Observed-Data Notation for Longitudinal TMLE

Why the Observation Record Needs Careful Indices

Longitudinal TMLE is written in terms of information recorded across a sequence of visits. The notation must say not only which covariates and treatments are present, but also when each is measured and whether it is available after a participant is censored. A misplaced time index can reverse the order of covariate measurement and treatment, while an omitted censoring indicator can make unavailable outcomes look like observed values.

We use a finite sequence of \(K\) treatment decisions. At visit \(t\), the time-varying covariate \(L_t\) is measured first, treatment \(A_t\) is then assigned or received, and censoring during the interval after that decision is recorded by \(C_t\). The terminal outcome \(Y\) is measured after the last decision, provided the participant remains uncensored through the end of follow-up. This order is a convention for describing the record; an application must define its visits to match the actual measurement schedule.

At-Risk Indicators and the Observed Record

Define \(R_t\) to indicate that the participant is uncensored and available at the start of visit \(t\). Thus \(R_1=1\) for a participant entering follow-up, and \(R_t=1\) means that \(L_t\) and \(A_t\) are observed under the ordering above. If \(R_t=1\), the next at-risk indicator is determined by the censoring status after that visit. If \(R_t=0\), the participant has already left observation and cannot re-enter this record.

The terminal outcome is observed exactly when \(R_{K+1}=1\). Equivalently, for a participant entering follow-up,

When a participant is censored, later covariates and treatments are not recorded in the observed data. It is useful to display the record as a fixed-length tuple, but mark those later entries as unavailable rather than treating them as values. Let \(*\) denote an unavailable entry. Then a schematic observed record is

where \(L_t\) and \(A_t\) are recorded only if \(R_t=1\), \(C_t\) is recorded only if \(R_t=1\), and \(Y\) is recorded only if \(R_{K+1}=1\). In a fixed-width data table, unavailable entries can be coded as \(*\), or as missing values with the indicators retained. The indicator \(R_t\) is essential: a missing \(Y\) when \(R_{K+1}=0\) is not the same data record as an observed outcome equal to zero.

Histories and the Information Available at a Decision

At treatment decision \(t\), the measured information that precedes the decision consists of baseline covariates, time-varying covariates through visit \(t\), and earlier treatments. Define the treatment history

where \(\bar A_0\) is the empty history. This definition encodes the chosen order: \(L_t\) is available before \(A_t\), so \(A_t\) is not part of \(H_t\). On observed records, \(H_t\) is available only when \(R_t=1\). Previous censoring does not appear as a separate component of \(H_t\) for an at-risk participant, because \(R_t=1\) already implies that all earlier censoring indicators were zero.

This notation describes what is observed before treatment; it does not itself specify a treatment model, a censoring model, or a causal parameter. Those objects require additional definitions and assumptions. In particular, the record alone does not imply that censoring is independent of the outcome, or that every treatment option is represented at every history.

Proof. For \(t=1\), the assertion about earlier censoring is vacuously true, and \(R_1=1\) by definition. Suppose it holds at time \(t\). If \(R_t=0\), the no-re-entry rule gives \(R_{t+1}=0\), and at least one earlier \(C_s\) equals one. If \(R_t=1\), the recursion gives \(R_{t+1}=1-C_t\). Thus \(R_{t+1}=1\) exactly when \(C_t=0\) and every earlier \(C_s=0\). This proves the equivalence by induction. The observation statement follows from the record ordering: each earlier variable was recorded at a visit with \(R_s=1\), and \(L_t\) is measured at the start of a visit with \(R_t=1\). \(\square\)

Indexing Choices, Comparisons, and Failure Modes

Some data sets use visits numbered \(0,\ldots,K-1\), while others use \(1,\ldots,K\). Either convention is valid if the ordering is stated and used consistently. Under zero-based indexing, the first covariate and treatment might be \(L_0,A_0\); this tutorial uses \(L_1,A_1\) for the first pair. A baseline covariate remains \(W\) rather than being silently relabeled \(L_0\). This distinction helps prevent confusion about whether a variable is measured before follow-up or during a treatment visit.

There are also two common ways to represent censoring. One may retain the interval indicators \(C_t\) and derive the at-risk indicators \(R_t\), as done here. Alternatively, one may present only the at-risk process or use a single end-of-follow-up indicator. The interval-specific representation preserves the timing of censoring, which matters when censoring can occur between successive decisions. A single indicator can say that a participant was censored but may not say when.

• Reversing the visit order. Putting \(A_t\) before \(L_t\) in the record conflicts with the convention that \(L_t\) is available before the decision. State a different ordering if the study measures covariates after treatment.
• Using zero for an unobserved value. A missing outcome is not an observed outcome of zero. Retain an observation indicator or an explicit missing-value marker.
• Allowing participants to re-enter without defining it. The recursion here assumes no re-entry after censoring. If follow-up can resume, the observation process needs a different definition.
• Forgetting the terminal index. With \(K\) decisions and \(K\) censoring intervals, the final at-risk indicator is \(R_{K+1}\), not \(R_K\).

These conventions are useful because they make the data structure auditable before any model is fit. They do not resolve problems such as intermittent missing measurements, multiple censoring processes, or competing event types. Those settings require the record and indicators to be expanded to represent the actual observation process. Nor does correct notation establish the causal assumptions needed for a TMLE analysis; it only states clearly what was measured and when.