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Causal Inference Course

10 Data-driven methods

10.1 Introduction

Nov 14 Slides.

For any given intervention, some subgroups of people will respond more than others. Ideas from machine learning can help us target human attention toward these subgroups.

Concrete example. Who responds most to a nudget to go for a walk? Imagine your first conduct a survey that asks people how much they love the fall, from (\(X = 1\) for least) to (\(X = 10\) for most). You then randomize them to a control condition (\(A = \texttt{untreated}\)) or a treatment condition (\(A = \texttt{treated}\)) that encourages them to go for a walk outside. The outcome \(Y\) is active minutes in the day, as recorded on an activity tracker.

Simulated data. In real data, it can be difficult to evaluate causal estimators because the truth is unknown. Today we will use data simulated from a known process in order to study the properties of estimators. The code below will prepare your R environment with a function simulate_sample() that will generate data with 50 observations.

library(tidyverse)
source("https://raw.githubusercontent.com/causal3900/causal3900.github.io/main/assets/data/simulate_sample.R")

Here is an example of the code to simulate data:

simulated <- simulate_sample()
##   X         A         Y
## 1 1 untreated  8.115603
## 2 1   treated 39.926003
## 3 1 untreated 49.945026
## 4 1   treated  7.510654
## 5 1 untreated 43.629752
## 6 2   treated 16.847279

Causal estimands. In this example, we would like to estimate \[\tau_x = E(\underbrace{Y^1 - Y^0}_{\substack{\text{effect of}\\\text{nudge to walk}\\\text{on active}\\\text{minutes}}}\mid \underbrace{X = x}_{\substack{\text{among those}\\\text{with love of}\\\text{fall = }x}})\] for each value \(x = 1,\dots,10\). These estimands are the average causal effect of a nudge to walk (\(A\)) on active minutes (\(Y\)) within subgroups defined by each value of the scale for love of fall (\(X\)).

Identification. In our simulate data, \(A\) is assigned at random. There are no backdoor paths between \(A\) and \(Y\).

Estimator. An estimator is a function that takes a dataset and returns estimates. Below is a nonparametric estimator for our setting.

estimator <- function(data) {
  data %>%
    # Group by treatment A and confounder X
    group_by(A, X) %>%
    # Summarize by the average outcome within groups
    summarize(Y = mean(Y),
              .groups = "drop") %>%
    # Reshape the data
    pivot_wider(names_from = "A",
                values_from = "Y",
                names_prefix = "y_") %>%
    # Estimate the effect within groups
    mutate(effect = y_treated - y_untreated)
}

You can apply this estimator as follows.

estimate <- estimator(simulated)

Task. Using a sample simulated on your computer, estimate the average causal effect of \(A\) on \(Y\) within subgroups defined by \(X\). Report two numbers to us.

  • for which value of \(X\) is the estimated effect of \(A\) most positive?
  • what is that effect estimate?

We will discuss the distribution of estimates that we get as a class.

If you are ready early, you could think about how you might evaluate performance of this approach over many repeated simulations.

10.2 Machine learning approaches

Nov 16 Slides.

Today we generalize the ideas from Tuesday. We will discuss how sample splitting makes it easier to

  • choose among many estimands
  • choose among many estimators
  • develop new data science approaches