Deriving Restricted Mean Survival Time Targets

Why Restricted Mean Survival Time Is an Area

A treatment can change not only whether an event occurs by a chosen time, but also how long people remain event-free. The restricted mean survival time (RMST) summarizes this duration up to a fixed horizon $\tau$. Unlike a hazard ratio, it is measured in units of time and does not require a proportional-hazards interpretation.

Let $T^g$ be the event time that would occur under treatment regime $g$, and define its survival function by $S^g(t)=P(T^g>t)$. The regime-specific RMST through $\tau$ is the expected event-free time accumulated before that horizon.

Definition: For a nonnegative event time $T^g$, the restricted mean survival time through $\tau>0$ is $$ \mu^g(\tau)=E[\min(T^g,\tau)]. $$ When the expectation is finite, it also has the survival-function representation $$ \mu^g(\tau)=\int_0^\tau S^g(t)\,dt. $$

The integral identity does not require a density or a continuous event-time distribution. It follows from expressing a nonnegative duration as the area under its indicator of still being in progress. The endpoint convention $P(T^g>t)$ is important: it agrees with event-free time accumulated over the interval from zero to $\tau$, even if the event-time distribution has point masses.

Identification Under a Longitudinal Regime

Consider discrete treatment decisions at times $k=0,\ldots,K-1$, with event status observed through time $K$. Write $H_k$ for the measured history available just before treatment $A_k$. A regime $g$ specifies the treatment $a_k(H_k)$ at each decision. Let $Y_j=I(T>j)$, so $Y_0=1$ for an event-free population at baseline and $Y_j$ indicates survival beyond time $j$.

The target survival probability at grid time $j$ is $S^g(j)=P(T^g>j)=E[Y_j^g]$. To identify it from observed data, assume consistency, no interference between individuals, sequential exchangeability of treatment given the measured histories, and positivity: each treatment required by $g$ has positive probability at histories that occur under $g$. For the derivations below, also assume event status is observed through the horizon. Dropout and censoring require additional methods and are not addressed here.

Under these assumptions, the longitudinal g-formula identifies each $S^g(j)$ from the observed conditional distributions. Longitudinal TMLE estimates these survival risks by recursively fitting outcome regressions, targeting them for the regime, and averaging the final baseline predictions. The earlier material on longitudinal TMLE for survival risk develops that targeting procedure; here its output is used as a component of the RMST estimate.

From Survival Probabilities to RMST

For a discrete event time supported on integer times, define $Y_k=I(T>k)$. For every realized event time $T$, $$ \min(T,K)=\sum_{k=0}^{K-1} I(T>k). $$ If $T=2$, for example, the indicators at $k=0,1$ equal one and all later indicators equal zero, so the sum is two. Taking expectations gives a finite-sum representation.

Result: If time is measured in unit intervals and follow-up extends through $K$, then $$ \mu^g(K)=E[\min(T^g,K)]=\sum_{k=0}^{K-1}P(T^g>k)=\sum_{k=0}^{K-1}S^g(k). $$ On a grid $0=t_0<t_1<\cdots<t_J=\tau$, if survival is represented as constant over each interval $[t_j,t_{j+1})$, the corresponding area is $\sum_{j=0}^{J-1}(t_{j+1}-t_j)S^g(t_j)$.

For a general event-time distribution, the integral identity remains exact. A finite grid sum is exact when it matches the time convention and survival representation being used; otherwise it approximates the area, with the approximation depending on grid width and how survival varies within intervals.

Worked Example: Checking the Integral Identity

Suppose survival under a regime is $S^g(t)=1-t/4$ for $0\leq t\leq2$, and $S^g(t)=1/2$ for $2<t\leq4$. Find the RMST through $\tau=4$.

The first part of the area is $$ \int_0^2(1-t/4)\,dt=\left[t-t^2/8\right]_0^2=2-\frac{4}{8}=\frac{3}{2}. $$

The second part is $$ \int_2^4\frac{1}{2}\,dt=\frac{1}{2}(4-2)=1. $$

Therefore $\mu^g(4)=3/2+1=2.5$ time units. This value lies between zero and the horizon four, as any RMST must. The calculation is an area under the survival curve, not the median event time or the mean among only those who experience the event before time four.

