Subclassification and Matching in Data Analysis
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Questions and Answers

What is the Conditional Independence Assumption (CIA) in the context of causal inference?

Y0i, Y1i ⊥ D | Xi, meaning that potential outcomes are independent of treatment status D given the covariates X

What is the goal of pruning in the context of regression adjustment?

To remove control units without similar treatment units, ensuring common support

How is the average treatment effect estimated in regression adjustment?

As a weighted average of cell-specific causal effects

What is the difference between the Average Treatment Effect (ATE) and the Average Treatment Effect on the Treated (ATET)?

<p>ATE refers to E[Y1i - Y0i], while ATET refers to E[Y1i - Y0i | Di = 1]</p> Signup and view all the answers

What is the purpose of subclassification in the context of causal inference?

<p>To define groups of units with similar covariate profiles, allowing for the estimation of treatment effects within each group</p> Signup and view all the answers

How are the predictions obtained in the regression Y = α + τD + βX + U?

<p>From the coefficients of the regression using the pruned data</p> Signup and view all the answers

What is the role of covariates X in the regression adjustment approach?

<p>To control for the differences between treatment and control groups, ensuring that the treatment effect is estimated conditional on X</p> Signup and view all the answers

What is the intuition behind the running example with two covariates, X1 and X2?

<p>To illustrate the idea of estimating causal effects using regression adjustment, with a focus on the difference between ATE and ATET</p> Signup and view all the answers

What is the estimate of τj when j is equal to 0, given the provided estimates?

<p>τ0 = 1.1955447</p> Signup and view all the answers

How is the variable x3 defined in the running example?

<p>x3 = 0 if x1 = x2 = 0; = 1 if x1 = 1, x2 = 0; =2 if x1 = 0, x2 = 1; =3 if x1 = x2 = 1</p> Signup and view all the answers

What is the resulting subclassification estimate of ATT in the running example?

<p>4.077</p> Signup and view all the answers

What is the difference between τATT and τATE?

<p>τATT is the average effect for the treated, while τATE is the average treatment effect for the entire population</p> Signup and view all the answers

How are the weights for estimating ATE obtained in the running example?

<p>By tabulating x3</p> Signup and view all the answers

What is the resulting subclassification estimate of ATE in the running example?

<p>3.070</p> Signup and view all the answers

What is an alternative approach to estimating the treatment effect, aside from subclassification?

<p>Running a regression</p> Signup and view all the answers

What is the general form of the regression equation for estimating the treatment effect?

<p>Yi = α + τ Di + γ1 X1i + γ2 X2i + γ3 X1i X2i + εi</p> Signup and view all the answers

What assumption allows us to eliminate selection bias after conditioning on $X$, enabling the estimation of the treatment effect on the treated?

<p>The Conditional Independence Assumption (CIA)</p> Signup and view all the answers

What is the formula for the average treatment effect on the treated ($τAT T$) in the discrete-only covariate setting?

<p>$τAT T = ∑ τx P(Xi = x|Di = 1)$</p> Signup and view all the answers

What is the name of the estimator used to calculate $τAT T$ in the discrete-only covariate setting?

<p>The subclassification estimator (or exact matching estimator)</p> Signup and view all the answers

What is the purpose of iterating expectations over $X$ in the estimation of $τAT T$?

<p>To eliminate selection bias and obtain an unbiased estimate of the treatment effect on the treated</p> Signup and view all the answers

What is the role of $E[Y0i |Xi , Di = 1]$ in the estimation of $τAT T$?

<p>It is a counterfactual term that represents the potential outcome for the treated unit if they had not received the treatment</p> Signup and view all the answers

How is the weighted average of $X$-specific differences in $Y$ calculated in the subclassification estimator?

<p>Using the empirical distribution of $X$ among the treated ($P(Xi = x|Di = 1)$)</p> Signup and view all the answers

What is the main advantage of using regression adjustment in estimating the treatment effect?

<p>It allows for the control of confounding variables ($X$) and the estimation of the treatment effect while accounting for the influence of $X$</p> Signup and view all the answers

What is the purpose of running separate regressions for each possible value of $X$ in the subclassification estimator?

<p>To estimate the $X$-specific treatment effects ($τx$) and eventually obtain the overall treatment effect on the treated ($τAT T$)</p> Signup and view all the answers

What is the purpose of weights w(X) in regression adjustment, and what property do they have?

<p>The purpose of weights w(X) is to minimize the sum of squared residuals, and they have the property w(X) &gt; 0.</p> Signup and view all the answers

What is the formula to compute AT E using regression adjustment?

<p>AT E = (1/n) * (Y1i - Y0i)</p> Signup and view all the answers

What is the difference between AT E and AT T in regression adjustment?

<p>AT E is the average of (Y1i - Y0i) in the full sample, while AT T is the average in the treated subsample.</p> Signup and view all the answers

How can you identify both AT E and AT T using regression adjustment?

