Choosing Fluctuation Submodels for Outcome Support

Why the Fluctuation Link Matters

In longitudinal TMLE, targeting modifies an initial estimate of a conditional mean so that it better satisfies an empirical estimating equation. The fluctuation submodel specifies which modifications are allowed. Its link and offset therefore determine not only how the update is calculated, but also whether the updated predictions remain compatible with the outcome’s support.

Consider an outcome \(Y\) and an initial conditional mean estimate \(Q_0(W)\), where \(W\) denotes the predictors or history at a particular stage. A one-dimensional offset fluctuation keeps \(Q_0\) fixed as the starting prediction and changes it according to a scalar parameter \(\epsilon\). In longitudinal targeting, a known clever covariate \(H\) can vary across observations and multiplies that parameter. The choice of \(H\) comes from the target and its estimating equation; the choice of link must also suit the range of possible conditional means.

Definition: An offset fluctuation submodel is a family \(Q_\epsilon\), indexed by a parameter \(\epsilon\), that contains the initial estimate \(Q_0\) at \(\epsilon=0\). A common form is \(g(Q_\epsilon(W))=g(Q_0(W))+\epsilon H(W)\), where \(g\) is a link function. The initial prediction is the offset: it is not re-estimated as an ordinary free intercept in the fluctuation fit.

The same principle applies at each sequential regression step. If the relevant conditional outcome or pseudo-outcome is binary, bounded, or unrestricted, its conditional mean has a corresponding range. A fluctuation should preserve that range without imposing unsupported restrictions on the target.

Binary and Bounded Outcomes: Logistic Fluctuations

For a binary outcome, the conditional mean is a probability in \([0,1]\). The logistic fluctuation uses the logit link:

\operatorname{logit}\{Q_\epsilon(W)\} = \operatorname{logit}\{Q_0(W)\}+\epsilon H(W), \qquad Q_\epsilon(W)=\operatorname{expit}\{\operatorname{logit}Q_0(W)+\epsilon H(W)\}.

Here \(\operatorname{logit}(u)=\log\{u/(1-u)\}\) and \(\operatorname{expit}(v)=1/(1+\exp(-v))\). If \(0<Q_0(W)<1\) and the linear predictor is finite, then \(0<Q_\epsilon(W)<1\) for every finite \(\epsilon\). The fluctuation can approach zero or one as \(\epsilon\) diverges, but cannot attain either boundary at a finite parameter value.

A usual fit minimizes the Bernoulli negative log-likelihood, using the offset \(\operatorname{logit}Q_0\) and the covariate \(H\). Its score is especially convenient. For a bounded continuous outcome \(Y\in[a,b]\), first scale it to \(Z=(Y-a)/(b-a)\in[0,1]\). The same logistic loss can be used as a cross-entropy loss for \(Z\), even though \(Z\) is not Bernoulli. This is a quasi-likelihood or proper-loss targeting device: it targets the conditional mean, not a claim that the continuous observations follow a Bernoulli distribution.

For this bounded-outcome use, the initial mean must lie in \((a,b)\) so that the scaled logit is defined. The update then stays strictly inside \((a,b)\) for finite \(\epsilon\). If the true conditional mean can equal an endpoint, a finite logistic fluctuation cannot represent it exactly. Clipping an initial prediction away from the boundary makes the computation defined but changes the starting estimate and should be disclosed.

Worked Example: Binary Outcome With a Constant Clever Covariate

Suppose five observations have binary outcomes \(Y=(1,1,1,0,0)\), each has \(H=1\), and the initial prediction is \(Q_0=0.30\) for everyone. The logistic offset model has a common updated probability because both the offset and \(H\) are constant.

The Bernoulli score equation sets the fitted mean equal to the observed proportion: \(\sum_{i=1}^5(Y_i-Q_\epsilon)=0\). Thus \(5(0.60-Q_\epsilon)=0\), so \(Q_\epsilon=0.60\). The fitted fluctuation parameter is the change in log odds:

\hat\epsilon = \operatorname{logit}(0.60)-\operatorname{logit}(0.30) = \log(1.5)-\log(3/7) = \log(3.5) \approx 1.253.

Checking the update, the initial odds are \(0.30/0.70=3/7\). Multiplying by \(\exp(\hat\epsilon)=3.5\) gives updated odds \(1.5\), which correspond to probability \(1.5/(1+1.5)=0.60\). The updated prediction remains a valid probability.

