Regression box

The Regression box is used to estimate numerical relationships between variables after a causal model (or adjustment strategy) has been selected. Unlike search or graph-editing boxes, the Regression box does not modify the graph. Instead, it provides regression-based summaries that help interpret causal effects implied by the current model.

The tools in this box are primarily used to:

  • estimate regression coefficients for selected variables,

  • compute total causal effects using valid adjustment sets,

  • sanity-check effect estimates under different graph orientations.

The Regression box currently provides four main tools:

  • Multiple Linear Regression\

  • Logistic Regression\

  • Adjustment Total Effects\

  • IDA Check

Each is described below.

Multiple Linear Regression

Purpose
Estimate linear relationships between a continuous outcome variable and one or more predictor variables.

Model For outcome Y and predictors X₁,…,Xₖ, the fitted model is: Y = β₀ + β₁X₁ + … + βₖXₖ + ε,

where ε is an error term.

Typical use - Exploratory analysis of associations. - Baseline comparison for causal effect estimates. - Regression-based effect summaries after choosing a graph.

Output - Estimated coefficients (β values). - Standard errors and test statistics. - Model fit diagnostics (when available).

Notes - This tool estimates associational regressions unless the predictors and covariates are chosen using a valid adjustment set. - Interpretation as a causal effect requires appropriate adjustment for confounding.

Logistic Regression

Purpose
Estimate the effect of predictors on a binary outcome variable.

Model For a binary outcome Y ∈ {0,1}, the model is: logit(P(Y=1)) = β₀ + β₁X₁ + … + βₖXₖ.

Typical use - Modeling binary outcomes (e.g., success/failure). - Estimating log-odds effects under adjustment.

Output - Regression coefficients on the log-odds scale. - Standard errors and test statistics.

Notes - As with linear regression, causal interpretation requires appropriate covariate adjustment. - Coefficients represent changes in log-odds, not probabilities.

Adjustment Total Effects

Purpose
Estimate total causal effects of one or more treatment variables on an outcome variable using valid adjustment sets derived from the graph.

Conceptual approach 1. A valid adjustment set is identified using the graph. 2. A regression model is fit with the treatment(s) and adjustment variables as predictors. 3. The coefficient(s) corresponding to the treatment variable(s) are reported as total effect estimates.

Typical use - Estimating causal effects implied by a DAG or PAG. - Comparing effect sizes across different adjustment sets. - Handling single or multiple treatments.

Notes - This tool relies on standard regression models (linear or logistic, depending on the outcome). - The validity of the effect estimates depends on the correctness of the adjustment set.

(See the Adjustment Total Effects detail page for full definitions and examples.)

IDA Check

Purpose
Evaluate the stability of effect estimates across Markov-equivalent graphs using the IDA (Intervention Calculus when the DAG is Absent) framework.

Conceptual approach - For each DAG consistent with the estimated equivalence class: - Compute a valid adjustment set. - Estimate the causal effect using regression. - Compare the resulting effect estimates across DAGs.

Typical use - Assess sensitivity of effect estimates to graph uncertainty. - Identify effects that are robust across equivalence classes.

Output - A range or set of effect estimates rather than a single number.

Notes - IDA Check does not assert that any single estimate is correct. - Instead, it highlights how much conclusions may vary given graph ambiguity.

(See the IDA Check detail page for definitions and interpretation.)

Interpretation and workflow notes

  • The Regression box is typically used after graph estimation or adjustment set selection.

  • Results should be interpreted in light of the assumptions encoded in the graph.

  • Regression estimates are only causal when the chosen covariates block all backdoor paths.

Summary

The Regression box provides regression-based tools for translating causal structure into numerical effect estimates. It bridges the gap between graphical causal models and quantitative interpretation, while making explicit the assumptions required for causal claims.