Inverse Probability Weighted Pseudo-Observations

• Inverse probability weighted pseudo-observations are defined as weighted individual contributions that serve as surrogates for complete-data quantities when data are censored, missing, or selectively observed.
• They integrate methodologies from causal inference, survival analysis, and missing data problems using constructions such as direct IPW pseudo-scores, jack-knife pseudo-values, and augmented corrections.
• This approach enables robust analysis through standard regression and estimating-equation techniques, while addressing bias and variance issues inherent in incomplete data settings.

Inverse probability weighted pseudo-observations are weighted individual contributions that reconstruct a target full-data, finite-population, or uncensored quantity from incomplete or selectively observed data. In causal IPW, the weighted term $\frac{A_iY_i}{G(1\mid W_i)}$ is a pseudo-observation of $Y^1$; in survival analysis, pseudo-observations may be jack-knife pseudo-values of an IPCW estimator or direct IPCW contributions such as $\frac{D_i I\{X_i\le t\}}{\widehat K(X_i-)}$; and in missing-data settings they appear as weighted full-data estimating functions or pseudo-scores such as $$\frac{\mathbf{1}(R_i=1)}{\pi_1(L_i;\hat\gamma)}M(L_i;\beta)$$ (Ertefaie et al., 2020, Overgaard, 2024, Strawderman et al., 2023, Sun et al., 2014). Across these formulations, the common objective is to construct a pseudo-population in which selection biases are eliminated, or to transform censored, missing, or selectively labeled observations into complete-data surrogates that can be analyzed with standard estimating-equation, regression, or survey-sampling tools (Zeng et al., 2021, Si et al., 2013).

1. Conceptual core

Inverse probability weighting replaces an unobserved or selectively observed contribution by a weighted observed contribution that is unbiased, or asymptotically unbiased, for the full-data quantity of interest. In the causal setting this takes the form

$$Y^{(1)}_{i,\text{pseudo}}=\frac{A_iY_i}{\hat G(1\mid W_i)},$$

so that treated units are up-weighted to represent units with the same covariate profile who were not treated. In survey sampling, each sampled unit is inflated by the inverse of its inclusion probability, and the weighted sample mimics the full finite population. In missing-data problems, a complete-case estimating function is multiplied by the inverse probability of being fully observed. These are all pseudo-observation constructions in the sense that each observed unit is made to stand in for a larger target population (Ertefaie et al., 2020, Si et al., 2013).

The term is used heterogeneously across literatures. Some papers reserve “pseudo-observation” for jack-knife pseudo-values, while others use an equivalent pseudo-score or pseudo-residual representation. This suggests a broad class of objects rather than a single formal definition: weighted outcomes, weighted estimating-function contributions, and jack-knife linearizations all serve as pseudo-observations when their empirical average targets a complete-data or full-population functional.

A useful unifying view is that pseudo-observations are individual-level building blocks for an estimating equation or plug-in estimator. If $\theta$ is identified through an unbiased complete-data moment $E\{U(O;\theta)\}=0$, then an inverse probability weighted pseudo-observation replaces $U(O;\theta)$ by a weighted version evaluable under the observed-data law. If $\theta$ is a functional such as a survival probability, cumulative incidence, or restricted mean survival time, a pseudo-observation may instead be a pseudo-outcome whose average reproduces the estimator of that functional.

2. Principal constructions

Three constructions recur in the literature.

Construction	Representative form	Role
Direct IPW pseudo-score	$\frac{R_i}{\pi_i}U_i$	Reweights observed contributions
Jack-knife pseudo-value	$Y^1$0	Linearizes a functional
Augmented pseudo-observation	$Y^1$1	Incorporates extra information

The jack-knife pseudo-value construction is explicit in fixed-time survival regression. If $Y^1$2 is an IPCW estimator of $Y^1$3 and $Y^1$4 is the leave-one-out estimator, then the pseudo-observation is

$Y^1$5

For Kaplan–Meier-based IPCW, this admits the decomposition

$Y^1$6

where the first term is the familiar IPCW-weighted outcome and the second term is the correction for how subject $Y^1$7 affects the estimated censoring distribution within stratum $Y^1$8 (Overgaard, 2024).

