Variance Estimation After Censoring Weighting

Why Weight Estimation Belongs in the Variance

Censoring weights let observed outcomes represent people whose outcomes are not observed at a chosen follow-up horizon. But the weights are usually estimated from the same sample. Their estimation adds uncertainty, and can also be correlated with the weighted outcome estimate. A standard error calculated as though the fitted weights were fixed may therefore be too large or too small.

To make the issue precise, consider a fixed horizon $t$. Let $Y_i(t)$ be the outcome of interest for subject $i$ at that horizon, and let $R_i(t)=1$ if that outcome is observed and $0$ otherwise. Write $Z_i$ for measured predictors of censoring. Suppose $G(Z_i)=P(R_i(t)=1\mid Z_i)$ is the probability of remaining observed through $t$, and estimate it with a model indexed by $\gamma$. The goal here is to quantify uncertainty in a censoring-weighted estimate, not to develop a particular censoring model or an event-type-specific risk estimator.

Definition: For a marginal mean $\theta=E\{Y(t)\}$, an inverse-probability-of-censoring-weighted ratio estimate is $$ \hat\theta= \frac{\sum_{i=1}^{n}R_i(t)Y_i(t)/\hat G(Z_i)} {\sum_{i=1}^{n}R_i(t)/\hat G(Z_i)}. $$ Only subjects with $R_i(t)=1$ contribute an observed outcome. The weight $1/\hat G(Z_i)$ compensates for their estimated probability of being observed.

This estimate has a useful estimating-equation form. Define $q_i(\theta,\gamma)=R_i(t)\{Y_i(t)-\theta\}/G(Z_i;\gamma)$. The estimate solves $\sum_i q_i(\hat\theta,\hat\gamma)=0$, where $\hat\gamma$ is obtained from the censoring model. For the weighted mean to target the marginal mean, a key condition is conditional independent censoring: given $Z_i$, observation through $t$ is independent of $Y_i(t)$. Also require positivity, $G(Z_i)>0$ in the target population, and a sufficiently well-specified censoring model. These conditions are not repaired by a variance estimator.

A Small Weighted-Mean Calculation

Worked Example: Computing the Weighted Estimate

Suppose a hypothetical sample has six subjects. The table gives their estimated observation probabilities; outcomes are recorded only when $R_i(t)=1$. Assume for this calculation that the censoring probabilities are known and that the censoring and positivity assumptions hold.

Subject	$R_i(t)$	$\hat G(Z_i)$	Observed $Y_i(t)$
A	1	0.80	10
B	1	0.50	14
C	0	0.70	Not observed
D	1	0.90	8
E	1	0.60	12
F	0	0.40	Not observed

The observed subjects have weights $1/0.80=1.25$, $1/0.50=2$, $1/0.90=1.1111$, and $1/0.60=1.6667$. Thus

\hat\theta= \frac{1.25(10)+2(14)+1.1111(8)+1.6667(12)} {1.25+2+1.1111+1.6667} = \frac{69.3892}{6.0278} \approx 11.51.

The estimate is about 11.51 outcome units at horizon $t$. This calculation alone gives no reliable standard error: the sample is tiny, and in a real analysis the probabilities would generally be estimated. The next sections address the uncertainty from that estimation.

Sandwich Variance With Estimated Weights

A direct way to include weight uncertainty is to treat the target estimate and the censoring-model estimate as one jointly estimated parameter. Let $s_i(\gamma)$ be the censoring-model estimating function for subject $i$, such as its likelihood score. Set $\eta=(\theta,\gamma^\mathsf{T})^\mathsf{T}$ and $U_i(\eta)=(q_i(\theta,\gamma),s_i(\gamma)^\mathsf{T})^\mathsf{T}$. Both fitted quantities solve the stacked equations $\sum_i U_i(\hat\eta)=0$.

