censoring | Chen Xing

Notes on Doubly Robust Censoring Unbiased Transformation

Tue, 17 Mar 2026 00:00:00 +0000

Predicting outcomes with right-censored survival data forces a choice: do we model the outcome distribution, or the censoring mechanism? Classical transformations require us to commit to one. Rubin and van der Laan (2007) tell us we don’t have to. Their doubly robust censoring unbiased transformation fuses both approaches, remaining valid as long as at least one of the two nuisance models is correctly specified. This post walks through the setup, the classical transformations, and how the doubly robust version combines them — drawing the analogy to AIPW along the way.

1. Setup

We observe an i.i.d. sample $\{O_i\}_{i=1}^n$ where each observation is

$$O = \bigl(W,\; \Delta = \mathbf{1}(Y \le C),\; \tilde{Y} = Y \wedge C\bigr)$$

$W$: covariates
$Y$: true (possibly unobserved) survival time
$C$: random censoring time
$\tilde{Y} = \min(Y, C)$ is what we actually record
$\Delta = 1$ if the event is observed ($Y \le C$), and $\Delta = 0$ if the outcome is right-censored ($Y > C$)

Our goal is to estimate the regression function

$$m(w) = \mathbb{E}[Y \mid W = w]$$

Two nuisance functions appear throughout:

$\bar{F}(\cdot \mid W)$: conditional survival function of the response $Y$ given $W$
$\bar{G}(\cdot \mid W)$: conditional survival function of the censoring time $C$ given $W$

We maintain the standard assumption $Y \indep C \mid W$.

2. The Challenge: Unidentifiability

When the censoring time $C$ corresponds to a fixed study endpoint, the true response $Y$ may exceed it. Beyond that endpoint, nothing can be learned about the tail of the survival distribution — making $m(W) = \mathbb{E}[Y \mid W]$ unidentifiable in general.

The fix: truncate the response at a known horizon $\tau$,

$$Y \longmapsto Y \wedge \tau = \min(Y, \tau)$$

and estimate $w \mapsto \mathbb{E}[Y \wedge \tau \mid W = w]$ instead.

The Surrogate-Response Strategy

The general approach to prediction with right-censored data is:

Replace the possibly unavailable responses $\{Y_i\}_{i=1}^n$ with surrogate values $\{Y^*(O_i)\}_{i=1}^n$ using an imputation map $Y^*(\cdot)$ built from observed data.
Plug the imputed dataset $\{W_i, Y^*(O_i)\}_{i=1}^n$ into any standard regression algorithm.

The imputation map $Y^*(\cdot)$ is called a censoring unbiased transformation (Fan and Gijbels 1996) if it satisfies

$$\mathbb{E}[Y^*(O) \mid W] = \mathbb{E}[Y \mid W] = m(W)$$

That is, the surrogate is an unbiased proxy for the true (latent) response, conditional on covariates.

3. Two Classical Transformations

A. The Buckley–James Transformation (depends on $\bar{F}$)

The Buckley–James transformation imputes a censored observation with its conditional mean given that it exceeds the censoring time:

$$Y^*(O) = \Delta Y + (1-\Delta) Q_{\bar{F}}(W, C)$$

where $\Delta = 1$ if $Y$ is observed and $\Delta = 0$ if right-censored, and

$$Q_{\bar{F}}(w, y) = \mathbb{E}[Y \mid W = w,\; Y > y] = \frac{1}{\bar{F}(y \mid W=w)} \int_y^{+\infty} u \; dF(u \mid W=w)$$

Intuitively: if we observe the event, we keep $Y$; if censored at $C$, we impute with the expected remaining survival time above $C$.

This transformation requires correctly estimating $\bar{F}(\cdot \mid W)$, i.e., the conditional distribution of the survival time.

B. The IPCW Transformation (depends on $\bar{G}$)

Inverse probability of censoring weighting (IPCW) up-weights the observed events to compensate for the censored ones:

$$Y^*(O) = \frac{Y \Delta}{\bar{G}(Y \mid W)}$$

This requires correctly estimating $\bar{G}(\cdot \mid W)$, the conditional survival function of the censoring time. It is the survival-analysis analogue of IPW in the causal inference literature.

