Why Clone Participants?
Suppose a target trial compares two strategies: initiate treatment within a specified grace period, or do not initiate during that period. At time zero, an eligible participant who has not yet started treatment may be consistent with either strategy. Their observed treatment decisions later in the grace period reveal which strategy their history follows, but using that future information to place them into a strategy group at baseline can create a misleading comparison.
Cloning represents this initial ambiguity directly. Each eligible participant is copied into one record for each strategy with which their treatment history is initially compatible. A clone is not a new person, a randomized assignment, or a second independent observation. It is a strategy-specific representation of the same participant in the emulated trial.
The strategies must be explicit enough to assess compatibility. For example, “initiate soon” is not operational; “initiate by the end of day 7 after time zero” is. The earlier material on treatment strategies and grace periods establishes how to define those rules. Here the focus is how to represent participants when treatment may begin during the grace period.
Defining Compatibility and the Clone Records
Let \(i=1,\ldots,n\) index eligible participants, \(k=1,\ldots,K\) index strategies, and \(t=0,\ldots,G\) index discrete times from time zero through the end of a grace period. Let \(A_{it}=1\) indicate that participant \(i\) initiates treatment at time \(t\), and \(A_{it}=0\) otherwise. Write \(H_i(t)=(A_{i0},\ldots,A_{it})\) for the observed treatment history through \(t\). Let \(d_k\) denote the treatment histories allowed by strategy \(k\).
For a rule requiring treatment initiation by day \(G\), starting by \(G\) is compatible and remaining untreated through \(G\) is not. For a rule prohibiting treatment through \(G\), the reverse is true. Before the deadline, an untreated participant can still be compatible with a strategy requiring initiation by the deadline: they have not yet missed it. The exact time at which histories cease to be compatible follows the operational rule, including how treatment on the deadline is classified.
A useful data structure has one row per participant, strategy, and follow-up time at which the clone is represented. It retains the original participant identifier, a strategy identifier, time, relevant covariates, treatment history, outcome information, and a compatibility indicator. At time zero, the strategy-specific records share the participant’s baseline information. Later records reflect the same observed participant’s course; they are not separate observed people.
Worked Examples of Cloning
Worked Example: Two Strategies and a Three-Day Grace Period
Four eligible participants are untreated at time zero. Compare \(d_1\), initiate by the end of day 3, with \(d_2\), do not initiate through day 3. Their observed initiation times are: A, day 2; B, day 4; C, never; and D, day 1. Create the initial clones and determine compatibility through day 3.
At time zero, each participant has one clone for each strategy. There are \(4\times2=8\) initial clone records. Treatment has not yet begun for any participant, so both strategy histories are initially possible for all four.
| Participant | Observed initiation | Clone under \(d_1\): initiate by day 3 | Clone under \(d_2\): no initiation through day 3 |
|---|---|---|---|
| A | Day 2 | Compatible through day 3 | Not compatible after initiation |
| B | Day 4 | Not compatible after day 3 passes untreated | Compatible through day 3 |
| C | Never | Not compatible after day 3 passes untreated | Compatible through day 3 |
| D | Day 1 | Compatible through day 3 | Not compatible after initiation |
By the end of day 3, A and D have observed histories compatible with \(d_1\); B and C have histories compatible with \(d_2\). This is not a baseline classification: all eight clones began at time zero. It records how each observed history compares with each strategy as the grace period unfolds.
Worked Example: Overlapping Initiation Deadlines
Consider three strategies for participants untreated at time zero: \(d_1\), initiate by day 3; \(d_2\), initiate by day 7; and \(d_3\), do not initiate through day 7. Three participants start on day 2, day 5, or never. How many clones are initially created, and which histories are compatible when each deadline has passed?
There are \(3\) participants and \(3\) strategies, so the initial data contain \(3\times3=9\) clones. The participant starting on day 2 is compatible with \(d_1\) and \(d_2\), but not with \(d_3\), because treatment began during the no-initiation period. The participant starting on day 5 is compatible with \(d_2\), but not \(d_1\), whose deadline has passed untreated, or \(d_3\), which prohibits initiation through day 7. The participant who never starts is compatible with \(d_3\), but not with either initiation-by-deadline strategy once its deadline has passed.
The observed histories therefore fit one, two, or more strategy histories according to the rules and the time being assessed; they are not forced into one strategy at baseline. In particular, before day 3 the day-5 starter’s \(d_1\) clone has not yet failed its deadline: the participant could still initiate by day 3.
Worked Example: Baseline Covariates in the Cloned Data
Four participants have a binary baseline covariate \(L_i\), with values \(1,1,0,0\). Each is cloned into two strategies that are both compatible at time zero. Find the mean \(L\) in each strategy’s baseline clone records.
