Add asymptotic confidence intervals for average treatment effect for linear regression with effect modifiers

[amit-sharma]

Refer to #335. DoWhy defaults to bootstrap confidence interval. It will be good to implement a computationally efficient confidence interval method.

With effect modifiers, the average treatment effect is a linear combination of parameters. For example, for y=a + b1t + b2t.x1 where x1 is the effect modifier, the ATE of t on y is b1+b2.mean(x1). Both b1 and b2 coefficients are random variables and so is mean(x1), so need to compute the standard error of the combined quantity.

This is a good start, for anyone who would like to contribute: https://scholar.princeton.edu/sites/default/files/jmummolo/files/interaction_models_jm.pdf

[kf-rahman]

Hey @amit-sharma , would love to take this on as my first contribution to this repo!

[github-actions]

🤖 This is an automated response from Repo Assist.

Welcome to the project, @kf-rahman — great to see your interest in this issue! 🎉

I've just opened a PR ([#aw_pr336fix]) that implements the asymptotic confidence intervals and standard errors for the effect-modifier case using the Delta method. Here's a brief summary of the approach in case it's useful to you:

The math: When effect modifiers are present, the ATE is a linear combination of OLS coefficients:

ATE = b_T + b_{T·X₁}·mean(X₁) + b_{T·X₂}·mean(X₂) + …

The Delta method gives us the variance of this linear combination directly from the OLS covariance matrix:

Var(ATE) = cᵀ · Σ · c

where c is the contrast vector that picks out the treatment and interaction coefficients weighted by the observed mean of each effect modifier, and Σ = model.cov_params() from statsmodels.

The PR also covers multiple treatments and multiple effect modifiers, and uses the t-distribution (with model.df_resid df) for exact finite-sample CIs rather than a normal approximation.

If you'd like to extend this further (e.g. for non-linear models, or a formula-based API), feel free to build on it!

Note

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[emrekiciman]

Hi @kf-rahman , the repo-assist AI suggested a PR for this issue: #1432

@amit-sharma and I are wondering if you would be willing to review the code and see if it is a good and correct solution for this issue?

[kf-rahman]

Hey @amit-sharma and @emrekiciman , yeah let me take a crack at it.

[kf-rahman]

Hey @amit-sharma and @emrekiciman, left a detailed review on #1432. The Delta method approach is correct but there's a bug with categorical variable encoding that needs fixing before merge. See my review comment for the fix and a branch with working tests.