library(readr); library(dplyr); library(knitr)
smoking = read_csv("smokingdata.csv") %>% mutate(smoke_bin = ifelse(smoke == "Never",0,1))
knitr::kable(smoking[1:4,c("ratio","smoke_bin")],format="pipe",digits=2,align="l",
caption = "A Subset of the Observed Data",
col.names=c("Lung function (Y)","Smoking status (A)"))| Lung function (Y) | Smoking status (A) |
|---|---|
| 0.94 | 0 |
| 0.92 | 0 |
| 0.81 | 1 |
| 0.84 | 0 |
Data: 2009-2010 National Health and Nutrition Examination Survey (NHANES).
library(ggplot2)
ggplot(smoking,aes(x=smoke,y=ratio)) + geom_boxplot() +
labs(x="Smoking Status (A)",
y="Lung Function (Y)",
title="") +
scale_x_discrete(labels=c("Daily (A=1)","Never (A=0)")) +
geom_hline(yintercept=0.8,linetype="dashed")
Daily smoking is strongly associated with reduction in lung function.
But, is the strong association evidence for causality? After all, association does not imply causation…
Conceptually, we say \(A\) is associated with \(Y\) if \(A\) is informative about \(Y\)
Formally, let \(f(\cdot)\) denote the density (or mass function) of a random variable.
\(A\) is associated with \(Y\) if and only if \(f_{Y \mid A}(y | a) \neq f_{Y}(y)\) for some \(y,a\) with \(f(a) > 0\).
We’ll use the notation \(A \not\perp Y\) to denote association between \(Y\) and \(A\). Similarly, we’ll use \(A \perp Y\) to denote no association between \(A\) and \(Y\).
There are other measures of association beyond the difference in means between daily smokers and never smokers, i.e., \(\mathbb{E}[Y | A=1] - \mathbb{E}[Y | A=0]\). Some measures of association for scalar \(A\) and scalar \(Y\).
Note that the population regression parameter does not assume that \(Y\) is a linear function of \(A\). Instead, \(\beta_0^*, \beta_1^*\) can be thought of as the intercept and slope of the ``best linear approximation’’ regression line between \(Y\) and \(A\). For more discussion related to this, see Section 3.1 of Buja et al. (2019) and Figure 1 of Suk and Kang (2022) for a visual illustration.
The derivation of \(\beta_1^*\) follows either from the theory of linear regression or multivariable calculus. For example, the solution to the minimize must satisfy the equation \(0 = \nabla_{\beta_0, \beta_1} \mathbb{E}[ (Y - \beta_0 - A\beta_1)^2] \bigg|_{\beta_0 = \beta_0^*, \beta_1 = \beta_1^*}\), which simplify to \[\begin{align*} 0 = \nabla_{\beta_0, \beta_1} \mathbb{E}[(Y-\beta_0)^2] + \beta_1^2 \mathbb{E}[A^2] - 2\beta_1 \mathbb{E}[(Y-\beta_0)A] \bigg|_{\beta_0 = \beta_0^*, \beta_1 = \beta_1^*} \end{align*}\] Solving for this equation leads to the familiar equation for the regression parameter: \[\begin{align*} \begin{pmatrix} \beta_0^* \\ \beta_1^* \end{pmatrix} &= \begin{pmatrix} 1 & \mathbb{E}[A] \\ \mathbb{E}[A] & \mathbb{E}[A^2] \end{pmatrix}^{-1} \begin{pmatrix} \mathbb{E}[Y] \\ \mathbb{E}[AY] \end{pmatrix} \end{align*}\]
We can estimate the population quantities by their sample counterparts (e.g., sample means, sample covariance, OLS).
Inspired by recent Marvel movies, I find the parallel universe analogy helpful to conceptualize causal effects.
Consider a particular snapshot in time (e.g., June 1, 2024, John’s 25th birthday) in two parallel universes.
Now suppose John’s lung functions are different between the two universes.
A helpful Youtube clip from movie Sliding Doors.
Let’s define the outcomes from the two parallel universes, often referred to as counterfactual or potential outcomes1.
Similar to the observed data table, we can create counterfactual/potential outcomes data table.
| \(Y(1)\) | \(Y(0)\) | |
|---|---|---|
| John | 0.54 | 0.94 |
| Sally | 0.91 | 0.91 |
| Kate | 0.81 | 0.60 |
| Jason | 0.60 | 0.84 |
Let’s take a look at \(Y_{\rm John}(1) - Y_{\rm John}(0) = -0.4\) and \(Y_{\rm Sally}(1) - Y_{\rm Sally}(0) = 0\).
Both numbers \(-0.4\) and \(0\) are individual causal effects as they reflect each person’s change in the outcome when their smoking status changes.
When causal effects differ from individual to individual, the causal effect is generally said to be heterogeneous. If the effects are the same for every individual, the causal effect is generally said to be homogeneous or constant.
