class: center, middle, inverse, title-slide # IDS 702: Module 6.3 ## Unconfoundedness and overlap ### Dr. Olanrewaju Michael Akande --- ## Observational studies - We will not focus on randomized experiments since most of the data you will have to analyze in practice are actually based on observational studies. -- - In observational studies, we do not control or know the assignment mechanism. -- - In addition, the presence of measured and unmeasured confounders can create unbalance between the groups. -- - Again, to do causal inference, we have to make some structural (often untestable) assumptions, e.g. on the treatment assignment, for identifying causal effects. -- - Once we have those general assumptions, we also usually have to make model assumptions to do the actual estimation. --- ## Estimands Once again, we will focus on the following estimands: - The .hlight[average treatment effect (ATE)]: .block[ .small[ $$ \tau = \mathbb{E}[Y_i(1) - Y_i(0)]. $$ ] ] -- - The .hlight[average treatment effect for the treated (ATT)]: .block[ .small[ $$ \tau = \mathbb{E}[Y_i(1) - Y_i(0) | W_i = 1]. $$ ] ] -- - The .hlight[average treatment effect for the control (ATC)]: .block[ .small[ $$ \tau = \mathbb{E}[Y_i(1) - Y_i(0) | W_i = 0]. $$ ] ] -- - For binary outcomes, .hlight[causal odds ratio (OR) or risk ratio (RR):]: .block[ .small[ $$ \tau = \dfrac{\mathbb{Pr}[Y_i(1) = 1]/\mathbb{Pr}[Y_i(1) = 0]}{\mathbb{Pr}[Y_i(0) = 1]/\mathbb{Pr}[Y_i(0) = 0]}. $$ ] ] --- ## Estimands - The relationship between ATE, ATT and ATC is given by .block[ .small[ $$ \textrm{ATE} = \mathbb{Pr}[W_i = 1] \cdot \textrm{ATT} + \mathbb{Pr}[W_i = 0] \cdot \textrm{ATC} $$ ] ] -- - In randomized experiments, ATT is equivalent to ATC because treatment and control groups are similar/comparable. -- - ATE is then also equivalent to ATT (and ATC). -- - In observational studies, ATE is usually different from ATT and ATC. -- - The above relation does not hold for ratio estimands. --- ## Assumptions: unconfoundedness We will need two major assumptions (in addition to SUTVA). The first, we already talked about, that is, -- Assumption 1: .hlight[Unconfoundedness] .block[ .small[ $$ \{Y_i(0), Y_i(1)\} \perp W_i | X_i, $$ ] ] or using the equivalent form from last class, .block[ .small[ $$ \mathbb{Pr}[W_i = 1 | X_i, Y_i(0), Y_i(1)] = \mathbb{Pr}[W_i = 1 | X_i] $$ ] ] -- - Assumes that within subgroups defined by values of observed covariates, the treatment assignment is random. -- - Rules out unobserved confounders. -- - Randomized experiments satisfy unconfoundedness. -- - Untestable in most observational studies, but sensitivity can be checked. --- ## Implications of unconfoundedness - Under unconfoundedness, it turns out that .block[ .small[ $$ \mathbb{Pr}[Y(w) | X] = \mathbb{Pr}[Y^{\text{obs}} | X, W =w] \ \ \ \ w = 0,1. $$ ] ] -- - That is, the observed distribution of `\(Y\)` in treatment arm `\(W = w\)` equals the distribution of the potential outcomes `\(Y(w)\)`. -- <div class="question"> Why does this matter or how does this help us? </div> -- - Well, the causal estimands are essentially expectations and probabilities. -- - Recall again that ATE is .block[ .small[ $$ \textrm{ATE} = \mathbb{E}[Y_i(1) - Y_i(0)]. $$ ] ] -- - ATE can then be estimated from the observed data using .block[ .small[ $$ \textrm{ATE} = \mathbb{E}_X\left(\mathbb{E}[Y^{\text{obs}} | X, W =1] - \mathbb{E}[Y^{\text{obs}} | X, W =0]\right). $$ ] ] -- - Note that we need to average out over the distribution of `\(X\)` since the original formula for ATE does not depend on any `\(X\)`. --- ## Assumptions: overlap Assumption 2: .hlight[Overlap (or positivity)] .block[ .small[ $$ 0 < \mathbb{Pr}[W_i = 1 | X_i] < 1, \ \ \ \textrm{for all}\ \ \ i. $$ ] ] -- - Notice that this is the probabilistic assignment from last class, that is, .block[ .small[ $$ 0 < \mathbb{Pr}[W_i = 1 | X_i, Y_i(0), Y_i(1)] < 1. $$ ] ] -- - However, we can exclude `\(\{Y_i(0), Y_i(1)\}\)` now because of the unconfoundedness assumption. -- - .block[ .small[ $$ e(x) = \mathbb{Pr}[W_i = 1 | X_i = x] $$ ] ] is usually called the .hlight[propensity score]. --- ## Implications of overlap - Overlap implies that, in large samples, for all possible values of the covariates, there are both treated and control units. -- - This is important within the potential outcomes (or counterfactual) framework both conceptually and operationally (variance inflation). -- - Unlike unconfoundedness, overlap can be directly checked from the data often using the estimated propensity scores. -- - Unconfoundedness and positivity jointly define the .hlight[strong ignorability] assumption. --- ## Acknowledgements These slides contain materials adapted from courses taught by Dr. Fan Li. --- class: center, middle # What's next? ### Move on to the readings for the next module!