4. The Doubly Robust Censoring Unbiased Transformation

The two classical transformations each stake everything on one nuisance model. The doubly robust approach combines them:

$$ Y^*(O) = \underbrace{\frac{Y\Delta}{\bar{G}(Y \mid W)}}_{\text{1st term}} + \underbrace{\frac{Q_{\bar{F}}(W,C)\,(1-\Delta)}{\bar{G}(C \mid W)}}_{\text{2nd term}} - \underbrace{\int_{-\infty}^{\tilde{Y}} \frac{Q_{\bar{F}}(W,c)}{\bar{G}^2(c \mid W)}\, dG(c \mid W)}_{\text{3rd term (correction)}} $$

where $\tilde{Y} = Y \wedge C = \min(Y,C)$ and $Q_{\bar{F}}(w, c) = \mathbb{E}[Y \mid W=w,; Y > c]$.

Theorem 1.

$\mathbb{E}[Y^*(O) \mid W] = \mathbb{E}[Y \mid W]$ whenever either $\bar{F}(\cdot \mid W)$ or $\bar{G}(\cdot \mid W)$ is correctly specified.

5. Intuition: An AIPW in Disguise

The three-term structure has a clean interpretation. Recall that $Q_{\bar{F}}(W,C) = \mathbb{E}[Y \mid W, Y > C]$ is the outcome regression for censored individuals.

Term	Role
1st: $Y\Delta \,/\, \bar{G}(Y\!\mid\! W)$	For observed outcomes — apply IPCW, the analogue of inverse propensity score weighting
2nd: $Q_{\bar{F}}(W,C)(1-\Delta) \,/\, \bar{G}(C\!\mid\! W)$	For censored outcomes — impute with the outcome regression $Q_{\bar{F}}$, then apply an IPCW-style weight
3rd: $-\int Q_{\bar{F}} / \bar{G}^2 \; dG$	Bias correction term

The first two terms together look exactly like IPW + imputation — the two ingredients of AIPW in the standard (binary treatment) setting. The third term is the survival-analysis counterpart of the augmentation correction in AIPW: it removes the bias that accumulates when both nuisance models are slightly off.

Takeaway

The doubly robust censoring unbiased transformation is a drop-in replacement for the classical Buckley–James or IPCW transformations. Once $Y^*(O_i)$ is computed for each observation, any off-the-shelf regression algorithm can be applied to the pairs $\{W_i, Y^*(O_i)\}$.

Reference

Rubin, Daniel and Mark J. van der Laan (2007), “A Doubly Robust Censoring Unbiased Transformation,” The International Journal of Biostatistics, 3 (1).

Fan, Jianqing and Irène Gijbels (1996), Local Polynomial Modelling and Its Applications, Chapman & Hall.

Notes on Propensity Score Methods

Tue, 03 Jun 2025 00:00:00 +0000

Introduction

Here are my notes on propensity scores, mainly from Prof. Ding’s textbook (2024).

The traditional propensity score analysis workflow is shown in the image below, which I will not cover in detail. Instead, I will summarize the key theorems and results from Ding’s textbook.

Figure 1: Traditional propensity score analysis workflow

I will also provide some connections with Riesz Representer (RR).

Why connect with the Riesz Representer (RR)? The connection provides a powerful generalization of the foundational Rosenbaum-Rubin (1983) result.
Rosenbaum and Rubin showed that conditioning on the propensity score is sufficient for removing confounding bias when estimating causal effects. The Riesz representer extends this principle: it suffices to regress on the Riesz representer to obtain unbiased estimates of the average treatment effect.
The key insight is that the Riesz representer, like the propensity score, serves as a sufficient statistic – it captures all the confounding information necessary for unbiased estimation of your target causal parameter.