Result: If subjects are independent and identically distributed, the stacked equations are smooth near the true parameter, their expected derivative is nonsingular, and the estimating functions have finite second moments, then the large-sample covariance estimate is $$ \widehat{\operatorname{Var}}(\hat\eta) = \frac{1}{n}\hat A^{-1}\hat B\hat A^{-\mathsf{T}}, \qquad \hat A=-\frac{1}{n}\sum_{i=1}^n \frac{\partial U_i(\hat\eta)}{\partial\eta^\mathsf{T}}, \qquad \hat B=\frac{1}{n}\sum_{i=1}^n U_i(\hat\eta)U_i(\hat\eta)^\mathsf{T}. $$ The estimated variance of $\hat\theta$ is the $\theta$-component on the diagonal of this covariance matrix.

The minus sign in the definition of $A$ matters for the influence calculation. A first-order expansion of the empirical equations around $\eta_0$ gives

0\approx \frac{1}{n}\sum_{i=1}^n U_i(\eta_0) -A(\hat\eta-\eta_0), \qquad \sqrt n(\hat\eta-\eta_0) \approx A^{-1}\frac{1}{\sqrt n}\sum_{i=1}^n U_i(\eta_0).

Thus the influence contribution is $A^{-1}U_i$, with a positive sign under this definition of $A$. Its covariance is $A^{-1}BA^{-\mathsf{T}}$, which explains the sandwich form. The “meat,” $B$, includes variation in both the target equation and the censoring-model score, as well as their covariance. This is what a calculation that holds fitted weights fixed omits.

How Nuisance Estimation Changes Uncertainty

The effect of estimating weights is not automatically an increase in variance. To see why, partition $A$ into target and nuisance blocks. When the censoring-model equation does not depend on $\theta$, its first-order equations give the target influence contribution

IF_{\theta,i} = A_{\theta\theta}^{-1} \left(q_i-A_{\theta\gamma}IF_{\gamma,i}\right), \qquad IF_{\gamma,i}=A_{\gamma\gamma}^{-1}s_i.

Here $A_{\theta\gamma}$ is the negative expected derivative of the target equation with respect to $\gamma$. The nuisance contribution is subtracted in this expression because the linearized target equation is $q_i-A_{\theta\theta}IF_{\theta,i}-A_{\theta\gamma}IF_{\gamma,i}=0$. Its covariance with $q_i$ can either raise or lower the variance.

Worked Example: A Covariance Contribution

Suppose a hypothetical analysis has scalar target and nuisance components with $A_{\theta\theta}=1.1$, $A_{\theta\gamma}=0.30$, $\operatorname{Var}(q_i)=0.090$, $\operatorname{Var}(IF_{\gamma,i})=0.040$, and $\operatorname{Cov}(q_i,IF_{\gamma,i})=0.010$. These values are assumed to be per-subject influence quantities from a valid stacked fit.

The target influence contribution is $(q_i-0.30IF_{\gamma,i})/1.1$. Therefore, using $\operatorname{Var}(X-cY)=\operatorname{Var}(X)+c^2\operatorname{Var}(Y)-2c\operatorname{Cov}(X,Y)$,

\operatorname{Var}(IF_{\theta,i}) = \frac{0.090+(0.30)^2(0.040)-2(0.30)(0.010)}{(1.1)^2} = \frac{0.0876}{1.21} \approx 0.07240.

For $n=200$ independent subjects, the estimated variance of $\hat\theta$ is $0.07240/200=0.0003620$, and its standard error is $\sqrt{0.0003620}\approx0.01903$ outcome units. If $\gamma$ were known, the corresponding variance under these same quantities would be $0.090/\{(1.1)^2(200)\}=0.0003719$, with standard error $0.01929$ outcome units. Here accounting for nuisance estimation slightly reduces the variance because the covariance term is positive. In other data, the covariance could have the opposite sign.

In practice, the sandwich is often called a robust variance estimator. “Robust” refers to its flexibility for the variance of the estimating equations; it does not mean robust to violations of conditional independent censoring, positivity, or a badly misspecified censoring model. Nor does it guarantee good small-sample coverage.

Bootstrap Inference and Practical Choices

A bootstrap offers another way to capture weight estimation. In each replicate, resample the independent sampling units, refit the censoring model, reconstruct the weights, and recompute the target estimate. The spread of the replicate estimates then reflects both outcome sampling and weight-model estimation. If subjects are sampled in clusters, resample clusters rather than individual records. If the design has strata or other sampling features, the resampling scheme should preserve them.

