Fluctuation Models for Continuous Outcomes

Why the Fluctuation Link Matters

A fluctuation submodel changes an initial outcome regression in a restricted direction so that the updated regression can respond to an empirical score equation. For continuous outcomes, the choice of link is consequential: an identity-link update is simple and directly adjusts predictions, while a bounded-outcome update can keep predictions inside a known range. Neither choice is universally preferable.

The setup here is one targeting step. For observation \(i\), let \(Y_i\) be the outcome or current regression target, \(Q_{0i}\) its initial prediction, and \(H_i\) a known targeting covariate. In longitudinal TMLE, \(H_i\) may be a clever covariate evaluated for the regression being targeted. Treating these quantities as fixed during the fluctuation makes the comparison precise; deriving the longitudinal sequence of targeting steps is a separate task.

The empirical score equation often has the form \(\sum_{i=1}^n H_i(Y_i-Q_{\hat\epsilon,i})=0\), or a constant multiple of it. This equation balances residuals after weighting them by \(H_i\). The fluctuation model determines how \(\epsilon\) changes the predictions and therefore whether this equation can be solved while retaining valid predictions.

Identity-Link Fluctuations

For an identity-link fluctuation, set \(Q_{\epsilon,i}=Q_{0i}+\epsilon H_i\). This is natural when the outcome is real-valued without a finite range constraint, or when it is acceptable for a working regression prediction to fall outside a physical range. A standard fitting criterion is squared error, equivalent to a Gaussian working likelihood with constant variance.

To verify this result, define \(R_i=Y_i-Q_{0i}\). The criterion is \(S(\epsilon)=\sum_i(R_i-\epsilon H_i)^2\). Differentiating gives \(S'(\epsilon)=-2\sum_i H_i(R_i-\epsilon H_i)\), and \(S''(\epsilon)=2\sum_i H_i^2>0\). Thus the unique minimizer solves \(\sum_i H_iR_i-\epsilon\sum_iH_i^2=0\), yielding the stated formula. Substitution gives the score equation. This is a finite-sample derivation; it does not require an asymptotic approximation. \(\square\)

The identity model is convenient because its fit has a closed form and directly expresses the score balance. However, if \(Y\) is known to lie in \([a,b]\), the updated value \(Q_{0i}+\hat\epsilon H_i\) can be less than \(a\) or greater than \(b\). The score equation alone does not prevent this.

Bounded Outcomes and Logistic Fluctuations

Suppose the outcome is known to lie in a finite interval \([a,b]\), with \(a<b\), and the initial predictions also lie in that interval. Rescale both to \([0,1]\): \(Y_i^*=(Y_i-a)/(b-a)\) and \(Q_{0i}^*=(Q_{0i}-a)/(b-a)\). When \(0<Q_{0i}^*<1\), a bounded-outcome fluctuation is \[ Q_{\epsilon,i}^*=\operatorname{expit}\{\operatorname{logit}(Q_{0i}^*)+\epsilon H_i\}, \qquad Q_{\epsilon,i}=a+(b-a)Q_{\epsilon,i}^*. \]

Here \(\operatorname{expit}(u)=1/(1+\exp(-u))\) and \(\operatorname{logit}(p)=\log\{p/(1-p)\}\). For a continuous bounded outcome, fitting this fluctuation by a Bernoulli log-likelihood expression is a working loss or quasi-likelihood criterion; it does not assert that the outcome is Bernoulli. The transformed outcome \(Y_i^*\) can be fractional.

For the derivation, write \(\eta_i(\epsilon)=\operatorname{logit}(Q_{0i}^*)+\epsilon H_i\). Since \(dQ_{\epsilon,i}^*/d\epsilon=H_iQ_{\epsilon,i}^*(1-Q_{\epsilon,i}^*)\), differentiating the \(i\)th summand gives \[ H_iQ_{\epsilon,i}^*(1-Q_{\epsilon,i}^*)\left\{\frac{Y_i^*}{Q_{\epsilon,i}^*}-\frac{1-Y_i^*}{1-Q_{\epsilon,i}^*}\right\} =H_i(Y_i^*-Q_{\epsilon,i}^*). \] Summing proves the score formula. Also, \[ \ell''(\epsilon)=-\sum_i H_i^2 Q_{\epsilon,i}^*(1-Q_{\epsilon,i}^*)\leq 0. \] If at least one \(H_i\ne0\), all initial predictions are strictly interior, and \(\epsilon\) is finite, then \(\ell''(\epsilon)<0\). The criterion is strictly concave, so any finite score root is its unique maximizer. A finite root is not guaranteed: if the score approaches zero only as \(\epsilon\) tends to infinity, the supremum can occur at an infinite coefficient. \(\square\)

For every finite \(\epsilon\) and finite \(H_i\), \(0<Q_{\epsilon,i}^*<1\), because the expit of a finite number is strictly between zero and one. Consequently, \(a<Q_{\epsilon,i}<b\). This is the support-preservation property of the bounded fluctuation. It does not put the predictions exactly at \(a\) or \(b\); exact boundary predictions require an infinite linear predictor.

Support Requirements and Practical Diagnostics

The bounded update requires a defensible finite range \([a,b]\), observed outcomes in that range, and initial predictions strictly inside it. If a prediction equals a boundary, its logit is not finite and the displayed submodel cannot be evaluated as written. A common numerical remedy is to replace the rescaled initial prediction by \(\min(1-\delta,\max(\delta,Q_{0i}^*))\) for a small prespecified \(\delta>0\). This is a deliberate modification of the initial regression, not an algebraic identity, and results can depend on the chosen truncation.

The identity fluctuation has no such logit-boundary problem, but it also has no automatic support protection. When predictions must represent probabilities, proportions, or measurements with known hard limits, inspect every updated prediction. Conversely, applying a bounded logistic fluctuation to an outcome that is genuinely unbounded requires an unjustified range choice and can distort the target.

Useful diagnostics include the fitted \(\hat\epsilon\), the score at the fitted value, the range of updated predictions, and the stability of the update under reasonable numerical tolerances. A very large absolute coefficient can indicate that the score root is near infinity, that a few observations dominate the fluctuation, or that the initial predictions are near a boundary. If all \(H_i=0\), neither model can move the predictions in the targeted direction: the identity denominator is zero and the logistic score is identically zero. For the logistic criterion, a score that does not cross zero at any finite \(\epsilon\) should not be reported as a finite converged fit.

The earlier material on logistic fluctuations for binary outcomes uses the same logit form, but a continuous bounded outcome is not thereby a binary variable. The rescaled fractional outcome enters the working criterion, and the support claim concerns the fitted conditional mean. In contrast to identity-link targeting, the bounded update trades a direct additive change for range preservation and a nonlinear response scale.

Both approaches are limited by the chosen one-dimensional fluctuation direction: they only alter the initial predictions through \(H\). Neither link repairs a misspecified initial regression or a poorly chosen targeting covariate. Selection should be based on the outcome’s support, the meaning of the predictions, and stable fitting—not on a preference for one link in isolation.