Setting & Notation

Binary treatment $Z$
Potential outcomes $\{Y(0), Y(1)\}$
Propensity score: $\P(Z = 1 \mid X)$, where $X$ represents covariates

Two approaches learning causal relationships:

Outcome process (via outcome regression)
Treatment assignment mechanism (via propensity score)

The following summarizes the key theorems and results related to propensity scores from Prof. Ding’s textbook.

1. The propensity score as a dimension reduction tool

Theorem 1 (pscore as dimension reduction tool).

$$ \text { If } Z \indep \{Y(1), Y(0)\} \mid X, \text { then } Z \indep \{Y(1), Y(0)\} \mid e(X) . $$

Covariates $X$ can be high dimensional, but the propensity score, $e(X) \in \R$, is a 1-dimensional scalar
We can view the propensity score as a dimensional reduction tool

2. Propensity score stratification

Idea: Discretize the estimated propensity score by its $K$ quantiles:
$$ Z \indep \{Y(1), Y(0)\} \mid \hat{e}^{\prime}(X)=e_k \quad(k=1, \ldots, K) . $$
Estimate ATE within each subclass and then average by the block size
Advantage: The propensity score stratification estimator only requires the correct ordering of the estimated propensity scores rather than their exact values, which makes it relatively robust compared with other methods

3. Propensity score weighting

Theorem 2 (Invese propensity score weighting (IPW)).

If $Z \indep \{Y(1), Y(0)\} \mid X$ and $ 0 < e(X) < 1$, then $$E\{Y(1)\}=E\left\{\frac{Z Y}{e(X)}\right\}, \quad E\{Y(0)\}=E\left\{\frac{(1-Z) Y}{1-e(X)}\right\}$$ and $$ \begin{aligned} \tau &=E\{Y(1)-Y(0)\}\\ &=E\left\{\frac{Z Y}{e(X)}-\frac{(1-Z) Y}{1-e(X)}\right\} \\ &=E\left\{HY \right\} \end{aligned} $$ where $H := \left[\frac{Z}{e(X)}-\frac{(1-Z) }{1-e(X)}\right]$ is called the Horvitz-Thompson transform.

Connection the Riesz Representer (RR)

Remark 1 (RR in the case of ATE).

In the case of ATE, the Riesz Representer, $\alpha(Z, X)$, has the same form as above Horvitz-Thompson transform, $$ \alpha(Z, X) = \left[\frac{Z}{e(X)}-\frac{(1-Z) }{1-e(X)}\right] $$

3.1 Estimation

The sample version of IPW is called the Horvitz–Thompson (HT) estimator,
$$ \hat{\tau}^{\mathrm{ht}}=\frac{1}{n} \sum_{i=1}^n \frac{Z_i Y_i}{\hat{e}\left(X_i\right)}-\frac{1}{n} \sum_{i=1}^n \frac{\left(1-Z_i\right) Y_i}{1-\hat{e}\left(X_i\right)} $$
HT estimator $\hat{\tau}^{\mathrm{ht}}$ has many problems
Problem: lack of invariance, i.e. if we replace $Y_i$ by $Y_i + c$, $\hat{\tau}^{\mathrm{ht}}$ changed because it depends on $c$. This is not reasonable.
Solution: normalizing the weights
$$ \hat{\tau}^{\text {hajek }}=\frac{\sum_{i=1}^n \frac{Z_i Y_i}{\hat{e}\left(X_i\right)}}{\sum_{i=1}^n \frac{Z_i}{\hat{e}\left(X_i\right)}}-\frac{\sum_{i=1}^n \frac{\left(1-Z_i\right) Y_i}{1-\hat{e}\left(X_i\right)}}{\sum_{i=1}^n \frac{1-Z_i}{1-\hat{e}\left(X_i\right)}} . $$
Hajek estimator is invariant to the location transformation

3.2 Strong overlap condition

Many asymptotic analyses require a strong overlap condition,

$$ 0<\alpha_{\mathrm{L}} \leq e(X) \leq \alpha_{\mathrm{U}}<1 $$ In practice,

Crump et al. (2009) suggested $α_L = 0.1$ and $α_U = 0.9$
Kurth et al. (2005) suggested $α_L = 0.05$ and $α_U = 0.95$

4. Balancing property

Theorem 3 (balancing property).