Worked Example: A Percentile Bootstrap Interval

Suppose a hypothetical study estimates a mean outcome of $\hat\theta=12.0$ units. In 1,000 subject-level bootstrap replicates, the analyst refits the censoring model and recalculates $\hat\theta$ each time. Suppose the 2.5th and 97.5th percentiles of the 1,000 estimates are 10.8 and 13.2 units.

The 95% percentile bootstrap interval is $[10.8,13.2]$ units. Under the sampling and model conditions that make this bootstrap procedure valid, it is an interval procedure intended to cover the target mean in 95% of repeated samples. It does not mean there is a 95% probability that the fixed target mean lies in this particular interval. The interval is only as trustworthy as the resampling scheme and the original censoring assumptions.

The sandwich and bootstrap are complementary, not universally interchangeable. A sandwich estimator is efficient to compute and makes the nuisance contribution explicit, but it relies on a credible derivative calculation and can be unreliable when the sample is small or weights are highly variable. A nonparametric bootstrap can accommodate complicated smooth estimation steps without deriving every derivative, but can be computationally expensive and may fail often when some resamples produce near-zero observation probabilities or nonconvergent censoring models. A bootstrap that keeps the original fitted weights fixed does not address the uncertainty at issue.

Procedure	What it captures	Main caution
Stacked sandwich	Target and censoring-model estimating-equation variation and covariance	Requires correct implementation of equations and derivatives; may be poor in small samples
Bootstrap with model refitting	Sampling variation across the full estimation procedure	Can be unstable with extreme weights, failed fits, or a resampling scheme that ignores the design
Variance conditional on fitted weights	Variation in the target equation while treating weights as fixed	Generally omits censoring-model estimation uncertainty

Diagnostics, Failure Modes, and Limitations

Before trusting either standard errors or intervals, check that the underlying analysis is stable. Examine censoring-model convergence and fitted observation probabilities; inspect the weighted estimate and its influence contributions for domination by a few subjects; and compare the sandwich and bootstrap results where feasible. These checks complement the earlier material on positivity diagnostics and weight influence. A close agreement between two variance procedures is reassuring but does not establish the censoring assumptions.

Holding weights fixed in the bootstrap. This reproduces uncertainty conditional on one fitted censoring model. Refit that model in every replicate to capture its estimation variability.
Assuming nuisance estimation must inflate variance. The nuisance influence can be correlated with the target contribution. Its covariance term can increase or decrease the variance, as the calculation above shows.
Expecting robust standard errors to cure bias. A sandwich variance quantifies sampling uncertainty around the estimator’s limiting target. It cannot turn informative censoring, lack of positivity, or a misspecified censoring model into valid identification.
Ignoring unstable weights or failed bootstrap fits. A few extreme weights can make both the estimate and its variance unstable. Frequent fit failures are a warning about the analysis, not a reason to silently discard replicates.

Weight truncation or stabilization may change the estimating procedure and, depending on the target and implementation, can introduce bias or alter the estimand. A variance estimate for the modified procedure describes its sampling variability; it does not measure that bias. For small samples, highly estimated weights, or data-adaptive model selection, ordinary sandwich and percentile intervals may have poor coverage. Report how weights were estimated, whether they were modified, which variance procedure was used, and any failed resamples or instability.

Check Your Understanding

Answer each question, then check the brief solutions.

Why should a bootstrap replicate refit the censoring model instead of reusing the original weights?
In the scalar influence calculation, what is the role of $\operatorname{Cov}(q_i,IF_{\gamma,i})$?
If a sandwich standard error is small, does that establish that censoring is conditionally independent of the outcome?
What should be resampled when the independent sampling units are clusters?

Solutions. (1) Refitting lets the replicate reflect sampling variation in the estimated censoring probabilities as well as variation in the weighted outcome. (2) It contributes the cross-term $-2A_{\theta\gamma}\operatorname{Cov}(q_i,IF_{\gamma,i})$ before division by $A_{\theta\theta}^2$, so it can raise or lower the variance. (3) No. The sandwich estimates sampling variability under the estimating procedure; it does not verify conditional independent censoring or positivity. (4) Resample clusters, preserving any relevant sampling design, rather than treating within-cluster observations as independent units.