This decomposition shows that pseudo-observations are not merely reweighted observed outcomes. They can also contain a correction for nuisance-parameter estimation. In the Kaplan–Meier-based comparison of weighted regression, weighted outcome regression, and pseudo-observation regression, the pseudo-observation method is defined precisely by regressing $Y^1$9 on covariates through a standard complete-data estimating equation. The corresponding pseudo-values are therefore IPCW-corrected outcomes plus a jack-knife correction for weight estimation (Overgaard, 2024).

A parallel construction arises in prediction-powered inference. There the rectifier term is built from residual pseudo-observations, and with informative labeling the correction becomes an inverse-probability weighted pseudo-residual,

$\frac{D_i I\{X_i\le t\}}{\widehat K(X_i-)}$0

rather than a weighted outcome. This suggests that pseudo-observations need not target the primary outcome itself; they can target a bias-correction quantity, provided their empirical mean reproduces the relevant functional.

3. Missing data and full-data estimating equations

For nonmonotone missing at random data, the full data for one individual are

$\frac{D_i I\{X_i\le t\}}{\widehat K(X_i-)}$1

with missingness pattern

$\frac{D_i I\{X_i\le t\}}{\widehat K(X_i-)}$2

where $\frac{D_i I\{X_i\le t\}}{\widehat K(X_i-)}$3 denotes complete cases. The MAR condition is

$\frac{D_i I\{X_i\le t\}}{\widehat K(X_i-)}$4

and the complete-case probability must satisfy

$\frac{D_i I\{X_i\le t\}}{\widehat K(X_i-)}$5

The missingness model is specified through pattern-specific probabilities

$\frac{D_i I\{X_i\le t\}}{\widehat K(X_i-)}$6

with

$\frac{D_i I\{X_i\le t\}}{\widehat K(X_i-)}$7

A concrete parametric submodel uses logistic regressions for each non-complete pattern, while leaving the full-data law of $\frac{D_i I\{X_i\le t\}}{\widehat K(X_i-)}$8 unrestricted (Sun et al., 2014).

If $\frac{D_i I\{X_i\le t\}}{\widehat K(X_i-)}$9 is defined by the full-data estimating equation

$$\frac{\mathbf{1}(R_i=1)}{\pi_1(L_i;\hat\gamma)}M(L_i;\beta)$$0

then under MAR and positivity the basic IPW estimating equation is

$$\frac{\mathbf{1}(R_i=1)}{\pi_1(L_i;\hat\gamma)}M(L_i;\beta)$$1

The inverse probability weighted pseudo-observation is therefore

$$\frac{\mathbf{1}(R_i=1)}{\pi_1(L_i;\hat\gamma)}M(L_i;\beta)$$2

which is zero for incomplete cases and inflated for complete cases. Each complete case is representing, in expectation, a number $$\frac{\mathbf{1}(R_i=1)}{\pi_1(L_i;\hat\gamma)}M(L_i;\beta)$$3 of individuals with similar covariates whose outcomes are missing (Sun et al., 2014).

The same framework admits augmentation. Any regular asymptotically linear estimator under MAR can be represented as the solution of

$$\frac{\mathbf{1}(R_i=1)}{\pi_1(L_i;\hat\gamma)}M(L_i;\beta)$$4

with

$$\frac{\mathbf{1}(R_i=1)}{\pi_1(L_i;\hat\gamma)}M(L_i;\beta)$$5

Now incomplete cases have nonzero contributions. The augmentation terms are pattern-specific pseudo-scores based only on observed components, designed to have mean zero under MAR and to reduce variance (Sun et al., 2014).

Two estimation issues are central in this setting. First, unconstrained maximum likelihood estimation of the missingness mechanism can encounter convergence problems because the implied complete-case probability may approach or cross zero in finite samples. Second, the paper therefore proposes constrained Bayesian estimation, which enforces the empirical positivity restriction through the posterior. The resulting IPW and AIPW estimators are consistent and asymptotically normal under correct specification of the missingness model, but the paper does not claim any double robustness (Sun et al., 2014).

