Paper review: Sensitivity Analysis for Inverse Probability Weighting Estimators via the Percentile Bootstrap

A review of Zhao, Small, and Bhattacharya (2019) on conducting robust sensitivity analysis for IPW estimators using marginal sensitivity models and the percentile bootstrap.

1. Sensitivity Analysis for Inverse Probability Weighting Estimators via the Percentile Bootstrap

   Qingyuan Zhao, Dylan S. Small, and Bhaswar B. Bhattacharya

   Journal of the Royal Statistical Society Series B: Statistical Methodology, Sep 2019

The Setup: The Data Generating Process

• Missing data problem is a common problem in causal inference literature, and share similarity with ATE estimation from observational data.
• For each subject $i \in {1, 2, \dots, n}$, we define the following variables:
  • $A_i$: An indicator variable where $A_i = 1$ if the response is observed, and $A_i = 0$ if it is missing.
  • $X_i \in \mathcal{X} \subset \mathbb{R}^d$: A vector of fully observed covariates.
  • $Y_i \in \mathbb{R}$: The target response variable.

• We assume that $(A_1, X_1, Y_1), \dots, (A_n, X_n, Y_n)$ are independent and identically distributed (IID) from a joint true data-generating distribution, $F_0$.

• Crucially, $Y_i$ is only observed if $A_i = 1$. In reality, the data we actually get to see is $(A_i, X_i, A_i Y_i)$.

• Our goal is to estimate the true mean response in the complete data:

$$\mu := \mathbb{E}_0[Y]$$

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The “Missing at Random” (MAR) Assumption

• Let’s define missingness mechanism as the probability that a data point is observed.

$$e_0(x, y) = \mathbb{P}_0(A = 1 \mid X = x, Y = y)$$

• Previous IPW methods for estimating $\mu$ rely on the Missing at Random (MAR) assumption:

$$e_0(x, y) = e_0(x)$$

• MAR assumes that the probability of missingness depends only on the observed covariates, not the unobserved response.

• However, MAR is a strong assumption, and in many real-world scenarios, it is easily violated.

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Sensitivity Model

• Because $e_0(x, y)$ cannot be identified from the data without the MAR assumption, we need a way to test how our estimate of $\mu$ changes if MAR is violated.
• To do this, we use a user-specified function for $e_0(x, y)$ known as a sensitivity model. It’s called a model because it is a user-specified function.

Definition: The Marginal Sensitivity Model

Fix a parameter $\Lambda \ge 1$. We assume that our missingness mechanism $e(x, y)$ belongs to a set $\mathcal{E}(\Lambda)$, defined as:

$$\mathcal{E}(\Lambda) = \left\{ e(x, y) \in [0, 1] : \frac{1}{\Lambda} \le \text{OR}(e(x, y), e_0(x)) \le \Lambda, \text{ for all } x \in \mathcal{X}, y \in \mathbb{R} \right\}$$

Where $\text{OR}$ is the odds ratio, defined as:

$$\text{OR}(p_1, p_2) = \frac{p_1 / (1 - p_1)}{p_2 / (1 - p_2)}$$

By adjusting $\Lambda$, you control the “budget” of unmeasured confounding. If $\Lambda = 1$, you are strictly enforcing the MAR assumption. As $\Lambda$ grows, you allow for more severe violations of MAR.

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The Logistic Representation

It is often much easier to write this model using the logistic (logit) scale.

Let’s define the logit functions for our probabilities:

$$g_0(x) = \text{logit}\{e_0(x)\} = \log \left( \frac{e_0(x)}{1 - e_0(x)} \right)$$

$$g_0(x, y) = \text{logit}\{e_0(x, y)\}$$

$$h_0(x, y) = g_0(x) - g_0(x, y)$$

So $h$ quantifies the degree of shift. We can now express our sensitivity model probability as a function of this shift, $h$:

$$e^{(h)}(x, y) = \left[ 1 + \exp\{h(x, y) - g_0(x)\} \right]^{-1}$$

Key Takeaway: The marginal sensitivity model simply puts a bound on the $L_\infty$-norm of $h$.

If we define $\lambda = \log(\Lambda)$, the sensitivity model translates to:

$$\mathcal{H}(\lambda) = \left\{ h : \mathcal{X} \times \mathbb{R} \to \mathbb{R} \text{ and } \|h\|_\infty \le \lambda \right\}$$

It is easy to see the model’s behavior: as $h(x, y) \to \pm\infty$, the selection probability $e^{(h)}(x, y)$ approaches $0$ or $1$.