Worked Example: Discrete Event-Time Convention

Suppose $T^g$ takes values $1,2,3$ with probabilities $0.2,0.3,0.5$, respectively. Compute the RMST through $K=3$.

The survival probabilities at the left endpoints are $S^g(0)=1$, $S^g(1)=P(T^g>1)=0.8$, and $S^g(2)=P(T^g>2)=0.5$. Thus $$ \mu^g(3)=S^g(0)+S^g(1)+S^g(2)=1+0.8+0.5=2.3. $$

Directly, $E[\min(T^g,3)]=1(0.2)+2(0.3)+3(0.5)=0.2+0.6+1.5=2.3$, confirming the result. Using $P(T^g\geq k)$ in place of $P(T^g>k)$ without changing the interval convention would shift terms at event times and can give the wrong answer.

Constructing the Longitudinal TMLE Plug-In Estimate

For each $j=0,\ldots,K-1$, apply the longitudinal survival-risk TMLE for the binary endpoint $Y_j=I(T>j)$ under regime $g$. Briefly, its initial outcome regressions are conditional expectations of the later survival outcome given the observed history and treatment. They are evaluated at the regime’s treatment choices, targeted through the longitudinal sequence using the treatment mechanism, and finally averaged over the empirical baseline distribution. Denote the resulting targeted baseline prediction for endpoint $j$ by $\hat Q_{j,0}^{*}(L_0)$. Then $$ \hat S^g(j)=\frac{1}{n}\sum_{i=1}^n \hat Q_{j,0}^{*}(L_{0i}). $$ At $j=0$, survival is one if everyone is event-free at baseline. The endpoint $j=K$ is not needed for the unit-grid sum through $K$.

Because the RMST is a finite weighted sum of survival probabilities, the plug-in estimate is the same weighted sum of their targeted estimates: $$ \hat\mu^g(K)=\sum_{j=0}^{K-1}\hat S^g(j) =\frac{1}{n}\sum_{i=1}^n\sum_{j=0}^{K-1}\hat Q_{j,0}^{*}(L_{0i}). $$ For a grid with interval widths $\Delta_j=t_{j+1}-t_j$, use $\hat\mu^g(\tau)=\sum_{j=0}^{J-1}\Delta_j\hat S^g(t_j)$. This is a plug-in estimate: targeted predictions are substituted into the defining functional for the parameter.

Worked Example: Aggregating Targeted Survival Predictions

At three baseline covariate values, suppose the targeted predictions for survival at times zero, one, and two are as follows. The horizon is $K=3$, with unit-width intervals.

Individual	$\hat Q_{0,0}^{*}$	$\hat Q_{1,0}^{*}$	$\hat Q_{2,0}^{*}$
1	1.00	0.90	0.70
2	1.00	0.80	0.60
3	1.00	0.70	0.50

Average each column to obtain $\hat S^g(0)=1$, $\hat S^g(1)=(0.90+0.80+0.70)/3=0.80$, and $\hat S^g(2)=(0.70+0.60+0.50)/3=0.60$. Therefore $$ \hat\mu^g(3)=1+0.80+0.60=2.40. $$

Equivalently, the three individual sums are $2.60,2.40,2.20$, whose average is $7.20/3=2.40$. This checks that averaging the targeted individual predictions and then summing gives the same result as summing the survival estimates.

Why the Plug-In Formula Is Valid

The first justification is algebraic and does not depend on how the survival probabilities were estimated. For any finite collection of real-valued estimates and fixed interval weights, sums and averages can be interchanged: $$ \sum_{j=0}^{K-1}\hat S^g(j) =\sum_{j=0}^{K-1}\frac{1}{n}\sum_{i=1}^n\hat Q_{j,0}^{*}(L_{0i}) =\frac{1}{n}\sum_{i=1}^n\sum_{j=0}^{K-1}\hat Q_{j,0}^{*}(L_{0i}). $$ The equality holds because both sums are finite, so no limit or integration interchange is involved.