4. Right-censoring, Kaplan–Meier weights, and survival pseudo-values

For fixed-time survival functionals under right-censoring, let

$$\frac{\mathbf{1}(R_i=1)}{\pi_1(L_i;\hat\gamma)}M(L_i;\beta)$$6

and define the inverse probability of censoring weight

$$\frac{\mathbf{1}(R_i=1)}{\pi_1(L_i;\hat\gamma)}M(L_i;\beta)$$7

Replacing $$\frac{\mathbf{1}(R_i=1)}{\pi_1(L_i;\hat\gamma)}M(L_i;\beta)$$8 by a stratified Kaplan–Meier estimator $$\frac{\mathbf{1}(R_i=1)}{\pi_1(L_i;\hat\gamma)}M(L_i;\beta)$$9 yields $$Y^{(1)}_{i,\text{pseudo}}=\frac{A_iY_i}{\hat G(1\mid W_i)},$$0. Three regression approaches are then compared: individual-weighted regression, outcome-weighted regression, and pseudo-observation regression based on jack-knife pseudo-values of an IPCW estimator (Overgaard, 2024).

The pseudo-observation regression method uses

$$Y^{(1)}_{i,\text{pseudo}}=\frac{A_iY_i}{\hat G(1\mid W_i)},$$1

where

$$Y^{(1)}_{i,\text{pseudo}}=\frac{A_iY_i}{\hat G(1\mid W_i)},$$2

Its asymptotic behavior differs from that of weighted-regression and weighted-outcome methods because the pseudo-values incorporate the influence of each subject on the censoring estimator. In terms of low asymptotic variance, a clear winner cannot be found. Which approach will have the lowest asymptotic variance depends on the censoring distribution. The usual sandwich variance estimator is conservative under the implied assumptions (Overgaard, 2024).

A complementary line of work gives a product-limit and Volterra-integral foundation for IPCW pseudo-observations. With right-censored data $$Y^{(1)}_{i,\text{pseudo}}=\frac{A_iY_i}{\hat G(1\mid W_i)},$$3, the product-limit estimator of the failure CDF can be written as

$$Y^{(1)}_{i,\text{pseudo}}=\frac{A_iY_i}{\hat G(1\mid W_i)},$$4

and the paper shows that

$$Y^{(1)}_{i,\text{pseudo}}=\frac{A_iY_i}{\hat G(1\mid W_i)},$$5

Thus the Kaplan–Meier estimator of the failure distribution is exactly an IPCW estimator, with weights $$Y^{(1)}_{i,\text{pseudo}}=\frac{A_iY_i}{\hat G(1\mid W_i)},$$6. This supplies an explicit pseudo-observation for the failure CDF at time $$Y^{(1)}_{i,\text{pseudo}}=\frac{A_iY_i}{\hat G(1\mid W_i)},$$7,

$$Y^{(1)}_{i,\text{pseudo}}=\frac{A_iY_i}{\hat G(1\mid W_i)},$$8

whose empirical mean reproduces the product-limit estimator (Strawderman et al., 2023).

The same work shows that ties require the censoring product-limit estimator to be built from the modified at-risk process

$$Y^{(1)}_{i,\text{pseudo}}=\frac{A_iY_i}{\hat G(1\mid W_i)},$$9

rather than a naive role-swapping of events and censorings. It also shows that Efron’s redistribution-to-the-right algorithm is essentially an IPCW estimator for the failure distribution, with jump sizes proportional to $\theta$0 (Strawderman et al., 2023).

5. Propensity score weighting with survival pseudo-observations

For comparative effectiveness studies with right-censored survival outcomes, a “once for all” strategy first constructs pseudo-observations and then applies propensity score weighting as if the outcomes were completely observed. Let

$\theta$1

These yield survival probability and restricted mean survival time functionals. For each subject,

$\theta$2

with $\theta$3 and $\theta$4, where $\theta$5 is the Kaplan–Meier estimator (Zeng et al., 2021).

Treatment weighting is introduced through balancing weights

$\theta$6

where $\theta$7 and $\theta$8 is a tilting function defining the target population. The pairwise causal contrast for treatment arms $\theta$9 and $E\{U(O;\theta)\}=0$0 is estimated by the difference of weighted mean pseudo-observations,

$E\{U(O;\theta)\}=0$1

This formulation is model free with respect to the survival outcome, and it is applicable to both binary and multiple treatments (Zeng et al., 2021).