<p>Regress Y on X separately in treatment and control groups, predict Y1 and Y0, and then compute AT E and AT T.</p> Signup and view all the answers

What is the purpose of regression adjustment in causal inference?

<p>To identify the average treatment effect (AT E) and the average treatment effect on the treated (AT T).</p> Signup and view all the answers

What is the relationship between the least squares regression coefficients and the average treatment effect (AT E)?

<p>The least squares regression coefficients do not identify AT E, unless the effect is constant.</p> Signup and view all the answers

What is the role of subclassification in regression adjustment?

<p>Subclassification is used to divide the data into subgroups based on the propensity score, which is then used in regression adjustment.</p> Signup and view all the answers

How does the teffects command in Stata implement regression adjustment?

<p>The teffects command in Stata implements regression adjustment by regressing Y on X separately in treatment and control groups, and then computing AT E and AT T.</p> Signup and view all the answers

What is the primary goal of subclassification and matching strategies in causal analysis?

<p>To control for selection bias</p> Signup and view all the answers

What is the conditional independence assumption (CIA) in the context of causal analysis?

<p>Selection on observables, but not on unobservables</p> Signup and view all the answers

What is the purpose of matching in causal analysis, besides generating data for regression analysis?

<p>To generate data that can be analyzed as if it is the result of a randomized experiment</p> Signup and view all the answers

Why is the average treatment effect on the treated (AT T) not equal to the average treatment effect (AT E) in a matching analysis?

<p>Because the covariates are not balanced between the treatment and control groups in the original non-experimental sample</p> Signup and view all the answers

What is the common support problem in matching, as illustrated in the scatterplot example?

<p>There are many control units for which there are no treatment units</p> Signup and view all the answers

What is the purpose of regression adjustment in causal analysis?

<p>To control for remaining differences between the treatment and control groups after matching or subclassification</p> Signup and view all the answers

What is the role of the propensity score in causal analysis?

<p>To estimate the probability of treatment assignment given the observed covariates</p> Signup and view all the answers

What is the advantage of using subclassification and matching strategies together?

<p>To combine the benefits of both approaches, controlling for selection bias and finding matches for each treatment unit</p> Signup and view all the answers

Study Notes

Subclassification and Matching

  • Subclassification and matching are strategies to control for selection bias, motivated by the conditional independence assumption (CIA).
  • CIA states that, within each cell defined by the values of X, treatment is as good as randomly assigned, i.e., no selection effect.

Introduction to Matching

  • Matching is a data cleaning process before regression analysis, also known as pruning or pre-processing.
  • Matching can be used to generate data that can be analyzed as if it is the result of a randomized experiment, without the need for regression.
  • However, in the original non-experimental sample, the covariates are not balanced, and treatment and control groups differ in their characteristics.

Purposes of Matching

  • One purpose is to prune the data, removing control units for which there are no similar treatment units.
  • Another purpose is to use matching to generate data that can be analyzed as if it is the result of a randomized experiment.

Subclassification and Regression

  • Subclassification is a method to do causal analysis in subgroups and aggregate.
  • The subclassification estimator (or exact matching estimator) is a weighted average of X-specific differences in Y using the empirical distribution of X among the treated.

Identification under CIA

  • The selection bias disappears after conditioning on X, so the treatment effect on the treated can be obtained by iterating expectations over X.
  • The treatment effect on the treated (AT T) can be written as the weighted average of X-specific differences in Y.

Average Treatment Effect on the Treated (AT T)

  • In the discrete-only covariate setting, AT T can be written as a weighted average of X-specific differences in Y using the empirical distribution of X among the treated.
  • The subclassification estimator is a weighted average of X-specific differences in Y using the empirical distribution of X among the treated.

Unconditional Average Treatment Effect (AT E)

  • AT E is the average treatment effect for the entire population, whereas AT T is the average effect for the treated.
  • The unconditional average treatment effect (AT E) can be written as the expectation of the X-specific differences in Y using the marginal distribution of X.

Running a Regression

  • Running a regression of Y on D, X, and their interactions does not identify AT E or AT T unless the effect is constant.
  • The least squares regression coefficients do not identify AT E or AT T, and the weights chosen by OLS have no meaningful intuition.

Identifying AT E and AT T with Regression

  • One way to identify both AT E and AT T with regression is to:
    1. Regress Y on X separately in the treatment and control groups.
    2. Predict Ŷ1 and Ŷ0 for each observation.
    3. Compute AT E and AT T using these predictions.

Regression with teffects

  • The teffects command in Stata can be used to implement regression adjustment to identify AT E and AT T.

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Causal Analysis Subclassification and Matching PDF

Description

Learn about subclasses and matching strategies to control for selection bias in data analysis, including the conditional independence assumption (CIA).

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