Cloning Strategies For Grace-Period Emulation

Why Clone Strategies?

A grace period allows treatment to begin after eligibility and time zero, rather than requiring initiation on that exact day. This creates a timing problem: before the grace period ends, a person who has not yet started treatment may still be following a strategy that requires initiation within the period. Classifying that person as treated from baseline uses future information; classifying the person only by treatment eventually received can also assign early follow-up selectively. Cloning makes the strategy comparison explicit from time zero.

The basic operation is to create a record for each eligible person under each strategy being compared. Each record follows that strategy’s rules. If observed care becomes incompatible with a clone’s assigned strategy, that clone is artificially censored at the point of incompatibility. Other clones for the same person may remain compatible. Because this censoring can select which outcomes remain observed, the comparison generally needs weights that account for the probability of remaining uncensored.

Definition: A strategy clone is a record for an eligible person assigned, for analysis, to one specified treatment strategy at the common time zero. Cloning a person across \(K\) strategies creates \(K\) records for that person. Artificial censoring ends a clone’s follow-up when the person’s observed care can no longer satisfy that clone’s assigned strategy. It is not a claim that the person left the study or that an outcome occurred.

Specify the Strategies Before Making Clones

The clone records are meaningful only if the strategies are operational. Specify eligibility and time zero, the allowed treatment actions and their timing, the follow-up horizon, and the outcome. State what happens when treatment is initiated during the grace period and what happens at its end. These are protocol choices, not decisions to make after examining outcomes.

For illustration, consider two strategies for eligible people who have not started treatment at time zero: (1) initiate treatment A within 30 days, and (2) do not initiate A during those 30 days. Suppose both strategies permit usual care after day 30, and the outcome is assessed through day 90. Under this definition, a person who starts A on day 10 is compatible with the first strategy until that initiation; the second strategy’s clone is censored at day 10. A person who has not started A by the end of day 30 is censored from the first strategy’s clone at the end of the grace period, but remains compatible with the second. A person who starts A after day 30 is not censored for that reason under either strategy, because the stated strategies constrain initiation only during the grace period.

A protocol could instead require continued treatment, forbid treatment after day 30, or define a different post-grace rule. Those choices would change when a clone becomes incompatible. Do not silently extend a grace-period rule beyond its specified window.

Worked Example: Creating Records From Treatment Timelines

Suppose three eligible people are followed under the two strategies just defined. Each person receives one clone for each strategy at day 0.

Person	Observed timeline	Initiate-within-30 clone	No-initiation-during-30 clone
1	Starts A on day 10	Remains compatible	Artificially censored on day 10
2	Does not start A by day 30	Artificially censored at the end of day 30	Remains compatible
3	Starts A on day 20; outcome on day 25	Remains compatible; outcome recorded	Artificially censored on day 20

For person 3, the outcome occurs after the no-initiation clone has become incompatible, so it is not an outcome observed under that clone. It is recorded for the initiate-within-30 clone, which was still compatible when the outcome occurred. By contrast, if an outcome occurred on day 5 before either person’s care had diverged from both strategies, it would be recorded in both clones. That duplication reflects the two strategy-specific records, not two independent people.

Artificial Censoring and Weighted Outcomes

Let \(r\) index a strategy, \(i\) a person, and \(t\) a follow-up interval. Let \(R_{ir}(t)=1\) if person \(i\)’s clone for strategy \(r\) remains uncensored and under observation through \(t\); otherwise \(R_{ir}(t)=0\). Artificial censoring occurs at the first time the observed treatment path violates strategy \(r\). A separately specified loss to follow-up can also end observation, but it is not itself the strategy-incompatibility rule.

To correct for selection caused by artificial censoring, define \(q_{irj}\) as the conditional probability that clone \(ir\), still uncensored at the start of interval \(j\), remains uncensored through that interval, given the measured history available at that point. For a clone observed through interval \(t\), an unstabilized inverse-probability weight is:

W_{ir}(t)=\prod_{j=0}^{t}\frac{1}{q_{irj}}

A clone with a low estimated probability of remaining compatible and observed receives a larger weight. Intuitively, it represents similar clones that were more likely to be censored. In practice, the probabilities are estimated from a model for remaining uncensored, using relevant measured baseline and time-varying history. The model must reflect the actual timing of treatment decisions, covariates, censoring, and outcomes.