Several target populations are represented by special choices of $E\{U(O;\theta)\}=0$2. The combined sample average treatment effect uses $E\{U(O;\theta)\}=0$3; the average treatment effect on the treated for arm $E\{U(O;\theta)\}=0$4 uses $E\{U(O;\theta)\}=0$5; and overlap weights use

$E\{U(O;\theta)\}=0$6

Under a generalized homoscedasticity condition, overlap weights minimize the total asymptotic variance of the collection of pairwise causal contrasts within the balancing-weights family. The paper also derives a closed-form variance that accounts for both pseudo-observation calculation and propensity score estimation, avoiding resampling (Zeng et al., 2021).

When censoring depends on covariates, the pseudo-observations themselves are modified by inverse probability of censoring weighting:

$E\{U(O;\theta)\}=0$7

The resulting estimator is therefore a layered construction: pseudo-observations handle censoring, while propensity score weights handle confounding.

6. Modern extensions, regularization, and alternatives

In survey sampling with externally supplied design weights, Bayesian nonparametric weighted sampling inference reinterprets IPW pseudo-observations through a hierarchical model on weight cells. Unique observed weight values define poststratification cells, the unknown cell sizes $E\{U(O;\theta)\}=0$8 are modeled through a multinomial sampling model, and outcomes are modeled by a Gaussian process regression on log-weights. The resulting finite-population estimator can be interpreted as a smoothed IPW pseudo-observation estimator in which both the represented counts and the represented outcomes are regularized rather than taken as raw $E\{U(O;\theta)\}=0$9 and $U(O;\theta)$0 (Si et al., 2013).

Prediction-powered inference extends the idea to partially labeled data. Under informative labeling, the unweighted rectifier becomes biased, and the residual correction is replaced by Horvitz–Thompson or Hájek pseudo-residuals,

$U(O;\theta)$1

Equivalently, the estimator averages a pseudo-complete outcome

$U(O;\theta)$2

Under correct propensity modeling, simulations show that IPW-adjusted PPI with estimated propensities closely matches the known-probability case, retaining both nominal coverage and the variance-reduction benefits of PPI (Datta et al., 13 Aug 2025).

In alternating recurrent events, pseudo-observations are combined with a time-varying at-risk model. For window start $U(O;\theta)$3, the restricted-mean pseudo-observation is

$U(O;\theta)$4

and the IPW pseudo-observation is

$U(O;\theta)$5

Here pseudo-observations handle right-censoring, whereas IPW corrects informative missingness induced by the secondary state. The proposed regression estimator solves a weighted GEE with random-forest estimates of $U(O;\theta)$6, and simulations show that unweighted GEE is biased while the PAIR-GEE framework attains very small bias when rich history variables are used (Loe et al., 27 Oct 2025).

A further development concerns how the weights themselves are estimated. Nonparametric inverse probability weighted estimators based on undersmoothing of the highly adaptive lasso retain the pseudo-observation interpretation

$U(O;\theta)$7

but estimate the weighting mechanism by HAL in a way that makes the resulting IPW estimator asymptotically linear with variance converging to the nonparametric efficiency bound, without specification of an outcome regression (Ertefaie et al., 2020). By contrast, biased-sample empirical likelihood weighting is proposed as an alternative to inverse probability weighting. Its weights $U(O;\theta)$8 sum to 1, avoid direct use of inverse probabilities, and are asymptotically more efficient than the IPW estimator and its stabilized version for missing data problems and unequal probability sampling without replacement (Liu et al., 2021).

Taken together, these developments show that inverse probability weighted pseudo-observations are not confined to a single methodology. They appear as survey-weighted pseudo-units, missing-data pseudo-scores, jack-knife pseudo-values of IPCW estimators, causal survival pseudo-outcomes, residual corrections in prediction-assisted inference, and time-varying pseudo-observations for alternating recurrent-event processes. The common structure is the replacement of an unavailable complete-data contribution by a weighted surrogate whose empirical average targets the estimand under an explicit sampling, censoring, or missingness model.