Why Target an Efficient-Score Equation?
A longitudinal estimator can use fitted outcome regressions and treatment and censoring mechanisms without ensuring that the resulting estimate satisfies the empirical efficient influence-function equation. Targeting adds a deliberate update to the outcome regressions: the update is chosen so that their residuals, weighted by the efficient influence function’s clever covariates, have empirical means close to zero. This is a particular estimating-equation goal, not a guarantee that every nuisance model is correct or that every source of bias has disappeared.
Let \(P_n f=n^{-1}\sum_{i=1}^n f(O_i)\) denote the sample average. For a regime-specific mean \(\psi\), write the efficient influence function in the sequential form
where \(Q_{K+1}=Y\), the \(Q_j\) are the successive outcome regressions evaluated along the regime, and \(H_j\) are the cumulative clever covariates. In a setting with treatment and censoring, a typical \(H_j\) contains indicators for following the regime and remaining observed through stage \(j\), divided by the corresponding treatment and censoring probabilities. If there is no censoring, omit the censoring factors. The exact histories and indexing depend on the longitudinal data structure; here the important point is that the \(H_j\) are the coefficients multiplying the sequential residuals in the influence function.
The plug-in estimate based on the targeted initial regression is \(\hat\psi=P_n Q_0^*\). Consequently, its empirical influence-function mean is
where \(\hat g\) and \(\hat\gamma\) denote the fitted treatment and censoring mechanisms used in the clever covariates. The \(Q_0^*-\hat\psi\) term averages to zero by the definition of the plug-in estimate. Thus, driving the weighted residual means toward zero drives the empirical influence-function mean toward zero as well.
Fluctuation Scores and the Clever Covariate
To see the link directly, consider an identity-link working fluctuation at stage \(j\):
Let \(Z_j\) be the stage-specific outcome used to fit this regression: \(Z_j=Q_{j+1}^*\) for a nonterminal stage, and \(Z_K=Y\) at the terminal stage. For a squared-error working loss, the empirical loss is \(P_n[(Z_j-Q_{j,\epsilon})^2/2]\). Differentiating this finite-sample expression gives
Therefore, a minimizer in the interior of the allowed parameter space satisfies the score equation \(P_n[H_j(Z_j-Q_{j,\epsilon})]=0\). This is a finite-sample calculation; it requires no interchange of a derivative and an expectation. For bounded outcomes or other outcome types, the fluctuation and its score must respect the outcome model. The general targeting principle is the same, but the loss and score need not be squared error.
The score is not an arbitrary weighted regression diagnostic. Its weight is chosen to match the residual component of the efficient influence function. In a longitudinal procedure, targeting is typically performed recursively: update a later regression, use that updated regression to form the next pseudo-outcome, and then solve the corresponding earlier-stage score equation. At the end, the sequential residuals in the empirical influence function are precisely the residuals whose clever-covariate-weighted means have been targeted.
Deriving the Empirical Influence-Function Equation
Proof. Substitute the targeted regressions into the displayed influence-function representation and take the sample average. By linearity of \(P_n\),
The plug-in definition makes the last two terms cancel. Each remaining term is zero by its stagewise score equation, proving the exact statement. If the \(j\)th score is \(r_j\) rather than zero, the same equality gives \(P_nD^*=\sum_j r_j\). In particular, small component scores imply a small empirical influence-function mean only if their signed sum is small; checking each component separately avoids relying on accidental cancellation. \(\square\)
For the identity-link squared-error fluctuation, the score equation can also be solved explicitly. If \(P_nH_j^2>0\), then
The second equality follows by substituting the first into the left side. This derivation assumes an unconstrained update and a positive denominator. If the update is constrained, or a numerical algorithm stops before reaching the minimizer, the score need not be zero.
Worked Example: One-Stage Identity-Link Targeting
Suppose the initial predictions are \(Q=(4,5,6)\), the observed outcomes are \(Z=(6,5,5)\), and the clever covariates are \(H=(1,2,1)\). The initial residuals are \(Z-Q=(2,0,-1)\). With \(n=3\), the score numerator and denominator are
Thus \(\hat\epsilon=(1/3)/2=1/6\), and the targeted predictions are \(Q^*=Q+\hat\epsilon H=(25/6,16/3,37/6)\). The updated residuals are \((11/6,-1/3,-7/6)\). Their weighted sum is
The plug-in estimate is \(\hat\psi=(25/6+16/3+37/6)/3=47/9\). Hence the empirical influence-function mean, consisting of the targeted weighted residual term plus \(P_nQ^*-\hat\psi\), is also zero.