Conditional on $e(X)$, the treatment and the covariates are independent
Within the same level of the propensity score, the covariate distributions are balanced across the treatment and control groups
Useful implication: we can check whether the propensity score model is specified well enough to ensure the covariate balance in the data

4.1 Propensity score is a balancing score

Definition 1.

Theorem 4 (Propensity score is a balancing score).

This is relevant in subgroup analysis
The conditional independence in (11.5) ensures unconfoundedness holds given the propensity score, within each level of $X_1$. Therefore, we can perform the same analysis based on the propensity score, within each level of $X_1$, yielding estimates for two subgroup effects

5. Doubly Robust or AIPW

The following Theorem is summarized from Prof. Wager’s lecture notes (2024).

Theorem 5 (strong double robustness of AIPW estimator).

Define the outcome regression as $$ \mu_{(z)}(x)=\mathbb{E}\left[Y_i(z) \mid X_i=x\right], $$ Define AIPW estimator as $$ \begin{aligned} \hat{\tau}_{A I P W} & =\underbrace{\frac{1}{n} \sum_{i=1}\left(\hat{\mu}_{(1)}\left(X_i\right)-\hat{\mu}_{(0)}\left(X_i\right)\right)}_{\text { outcome regression estimator }} \\ & +\underbrace{\frac{1}{n} \sum_{i=1}^n\left(\frac{Z_i}{\hat{e}\left(X_i\right)}\left(Y_i-\hat{\mu}_{(1)}\left(X_i\right)\right)-\frac{1-Z_i}{1-\hat{e}\left(X_i\right)}\left(Y_i-\hat{\mu}_{(0)}\left(X_i\right)\right)\right)}_{\text { applying IPW to the regression residuals }} \end{aligned} $$ If we use estimators $\hat{\mu}_{(z)}(x)$ and $\hat{e}(x)$ that are both consistent with root-mean squared error (RMSE) decaying faster than $n^{-\alpha_\mu}$ and $n^{-\alpha_e}$ respectively, and if furthermore $\alpha_\mu+\alpha_e \geq 1 / 2$, then $$ \begin{aligned} & \sqrt{n}\left(\hat{\tau}_{A I P W}-\tau\right) \Rightarrow \mathcal{N}\left(0, V_{A I P W}\right) \\ & V_{A I P W}=\operatorname{Var}\left[\tau\left(X_i\right)\right]+\mathbb{E}\left[\frac{\sigma_0^2\left(X_i\right)}{1-e\left(X_i\right)}\right]+\mathbb{E}\left[\frac{\sigma_1^2\left(X_i\right)}{e\left(X_i\right)}\right] \end{aligned} $$ where $$ \sigma_{(z)}^2(x)=\operatorname{Var}\left[Y_i(z) \mid X_i=x\right] $$

Check my previous post: Intuition for Doubly Robust Estimator
AIPW provides a natural starting point for understanding Double Machine Learning
Key insight of RR in DML framework: Leverage the Riesz Representer, a “generalized version of propensity score” to “correct the bias”

More general, Li et al. (2018a) gave a unified discussion of the causal estimands in observational studies.

Theorem 6 (Ding (2024), Section 13.4).

Summary Table of common estimands:

This table provides us a good way to understand and remember IPW estimator for ATT
How to remember $\tau^h$? Apply IPW on “pseudo outcome” $Yh(X)$ then divide by $E(h(X))$
When the parameter of interest is ATT, then $$E(h(X)) = E(e(X)) = E(E(Z \mid X)) = E(Z) = \P(Z = 1) = e$$
Use it to better understand IPW for ATT

7. Propensity Score in Regression

PS as a covariate

Theorem 7 (regression with pscore as a covariate).

