1. Introduction

2. Inverse pobability weights and the Cox proportional hazards model

3. Example: Ewing's sarcoma data set

• Statistical problem:
  • We want to compare Kaplan-Meier (KM) survival curves for observational studies.
  • There is almost certainly selection bias and/or confounding.
  • Using Cox proportional hazard (PH) model to adjust KM survival curves requires evaluation at specific covariate values.
• Statistical solution (Cole and Hernán 2004):
  • Estimate probability of receiving treatment conditional on baseline confounders.
  • Use these probability weights to estimate a weighted Cox PH model with a single treatment covariate.

• Consider a sample of \(N\) individuals and for the \(i^{th}\) person denote:
  • \(x_i\): a binary treatment option
  • \(\bzi\): vector of baseline features (possibly confounders)
  • \(w_i\): inverse of the probability of receiving person \(i\)'s treatment \(x_i\) conditional on the observed covariate vector \(\bzi\)
  • Specifically, \(w_i = [f_{X|Z}(x_i|\bzi)]^{-1}\)
• While \(w_i\) is unknown, it can be estimated parametrically: \(\hat{w}_i\)

• Fit a logistic regression model of \(x_i\) on \(\bzi\):

\[ \begin{align*} \log \Bigg( \frac{Pr(X_i=x_i|\bZi=\bzi)}{1-P(X_i=x_i|\bZi=\bzi)} \Bigg) &= \bmu^T\bzi \\ \end{align*} \]

• Use fitted coefficients to get estimate of the (inverse) of person \(i\) receiving their treatment:

\[ \begin{align*} \hat{w_i} &= [\hat{Pr}(X_i=x_i|\bZi=\bzi)]^{-1} \\ &= \begin{cases} 1+\exp\{-(\hat{\bmu}^T\bzi) \} & \text{ if } x_i=1 \\ 1+\exp\{\hat{\bmu}^T\bzi \} & \text{ if } x_i=0 \\ \end{cases} \end{align*} \]

• If a person has treatment \(x_i\) but was highly unlikely to have received it, then their observations receive a very large weight.
• This may lead to too much variance, so we use stabilized weights instead:

\[ \begin{align*} \hat{sw}_i &= \frac{\hat{Pr}(X_i=x_i)}{\hat{Pr}(X_i=x_i|\bZi=\bzi)} \\ \end{align*} \]

• A Cox PH model uses covariates to model the hazard rate:

\[h(t;\boldsymbol z) = h_0(t) \exp\{\boldsymbol\gamma^T \boldsymbol z \} \]

• Results can be combined with a non-parametric survival function: \(\hat{S}_j = [\hat{S}_0(t_j)]^{\exp\{\hat{\boldsymbol\gamma}^T \boldsymbol z\}}\)

• Disease-free survival for 76 Ewing's sarcoma patients, 47 of whom received a novel treatment, while 29 received a standard treatment, as well as information on whether a patient had abnormally high or normal serum lactic acid dehydrogenase (LDH) enzyme levels.

• Stratified approach: \(\hat{S}_{j,x} = \hat{S}_{0,x}(t_j)\), for \(x=\{0,1\}\).
• Marginal approach: \(\hat{S}_j = [\hat{S}_0(t_j)]^{\exp\{\hat{\beta} x \}}\), for \(x=\{0,1\}\).