Suppose we add additional information about the individuals
| \(Y(1)\) | \(Y(0)\) | Age \((X_1)\) | Graduated HS? \((X_2)\) | |
|---|---|---|---|---|
| John | 0.54 | 0.94 | 23 | Yes |
| Sally | 0.91 | 0.91 | 27 | No |
| Kate | 0.81 | 0.60 | 32 | No |
| Jason | 0.60 | 0.84 | 30 | Yes |
These are examples of causal estimands/parameters because they are functions of the counterfactual outcomes.
Similar to the observed data \((Y,A)\), you can think of the counterfactual data table as an i.i.d. from some population distribution of \(Y(1),Y(0)\).
Formally, let \(F(\cdot)\) denote the cumulative distribution function of a random variable. Then, \(Y_i(1), Y_i(0) \overset{\text{i.i.d.}}{\sim} F_{Y(1),Y(0)}\))
Or, you can think of \(n=4\) as the entire population.
The latter framework is uncommon in typical statistics courses. However, it’s very popular among some circles of causal inference researchers (e.g. Rubin, Rosenbaum and their students). The appendix of Erich Leo Lehmann (2006), P. Rosenbaum (2002b), and Li and Ding (2017) provide a list of technical tools to conduct this type of inference.
There has been a long debate about which the “right” framework for inference. My understanding is that it’s now (i.e. Jan. 2025) a matter of personal taste. Also, as Paul Rosenbaum puts it:
In most cases, their disagreement is entirely without technical consequence: the same procedures are used, and the same conclusions are reached…Whatever Fisher and Neyman may have thought, in Lehmann’s text they work together. (Page 40, P. Rosenbaum (2002b)
The textbook that Paul is referring to is (now) Erich L. Lehmann and Romano (2006). Note that this quote touches on another debate in the literature in finite-sample inference, which is what is the correct null hypothesis to test. In general, it’s good to be aware of the differences between the two frameworks and as Lehmann did (see full quote), use the strengths of each different frameworks. For some interesting discussions on this topic as it relates to causal inference, see Robins (2002), P. Rosenbaum (2002a), Chapter 2.4.5 of P. Rosenbaum (2002b), and Abadie et al. (2020). For more well-known works in the area of finite population inference, see Neyman (1923), Freedman and Lane (1983), Freedman (2008), and Lin (2013).
Let’s consider the ATE \(\mathbb{E}[Y(1) - Y(0)]\), by far the most popular causal estimand/measure of a causal effect.
If this average is zero, then on average, the causal effect of smoking on lung function is zero.
If this average is negative, then being a daily smoker is, on average, cause decrease in lung function compared to being a never-smoker.
By linearity of expectations, \(\mathbb{E}[Y(1) - Y(0)] = \mathbb{E}[Y(1)] - \mathbb{E}[Y(0)]\).
What’s the difference between \(\mathbb{E}[Y | A=1]\) versus \(\mathbb{E}[Y(1)]\)?
Causal effects of a treatment is often defined as a comparison to another (possibly inactive) treatment.
We focus on the effects of causes (e.g., effect of daily smoking on lung function) rather than the causes of effects (e.g., why does John have poor lung function?). The causes of effects are hard to define because of the problem of infinite regress.
Example (from Don Rubin): He got lung cancer because he smoked cigarettes. The real reason he smoked is because his parents smoked, and they smoked because they hated each other and they hated each other because…
Cause-effect relationships have a natural temporal ordering where the treatment variable (i.e., smoking status) always precedes the outcome variable (i.e., lung function)
(Discussed more later) The notation currently does not make a distinction between different kinds of daily smoking on lung function (e.g., John smokes 10 packs of cigars per day versus 1 cigar per day). The notation assumes no multiple versions of treatment.
To be more precise in the smoking example, we need to further define what it means to be a daily smoker and a never-smoker. For example,
Each variation in the definition of daily smoking can change the parallel universe 1 and subsequently, the difference between John’s lung function between the two parallel universes. For example, under definition (a), the difference between John’s lung function may be \(-0.4\). But under definition (b), the difference may be \(0\), i.e., smoking status does not change John’s lung function. The two definitions yield scientifically different causal conclusions about John’s smoking status on his lung function.
Also, it’s equally important (in my personal opinion, far more important) to precisely define the never smokers, or more generally, the control group. As mentioned in the subtle points, causal effects are usually contrasts of outcomes and in the smoking example, the control group is a never smoker. If we choose an “extreme” control group, say our control group are those who not only never smoked, but never been near someone who smoked cigarettes, the difference between the outcomes among treated and control groups may be too dramatic than in most realistic settings. In fact, as we’ll discuss later, you can study biases from unmeasured confounders/selection on unobservables/etc. by selecting a good control group (e.g., P. R. Rosenbaum (1987)).