Assumptions for a causal interpretation: The strategy definitions and time zero are well specified; observed outcomes under a person’s followed strategy correspond to the relevant potential outcomes (consistency); one person’s strategy does not affect another’s outcome (no interference); measured history is sufficient to control common causes of remaining uncensored and the outcome (conditional exchangeability for censoring); and each relevant history has a positive probability of remaining uncensored under the strategy (positivity). The censoring-weight model must also be adequately estimated. These assumptions are not guaranteed by cloning.

The exchangeability condition is especially important: after conditioning on the measured history used in the weight model, the outcome experience of clones artificially censored at a given interval must represent that of comparable clones who remain uncensored. If prognosis-related information affecting treatment decisions is omitted, weighting may not remove the resulting selection. Positivity requires that the relevant histories have some chance of following the strategy; no finite weight can recover information for a history where that chance is zero.

Why the Weighting Works

The basic justification is that, within a history stratum, weighting uncensored clones by the inverse of their probability of remaining uncensored offsets their under-representation. If a clone has probability \(q\) of remaining uncensored, an observed clone receives weight \(1/q\). The expected weighted contribution from that stratum is proportional to \(q(1/q)=1\), rather than \(q\). Applying this logic at each interval yields the product weight.

P(\text{remains observed})\times W = \left(\prod_{j=0}^{t}q_{irj}\right) \left(\prod_{j=0}^{t}\frac{1}{q_{irj}}\right) =1

This cancellation is a justification under the stated conditional exchangeability and positivity assumptions, not a guarantee that an estimated analysis is unbiased. Estimated probabilities can be wrong, histories may be inadequately measured, and very small probabilities can produce unstable weights.

For a binary outcome \(Y_{ir}\) by a fixed horizon \(T\), a simple weighted risk estimate within strategy \(r\) is the weighted proportion among clones with outcome ascertainment:

\widehat{R}_r(T)= \frac{\sum_i R_{ir}(T)W_{ir}(T)Y_{ir}} {\sum_i R_{ir}(T)W_{ir}(T)}

A risk difference can then be formed by subtracting the estimated risk under one strategy from the other, with the order of subtraction specified in advance. More complex outcomes or follow-up summaries require an estimator appropriate to the outcome and censoring structure. Inference must also account for the fact that multiple clones from one person are dependent; treating all clone records as independent people generally understates uncertainty.

Worked Example: Calculating a Clone’s Weight

Suppose a clone remains compatible and observed across two intervals. Based on its measured history, the estimated conditional probabilities of remaining uncensored are \(q_{ir0}=0.80\) and \(q_{ir1}=0.75\). Its unstabilized weight through the second interval is:

W_{ir}(1)=\frac{1}{0.80}\times\frac{1}{0.75} =1.25\times1.333 =1.667

The corresponding probability of remaining uncensored across both intervals is \(0.80\times0.75=0.60\). The weight is therefore \(1/0.60=1.667\). This clone contributes about 1.667 weighted records to the relevant strategy-specific calculation, rather than one. This arithmetic illustrates the weight only; it does not show that the probabilities are correctly estimated or that the assumptions hold.

Worked Example: Comparing Weighted Risks

Consider a hypothetical analysis with four outcome-ascertained clones per strategy at day 90. The table gives each clone’s outcome \(Y\) (1 for the outcome, 0 otherwise) and its censoring weight. These values are illustrative and do not represent a real study.