Each strategy has the same four participants’ baseline values. Thus, for either strategy, the mean is \( (1+1+0+0)/4=2/4=0.5 \). There are eight clone records overall, but the mean in each strategy’s baseline records remains \(0.5\). The strategy-specific records share the same baseline covariates because each is a copy of an eligible participant, not a new sample selected by their future treatment.
This arithmetic does not establish that later outcomes are equal under the strategies, nor does it make the two sets of clone records independent. It only describes the baseline construction.
Two Properties of the Construction
The first property is that cloning avoids assigning a participant to a single strategy using treatment that occurs after time zero. For strategies that are all initially compatible, the construction is mechanical: start with the eligible cohort, then create one record per participant per strategy. The compatibility assessment uses observed history as it accrues.
Proof. Index the eligible participants by \(i=1,\ldots,n\) and the strategies by \(k=1,\ldots,K\). The construction creates the pair \((i,k)\) once for every such participant-strategy pair. There are \(K\) pairs for each of \(n\) participants, hence \(nK\) pairs in total. Since each pair includes each participant once for every strategy, each participant has one baseline clone in each strategy. No part of this enumeration depends on treatment occurring after time zero. \(\square\)
The second property concerns baseline covariate distributions. Let \(L_i\) be any measured baseline quantity and \(S_k\) the set of time-zero clone records for strategy \(k\). When every eligible participant is cloned into every strategy, the empirical distribution of \(L\) is identical across strategies by construction.
Proof. For a fixed strategy \(k\), \(S_k\) contains precisely one baseline record for each of the \(n\) participants, so \(|S_k|=n\). The value \(L_i\) in that record is the original participant’s baseline value. Consequently, the right-hand sum contains exactly the same \(n\) indicator terms as the left-hand sum, with the same denominator. This proves equality for every \(B\), so the empirical distributions agree. The statement applies at baseline; later records may differ in which clones remain compatible. \(\square\)
These properties describe the data structure, not a causal effect. Causal interpretation of a contrast between strategies still depends on a well-defined target trial, valid treatment and outcome measurement, consistency, and the assumptions required by the analysis. Cloning does not randomize treatment or remove confounding.
Diagnostics, Alternatives, and Limitations
A practical audit starts with the expected number of baseline rows. If there are \(n\) eligible participants and \(K\) strategies initially compatible with all, there should be \(nK\) time-zero records. Check that every expected participant-strategy pair appears exactly once, all copies of baseline variables agree within participant, and the strategy labels correspond to the intended rules. Also verify the timing convention at each deadline: an initiation on day \(G\) must be treated consistently with the rule “by day \(G\).”
A common mistake is to place each participant in the “initiate” or “do not initiate” group at time zero according to what they eventually did. That reasoning uses future information: a person who has not yet started treatment may still start before the deadline. The correction is to create the initially compatible clones at time zero and assess compatibility as the observed history develops. A second mistake is to treat clone records as independent participants. They share the same original person and often the same measurements; analyses must preserve the original participant identifier and account for this dependence where required.
An alternative is to classify people by observed treatment at the end of the grace period. That can discard the time-zero comparison and select groups using post-baseline information. A landmark analysis instead starts follow-up at a later landmark among those eligible at that time; it addresses a different target population and time zero. Cloning is useful when the target trial’s strategies are assigned at the original time zero but allow decisions to unfold during a grace period.
Cloning also has limits. It cannot make a vague strategy operational, recover treatment times or outcomes that were not measured, or guarantee that every strategy is supported by the observed data. If some strategies are not initially compatible for certain participants, only the compatible strategy clones should be created, and the simple \(nK\) count and identical-baseline-distribution result need not hold. In the usual grace-period comparison, clones that cease to follow their assigned rule must be handled in subsequent analysis; cloning alone does not specify that analysis. In particular, it is not itself an estimator of a per-protocol effect.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- There are 120 eligible participants and two strategies, both compatible at time zero. How many time-zero clone records should there be?
- A strategy requires initiation by day 5. A participant is untreated through day 3. Is their clone already incompatible with that strategy? Explain.
- Four participants have baseline values \(0,0,1,1\), and each is cloned into three initially compatible strategies. What is the baseline mean in each strategy’s clone records?
- Why is assigning participants to a strategy based on whether they eventually initiate during the grace period not the same as cloning them at time zero?
Solutions. (1) There are \(120\times2=240\) records, because each participant contributes one clone to each strategy. (2) No. The deadline has not passed, so initiation by day 5 remains possible and the observed history is still compatible. (3) In each strategy the four values are \(0,0,1,1\), so the mean is \((0+0+1+1)/4=0.5\). (4) Eventual initiation is information from after time zero. Assigning by that future event puts each person into a single observed-treatment group; cloning instead creates the strategy-specific records that are initially compatible and evaluates their histories as time unfolds.