From a Candidate Blip to a G-Estimate
In failure-time g-estimation, the unknown acceleration parameter is not obtained by simply fitting a regression of observed survival time on treatment. Instead, each candidate parameter value is used to remove the modeled effect of treatment from the observed failure time. The candidate is supported when the resulting blipped-down time has no residual association with the treatment decision, conditional on the measured history that informed that decision.
This example uses two treatment decisions. The first is at time 2, and the second is at time 3. We focus on a single parameter \(\psi\) for the first treatment decision; the second decision is included to show how artificial censoring protects a later risk set. All individuals are alive and uncensored at time 2. Write \(T\) for the failure time, \(C\) for the right-censoring time, \(Y=\min(T,C)\) for the observed time, and \(\Delta=I(T\le C)\) for the failure indicator. The first treatment is \(A_1\), and \(H_1\) denotes the measured history just before that decision.
The sign in the transformation matters. If \(\psi>0\), treatment lengthens the post-treatment survival time under this model. Removing that effect therefore multiplies the treated person’s observed remaining time by \(\exp(-\psi)\), not \(\exp(\psi)\). This transformation changes only the time after treatment begins; it leaves the time origin and the time before treatment fixed.
A Small Right-Censored Longitudinal Dataset
Suppose \(A_1\) is assigned with probability one half at time 2, independently of future failure time. The second decision \(A_2\) is made at time 3 among people still at risk. There is no measured baseline history in this deliberately small example, so the treatment probability at the first decision is \(e(H_1)=P(A_1=1\mid H_1)=0.5\). The table lists six observed failures and one right-censored individual. The censoring time of 12 occurs after the listed failures.
| Person | \(A_1\) | Observed time \(Y\) | \(\Delta\) | \(A_2\) at time 3? |
|---|---|---|---|---|
| 1 | 0 | 3 | 1 | No: failed at 3 |
| 2 | 0 | 4 | 1 | 1 |
| 3 | 0 | 5 | 1 | 0 |
| 4 | 1 | 4 | 1 | 0 |
| 5 | 1 | 6 | 1 | 1 |
| 6 | 1 | 8 | 1 | 0 |
| 7 | 1 | 12 | 0 | 1 |
For the first decision’s estimating equation, the transformed failure times for people 1–6 are used. Person 7 contributes a censoring bound, not a failure time. Since censoring occurs at 12 and all six failures occur by 8, the example’s censoring survival probability is one at each observed failure time. The inverse-probability-of-censoring weights for these six failures are therefore one. This convenient feature makes the arithmetic transparent; it does not make it generally valid to ignore right censoring.
Worked Example: Transforming Failure Times and Finding the Candidate
Evaluate three candidates: \(\psi=0\), \(\psi=\log(2)\), and \(\psi=\log(4)\). For controls, \(U_\psi=Y\). For treated people, compute \(U_\psi=2+(Y-2)\exp(-\psi)\).
At \(\psi=0\), \(\exp(-\psi)=1\), so the six failure times after blip-down are \(3,4,5\) for \(A_1=0\) and \(4,6,8\) for \(A_1=1\). The control mean is \((3+4+5)/3=4\); the treated mean is \((4+6+8)/3=6\). Treatment remains associated with the transformed time.
At \(\psi=\log(2)\), \(\exp(-\psi)=1/2\). The treated times become \(2+(4-2)/2=3\), \(2+(6-2)/2=4\), and \(2+(8-2)/2=5\). The control times remain \(3,4,5\). Both group means equal 4, so the empirical treatment–time association is removed.
At \(\psi=\log(4)\), \(\exp(-\psi)=1/4\). The treated times are \(2.5,3,3.5\), with mean 3; the control mean remains 4. The association has changed direction. Thus the candidate that balances the transformed means in this example is \(\hat\psi=\log(2)\).
