Initializing the Terminal Outcome Regression

What the Initial Terminal Regression Represents

In longitudinal targeted maximum likelihood estimation (TMLE), the terminal outcome regression is the starting estimate of the expected outcome given what has been observed before that outcome. Its job is specific: describe the conditional mean across the observed histories. It is not itself a causal effect, and fitting it does not yet impose a treatment regime or update the fit through targeting.

Use \(W\) for baseline covariates, \(L_t\) for covariates measured at time \(t\), \(A_t\) for treatment at time \(t\), and \(Y\) for the terminal outcome. With \(K\) treatment decisions, define the complete pre-outcome history as \(H_K=(W,L_1,A_1,\ldots,L_K,A_K)\), with entries ordered according to when they were measured. The observed terminal regression is the function \(Q_{K+1,0}(h)=E[Y\mid H_K=h]\). The subscript 0 indicates an initial, not yet targeted, regression; \(K+1\) labels the terminal outcome step in the backward sequence.

“Complete history” means all relevant recorded pre-outcome variables in the longitudinal data structure, not every conceivable predictor. If the outcome is subject to censoring, an unobserved \(Y\) is missing, not a zero. Which observations can contribute and how censoring is addressed depend on the data structure and analysis assumptions; do not silently code missing outcomes as observed values.

Specification and Fitting

First decide the outcome scale and the role of the history variables. For a binary \(Y\), a common working model is logistic:

Here \(f(H_K)\) is a selected set of history features: main effects, nonlinear terms, and interactions when scientifically justified. The fitted probability is the inverse-logit of the linear predictor, so it lies between zero and one. For a continuous \(Y\), one may use a linear conditional-mean model \(Q_{K+1,0}(H_K)=\beta_0+\beta^\top f(H_K)\), or another model whose mean and range suit the outcome. The fitting criterion should match the model and the outcome: for example, Bernoulli log likelihood for binary data and squared-error loss for an identity-link linear mean model.

The history must be represented in its temporal order. Omitting an earlier covariate or treatment may make the model too restrictive; including a variable measured after \(Y\) changes the conditioning set and can introduce information unavailable at the outcome time. The aim is not to include every possible interaction automatically. Instead, choose a defensible feature set based on subject-matter knowledge, sample size, and the need to represent important variation in the conditional mean.

Why These Losses Estimate a Conditional Mean

The population target is a conditional expectation. The following results explain why common fitting criteria are aligned with that target when their models allow the relevant conditional mean or probability.

Proof. Write \(m=E[Y\mid H_K]\). For any eligible \(g(H_K)\), expand the square:

The last term is zero. Indeed, \(m-g\) is a function of \(H_K\), so the defining property of conditional expectation gives \(E[(Y-m)(m-g)]=E[E[Y-m\mid H_K](m-g)]=0\). Therefore,

Equality holds exactly when \(E[(m-g)^2]=0\), which means \(g=m\) almost surely. This establishes the minimization and uniqueness claim. \(\square\)

Proof. Conditional on \(H_K=h\), the expected negative log likelihood for a proposed probability \(q\) is \(r(q)=-p\log(q)-(1-p)\log(1-q)\). For \(0<p<1\) and \(0<q<1\), \(r'(q)=-p/q+(1-p)/(1-q)=(q-p)/\{q(1-q)\}\). The denominator is positive, so the derivative is negative for \(q<p\), zero at \(p\), and positive for \(q>p\). Thus the unique minimum is \(q=p\). If \(p=0\) or \(p=1\), the minimum over the closed interval \([0,1]\) is attained at the corresponding endpoint. Minimizing this conditional loss at almost every history minimizes its expectation, proving the claim. \(\square\)

These are population statements, not guarantees that a particular finite-sample model is correct. If the chosen model excludes the true conditional mean, its fitted values generally converge to a best approximation under the selected loss, rather than to the exact regression.

Diagnostics, Alternatives, and Limitations

Assess whether the initial fit is credible on the histories where it will be used. For a binary outcome, inspect predicted probabilities for impossible values, extreme fitted values, sparse combinations of history variables, and systematic residual patterns. For continuous outcomes, inspect residuals against fitted values and important history features, and check for implausible predictions on the outcome scale. Calibration summaries can compare observed and predicted outcome frequencies across groups of predicted risk. These checks do not prove correctness, but patterns such as persistent underprediction in one history group signal that the model may be missing structure.

A saturated model can represent every cell mean for a finite categorical history, as in the first example, but it becomes impractical when the history has many variables: cells may be empty or contain too few observations. A parametric regression is more compact, but its adequacy depends on its functional form and interactions. Flexible regression or machine-learning methods can represent nonlinearities and interactions with less manual specification, but they still require adequate data, appropriate outcome handling, and careful assessment. No method can learn a well-supported conditional mean for histories with no or very little observed data.

The initial regression differs from an inverse-probability-weighted approach, which represents treatment or censoring processes through weights rather than beginning with this outcome regression. It also differs from the targeted update used later in longitudinal TMLE: the initial fit is an input to that procedure, not its final plug-in estimate. Correctly fitting \(Q_{K+1,0}\) is important, but by itself it does not address all identification assumptions, guarantee support for every history relevant to a regime, or establish causal interpretation.