Why a Candidate Blip Changes Who Is at Risk
Failure-time g-estimation asks which candidate treatment-effect parameter makes the treatment assignment compatible with the model for the failure time that would have been observed without treatment. A candidate parameter does more than rescale an outcome: it changes the time at which a treated subject is placed on the untreated scale. That change can alter whether the subject is counted as at risk at a given time. Artificial censoring is a bookkeeping device for this candidate-specific change; it is not a subject’s actual loss to follow-up.
For clarity, consider a single binary treatment \(A\in\{0,1\}\) assigned at baseline, a positive observed failure time \(T\), and a log-acceleration parameter \(\psi\). In the rank-preserving accelerated failure-time version of the model, treatment multiplies the untreated failure time by \(\exp(\psi)\). Thus, a positive \(\psi\) represents treatment that lengthens survival. The candidate blip-down transformation maps the observed time back to the untreated scale.
This definition presumes the model’s transformation is meaningful for the treatment and time scale in the analysis. It also presumes that the relevant failure times are observed: ordinary right censoring is set aside here. If a model is distributional rather than rank-preserving, the transformation must be interpreted according to that model; one should not automatically claim that it recovers each person’s individual counterfactual time.
Artificial Follow-Up and Risk-Set Definitions
To express the candidate transformation as a survival-data construction, compare the observed clock \(T\) with the blipped-down clock \(U(\psi)\). Stop artificial follow-up at the earlier of those two times. If the observed failure occurs first, it is recorded as a failure in this construction. If the blipped-down time occurs first, follow-up ends there by artificial censoring.
The risk-set identity follows directly: \(\min\{T,U(\psi)\}\geq t\) holds if and only if both \(T\geq t\) and \(U(\psi)\geq t\). The indicator \(\Delta(\psi)\) labels the earlier clock; it does not mean the artificial censoring time is an observed death. In particular, a subject may remain informative for comparisons before \(X(\psi)\) and cease contributing afterwards.
Proof. If \(A=0\), the definition gives \(U(\psi)=T\), so \(X(\psi)=\min(T,T)=T\), \(\Delta(\psi)=I(T\leq T)=1\), and \(Y(\psi,t)=I(T\geq t)\). If \(A=1\) and \(\psi>0\), then \(\exp(-\psi)<1\), hence \(U(\psi)=T\exp(-\psi)<T\). It follows that \(X(\psi)=U(\psi)\) and \(I\{T\leq U(\psi)\}=0\). The risk-set statement then follows from its definition. \(\square\)
The positive-\(\psi\) result gives the familiar artificial-censoring picture when treatment is assumed to prolong survival: treated subjects’ observed failures occur later than their candidate untreated-scale failures, so their artificial follow-up stops at the latter. If \(\psi=0\), the clocks tie and the stated tie convention records a failure. If \(\psi<0\), then \(U(\psi)>T\) for treated subjects; the observed failure comes first and \(\Delta(\psi)=1\). The formulas remain valid, but the verbal description “treated subjects are artificially censored first” no longer applies.
Worked Examples
Worked Example: Constructing Artificial Follow-Up
Suppose the candidate is \(\psi=\log(2)\), so \(\exp(-\psi)=1/2\). Four subjects have the following observed treatment and failure times. Compute \(U(\psi)\), \(X(\psi)\), \(\Delta(\psi)\), and membership in the artificial risk set at \(t=4\).
| Subject | \(A\) | \(T\) | \(U(\psi)\) | \(X(\psi)\) | \(\Delta(\psi)\) | \(Y(\psi,4)\) |
|---|---|---|---|---|---|---|
| 1 | 0 | 3 | 3 | 3 | 1 | 0 |
| 2 | 0 | 7 | 7 | 7 | 1 | 1 |
| 3 | 1 | 6 | 3 | 3 | 0 | 0 |
| 4 | 1 | 10 | 5 | 5 | 0 | 1 |
For subjects 1 and 2, \(A=0\), so \(U=T\). Their artificial follow-up equals their observed time, and both are failures under the construction. For subject 3, \(U=6/2=3\), so \(X=3\) and \(\Delta=I(6\leq3)=0\); the subject is artificially censored before time 4. For subject 4, \(U=10/2=5\), so \(X=5\) and \(\Delta=I(10\leq5)=0\); this subject remains at risk at time 4 and is artificially censored later. The risk set at 4 consists of subjects 2 and 4.