This is a small-sample empirical calculation, not a claim that every possible measure of association is zero. With a constant treatment probability and no history, the corresponding centered score is $$ \sum_{i:\Delta_i=1}(A_{1i}-0.5)U_{\psi i}. $$ At \(\psi=\log(2)\), it equals \((3+4+5)/2-(3+4+5)/2=0\). At \(\psi=0\), it equals \((4+6+8)/2-(3+4+5)/2=3\). The root therefore agrees with the equal-means calculation.
Why the Score Targets the Blip
The identifying idea is conditional treatment exchangeability at the decision: given \(H_1\), the treatment choice is unrelated to the failure time that would remain after the modeled treatment effect is removed. If the model is correct at \(\psi_0\), then \(U_{\psi_0}\) is independent of \(A_1\) conditional on \(H_1\). Let \(e(H_1)=P(A_1=1\mid H_1)\), with \(0<e(H_1)<1\).
To verify the moment, condition on \(H_1\). Conditional exchangeability gives \(E(U_{\psi_0}\mid A_1,H_1)=E(U_{\psi_0}\mid H_1)\). Also, \(E(A_1-e(H_1)\mid H_1)=0\). Therefore $$ E[\{A_1-e(H_1)\}U_{\psi_0}\mid H_1] =E(U_{\psi_0}\mid H_1)E[A_1-e(H_1)\mid H_1]=0. $$ Taking expectations over \(H_1\) proves the result. The argument explains why the score centers treatment by its conditional probability rather than comparing raw treatment groups when treatment probabilities differ across histories.
Right censoring requires an additional condition for the observed-data version of this moment. Suppose censoring is independent of failure time conditional on the treatment and measured history, and define \(G(t\mid A_1,H_1)=P(C\ge t\mid A_1,H_1)\), with \(G(t\mid A_1,H_1)>0\) over the failure-time range used. The weight for an observed failure at \(Y\) is \(1/G(Y\mid A_1,H_1)\).
For the proof, condition on \(T,A_1,H_1\). Conditional independent censoring implies \(P(C\ge T\mid T,A_1,H_1)=G(T\mid A_1,H_1)\), and \(\Delta=I(C\ge T)\), up to the chosen convention for ties. Hence $$ E\left[\left.\frac{\Delta}{G(T\mid A_1,H_1)}q(T,A_1,H_1)\right|T,A_1,H_1\right] =\frac{q(T,A_1,H_1)}{G(T\mid A_1,H_1)}G(T\mid A_1,H_1) =q(T,A_1,H_1). $$ Taking expectations proves the identity. In practice \(T\) is not observed when \(\Delta=0\); the displayed weighted expression is evaluated only for failures, using \(Y=T\) on those records. This identity justifies censoring-weighted score equations, not treating censored times as failures. \(\square\)
Artificial Censoring at the Later Decision
Blipping down can change who belongs in a later treatment decision’s risk set. At time 3, a person is eligible for \(A_2\) in the blipped-down world only if their reconstructed failure time is strictly greater than 3. The strict inequality means that a failure at time 3 is not considered at risk immediately after that decision time. For a candidate \(\psi\), those with \(U_\psi\le3\) are artificially censored at 3 for analyses of the later decision. This is not ordinary loss to follow-up; it is a candidate-dependent construction needed to avoid assigning a later treatment to someone who would already have failed in the blipped-down world.
Worked Example: Applying Artificial Censoring at Time 3
Use the candidate \(\psi=\log(2)\). For the three treated failures, the blipped-down times are 3, 4, and 5. The first treated person, with observed failure time 4, has \(U_\psi=3\). Although this person actually survived to the second decision, the reconstructed failure is at the decision time, so this record is artificially censored at 3 for the \(A_2\) analysis. The treated people with transformed times 4 and 5 remain eligible.
Among controls, the transformed times are still 3, 4, and 5. The person who fails at 3 is not in the actual time-3 risk set. The other two remain eligible. The right-censored person has \(Y=12\), \(\Delta=0\), and transformed censoring bound \(2+(12-2)/2=7\). That bound exceeds 3, so this record is not artificially censored before the second decision; it remains right-censored rather than being classified as a failure.
