Longitudinal Doubly Robust Estimation and LTMLE

Definition and scope

Longitudinal causal estimation combining sequential outcome and treatment/observation models for static or dynamic treatment strategies, with estimator-specific robustness and inference conditions. The target population, use, and analysis remain protocol decisions; this card explains the choices and their consequences rather than approving a study.

Operational specification

Specify ordered time points, baseline and time-varying confounders, treatment decisions, observation/censoring, absorbing events, intervention rule, outcome, and target population. All tailoring information must be available before each strategy decision. Distinguish a longitudinal g-formula plug-in from targeted or augmented estimators and identify the exact robustness theorem applicable to the chosen method.

Analysis and diagnostics

Fit sequential regressions and treatment/observation mechanisms using an implementation that matches the longitudinal data structure, with appropriate sample splitting/cross-fitting when needed. Check longitudinal positivity, cumulative weights, intervention support, model convergence, and valid influence-function or other uncertainty estimation. Point-estimate double robustness does not automatically validate inference or permit arbitrary combinations of misspecified stages.

When NOT to use and common errors

Do not adjust for all post-baseline variables in a standard regression and call it LTMLE, ignore joint support across repeated treatment decisions, or claim consistency whenever any one nuisance model is correct. The code below checks only a path-probability calculation.

Worked example and implementation boundary

A synthetic two-decision history has conditional treatment probabilities 0.5 and 0.4 and observation probability 0.9. Calculate the probability of that observed treatment/observation path under the stated factorization. The path probability is 0.18. This is a diagnostic preparation calculation, not a fitted longitudinal causal effect. The code performs that explicitly stated calculation with synthetic inputs. It is not a complete fitted analysis and does not demonstrate the identification assumptions, estimate sampling uncertainty, or certify application in another population.

Data-source and reporting requirements

Preserve the provenance, dates, units, and observation rule for each input. Claims capture services and transactions under observable benefits; absence of a record is not proof that an event or resource use did not occur. EHR and registry measurements depend on the collection process and may miss care outside the system. Linked sources add reconciliation and selection questions. Primary collection can improve a missing domain but still requires valid measurement and follow-up. Report which of these limitations affects the stated use, which evidence supports the chosen inputs, and what remains uncertain.

Worked example

Scenario

A synthetic two-decision history has conditional treatment probabilities 0.5 and 0.4 and observation probability 0.9. Calculate the probability of that observed treatment/observation path under the stated factorization.

Dataset

Synthetic inputs with units defined in the scenario; no patient data.

Steps

Result

The path probability is 0.18. This is a diagnostic preparation calculation, not a fitted longitudinal causal effect.

Trade-offs

Code example

p_a0 = 0.5
p_a1 = 0.4
p_observed = 0.9
result = p_a0*p_a1*p_observed
assert abs(result - 0.18000000000000002) < 1e-9
print(result)

p_a0 <- 0.5
p_a1 <- 0.4
p_observed <- 0.9
result <- p_a0*p_a1*p_observed
stopifnot(abs(result - 0.18000000000000002) < 1e-9)
print(result)

data worked_example;
  p_a0 = 0.5;
  p_a1 = 0.4;
  p_observed = 0.9;
  result = p_a0*p_a1*p_observed;
  put result=;
run;

Citations

1. [1]Petersen M, et al. Targeted Maximum Likelihood Estimation for Dynamic and Static Longitudinal Marginal Structural Working Models. Journal of Causal Inference. 2014.
2. [2]Hernán MA, Robins JM. Using big data to emulate a target trial when a randomized trial is not available. American Journal of Epidemiology. 2016;183(8):758-764.