Between the control and the treatment, the treatment is usually well-defined in most policy evaluations (e.g., implementation/enactment of policy) and randomized experiments (e.g., giving a particular drug).
Finally, while not apparent right now, the definition of treatment/intervention/exposure must also be consistent across all study units; see later lecture notes. In general, causal inference requires, to some extent, a well-defined intervention/treatment/exposure to yield useful, practically meaningful causal conclusions; see fine point 1.2 in Hernán and Robins (2020). Otherwise, if there are variations in the definition of the treatment across individuals, say John’s definition of daily smoking differs from Sally’s definition of daily smoking, and we conduct a causal analysis that includes John and Sally, it would be difficult to make useful, practically meaningful causal claims about the causal effect of daily smoking.
In short, when interpreting causal effects, it is important to be cognizant of how the treatment/intervention is defined to yield unambiguous conclusions about causality.
A healthy majority of people in causal inference argue that the counterfactual outcome of race and gender are ill-defined. For example, suppose we are interested in whether being a female causes lower income. We could define the counterfactual outcomes as
Similarly, we are interested in whether being a black person causes lower income, we could define the counterfactual outcomes as
But, if Jamie is a female, can there be a parallel universe where Jamie is a male? That is, is there a universe where everything else is the same (i.e. Jamie’s whole life experience up to 2025, education, environment, maybe Jamie gave birth to kids), but Jamie is now a male instead of a female? This question is related to the question of manipulability of treatment.
In the intuitive understanding of experimentation that most people have, it makes sense to say, “Let’s see what happens if we require welfare recipients to work”; but it makes no sense to say “Let’s see what happens if I change this adult male into a three-year-old girl.” And so it is also in scientific experiments. Experiments explore the effects of things that can be manipulated, such as the dose of a medicine, the amount of a welfare check, the kind or amount of psychotherapy or the number of children in a classroom. Nonmanipulable events (e.g., the explosion of a supernova) or attributes (e.g., people’s ages, their raw genetic material, or their biological sex) cannot be causes in experiments because we cannot deliberately vary them to see what then happens. Consequently, most scientists and philosophers agree that it is much harder to discover the effects of nonmanipulable cause. (Page 8 of Shadish, Cook, and Campbell (2002)).
Note that we can still measure the association of gender on income, for instance with a linear regression of income (i.e \(Y\)) on gender (i.e. \(A\)). This is a well-defined quantity.
There is an interesting set of papers on this topic: VanderWeele and Robinson (2014), Vandenbroucke, Broadbent, and Pearce (2016), Krieger and Davey Smith (2016), VanderWeele (2016). See Volume 45, Issue 6, 2016 issue of the International Journal of Epidemiology. Some even take this example further and argue whether counterfactual outcomes are well-defined in the first place; see Dawid (2000) and a counterpoint in Sections 1.1, 2 and 3 of Robins and Greenland (2000).
| \(Y(1)\) | \(Y(0)\) | |
|---|---|---|
| John | 0.54 | 0.94 |
| Sally | 0.91 | 0.91 |
| Kate | 0.81 | 0.60 |
| Jason | 0.60 | 0.84 |
| \(Y\) | \(A\) | |
|---|---|---|
| John | 0.94 | 0 |
| Sally | 0.91 | 0 |
| Kate | 0.81 | 1 |
| Jason | 0.84 | 0 |
In the counterfactual table, we see what everyone’s lung function would be if they are never-smokers and daily smokers.
In the observed table, we only see everyone’s lung function under one particular status of smoking status.
Without additional information, it’s impossible to study causal effects from the observed data table because we don’t get to observe all counterfactual outcomes. This is the fundamental problem of casual inference (Holland 1986).
A key goal in causal inference is to learn about both counterfactual outcomes \(Y(1), Y(0)\) when you only observe one of them.
The fundamental problem is closely related to a missing data problem; we’ll explore this later.
There are situations in the real world where you can observe all counterfactual outcomes. Most of them take place in lab experiments or in manufacturing and all of them fundamentally rely on some domain knowledge to claim that all counterfactual outcomes are observable.
Suppose we want to determine the causal effect of putting a chocolate bar over a candle.
We put one chocolate bar over a candle and another bar away from the candle, resulting in the following table.
| \(Y(1)\) | \(Y(0)\) | |
|---|---|---|
| 1st chocolate bar | 1 | NA |
| 2nd chocolate bar | NA | 0 |
Despite the missing values in the potential outcomes, we can impute them from our daily experiences.
This phenomena is known as the unit homogeneity assumption and is formalized as follows
\[Y_{i}(1) = Y_j(1) \text{ and } Y_i(0) = Y_j(0) \quad{} \forall i\neq j \] Note that we don’t even have to randomize which chocolate bar is exposed to heat or not to identify the causal effect of heat on the chocolate bar. We also don’t have to sample \(10\) or \(100\) chocolate bars to understand the causal effect of exposing chocolate to heat on melting as all chocolate bar.