp(-0.090)-\exp(-0.120)=0.02701\), about 2.70 percentage points.

The survival difference grows between these evaluation times because additional post-lag cumulative hazard accrues. It does not follow that the difference must grow at the same rate at all times.

Worked Example: The Hazard Ratio Is Not the Cumulative Hazard Ratio

Suppose, hypothetically, that the control hazard is constant at \(0.060\) events per person-year, \(\tau=1\) year, and \(\theta=0.50\). At \(t=3\) years, calculate the instantaneous hazard ratio, cumulative hazard ratio, and survival difference.

The model has equal hazards during the first year and a treatment hazard of \(0.50(0.060)=0.030\) events per person-year from year 1 to year 3. Thus \(H_0(3)=0.060(3)=0.180\), whereas \(H_1(3)=0.060(1)+0.030(2)=0.120\).

At year 3 the instantaneous hazard ratio is \(0.030/0.060=0.50\). The cumulative hazard ratio is \(H_1(3)/H_0(3)=0.120/0.180=0.6667\), because the first year contributed equally to both cumulative hazards. The survival probabilities are \(S_0(3)=\exp(-0.180)=0.8353\) and \(S_1(3)=\exp(-0.120)=0.8869\). The survival difference is \(\exp(-0.120)-\exp(-0.180)=0.05165\), or about 5.17 percentage points. These are distinct summaries: the post-lag instantaneous hazard ratio is 0.50, the cumulative hazard ratio at year 3 is 0.6667, and the survival difference is 5.17 percentage points.

Assumptions, Diagnostics, and Alternatives

The formulas describe a population model under several assumptions that should be made explicit. The lag \(\tau\) is prespecified on a defined time scale, ordinarily time since randomization for this model. The treatment effect is absent during the lag and has a constant multiplicative effect \(\theta\) afterward. The baseline hazard is specified adequately, including any piecewise intervals and units. If the model is used to interpret randomized trial data, treatment assignment, outcome ascertainment, and censoring must also be handled appropriately; in particular, independent censoring is needed for standard survival estimates to represent the event-time survival distribution.

Diagnostics should ask whether the modeled pattern is plausible, not merely whether the algebra is correct. Compare observed survival patterns by arm over time, examine event counts and numbers at risk in the periods before and after the proposed lag, and assess whether the apparent onset is supported by enough follow-up. A thin risk set late in follow-up can make the post-lag pattern unstable. Also check that a chosen piecewise baseline does not hide an important change in the control event rate.

A common alternative is a proportional-hazards model with one treatment hazard ratio applying from time zero. That alternative is simpler, but it cannot represent an exact period of no treatment effect followed by a different ratio. A delayed piecewise model is more explicit about that timing, but it imposes a sharp transition and a constant post-lag ratio. More flexible models can allow gradual onset or other time-varying effects; those require additional modeling choices and are not interchangeable with a prespecified lag model.

In a group sequential setting, the assumed effect timing can influence how much treatment contrast is present by each planned analysis. The equations here describe that time pattern; they do not by themselves determine sequential boundaries or guarantee a particular power. Those operating characteristics require a design calculation under the assumed event and follow-up process.

Common Pitfalls and Limitations

Calling \(\theta\) the hazard ratio at every time. In this model, the hazard ratio is 1 through the lag and \(\theta\) afterward.
Using \(\theta\) as the cumulative hazard ratio. The cumulative hazard includes both the pre-lag and post-lag periods, so its ratio generally differs from \(\theta\).
Subtracting rounded survival values. Compute the difference from unrounded values, then round the result. Rounding each survival first can alter the reported contrast.
Treating a sharp onset as a fact established by the model. The lag and abrupt change are assumptions. Clinical rationale and observed data should be considered, and uncertainty about onset may matter.
Ignoring the time scale. A lag of six months since randomization is not necessarily the same as six months since treatment initiation if those dates differ.

The main limitation is structural: a single \(\tau\) and \(\theta\) cannot capture every possible effect pattern. The model is useful when the prespecified delay and subsequent multiplicative effect are meaningful approximations. If the effect is gradual, changes more than once, or varies substantially across patients, the predicted survival contrast may be misleading even when each calculation is performed correctly.

Check Your Understanding

Answer each question using the delayed-effect model defined above. Full solutions are provided below.

With \(\tau=2\) years, what does the model imply about treatment and control hazards and survival at \(t=1.5\) years?
For a constant control hazard of \(0.040\) events per person-year, \(\tau=1\) year, and \(\theta=0.75\), calculate \(H_0(2)\), \(H_1(2)\), and both survival probabilities at two years.
Why can the cumulative hazard ratio at a time after the lag differ from the post-lag hazard ratio?
Name one diagnostic concern if only a few participants remain at risk after the specified lag.

Solutions. (1) At \(1.5<2\) years, the hazards are equal, both survival curves are equal under the model, and the treatment contrast is zero. (2) \(H_0(2)=0.040(2)=0.080\). For treatment, \(H_1(2)=0.040(1)+0.75(0.040)(1)=0.070\). Thus \(S_0(2)=\exp(-0.080)=0.9231\) and \(S_1(2)=\exp(-0.070)=0.9324\). (3) The cumulative hazards include the common pre-lag contribution as well as the post-lag contribution multiplied by \(\theta\); their ratio therefore averages both periods. (4) The estimated post-lag pattern may be unstable because few individuals remain under observation and the late risk set provides limited information.