Causal Inference

Causal inference is the study of methods for determining how an intervention changes an outcome. It combines statistics, study design, and assumptions about causation to distinguish causal effects from observed associations. Its central questions concern what would happen if a treatment, policy, or other exposure were changed, rather than simply whether variables occur together. Applications include economics, epidemiology, and the social sciences. Causal conclusions depend on both evidence and assumptions that ordinary statistical analysis cannot supply by itself. (web.cs.ucla.edu)

Association and intervention

An association describes how outcomes differ across observed groups. A causal effect compares outcomes under alternative interventions. Correlation alone cannot establish this comparison: an association may reflect a causal relationship, reverse causation, or confounding by common causes. For example, an association between participation in a training program and subsequent earnings might partly reflect differences in participants’ prior qualifications. (web.cs.ucla.edu)

Causal analysis separates three tasks. First, the research question defines the target effect, or estimand. Second, identification determines whether that effect can be expressed using the observed data distribution under specified assumptions. Third, estimation uses a finite sample to calculate its value. A precise estimate of an association need not identify a causal effect. (web.cs.ucla.edu)

Potential outcomes

The potential-outcomes framework represents each unit’s outcome under alternative treatment conditions. For a binary treatment, let $Y_i(1)$ denote unit $i$’s outcome under treatment and $Y_i(0)$ its outcome without treatment. The individual causal effect is

Only the outcome corresponding to the unit’s actual treatment is observed; the other is a counterfactual. This missing comparison is often called the fundamental problem of causal inference. Consequently, studies usually estimate population or subgroup effects rather than directly observing individual effects. (hsph.harvard.edu)

The average treatment effect is

where $E$ denotes expected value. Other targets include the average effect among treated units and effects conditional on baseline characteristics. An average effect can conceal substantial variation between units, and effects defined for different populations need not coincide. (hsph.harvard.edu)

Causal graphs and structural models

A structural causal model describes how variables are generated by other variables and background factors. Its graphical representation often uses a directed acyclic graph, with arrows encoding hypothesized direct causal relationships. Such graphs make assumptions explicit and help distinguish common causes from intermediate variables and selection mechanisms. (web.cs.ucla.edu)

An intervention is commonly written $do(A=a)$. Unlike conditioning on the observation $A=a$, it replaces the mechanism assigning $A$ with a fixed value. Thus $P(Y\mid do(A=a))$ generally differs from the conditional probability $P(Y\mid A=a)$. Graphical criteria can identify adjustment sets that block noncausal paths. Adjusting indiscriminately for every available variable can instead introduce bias, notably when conditioning on a common effect of two variables. (web.cs.ucla.edu)

Identification assumptions

For observational comparisons of binary treatments, three standard assumptions support identification:

• Consistency: a unit’s observed outcome equals its potential outcome under the treatment actually received, with treatment versions sufficiently specified.
• Conditional exchangeability: given appropriate pretreatment covariates $X$, treatment assignment has conditional independence from the potential outcomes. This is commonly interpreted as no unmeasured confounding.
• Positivity: each treatment has nonzero probability at every covariate configuration relevant to the target population.

Under these assumptions, standardization identifies each potential-outcome mean:

Simple formulations also assume that one unit’s treatment does not affect another unit’s outcome. Spillovers require an expanded treatment definition or a model explicitly accommodating interference. (hsph.harvard.edu)

Study designs and estimation

A randomized controlled trial uses random assignment to make treatment groups comparable in their potential outcomes, in expectation. It does not guarantee exact balance in a realized sample. Nonadherence, missing outcomes, and selective enrollment can complicate interpretation; effects of assignment and effects of treatment received are distinct targets. The credibility of an analysis therefore depends on experimental design as well as its statistical model. (arxiv.org)

Observational adjustment methods include matching, stratification, outcome regression, and weighting. The propensity score, $P(A=1\mid X)$, summarizes treatment-assignment probabilities given measured covariates. Under the relevant assumptions, it supports comparisons balanced on those covariates. Neither propensity-score adjustment nor linear regression automatically removes unmeasured confounding. (stat.cmu.edu)

Other designs exploit particular assignment mechanisms:

• Instrumental variables use a variable that shifts treatment while satisfying independence and exclusion restrictions; additional assumptions determine the effect identified.
• Regression discontinuity exploits treatment changes at a threshold, typically identifying a local effect under continuity and assignment assumptions.
• Difference-in-differences compares outcome changes between groups, relying on assumptions about their untreated trends rather than equality of their initial outcome levels. (nber.org)

Machine learning and uncertainty

Machine learning can estimate flexible outcome models or treatment probabilities within a causal analysis. Double machine learning combines orthogonal estimating equations with sample splitting or cross-fitting to reduce sensitivity to nuisance-model estimation errors. These techniques improve estimation under specified conditions; they do not replace causal identification assumptions. (nber.org)

Uncertainty also concerns research design, not only sampling variation. Sensitivity analyses examine how conclusions change under departures from assumptions. Lack of covariate overlap can make a population effect difficult to estimate without extrapolation, while inadequate measurement of important covariates can undermine comparability even in very large datasets. (arxiv.org)