Worked Example: Bounded Continuous Outcome

Suppose \(Y\) is a continuous outcome measured between 10 and 20, with observed values \(14,16,18,12\). Let \(H=1\) for all four observations, and let the initial conditional mean be 15. Scale to \(Z=(Y-10)/10\). The scaled observations are \(0.4,0.6,0.8,0.2\), whose mean is \(0.5\); the initial scaled mean is also \(0.5\).

With a logistic cross-entropy fluctuation, the score equation is \(\sum_i(Z_i-Q_\epsilon)=0\). The observed scaled mean is \(0.5\), so the solution is \(\hat\epsilon=0\): the initial prediction already satisfies this equation. Transforming back gives \(10+10(0.5)=15\), within the outcome range.

If instead the initial scaled mean were \(0.4\), the corresponding common update would be \(\hat\epsilon=\operatorname{logit}(0.5)-\operatorname{logit}(0.4)=\log(1.5)\approx0.405\). The updated scaled mean is \(0.5\), hence the updated mean on the original scale is 15. This calculation targets a mean; it does not assert that the outcome distribution itself is logistic or binary.

Unrestricted Outcomes: Identity Fluctuations

If the outcome is genuinely unrestricted on the real line, an identity fluctuation is often appropriate:

Q_\epsilon(W)=Q_0(W)+\epsilon H(W).

This form does not constrain predictions to a bounded interval. Under squared-error loss, its score equation has a closed-form solution when \(\sum_i H_i^2>0\). The fitted parameter is the least-squares projection of the initial residuals onto the clever covariate.

Worked Example: Identity-Link Update

Suppose four observations have pseudo-outcomes \(Y=(3,5,4,8)\), initial predictions \(Q_0=(2,4,5,6)\), and clever covariates \(H=(1,1,2,0)\). The residuals \(Y-Q_0\) are \((1,1,-1,2)\). Minimizing \(\sum_i\{Y_i-Q_0(W_i)-\epsilon H_i\}^2\) gives

\hat\epsilon = \frac{\sum_i H_i(Y_i-Q_0(W_i))}{\sum_i H_i^2} = \frac{1+1-2+0}{1+1+4+0} = 0.

The score numerator is zero, so this particular update makes no change. To check the calculation, the residuals already have zero inner product with \(H\): \(1(1)+1(1)+2(-1)+0(2)=0\). The denominator is 6, which is positive, so the minimizer is uniquely defined.

Two Derivations for Choosing and Checking a Fluctuation

The score equation explains why logistic targeting works for both binary and scaled bounded outcomes. Assume finite observations \((Y_i,H_i,Q_{0i})\), with \(Y_i\in[0,1]\), \(0<Q_{0i}<1\), and a finite scalar \(\epsilon\). Define the empirical cross-entropy loss \(\mathcal L(\epsilon)=-\sum_i[Y_i\log Q_{\epsilon i}+(1-Y_i)\log(1-Q_{\epsilon i})]\). Differentiating the logistic model gives \(dQ_{\epsilon i}/d\epsilon=H_iQ_{\epsilon i}(1-Q_{\epsilon i})\). Therefore,

\frac{d\mathcal L(\epsilon)}{d\epsilon} = -\sum_i H_i(Y_i-Q_{\epsilon i}).

A finite interior minimizer consequently satisfies \(\sum_iH_i(Y_i-Q_{\hat\epsilon i})=0\). This is the empirical score equation, provided the minimizer exists and the loss is differentiable there. It does not guarantee that a finite minimizer exists: separation or a boundary solution can send the estimate toward infinite \(\epsilon\).

For the identity update, assume \(\sum_iH_i^2>0\) and minimize squared error. Expanding the objective gives

S(\epsilon)=\sum_i\{Y_i-Q_{0i}-\epsilon H_i\}^2,\qquad S'(\epsilon)=-2\sum_iH_i(Y_i-Q_{0i})+2\epsilon\sum_iH_i^2.

Setting this derivative to zero yields \(\hat\epsilon=\sum_iH_i(Y_i-Q_{0i})/\sum_iH_i^2\). Since \(S''(\epsilon)=2\sum_iH_i^2>0\), this is the unique global minimum. Unlike the logistic fluctuation, this update does not impose a support bound.

