Causal Inference - Model Assumptions

Hidden Assumption for Potential Outcome Framework

Hidden assumptions for potential outcome framework is Stable Unit Treatment Value Assumption (SUTVA).

SUTVA: no interference & consistency

Assumption (no interference): Unit i’s potential outcomes do not depend on other units’ treatments. This is sometimes called the no-interference assumption.

Assumption (consistency): There are no other versions of the treatment. Equivalently, we require that the treatment levels be well-defined, or have no ambiguity at least for the outcome of interest. This is sometimes called the consistency assumption.

Outcome Regression

Under the following assumptions (i.e., unconfoundedness and overlap), we have the following identification formuli for ATE:

$$\tau^{\text{ATE}} = E\left [\mu_1(x) - \mu_0(x)\right ],$$

$$\text{where, } \mu_{x} = E(Y(z)|X) = E(Y|Z = z, X) \text{ is the outcome model under treatment } Z.$$

Assumptions

• Rules out unobserved confounders, also known as the assumption of “no unmeasured confounders”.

• Untestable in most observational studies, but sometimes can be indirectly tested, and sensitivity can be checked.

$$0 < e(X) < 1$$

• This implies that, for all possible values of the covariates there are both treated and control units.

• Testable: Overlap can be directly checked from the data.

Propensity Score & IPW & AIPW

Besides outcome regression, another way to estimate ATE is to use Inverse Propensity Weighting (IPW). This approach also requires unconfoundedness and overlap assumptions. Under such assumptions, we have

$$\tau^{\text{ATE}} = E[Y(1)-Y(0)]=E\left[\frac{Z Y}{e(X)}-\frac{(1-Z) Y}{1-e(X)}\right]$$

Assumptions

• The definition of propensity score requires strong ignorability (see Definition 11.1 (propensity score) in Peng Ding’s textbook).

AIPW

The doubly robust estimator, augmented inverse propensity score weighting (AIPW) is a combination of outcome regression estimator and IPW estimator. Therefore, AIPW also requires Unconfoundedness and Overlap assumptions.

Estimation of heterogeneous treatment effects (HTE)

Assumptions

The identifying assumptions are the same:

Checking Unconfoundedness

Checking whether the Unconfoundedness assumption holds is a critical step in causal inference, yet it presents a significant challenge because unconfoundedness is a fundamentally untestable assumption .

Checking Unconfoundedness

1. Examine Study Design: In randomized controlled trials, unconfoundedness is typically assumed to hold by design, as the randomization process should, in theory, balance both observed and unobserved confounders across treatment groups. However, in observational studies, this assumption is more problematic as treatment assignment is not random.

2. Covariate Balance Checks: In observational studies, one approach to assess the plausibility of the unconfoundedness assumption is to check the balance of observed covariates between treatment groups. If the treatment and control groups are similar in terms of observed covariates, it increases confidence in the unconfoundedness assumption. However, this does not guarantee that unobserved confounders are also balanced.

3. Sensitivity Analysis: Conduct sensitivity analysis to determine how robust your findings are to potential unmeasured confounding. This involves varying the assumptions about the presence and impact of unmeasured confounders and observing how these changes affect the estimated causal effect.

4. Instrumental Variable Analysis: If unconfoundedness is suspected to be violated, an instrumental variable (IV) can be used, provided it meets the necessary criteria (relevance, independence, and exclusion restriction). IV methods can help to address unmeasured confounding.

Actions if Unconfoundedness Does Not Hold

1. Use of Instrumental Variables (IV): As mentioned, IV methods can be employed when the unconfoundedness assumption is violated. This requires finding a variable that influences treatment but is not related to the outcome except through the treatment.

2. Propensity Score Methods: These methods, including matching, weighting, and stratification based on propensity scores, can help balance observed covariates between treatment groups, although they do not address unmeasured confounding directly.

3. Regression Discontinuity Design: If applicable, this design can be employed where treatment assignment is determined by whether an observed covariate crosses a certain threshold. It can provide causal inference insights in the presence of unmeasured confounding.

4. Difference-in-Differences (DiD): If data is available over time, DiD methods can be used, assuming that the parallel trends assumption holds. This method can control for unmeasured confounders that are time-invariant.

5. Marginal Structural Models and Structural Nested Models: These advanced statistical models are used to address time-varying confounding and can be useful if unconfoundedness does not hold in longitudinal studies.

6. Sensitivity Analysis for Unmeasured Confounding: Conducting a sensitivity analysis allows you to explore how sensitive your results are to potential unmeasured confounders. This doesn’t solve the problem of unmeasured confounding but provides a way to assess its potential impact.