A second useful fact is that consistency of the component survival estimates implies consistency of the RMST estimate when the horizon is fixed. Suppose $K$ is fixed and, for every $j=0,\ldots,K-1$, $\hat S^g(j)\xrightarrow{p}S^g(j)$. For any $\epsilon>0$, if the absolute error in the sum exceeds $\epsilon$, at least one component error must exceed $\epsilon/K$. Hence $$ P\left(\left|\hat\mu^g(K)-\mu^g(K)\right|>\epsilon\right) \leq \sum_{j=0}^{K-1}P\left(\left|\hat S^g(j)-S^g(j)\right|>\epsilon/K\right)\longrightarrow 0. $$ Each probability tends to zero by the assumed componentwise convergence, and there are only finitely many terms. Thus $\hat\mu^g(K)\xrightarrow{p}\mu^g(K)$. This argument establishes the aggregation step; it does not by itself establish that any particular nuisance-model fitting procedure yields consistent component estimates. $\square$

Worked Example: Unequal Grid Intervals

Suppose a follow-up grid is $0,1,3,5$, and the estimated survival probabilities at the left endpoints are $1,0.8,0.5$. The interval widths are $1,2,2$, so the grid-based RMST estimate through time five is $$ 1(1)+2(0.8)+2(0.5)=1+1.6+1=3.6. $$

A simple unweighted sum would give $2.3$, which is incorrect for this grid because the second and third survival values each represent two time units. The weighted estimate is between zero and the horizon five, as expected.

Diagnostics, Alternatives, and Limitations

Several checks follow directly from the target’s definition. Every estimated survival probability should lie between zero and one, and an estimated RMST should lie between zero and $\tau$. A survival curve should also be nonincreasing in time. If survival risks are targeted independently for each endpoint, that last property is not guaranteed by the sum formula itself. A nonmonotone collection of estimates may still produce a numerical sum, but it is a warning that the estimated curve is not a coherent survival distribution. Inspect the endpoint-specific estimates and the fitted predictions rather than silently interpreting such a curve as valid.

The regime must also have support in the observed treatment process: very small probabilities of receiving a required treatment can make targeted updates unstable and predictions highly sensitive to a few observations. Check the distribution of estimated treatment probabilities and the targeted survival predictions. Estimates near zero or one can also indicate sparse support or boundary behavior. These diagnostics flag problems; they do not repair a failure of positivity.

An alternative is to define the bounded outcome $Z^g=\min(T^g,\tau)$ and target its mean directly. This has the same population target by the integral identity, but its longitudinal outcome regression and targeting construction must match that outcome and the observed follow-up structure. Estimating each survival risk and summing is particularly transparent when the survival-risk TMLE is already specified. Direct estimation may avoid fitting many separate endpoint regressions, but is not automatically superior or more robust.

The horizon is part of the estimand: two RMSTs at different $\tau$ answer different questions. Results are also limited by the identification assumptions and the available follow-up. If event status is not observed through $\tau$, treating censored individuals as event-free to the horizon is generally wrong; handling dropout requires methods beyond this derivation. Finally, RMST is an absolute time summary, so it does not describe every feature of the survival curves or imply a constant treatment effect over time.

Common pitfall: Summing survival probabilities at integer times without checking the time intervals can misstate the area. On a unit grid, each term has weight one; on an unequal grid, each term must be multiplied by its interval width. Likewise, $P(T>k)$ and $P(T\geq k)$ are not interchangeable when events can occur exactly at grid points.

Check Your Understanding

Try each question, then check the solutions below.

If $S^g(t)=0.9$ from time zero to one and $0.6$ from time one to three, what is the RMST through time three under this piecewise-constant representation?
On a unit grid through $K=4$, suppose $S^g(0)=1$, $S^g(1)=0.75$, $S^g(2)=0.5$, and $S^g(3)=0.25$. Calculate the RMST.
Why does componentwise convergence of a fixed finite set of estimated survival probabilities imply convergence of their RMST sum?
Name two checks that can reveal a problem in an estimated RMST analysis.

Solutions. (1) The area is $1(0.9)+2(0.6)=2.1$ time units. (2) The sum is $1+0.75+0.5+0.25=2.5$ time units. (3) The absolute error in a finite sum is bounded by the sum of the absolute component errors; with a fixed number of terms, convergence in probability of each term implies convergence in probability of the sum. (4) Check that the RMST lies between zero and the horizon and that estimated survival probabilities lie in $[0,1]$ and do not increase over time; also inspect treatment-probability support and follow-up through the horizon.