Modeling Delayed Effects with Change-Point Hazards

Representing a Delayed Treatment Effect

A delayed treatment effect means that the treatment and control groups have the same event hazard initially, with a difference emerging after some time. A change-point model represents this pattern with a specified time $\tau$: before $\tau$, the treatment hazard equals the control hazard; after $\tau$, it is a constant multiple of the control hazard. This makes the assumed timing and size of the effect explicit, and gives a direct route from hazards to survival probabilities.

Let $t$ denote time since randomization, and let $\lambda_0(t)$ and $\lambda_1(t)$ be the hazard functions in the control and treatment arms. The hazard is the instantaneous event rate among individuals who have not yet experienced the event. Assume $\tau>0$ is a fixed, prespecified change point and $\theta>0$ is the post-change hazard ratio.

Definition: A delayed-effect change-point hazard model is $$ \lambda_1(t)= \begin{cases} \lambda_0(t), & 0\le t\le \tau,\\ \theta\lambda_0(t), & t>\tau. \end{cases} $$ The hazard ratio is $1$ before the change point and $\theta$ afterward. When $\theta<1$, the model represents a lower treatment-arm hazard after $\tau$; when $\theta>1$, it represents a higher treatment-arm hazard. The value at the single time $t=\tau$ does not affect survival probabilities for continuous event times.

The model allows the control hazard to vary freely over time. The restriction is on the ratio of the arm-specific hazards, not on the shape of the control hazard. In particular, after $\tau$, the treatment hazard follows any changes in $\lambda_0(t)$, scaled by $\theta$.

Assumptions and Survival-Curve Derivation

This model is appropriate as a description when the event endpoint and time origin are well defined, the delay is common to both arms on the chosen time scale, and there is no treatment effect on the hazard before $\tau$. After $\tau$, it assumes a constant hazard ratio, even if the baseline hazard itself changes. The model also presumes that the treatment effect is adequately represented by the hazard ratio for the event being studied; it does not by itself describe treatment effects on competing events.

For interpretation of survival curves estimated from data, the usual survival-analysis conditions also matter. In particular, censoring should be independent of the event time within treatment arm, or conditionally independent given variables included in the analysis. Random assignment supports comparison of arms but does not guarantee independent censoring, a correct change point, or a constant post-change hazard ratio.

Write the control cumulative hazard as $H_0(t)=\int_0^t\lambda_0(u)\,du$. Survival is the probability of remaining event-free through time $t$; under a hazard function $\lambda(t)$, it is $S(t)=\exp[-\int_0^t\lambda(u)\,du]$. The cumulative treatment hazard is obtained by integrating the two hazard segments separately:

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

The second line adds the shared cumulative hazard through $\tau$ to the post-change control cumulative hazard multiplied by $\theta$. Applying the survival definition gives the arm-specific curves:

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

An equivalent post-change expression is $S_1(t)=S_0(\tau)\{S_0(t)/S_0(\tau)\}^{\theta}$. It separates survival up to the change point, which is shared, from survival conditional on having reached that point. For $t>\tau$, the ratio of the survival probabilities is

\frac{S_1(t)}{S_0(t)} =\exp\{(1-\theta)[H_0(t)-H_0(\tau)]\}.

Thus, if $\theta<1$, treatment survival is at least as high as control survival after the change point, with the difference depending on the accumulated baseline hazard since $\tau$. The survival curves meet at $\tau$, because the cumulative hazards are identical through that time. The hazard ratio is a statement about instantaneous rates, not a direct statement about the difference in survival probabilities.

Worked Example: Constant Baseline Hazard

Suppose, hypothetically, that the control hazard is constant at $0.12$ events per person-year, the change point is $\tau=0.5$ years, and the post-change hazard ratio is $\theta=0.65$. Calculate both survival probabilities at $0.5$ years and at $2$ years.

The model assumptions for this calculation are that the control hazard remains $0.12$ per person-year, treatment and control hazards are equal through $0.5$ years, and the treatment hazard afterward is $0.65$ times the control hazard. Therefore, the treatment hazard after the change point is $0.65(0.12)=0.078$ events per person-year.