The artificial-censoring rule is reapplied for each candidate \(\psi\). It is part of constructing the appropriate risk set, not an optional deletion based on whether a result appears inconvenient. In a full analysis, subsequent treatment and censoring mechanisms must also be handled under their respective assumptions.
Checks, Alternatives, and Limits
A practical calculation searches over candidate values, transforms the observed failure times, applies the artificial-censoring rule at each later decision, and evaluates a treatment-residual score with censoring adjustment. With measured history, the score uses \(A_1-\hat e(H_1)\); a conditional model or stratified score is needed rather than an unadjusted comparison. The estimate is a root or a value minimizing the absolute score if a finite-sample equation has no exact root.
Choose \(\psi\) and apply the failure-time blip-down to failures and censoring bounds.
At each later decision, artificially censor records whose candidate blipped-down failure time is at or before that decision time.
Use treatment residuals conditional on history and account for right censoring, for example with inverse-probability-of-censoring weights under the required assumptions.
Search candidate values for a score near zero, then inspect score behavior, positivity, censoring support, and sensitivity to model choices.
In the worked data, the score changes from positive at \(\psi=0\), to zero at \(\log(2)\), to negative at \(\log(4)\). A failure to see a stable crossing can signal weak information, poor overlap in treatment histories, a misspecified blip function, or censoring so heavy that the relevant transformed failure times are not supported by observed follow-up. The score may also have multiple roots in more complicated models; a numerical zero alone does not establish that the structural model is correct.
Compared with a standard accelerated failure-time regression, g-estimation directly targets the structural nested blip and uses treatment assignment conditional on history. It does not require the same outcome-distribution specification as a fully parametric survival regression, but it relies on adequate treatment and censoring models and conditional exchangeability. A weighted marginal structural model is another approach to treatment-confounding adjustment; it targets a different parameterization and does not replace the candidate-specific blip-down and risk-set construction shown here. None of these approaches repairs unmeasured confounding or a lack of positivity.
A frequent error is to transform every observed time and then treat all transformed values as uncensored failures. That is wrong for two separate reasons: a censored record supplies only a lower bound on its failure time, and a candidate blip can move a record out of a later decision’s risk set. The correction is to preserve the failure indicator, use an appropriate censoring adjustment under its assumptions, and apply artificial censoring at each later decision using the candidate transformed time. Another error is to reverse the sign in the exponential; checking that a positive acceleration effect is undone by multiplying the treated residual time by \(\exp(-\psi)\) prevents that mistake.
The numerical root is informative only under the model’s assumptions: consistency, a correctly specified blip transformation, treatment exchangeability given the measured history, positivity, and appropriate handling of right censoring. It is an estimation procedure, not a guarantee that these assumptions hold. The introductory material on structural nested failure-time models and g-estimation provides the broader framework; the calculation here emphasizes how candidate transformations, censoring, and later risk sets interact in an actual example.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- For a treated person who fails at time 10 after treatment began at time 2, calculate \(U_\psi\) when \(\psi=\log(2)\).
- Why does a right-censored observation at time 7 not contribute a failure time of 7 to the g-estimation score?
- At a later decision time 3, what happens to a record with candidate blipped-down failure time \(U_\psi=2.8\)?
- In the worked dataset, what is the treated mean transformed failure time at \(\psi=\log(4)\), and how does it compare with the control mean?
- Name two conditions needed for the censoring-weighted score to represent its full-data counterpart.
Solutions. (1) \(U_\psi=2+(10-2)/2=6\). (2) The observation says only that failure occurs after the censoring time; it is a censoring bound, not an observed failure, and must be handled under a censoring model. (3) It is artificially censored at time 3 for the later-decision analysis because \(2.8\le3\). (4) The treated times are \(2.5,3,3.5\), whose mean is 3; the control mean is 4. (5) Conditional independent censoring given treatment and measured history, and positive censoring survival over the failure-time range, are both required.