Worked Example: How the Candidate Value Changes Membership
A treated subject has observed failure time \(T=12\). Compare candidates \(\psi_1=\log(1.5)\) and \(\psi_2=\log(3)\), and determine whether this subject is in the artificial risk set at \(t=5\).
At \(\psi_1\), \(\exp(-\psi_1)=1/1.5=2/3\), so \(U(\psi_1)=12(2/3)=8\). Consequently, \(X(\psi_1)=\min(12,8)=8\), and \(Y(\psi_1,5)=I(8\geq5)=1\).
At \(\psi_2\), \(\exp(-\psi_2)=1/3\), so \(U(\psi_2)=12/3=4\). Now \(X(\psi_2)=4\), and \(Y(\psi_2,5)=I(4\geq5)=0\). The same observed record is informative at time 5 for the first candidate but not the second. A candidate-specific risk set must therefore be recomputed when \(\psi\) changes.
Worked Example: Checking Risk-Set Membership at a Decision Time
Suppose a treatment decision is made at time \(\tau=6\), and a treated subject has \(T=9\). For candidate \(\psi=\log(1.5)\), calculate the clocks and decide whether the subject can contribute to a risk-set comparison at the decision time.
Here, \(U(\psi)=9\exp\{-\log(1.5)\}=9/1.5=6\). Thus \(X(\psi)=\min(9,6)=6\). Under the convention that a subject is at risk at time \(t\) when \(X(\psi)\geq t\), \(Y(\psi,6)=I(6\geq6)=1\). The subject is present at the boundary time, but \(\Delta(\psi)=I(9\leq6)=0\): the artificial follow-up ends there by censoring, not by observed failure. Immediately after time 6, the subject is no longer in the risk set. If the analysis instead defines eligibility as strict survival beyond the decision time, it must use \(X(\psi)>\tau\); the boundary convention must be stated and applied consistently.
How These Risk Sets Enter G-Estimation
At each candidate \(\psi\), the transformation produces a different set of artificial follow-up records and hence different risk sets. G-estimation uses the candidate blipped-down outcome in a treatment-assignment comparison: the candidate is assessed by whether the treatment decision is compatible with the structural model and the assumed treatment-assignment mechanism, conditional on the relevant measured history. For a baseline decision, that history may be baseline covariates; for a decision made after follow-up has begun, eligibility must also respect survival up to that decision. The precise estimating equation depends on the treatment model and design. The risk-set construction alone is not an estimating equation and does not, by itself, establish identification.
A useful consequence is that candidate values can change both the transformed times and the records available at a comparison time. In the positive-\(\psi\) case, treated subjects with larger candidate acceleration factors are blipped to earlier times. Their artificial follow-up can therefore end before a comparison time at which they would have remained eligible under a smaller candidate. A g-estimation procedure must use the risk set associated with the candidate being evaluated, rather than holding the risk set fixed across the search.