Under unconfoundedness, the coefficient of $Z$ in the population OLS fit of $$ Y \sim 1+Z + e(X) $$ equals $\tau_{\mathrm{O}}$, $$ \tau_{\mathrm{O}}=\frac{E[e(X)\{1-e(X)\} \tau(X)]}{E[e(X)\{1-e(X)\}]} ,$$ which is the overlap-weighted average treatment effect

Based on above Theorem, we also have:

Corollary 1.

Under unconfoundedness,
1. the coefficient of $Z$ in the population OLS fit of $$ Y \sim 1+Z + e(X) + X $$ also equals $\tau_{\mathrm{O}}$,

2. the coefficient of $Z-e(X)$ in the population OLS fit of $$ Y \sim [Z - e(X)] \quad \text{or} \quad Y \sim 1 + [Z - e(X)] $$ also equals $\tau_{\mathrm{O}}$

PS as a weight

There is a convenient way to obtain $\hat{\tau}^{\text{hajek}}$ based on WLS.

Proposition 1 (convenient to obtain $\hat{\tau}^{\text{hajek}}$ based on WLS).

Need to use bootstrap for standard error
Why does the WLS give a consistent estimator for $\tau$ ?
In RCT with a constant propensity score, we can simply use the coefficient of $Z_i$ in the OLS fit of $Y_i$ on ( $1, Z_i$ ) to estimate $\tau$
In observational studies, we need to deal with the selection bias. The key idea is:
- If we weight the treated units by $\frac{1}{e(X_i)}$ and the control units by $\frac{1}{1-e(X_i)}$, then both treated and control groups can represent the whole population
- Thus, by weighting, we effectively have a pseudo-randomized experiment
- Remark 2 (IPCW).
  
  Inverse Probability of Censoring Weighting (IPCW) follows the same idea — it adjusts for censoring bias by reweighting observations based on their probability of being uncensored.
Consequently, the difference between the weighted means is consistent for $\tau$. The numerical equivalence of $\hat{\tau}^{\text {hajek }}$ and WLS is not only a fun numerical fact itself but also useful for motivating more complex estimators with covariate adjustment

Reference

Ding, Peng (2024), A First Course in Causal Inference, CRC Press.

Wager, S. (2024). Causal inference: A statistical learning approach. https://web.stanford.edu/~swager/causal_inf_book.pdf

Adjust Censoring and Confounding Bias by IP Weighting

Tue, 01 Oct 2024 00:00:00 +0000

Introduction

In the context of causal inference, adjusting for both censoring bias and confounding bias is crucial, particularly in survival analysis where right-censored data often complicates causal effect estimation. Right censoring occurs when the outcome of interest (e.g., time to an event) is not observed within the study period for some subjects, making it challenging to correctly assess the causal effect of a treatment. Moreover, confounding bias arises when treatment assignment is influenced by pre-treatment covariates, potentially leading to biased estimates of treatment effects if not properly accounted for.

To address these issues, inverse probability weights (IPW) are commonly employed. IPW adjusts for confounding by reweighting observations based on their treatment probabilities given covariates. In addition, inverse probability of censoring weights (IPCW) further adjust for censoring by reweighting observations based on their probabilities of being uncensored. Together, these techniques allow us to estimate causal effects in the presence of both confounding and censoring biases, providing more accurate insights into the relationships between treatments and outcomes.

This post explains how to apply IP weights to adjust for these biases, with references to key resources.

Hernán and Robins’ Causal Inference: What If (2020), especially Chapter 8.5 and Chapter 12.6, covers censoring and missing data in detail
Zubizarreta et al.’s Handbook of Matching and Weighting Adjustments for Causal Inference (2023), Chapter 21, discusses treatment heterogeneity in survival outcomes. These resources form the basis for our explanation on using IP weights in survival analysis

IP Weighting

Imagine we want to estimate the causal effect of eco-friendly packaging on a product’s selling price. However, the selling price is right-censored — meaning we only observe the price for items that have been sold, while unsold items remain censored (i.e., their final selling price is unknown).