At the change point, each arm has cumulative hazard $0.12(0.5)=0.060$, so

S_0(0.5)=S_1(0.5)=\exp(-0.060)=0.9418.

At $2$ years, the control cumulative hazard is $0.12(2)=0.240$. The treatment cumulative hazard includes $0.060$ before the change point and $0.078(2-0.5)=0.117$ afterward:

H_1(2)=0.060+0.117=0.177,\qquad S_1(2)=\exp(-0.177)=0.8378,\qquad S_0(2)=\exp(-0.240)=0.7866.

Under this hypothetical model, the event-free probabilities at two years are about $83.78\%$ in the treatment arm and $78.66\%$ in the control arm. The hazard ratio is $1$ through $0.5$ years and $0.65$ after that point. The survival difference at two years is $0.8378-0.7866=0.0512$, or 5.12 percentage points; it is not itself the hazard ratio.

Piecewise Baseline Hazards

A change point in the treatment effect need not coincide with changes in the control hazard. When the control hazard is piecewise constant, compute cumulative hazard by adding the rate multiplied by the duration in each interval. For the treatment arm, apply the hazard ratio only to the portions of those intervals after $\tau$. This segment-by-segment calculation prevents the common error of multiplying the entire follow-up cumulative hazard by $\theta$.

Worked Example: Change Point Within a Baseline-Hazard Segment

Consider a hypothetical model with control hazard $0.10$ events per person-year from time 0 to year 1 and $0.08$ per person-year after year 1. Let the treatment effect begin at $\tau=0.5$ years, with post-change hazard ratio $\theta=0.60$. Find the survival probabilities at $1.5$ years.

The treatment hazard equals the control hazard from time 0 to $0.5$ years. From $0.5$ to $1$ year, its rate is $0.60(0.10)=0.060$ per person-year. From year 1 to $1.5$, its rate is $0.60(0.08)=0.048$ per person-year. The cumulative hazards are therefore

$$ H_0(1.5)=0.10(1)+0.08(0.5)=0.140, $$

$$ H_1(1.5)=0.10(0.5)+0.060(0.5)+0.048(0.5)=0.104. $$

Thus $S_0(1.5)=\exp(-0.140)=0.8694$ and $S_1(1.5)=\exp(-0.104)=0.9012$. The treatment arm has higher model-based event-free survival at this time. The hazard ratio is $0.60$ throughout the post-change interval, including after the baseline hazard decreases at year 1; the baseline change alters both hazards, not their ratio.

Worked Example: Conditional Risk After the Change Point

Suppose, hypothetically, that the control hazard is $0.04$ events per person-month through month 3 and $0.06$ per person-month after month 3. A treatment effect begins at month 2 with $\theta=0.50$. Calculate the probability of experiencing the event between months 2 and 5, conditional on being event-free at month 2, for each arm.

For an individual event-free at month 2, the control cumulative hazard from month 2 to month 5 is $0.04(1)+0.06(2)=0.16$. The corresponding treatment cumulative hazard is $0.50(0.04)(1)+0.50(0.06)(2)=0.08$. For a time interval with cumulative hazard increment $\Delta H$, conditional survival through the interval is $\exp(-\Delta H)$, so conditional event risk is $1-\exp(-\Delta H)$. Hence

P_0(2<T\le 5\mid T>2)=1-\exp(-0.16)=0.1479,

P_1(2<T\le 5\mid T>2)=1-\exp(-0.08)=0.0769.

Under these assumptions, the conditional event risks from month 2 through month 5 are 14.79% and 7.69%, respectively. These are interval risks among those event-free at month 2, not risks among everyone randomized. Their difference is not $1-\theta$: a hazard ratio compares instantaneous rates, while a cumulative event risk depends on the duration and baseline hazard over the interval.

How to Check the Model and Choose an Alternative

The model is useful when a delayed effect is scientifically plausible and a prespecified onset time gives a clear summary. Its sharp transition is also a strong assumption. A practical assessment should ask whether the observed pattern is consistent with no early arm difference and a reasonably stable post-change hazard ratio. In randomized data, compare Kaplan-Meier curves and arm-specific event patterns over time, while remembering that such plots can be imprecise when few individuals remain at risk. Schoenfeld-residual and related proportionality diagnostics can indicate time variation, but they do not by themselves establish that the time variation follows this particular change-point form.