Strategy	Outcome values \(Y\)	Corresponding weights \(W\)
Initiate within 30 days	1, 0, 1, 0	1.00, 1.25, 2.00, 1.50
No initiation during 30 days	0, 1, 0, 0	1.00, 1.20, 1.50, 1.00

For initiation within 30 days, the weighted outcome total is \(1.00(1)+1.25(0)+2.00(1)+1.50(0)=3.00\), and the total weight is \(1.00+1.25+2.00+1.50=5.75\). Thus:

\widehat{R}_{\text{initiate}}(90)=\frac{3.00}{5.75}=0.5217

For no initiation during the 30 days, the weighted outcome total is \(1.00(0)+1.20(1)+1.50(0)+1.00(0)=1.20\), and the total weight is \(4.70\). Therefore:

\widehat{R}_{\text{no-init}}(90)=\frac{1.20}{4.70}=0.2553

The weighted risk difference, defined here as initiate-within-30 minus no-initiation-during-30, is \(0.5217-0.2553=0.2664\), or 26.64 percentage points. Under the assumptions and the stated strategy definitions, this estimates a higher day-90 outcome risk under the initiate-within-30 strategy. The calculation alone does not establish those assumptions, and it does not provide a confidence interval.

Diagnostics, Alternatives, and Limitations

Check the clone records against a small set of hand-worked treatment timelines before fitting any models. Confirm that every eligible person has the intended strategy records, that all start at the same time zero, and that censoring occurs at the first incompatibility—not before it and not after it. Check whether outcomes that happen before incompatibility are retained, and whether follow-up after the grace period follows the protocol’s stated rules.

Inspect the estimated probabilities and weights by strategy and over time. Probabilities near zero produce extreme weights; a few highly weighted clones can dominate the estimate. Review weight distributions, effective sample size, and the histories associated with large weights. Truncating extreme weights can improve numerical stability but changes the analysis and may introduce bias; it should be justified and reported, not used to conceal poor positivity. Also verify that records censored because of loss to follow-up are handled under an appropriate model rather than mistaken for artificial strategy censoring.

Cloning with artificial censoring and weighting is useful when strategies permit a grace period and treatment paths can be classified as compatible or incompatible over time. A simpler baseline comparison may be easier to communicate, but assigning people to strategies according to treatment they receive later risks using future information and can distort the comparison from time zero. Restricting analysis to people who eventually initiate treatment also changes who is included and can select on post-baseline events. Cloning does not eliminate confounding: it makes the strategy assignment and compatibility rules explicit, while weighting addresses selection from censoring only under its assumptions.

The method is limited by the quality and timing of observational data. If treatment dates are recorded imprecisely, the first incompatible day may be misclassified. If covariates that predict both treatment decisions and the outcome are unavailable, censoring weights may not remove selection. If a strategy is rarely followed in some histories, the resulting weights may be unstable or the comparison unsupported. Finally, the clone records are not independent replicates: variance estimation must account for clustering by original person.

Common Mistake

Common mistake: “This person started treatment on day 20, so I will label them treated from day 0 and count their outcome from day 0.” That classification uses future treatment to define baseline strategy membership and can assign untreated grace-period time according to what happens later. Instead, create both strategy clones at time zero, keep each compatible clone under observation until its own first incompatibility, and weight for the resulting artificial censoring under the required assumptions.

Check Your Understanding

Answer each question using the specified strategy rules. Solutions follow the questions.

Under “initiate A within 30 days” versus “do not initiate A during 30 days,” what happens to each clone when a person starts A on day 8?
A clone has conditional probabilities of remaining uncensored of 0.90 and 0.80 over two intervals. Calculate its unstabilized weight through both intervals.
Why should an outcome on day 4 be recorded in both strategy clones if neither clone has become incompatible by day 4?
Name two assumptions needed to interpret the weighted strategy comparison causally, and explain what each contributes.

Solutions. (1) The initiate-within-30 clone remains compatible; the no-initiation-during-30 clone is artificially censored on day 8. (2) \(W=(1/0.90)(1/0.80)=1/0.72=1.389\). (3) Both clones still represent compatible strategies at day 4, so the observed outcome is available under each clone; the duplicate records are dependent, not separate people. (4) Conditional exchangeability requires measured history to control common causes of censoring and outcome; positivity requires a nonzero probability of remaining uncensored for relevant histories, so the weighted comparison is supported by observed data rather than extrapolation from impossible paths.