Influence-Curve Standard Errors

Why the Influence Curve Gives a Standard Error

A longitudinal TMLE produces a point estimate, but the estimate alone does not indicate how much it would vary across repeated samples. Its efficient influence function (EIF) provides the subject-level contributions used to estimate that sampling variability. The calculation is subject-level even when each subject contributes measurements at several treatment and covariate times: the repeated observations are combined into one EIF value per subject.

For a regime \(d\), write \(\psi_d\) for the regime-specific mean and \(D_d^*(P)(O)\) for its EIF, evaluated at the true data distribution \(P\) for a subject's full observed data \(O\). The EIF has mean zero and, under the usual regularity conditions, describes the first-order sampling variation of a regular estimator. In practice \(P\) is unknown, so we evaluate the corresponding EIF expression using the fitted nuisance functions, targeting updates, and plug-in estimate.

The division by \(n\) is essential: \(\hat\sigma_d^2\) estimates the variance of one subject's influence value, while the variance of the sample mean of those contributions is approximately \(\operatorname{Var}(D_d^*)/n\). Centering by the empirical mean is appropriate even if the fitted EIF values do not sum to exactly zero.

Conditions and Justification

Two claims are involved in using this standard error. First, the estimator must have an asymptotically linear representation. Second, the empirical variance of its estimated EIF values must consistently estimate the variance of the true EIF. The first claim does not follow merely from computing an EIF; it depends on the estimator, nuisance estimation, and regularity conditions.

Proof of the distributional claim. The asymptotic linear representation gives $$ \sqrt n(\hat\psi_d-\psi_d)=\frac{1}{\sqrt n}\sum_{i=1}^nD_d^*(P)(O_i)+o_p(1). $$ The summands are independent and identically distributed, have mean zero, and have finite variance \(\sigma_d^2\). The central limit theorem therefore gives convergence of the sum to \(N(0,\sigma_d^2)\). Slutsky's theorem allows the \(o_p(1)\) remainder without changing that limit.

Proof of variance consistency. Abbreviate \(D_i=D_d^*(P)(O_i)\), \(e_i=\hat D_{d,i}-D_i\), and let \(A_n=n^{-1}\sum_i e_i^2\) and \(B_n=n^{-1}\sum_iD_i^2\). By assumption \(A_n\xrightarrow{p}0\), and by the law of large numbers \(B_n\xrightarrow{p}\sigma_d^2\). Cauchy–Schwarz gives $$ \left|\frac{1}{n}\sum_{i=1}^n\hat D_{d,i}^2-\frac{1}{n}\sum_{i=1}^nD_i^2\right| \leq A_n+2\sqrt{A_nB_n}\xrightarrow{p}0. $$ Also, \(n^{-1}\sum_iD_i\xrightarrow{p}0\), and $$ \left|\frac{1}{n}\sum_i e_i\right|\leq \sqrt{A_n}\xrightarrow{p}0, $$ so \(\bar{\hat D}_d\xrightarrow{p}0\). The empirical variance using divisor \(n\) is \(n^{-1}\sum_i\hat D_{d,i}^2-\bar{\hat D}_d^{\,2}\), which thus converges to \(\sigma_d^2\). Replacing divisor \(n\) by \(n-1\) multiplies it by \(n/(n-1)\to1\). This proves consistency of \(\hat\sigma_d^2\). \(\square\)

The conditions in this theorem are substantive. In particular, asymptotic linearity and convergence of the estimated EIF values require appropriate behavior of the fitted nuisance functions and targeting procedure. They are not guaranteed by a small empirical mean of the EIF values alone.

Worked Standard Errors for Means and Contrasts

For a contrast, estimate the influence value on each subject by applying the same contrast to that subject's EIF values. This preserves the dependence between regime-specific estimates, because both are calculated from the same subjects.

For a risk ratio, it is often convenient to work on the log scale. The delta method says that if a differentiable transformation \(g\) is applied to an asymptotically linear estimator of a vector of means, its EIF is the derivative of \(g\) applied to the vector of EIFs. For \(g(\psi_1,\psi_0)=\log(\psi_1/\psi_0)\), this gives \(D_{\log RR}^*=D_1^*/\psi_1-D_0^*/\psi_0\). This requires \(\psi_1,\psi_0>0\) and regular estimation of both means. Applying the derivative of \(\exp\) gives the EIF for the risk ratio itself, \((\psi_1/\psi_0)(D_1^*/\psi_1-D_0^*/\psi_0)\).

Diagnostics, Alternatives, and Limitations

Several checks help detect mistakes or unstable estimates. Confirm that there is one EIF value per independent subject and that each value includes all longitudinal contributions in the EIF, not only the terminal outcome residual. Check the empirical mean of the fitted EIF values: a large value can signal an implementation or targeting problem, although a mean near zero does not prove that the nuisance estimates or asymptotic approximation are adequate. Inspect the distribution of the values, especially unusually large magnitudes. A few extreme values can dominate the empirical variance and indicate unstable estimation; the resulting standard error may then be imprecise or poorly approximated by its large-sample justification.

The empirical EIF standard error is a nonparametric, first-order estimate of sampling variability. It differs from a model-based standard error that relies on a specified parametric covariance model. A bootstrap is another alternative, but it must resample independent subjects and repeat the complete estimation procedure, including nuisance fitting and targeting. Treating the repeated rows within a subject as independent is not an alternative valid calculation: it changes the sampling unit and generally understates or otherwise misrepresents uncertainty.

The calculation can be unreliable when the EIF has very large or infinite variance, when the sample is too small for a first-order approximation to be useful, or when the estimator lacks the required asymptotic linearity. The plug-in variance formula does not fix such problems. Nor does it by itself determine an appropriate confidence interval, particularly when the parameter is near a boundary.

• Common mistake: Calculate a variance for each regime separately and add the two estimated variances to obtain the risk-difference variance. This silently assumes zero covariance between estimates.
• Correction: Subtract the two regimes' EIF values within each subject, then calculate the sample variance of those paired differences. This automatically retains their covariance.
• Common mistake: Divide the sample variance of EIF values by \(n\), then report its square root as the standard error.
• Correction: Divide by \(n\) to estimate the estimator's variance; take the square root only afterward. Equivalently, the standard error is the EIF sample standard deviation divided by \(\sqrt n\).