Setting:

Let $W \in \{0,1\}$ be a binary treatment variable (e.g., $W= 1$, if the item has a eco-friendly package; $W = 0$, otherwise)
Let $Y(w, s)$ be potential outcomes
Let $S = \textbf{1} \{C = 0\}$ be non-censoring indicator, where $C \in \{0,1\}$ is the censoring indicator
- e.g. $S = 1$, if the item is non-censored (in this case sold); $S = 0$, if the item is censored (in this case on-sale)
Let $X, L \in \mathcal{X}$ be two sets of covariates and $X \subset L$. Then there exist a function $f(\cdot)$ such that $X = f(L)$

Considering $Y(W = w, S = s)$, our analysis was necessarily restricted to uncensored individuals, i.e., those with $S = 1$, because those were the only ones with known values of the outcome $Y$. Thus, the causal effect of interest is:

$$ \tau = \mathbb{E}\{Y(w = 1, s = 1)\} - \mathbb{E}\{Y(w = 0, s= 1)\} $$

Assumptions:

Recall the three identification conditions in Chapter 8.5 (Hernán and Robins, 2020, p. 113-114). Note that, the book uses $Y, A, C, L$ to represent the outcome, treatment, censoring indicator and covariates respectively.

“First, the average outcome in the uncensored individuals must equal the unobserved average outcome in the censored individuals with the same values of $A$ and $L$. This provision will be satisfied … if the variables in $A$ and $L$ are sufficient to block all backdoor paths between $C$ and $Y$.”

“Second, IP weighting requires that all conditional probabilities of being uncensored given A and the variables in L must be greater than zero.”

“The third condition is consistency, including sufficiently well-deﬁned interventions. IP weighting is used to create a pseudo-population in which censoring $C$ has been abolished, and in which the effect of the treatment $A$ is the same as in the original population.”

Following above three identification conditions, we make the following assumptions. Denote the propensity score as $e(\cdot)$ and censoring score as $c(\cdot)$,

$$ \begin{alignat}{2} &\text{1. (Unconfoundedness):} \quad && \{Y(w= 0, s = 1), Y(w= 1, s = 1)\} \perp \!\!\! \perp W \mid X \\[1.5em] &\text{2. (Overlap):} \quad && 0 < e(X) := \mathbb{P}(W = 1 \mid X) < 1 \\[1.5em] &\text{3. (Ignorable censoring):} \quad && \{Y(w= 0, s = 1), Y(w= 1, s = 1)\} \perp \!\!\! \perp S \mid (W, L) \\[1.5em] &\text{4. (Positivity):} \quad && c(W, L) := \mathbb{P}(S = 1 \mid (W, L)) > 0 \end{alignat} $$

Claim:

$$ \begin{align} \tau &= \mathbb{E}\left[Y(w= 1, s = 1) - Y(w= 0, s = 1)\right] \\[1em] &= \mathbb{E}\left[\frac{S W Y}{c(W, L) e(X)} - \frac{S (1-W) Y}{c(W, L) (1-e(X))}\right] \quad \tag{2} \end{align} $$

Proof:

First, note that the eqn (2) is well defined because of A2 and A4.

I only give the proof for, $$\mathbb{E}\left[Y(w= 1, s = 1)\right] = \mathbb{E}\left[\frac{S W Y}{c(W, L) e(X)}\right],$$ as the $\mathbb{E}\left[Y(w= 0, s = 1)\right]$ part follows the same logic.