Proof. For a treated subject, \(U(\psi)=T\exp(-\psi)\). Since the exponential function is increasing, \(\psi_2\geq\psi_1\) implies \(\exp(-\psi_2)\leq\exp(-\psi_1)\), and hence \(U(\psi_2)\leq U(\psi_1)\). Taking the minimum with the same \(T\) preserves this ordering: $$ X(\psi_2)=\min\{T,U(\psi_2)\} \leq \min\{T,U(\psi_1)\}=X(\psi_1). $$ Therefore \(X(\psi_2)\geq t\) implies \(X(\psi_1)\geq t\), which gives \(Y(\psi_2,t)\leq Y(\psi_1,t)\). For untreated subjects, \(X(\psi)=T\) for every candidate, so their risk-set membership does not change. \(\square\)
Assumptions, Diagnostics, and Limitations
The construction is most straightforward when the failure time is fully observed and the candidate structural model defines a valid blip-down transformation. Its causal interpretation additionally relies on consistency, an adequate treatment-assignment model or an appropriate randomization design, and the exchangeability and positivity conditions required by the g-estimation strategy. In a longitudinal setting, treatment must be conditionally unconfounded given the measured history used at each decision; the risk-set definition cannot repair unmeasured confounding or a misspecified blip function.
Several checks help reveal a faulty implementation. Verify that untreated records are unchanged for every candidate. For a positive candidate and a treated record, check that \(U(\psi)<T\), \(\Delta(\psi)=0\), and \(X(\psi)=U(\psi)\). Confirm that risk-set membership is nonincreasing in \(\psi\) for treated records when candidates are ordered as in the result above. Tabulate the number of treated and untreated subjects at risk across candidate values and times; abrupt or nearly empty risk sets may indicate weak information or a coding error. At \(\psi=0\), the clocks tie, so the stated tie convention should be visible in the implementation.
Artificial censoring differs from ordinary right censoring. Ordinary censoring occurs because the actual failure time is not observed after a follow-up limit or loss to follow-up. Artificial censoring is deterministically generated from observed data and the candidate \(\psi\), and it changes as \(\psi\) changes. If actual right censoring is present, one cannot generally substitute the censoring time for \(T\) in the transformation and proceed as if the failure time were known; additional methods are needed to address that problem.
The alternative of directly transforming every fully observed failure time to \(U(\psi)\) is useful for understanding the blip-down model, but it does not by itself spell out which records contribute to a survival-risk-set comparison. Artificial follow-up makes that bookkeeping explicit. Conversely, artificial censoring is not a replacement for modeling the treatment mechanism, checking positivity, or adjusting for actual censoring. Risk-set counts and treatment balance are diagnostics, not proofs that the causal assumptions hold.
The key practical rule is to recompute \(U(\psi)\), \(X(\psi)\), \(\Delta(\psi)\), and \(Y(\psi,t)\) at each candidate value, while keeping the event, censoring, and boundary conventions fixed. The construction identifies which records are available to a candidate-specific comparison; the g-estimation procedure then uses those records together with its treatment-assignment model.
Check Your Understanding
Work through each question, then compare your reasoning with the solutions.
- A treated subject has \(T=8\) and candidate \(\psi=\log(2)\). Find \(U(\psi)\), \(X(\psi)\), and \(\Delta(\psi)\).
- State the condition for a subject to be in the artificial risk set at time \(t\), using both \(T\) and \(U(\psi)\).
- For a treated subject with \(T=10\), compare artificial follow-up at \(\psi=0\) and at \(\psi=\log(2)\).
- Why is artificial censoring not the same as ordinary right censoring?
Solutions. (1) \(U=8\exp\{-\log(2)\}=4\), so \(X=\min(8,4)=4\) and \(\Delta=I(8\leq4)=0\). (2) Membership is \(Y(\psi,t)=I\{T\geq t,\ U(\psi)\geq t\}\), equivalently \(I\{X(\psi)\geq t\}\). (3) At \(\psi=0\), \(U=10\), \(X=10\), and the tied clocks give \(\Delta=1\) under the convention used here. At \(\psi=\log(2)\), \(U=5\), \(X=5\), and \(\Delta=0\); the record is artificially censored at time 5. (4) Ordinary right censoring reflects incomplete observation of the actual failure time. Artificial censoring is computed from the observed time and candidate blip value, so it changes with the candidate and is part of the risk-set construction.