Warning signs include a visible treatment-control separation well before $\tau$, a post-$\tau$ divergence that changes direction, or evidence that the post-change hazard ratio itself varies substantially. A single threshold can also be a poor approximation when the treatment effect rises gradually, differs across individuals, or begins at different times for different individuals. If $\tau$ is selected after inspecting the outcomes, ordinary inference that treats it as fixed can understate uncertainty. A prespecified change point avoids that particular selection step, but does not ensure that the assumed value is correct.

Approach	What it assumes or represents	When it may be preferable
Proportional-hazards model	A single hazard ratio applies throughout follow-up.	When a constant treatment-control hazard ratio is credible.
One change-point model	Equal hazards before a specified time and a constant ratio afterward.	When there is a clear, prespecified delay followed by a sustained effect.
Smoother time-varying effect	The hazard ratio can change gradually over time.	When a sudden transition is implausible and enough data support a more flexible pattern.
Multiple change points or time segments	The effect can differ across several defined periods.	When distinct phases are clinically motivated and the data can support the added complexity.

These alternatives answer different modeling needs. The ordinary proportional-hazards model is simpler but cannot represent a true delayed effect with ratio $1$ early and a different ratio later. More flexible models can describe gradual or complex changes, but require more choices and generally more information to estimate reliably. Weighted logrank procedures are alternatives for testing treatment differences under nonproportional hazards; they are not themselves a specification of the arm-specific hazard and survival curves developed here.

A change-point model is a compact description, not proof of a biological onset at exactly $\tau$. Its survival curves depend on the assumed baseline hazard as well as $\theta$, and results beyond the observed follow-up rely on extrapolating the model. It also does not directly provide a single effect summary that captures all features of a time-varying treatment effect. Those limits matter when translating a hazard model into clinical interpretation.

Common Mistake

Common mistake: “The treatment hazard ratio is $0.60$ after the change point, so treatment survival is $60\%$ of control survival after that time.” This confuses a ratio of instantaneous hazards with a ratio of survival probabilities. The correct survival calculation accumulates each arm’s hazard over time: after $\tau$, $S_1(t)=\exp\{-H_0(\tau)-\theta[H_0(t)-H_0(\tau)]\}$. The treatment and control survival curves are equal at $\tau$, and their later separation depends on the accumulated baseline hazard after $\tau$.

Key takeaway: Specify the time origin, change point, and post-change hazard ratio before calculating survival. Accumulate the shared hazard through $\tau$, then apply $\theta$ only to the baseline cumulative hazard accrued after $\tau$.

Check Your Understanding

Answer each question using the change-point model and the stated assumptions. Short solutions follow.

If $\theta=0.80$, what is the hazard ratio before and after $\tau$, and what does the post-change value mean?
Let the control hazard be constant at $0.05$ per month, with $\tau=4$ months and $\theta=0.50$. Find treatment survival at month 8.
Why is it incorrect to calculate treatment cumulative hazard at $t>\tau$ as $\theta H_0(t)$?
With a positive baseline hazard after $\tau$ and $\theta<1$, which arm has higher modeled survival after $\tau$? State the reasoning.

Solutions. (1) The hazard ratio is $1$ through $\tau$, then $0.80$. After $\tau$, the treatment instantaneous event rate is 80% of the control rate among individuals still event-free. (2) The shared cumulative hazard through month 4 is $0.05(4)=0.20$. The treatment cumulative hazard from month 4 to month 8 is $0.50(0.05)(4)=0.10$. Thus $S_1(8)=\exp[-(0.20+0.10)]=\exp(-0.30)=0.7408$. (3) The treatment and control cumulative hazards are identical before $\tau$, so multiplying the full control cumulative hazard by $\theta$ incorrectly changes the pre-$\tau$ hazard as well. The correct expression is $H_1(t)=H_0(\tau)+\theta[H_0(t)-H_0(\tau)]$. (4) Treatment survival is higher: for $t>\tau$, $S_1(t)/S_0(t)=\exp\{(1-\theta)[H_0(t)-H_0(\tau)]\}>1$ when the post-change cumulative hazard is positive and $\theta<1$.