Worked Example: Two Sequential Score Equations
Consider two stages with four observations. Suppose the stage-specific clever covariates and updated residuals \(Q_{j+1}^*-Q_j^*\) are \(H_0=(1,1,2,2)\), \(R_0=(1,-1,1/2,-1/2)\), and \(H_1=(1,2,1,2)\), \(R_1=(2,-1,1,-1)\). At stage zero, the weighted residual sum is \(1-1+2(1/2)+2(-1/2)=0\). At stage one, it is \(2+2(-1)+1+2(-1)= -1\), not zero; so stage one has not yet been targeted.
To complete the example, change the last stage-one residual from \(-1\) to \(-1/2\). Then its weighted residual sum is \(2-2+1-1=0\), and its empirical score is zero after division by \(n=4\). Both stagewise empirical scores are therefore zero. If the four targeted initial predictions are \(Q_0^*=(0.3,0.4,0.5,0.8)\), then \(\hat\psi=P_nQ_0^*=2/4=0.5\). The influence-function mean is \(0+0+(0.5-0.5)=0\).
The initial calculation deliberately identifies a failed stage before correcting it: a near-zero total influence-function mean caused by cancellation between a positive and negative stage score would not establish that both regressions were successfully targeted.
Worked Example: A Small Numerical Score Error
Return to the one-stage data in the first example, but suppose an optimizer stops at \(\tilde\epsilon=0.15\) instead of \(1/6\). The score at this value is
The targeted plug-in estimate is \(P_n(Q+0.15H)=5+0.15(4/3)=5.2\). Its empirical influence-function mean is \(1/30\), not zero: the centered plug-in term still cancels, but the score residual remains. Whether \(0.0333\) is acceptably small depends on the scale and the desired numerical tolerance; the word “approximately” should be supported by a reported score, not assumed from optimizer convergence alone.
Diagnostics, Alternatives, and Limits
At minimum, inspect the stage-specific values \(P_n[H_j(Q_{j+1}^*-Q_j^*)]\), the total empirical influence-function mean, and the size of the update. Also inspect the distribution of the clever covariates. Since they involve inverse probabilities, near-zero fitted treatment or observation probabilities can produce very large \(H_j\); a few observations may then dominate the score and the estimate. A numerically tiny score does not rule out this instability. Check fitted probabilities, extreme weights, convergence warnings, and whether the fluctuation has reached a boundary or been constrained.
Targeting differs from simply refitting an outcome regression: a conventional regression fit need not make the efficient-influence-function-weighted residual mean small. It also differs from changing the treatment or censoring model: the fluctuation uses the fitted mechanisms through \(H_j\), but the score update described here changes the outcome regression. Estimating nuisance mechanisms well remains important; solving the empirical score does not repair every error in those mechanisms.
The empirical equation is a finite-sample property of the fitted quantities, not a proof of identification, consistency, or asymptotic normality. Those conclusions require their own assumptions, including the causal identification conditions and suitable control of nuisance estimation and positivity. Further, an unconstrained identity-link update can yield predictions outside a permissible outcome range. The fluctuation must be appropriate to the outcome and the estimand; an exact score equation obtained from an inappropriate submodel is not a substitute for a valid analysis.
Check Your Understanding
Work through each question, then check the solutions.
- Why does \(P_n(Q_0^*-\hat\psi)=0\) for the targeted plug-in estimate?
- For \(H=(1,2)\) and initial residuals \(Z-Q=(2,-1)\), find the identity-link update and verify the updated score.
- If two stage scores are \(0.04\) and \(-0.04\), is the total empirical influence-function mean necessarily large? Why should the stage scores still be checked separately?
- What is one reason a zero empirical score may coexist with an unstable estimate?
Solutions. (1) By definition, \(\hat\psi=P_nQ_0^*\), so subtracting \(\hat\psi\) from every sample value and averaging gives zero. (2) The numerator is \(P_n[H(Z-Q)]=(2-2)/2=0\), while \(P_nH^2=(1+4)/2=2.5>0\). Thus \(\hat\epsilon=0\), and the updated score remains zero. (3) Their sum is zero, so the total can be zero, but cancellation hides two nonzero stage scores; separate checks show whether each regression’s targeting equation was met. (4) Extreme inverse-probability weights can make the estimate sensitive to a few observations even when the weighted residual score is zero.