Longitudinal Likelihood Factorization for Nuisance Estimation

Why factor the likelihood?

A longitudinal record contains several distinct processes: covariates evolve, treatment is assigned repeatedly, some participants are censored, and an outcome is eventually observed. Factoring the observed-data likelihood makes these processes explicit. It also clarifies an important distinction for longitudinal targeted maximum likelihood estimation (TMLE): a factor can belong to the likelihood without requiring a separate fitted nuisance model in the estimator.

We use a fixed ordering. There is a baseline covariate \(W\), and at each time \(t=0,\ldots,K\), a time-varying covariate \(L_t\) is measured, treatment \(A_t\) is assigned, and censoring by the next time is recorded. The terminal outcome \(Y\) is observed only if the person remains uncensored through the last interval. Define \(C_{t+1}=1\) to mean newly censored after \(A_t\), and let \(R_t=1\) indicate that the person is uncensored and at risk at the start of time \(t\). Thus \(R_0=1\) and \(R_{t+1}=R_t(1-C_{t+1})\).

Bars denote histories up to the indicated time, such as \(\bar L_t=(L_0,\ldots,L_t)\). Conditional densities below may be probability mass functions when variables are discrete. Each factor is evaluated at the individual’s observed value and history.

The observed-data likelihood

Write \(f_W\) for the baseline density, \(f_{L_t\mid H^L_t,R_t=1}\) for a covariate factor among those at risk, \(g_{A_t\mid H^A_t,R_t=1}\) for a treatment factor, and \(g_{C_{t+1}\mid H^C_t,R_t=1}\) for a censoring factor. Write \(f_{Y\mid \bar L_K,\bar A_K,R_{K+1}=1}\) for the terminal outcome factor. The likelihood contribution for one observed record is

The exponents express which values are observed. When \(R_t=0\), the person was censored before time \(t\), so the later covariate, treatment, and censoring factors are absent from that person’s observed-data likelihood. The outcome factor appears only when \(R_{K+1}=1\). For an uncensored person, all factors appear. For a censored person, the likelihood still includes the probability of the censoring event that occurred.

To see the justification, first condition on \(W\). For someone still at risk at time \(t\), the observed sequence is \(L_t,A_t,C_{t+1}\), so its conditional joint density factors as

Multiplying these conditional densities over the observed times gives the product over \(t\), with \(R_t\) indicating whether the factors at that time are present. If the person remains at risk through \(K\), append the conditional density of \(Y\); otherwise \(Y\) is unobserved and contributes no outcome-density factor. This is the chain rule applied to the observed sequence, not an assumption that the processes are independent. In particular, each factor may depend on the entire history listed in its conditioning set. \(\square\)

Which factors longitudinal TMLE uses

For a regime \(\bar d=(d_0,\ldots,d_K)\), the target is the mean outcome under the specified treatment choices, often written \(\psi=E[Y^{\bar d}]\). Under the causal identification assumptions—including consistency, sequential exchangeability, positivity, and suitable conditional independent censoring—the likelihood factorization helps organize the nuisance quantities needed to estimate this mean.

• Baseline distribution: \(f_W\) is part of the observed-data likelihood. In the usual empirical TMLE, integration over the baseline population is represented by averaging over the observed \(W_i\); a separate parametric model for \(f_W\) is not required.
• Time-varying covariate distributions: the factors \(f_{L_t\mid H^L_t,R_t=1}\) are also in the likelihood. Standard longitudinal TMLE for a mean does not generally fit them as nuisance models. The observed covariate histories supply the empirical distribution used in the estimation procedure, while outcome regressions account for the relationship between histories and future outcomes.
• Treatment mechanisms: the factors \(g_{A_t\mid H^A_t,R_t=1}\) describe treatment assignment given observed history. They are nuisance quantities used in longitudinal TMLE, including in the construction of its targeting steps.
• Censoring mechanisms: the factors \(g_{C_{t+1}\mid H^C_t,R_t=1}\) describe remaining uncensored conditional on history. Their relevant probabilities are nuisance quantities when censoring must be adjusted for.
• Outcome regression: the outcome factor contributes conditional outcome information. For a mean target, the relevant feature is a conditional mean, rather than the entire conditional distribution of \(Y\).

The fact that \(f_W\) and the covariate densities are not ordinarily fitted as separate nuisance models does not make them irrelevant. They help determine the distribution over which the counterfactual mean is defined. The empirical distribution can represent that distribution without estimating a high-dimensional density. By contrast, treatment and censoring probabilities describe the observed-data assignment and observation processes that must be accounted for.

To verify this, fix any distribution \(P^{\bar d}\) of histories \(H\) under the regime. The mean outcome under that distribution is obtained by integrating the terminal conditional density:

For a discrete outcome, replace the inner integral by a sum. The calculation shows why a mean-target procedure needs the conditional mean of the terminal outcome, not a fully specified distribution for every possible value of \(Y\). It does not remove the need to represent the relevant history distribution or to adjust for treatment and censoring. \(\square\)

Worked examples

Diagnostics, alternatives, and limitations

The factorization is useful for checking data processing before fitting any nuisance model. Confirm that the event defining censoring matches the actual visit schedule, that a censoring probability is conditioned on information available before censoring, and that later variables are missing rather than inadvertently coded as observed. Each conditional factor should use only its stated history. Conditioning a treatment model on future information, for example, would not represent the treatment-assignment mechanism at that time.

Sparse histories can make estimated treatment or censoring probabilities unstable, and probabilities near zero signal potential positivity problems. A probability of zero for a treatment choice required by the regime means that the corresponding conditional factor cannot support that intervention at that history. Very small probabilities can produce highly variable adjustments. Inspect fitted probabilities and the history strata in which they occur; a syntactically valid product does not guarantee stable estimation.

The factorization also helps compare approaches without confusing their modeling choices. In g-computation, one can model outcome regressions and, in some implementations, the covariate evolution needed to integrate over future histories. In inverse-probability weighting, treatment and censoring factors are used to construct weights, while the outcome regression need not be modeled in the same way. Longitudinal TMLE combines outcome-regression information with treatment and censoring nuisance information and targets a plug-in estimate. These approaches use different parts of the factorization explicitly; none changes which variables were observed.

A common mistake is to conclude that every likelihood factor must be fitted because it appears in the product. That reasoning confuses a factorization of the data law with an estimator’s nuisance-model requirements. For the usual mean-target longitudinal TMLE, the baseline distribution is commonly represented by the empirical sample, the covariate distributions are not generally fit as separate density nuisances, and the outcome distribution is reduced to conditional means. Treatment and censoring mechanisms, by contrast, are nuisance quantities that must be addressed. Exact implementation choices can vary, and empirical representation does not cure violations of exchangeability, positivity, or correct temporal ordering.