Recall that,

By our setting, $X = f(L)$ for some known function $f$
$c(W, L) := \mathbb{P}(S = 1 \mid W, L) = \mathbb{E}\left[S \mid W,L \right]$

Then we have,

\begin{aligned} \mathbb{E}\left[\frac{S W Y}{c(W, L) e(X)}\right] &= \mathbb{E}\left\{\frac{W}{e(f(L))c(W,L))} \cdot \mathbb{E}\left[SY(w, s=1) \,\middle|\, W,L \right] \right\} &&\text{(by LIE)} \\[1.5em] & = \mathbb{E}\left\{\frac{W}{e(f(L))c(W,L))} \cdot \mathbb{E}\left[S \,\middle|\, W,L \right] \cdot \mathbb{E}\left[Y(w, s=1) \,\middle|\, W,L \right] \right\} &&\text{(by A3)} \\[1.5em] & = \mathbb{E}\left\{\frac{W}{e(f(L))} \cdot \mathbb{E}\left[Y(w, s=1) \,\middle|\, W,L \right] \right\} \\[1.5em] & = \mathbb{E}\left\{\mathbb{E}\left[\frac{W}{e(f(L))} \cdot Y(w=1, s=1) \,\middle|\, W,L \right] \right\} &&\text{(by SUTVA)} \\[1.5em] & = \mathbb{E}\left[\frac{W}{e(X)} \cdot Y(w=1, s=1) \right] \\[1.5em] & = \mathbb{E}\left[Y(w=1, s=1) \right] &&\text{(by A1 and IPW)} \end{aligned}

The last equation holds by the classical proof of the unbiasedness of inverse propensity score weighting (IPW) estimator. For more details, one can check Theorem 11.3 at page 158-159 in Peng Ding’s textbook (Ding, 2023).

Q.E.D.

Future Work

How to create a doubly robust estimator based on above setting? One related literature is the causal survival forest model (Cui et al., 2023), which estimates heterogeneous treatment effects in time-to-event setting and obtains doubly robustness property. But sometimes we are more interested in the downstream outcomes (e.g. selling price) after the event (e.g. being sold). This motivates us to create a new estimator that is doubly robust…

References

Hernán MA, Robins JM (2020). Causal Inference: What If. Boca Raton: Chapman & Hall/CRC.

Zubizarreta, J. R., Stuart, E. A., Small, D. S., & Rosenbaum, P. R. (2023). Handbook of Matching and Weighting Adjustments for Causal Inference. CRC Press.

Cheng, C., Li, F., Thomas, L. E., & Li, F. (Frank). (2022). Addressing Extreme Propensity Scores in Estimating Counterfactual Survival Functions via the Overlap Weights. American Journal of Epidemiology, 191(6), 1140–1151. https://doi.org/10.1093/aje/kwac043

YouTube tutorial about IPCW: “Survival Analysis, Censoring and Time Scales”

Cui, Y., Kosorok, M. R., Sverdrup, E., Wager, S., & Zhu, R. (2023). Estimating heterogeneous treatment effects with right-censored data via causal survival forests. Journal of the Royal Statistical Society Series B: Statistical Methodology, 85(2), 179–211. https://doi.org/10.1093/jrsssb/qkac001

Ding, Peng. “A First Course in Causal Inference.” arXiv, October 3, 2023. https://doi.org/10.48550/arXiv.2305.18793.

censoring | Chen Xing

Notes on Doubly Robust Censoring Unbiased Transformation

1. Setup

2. The Challenge: Unidentifiability

The Surrogate-Response Strategy

3. Two Classical Transformations

A. The Buckley–James Transformation (depends on $\bar{F}$)

B. The IPCW Transformation (depends on $\bar{G}$)

4. The Doubly Robust Censoring Unbiased Transformation

5. Intuition: An AIPW in Disguise

Takeaway

Reference

Notes on Propensity Score Methods

Introduction

Setting & Notation

1. The propensity score as a dimension reduction tool

2. Propensity score stratification

3. Propensity score weighting

3.1 Estimation

3.2 Strong overlap condition

4. Balancing property

4.1 Propensity score is a balancing score

5. Doubly Robust or AIPW

6. Other Estimands related to IPW

7. Propensity Score in Regression

PS as a covariate

PS as a weight

Reference

Adjust Censoring and Confounding Bias by IP Weighting

Introduction

IP Weighting

Setting:

Assumptions:

Claim:

Proof:

Future Work

References