Support check: Before fitting, identify the support of the outcome or pseudo-outcome whose conditional mean is being updated. Use a link whose inverse maps finite predictors into the allowed mean range, and verify that the initial prediction is in the link’s domain. A support-preserving update cannot repair a misspecified support definition or a wrongly constructed pseudo-outcome.

Assumptions, Diagnostics, and Alternatives

A link choice encodes assumptions. Logistic fluctuation assumes the relevant conditional mean lies in \([0,1]\), and the finite-parameter model represents only interior values. Scaling a bounded outcome assumes its endpoints \(a,b\) are known and appropriate for the target population. Identity fluctuation assumes no support restriction is needed; it can be computationally convenient but may yield predictions outside a known range.

In longitudinal targeting, these choices apply to the conditional mean being updated at that step, not automatically to every variable in the data. For instance, a backward-regression pseudo-outcome formed by averaging future binary outcomes remains in \([0,1]\), so a bounded link can be suitable. If a pseudo-outcome includes an unbounded correction term, however, it may no longer be bounded even when the original outcome is binary. Inspect the actual update target rather than choosing a link from the final outcome label alone.

Useful diagnostics include checking the minimum and maximum initial predictions, verifying the outcome range and any scaling constants, inspecting the fitted \(\hat\epsilon\), and evaluating the empirical score after updating. A score that is not close to zero can indicate failed optimization, numerical overflow, or a mismatch between the fitted fluctuation and intended clever covariate. For logistic targeting, extreme offsets, predictions near zero or one, and large fitted parameters warrant attention; separation can prevent a finite optimum. These problems connect to the earlier material on separation and boundary predictions and to the earlier material on positivity failures, but they are not resolved merely by changing the link.

Choice	Typical use	Support implication	Main caution
Logistic offset	Binary outcome or mean in \([0,1]\)	Finite updates lie strictly between 0 and 1	Boundary means and separation may obstruct a finite fit
Scaled logistic offset	Bounded continuous outcome in \([a,b]\)	Finite updates lie strictly between \(a\) and \(b\)	Requires defensible bounds and scaling
Identity offset	Unrestricted real-valued outcome	No range restriction	Predictions can leave a known bounded support

A more general inverse link \(g^{-1}\) can be used when it maps predictors into the desired mean range. But changing the link changes the derivative of the loss and generally changes the score equation; the clever covariate and loss must be chosen so the intended estimating equation is actually represented. A bounded link is not automatically preferable: imposing bounds on an unrestricted mean is a substantive restriction, while an identity link is not automatically acceptable for a probability or a bounded mean.

Common pitfall: “The final outcome is binary, so every longitudinal pseudo-outcome is binary and a logistic update is always justified.” This is false: a pseudo-outcome can include correction terms and lie outside \([0,1]\). Check the range of the quantity entering the particular fluctuation. Conversely, if it is a conditional mean known to lie in \([0,1]\), an identity update can produce an impossible prediction even though its score equation is easy to solve.

Check Your Understanding

Answer each question using the support of the quantity being updated, not just the name of the final outcome.

A mean is known to lie in \([0,1]\), and the initial predictions are all strictly interior. What range does a finite logistic offset fluctuation produce?
A bounded outcome lies in \([5,15]\). Write its scaled outcome and describe the corresponding scaled-logistic update on the original scale.
For identity targeting, \(\sum_iH_i^2=0\). Can the least-squares formula identify a unique fluctuation parameter? Explain.
Why can a finite logistic fluctuation fail to represent a conditional mean equal to zero?
A final outcome is binary, but an update target includes an unbounded correction term. What should be checked before choosing a logistic fluctuation?

Solutions. (1) Every updated prediction is strictly between 0 and 1 because the inverse-logit of a finite real number is interior. (2) Set \(Z=(Y-5)/10\); model \(Q_{\epsilon,Z}=\operatorname{expit}\{\operatorname{logit}Q_{0,Z}+\epsilon H\}\), then transform back as \(Q_{\epsilon,Y}=5+10Q_{\epsilon,Z}\), which is strictly between 5 and 15 for finite predictors. (3) No. A sum of squares equal to zero implies every \(H_i=0\), so the objective is constant in \(\epsilon\) and the parameter is not identified by this fluctuation. (4) The inverse-logit is strictly positive for every finite predictor, so zero is only approached at an infinite negative predictor. (5) Check the actual range of the pseudo-outcome and the conditional mean being targeted. If the correction makes that target unrestricted, a logistic mean model may be incompatible; select a fluctuation and loss appropriate to its support and intended score.