Lecture 15

Application: Profit vs. Outcomes in Heart Attack Patients

Summary

  • We’re going to get started looking at real question in health policy.
  • Vague Question. Do for-profit hospitals prioritize profit over patient outcomes?
  • We’ll read two papers that address this question in the context of treatment for heart attacks.
  • We’ll discuss them in class Friday.
  • Today, our goal is to get you prepared to read them.
    • We don’t need to cover any new tools or ideas. What you need, you have.
    • But not everyone talks about this stuff in the same language we use in class.
    • What we do need to focus on is translation.

The Heart-Attack Context

About the Real Data

Fake Data from Kim et al. (2023)

Binary.

The actual outcome we’re talking about is binary—it’s whether a patient gets the treatment of interest (PCI).

Continuous.

But it’s easier to plot binary outcomes, so we’ll—for a minute anyway—look at binary outcomes with the same within-group means \(\mu(w,x)\). These are what Kim et al. (2023) use.

Plots

\[ \begin{aligned} Y_i &= \mu(W_i,X_i) + \varepsilon_i \qfor \mu(w,x) = (x_1 + x_2 + x5)^2 \qand \varepsilon \sim N(0,1) \\ \pi(x) &= \Phi^{-1}\left(x_1^2 + 2x_2^2 - 2 x_3^2 - (x_4+1)^3 - .05*\log(x_5+10) + x_6 - 1.5 \right) \end{aligned} \]

  • This sample is ‘sampled with replacement’ from an infinite population.
    • Each observation \((W_i,X_i,Y_i)\) is an independent random variable.
    • The mean of \(Y_i\) among people in the population with the same \((W_i,X_i)\) is \(\mu(W_i,X_i)\).
    • The probability that a person receives treatment (i.e. that \(W_i=1\)) is \(\pi(X_i)\) where \(\Phi\) is the CDF of a standard normal random variable.
  • When you read, don’t worry about this code or the mathematical description above.

About Observational Causal Inference

The Machine Learning Perspective

How People Think

Workflow

Balancing Weights

How People Think

Workflow

The Linear Probability Model

Log Odds Ratios

Tables

These Things Are Everywhere. What are they communicating?

The Target Population

Covariate Shift

What to Read

Shah et al. (2007)

  • This is short, so you should be able to read it straight through.
  • They report a bunch of stuff.
  • Focus on what they say is the effect of (W: profit vs. non-profit) on whether PCI is given (Y).
  • Think about this stuff.
    • What do you think they want to know?
    • What do you think their conclusions are?
    • Do you believe them?

Kim et al. (2023)

  • A lot of it focuses on computing their (IPW-like) estimator. Ignore all that.
    • Don’t bother reading Sections 3 or 4 at all.
  • Focus on their methods and their conclusions.
  • I’m throwing you into the deep end with this one.
  • Don’t stress if you don’t follow everything. Try to get the general idea.
  • Here’s two things to look for.
    • Their estimator for the ATT (\(\Delta_1\)) compares the sample average outcome of the treated group to the weighted sample average outcome of the control group. That’s weighted least squares in the horizontal lines model.
    • Their weights (they write \(w_i\) where we write \(\gamma(W_i,X_i)\)) aren’t defined as a ratio of covariate distributions \(\textcolor[RGB]{248,118,109}{p_{x\mid 1}}/\textcolor[RGB]{0,191,196}{p_{x \mid 0}}\). But they’re chosen to do exactly what weighting using that ratio does. That is, to make the distribution of the covariate in the weighted sample match match the distribution we’re averaging over when we calculate \(\Delta_1\).