Hierarchical Clustering As a Novel Solution to the Notorious Multicollinearity Problem in Observational Causal Inference

Multicollinearity is a long lasting challenge in observational causal inference, especially in regressions -- highly correlated independent variables make it hard to isolate their individual impacts on outcomes of interest. While common solutions such as shrinkage estimators and principal component regressions are helpful in prediction problems, a crucial limitation hinders their applicability to causal inference problems -- they cannot provide the original causal relationships. To fill the gap, we present an innovative and intuitive solution, by employing hierarchical clustering to aggregate data in a way that effectively alleviates collinearity. This method is generally applicable to causal problems featuring multicollinearity. We use a marketing application to demonstrate how and why it works. Expenditures on different advertising channels often exhibit correlations, making it exceedingly difficult to separately measure their impact. Many previous studies proposed to leverage granular cross-sectional data for better identification but, to our knowledge, none explicitly addressed multicollinearity, which undermines causal identification even with granular data. We propose to hierarchically cluster geographic units based on marketing spend correlation to reduce collinearity, and to implement a Bayesian Marketing Mix Model with cluster-level data. Such clustering happens in two steps -- we first normalize and demean geo-level data to establish a common scale and to eliminate the common trends; we then calculate pairwise distance to summarize marketing spend correlation between geos and cluster the ones with moderate to strong correlation. Both descriptive evidence and regression analysis affirm that such hierarchical clustering effectively mitigates collinearity and facilitates the separate identification of the impact of different marketing channels.

An efficient, accurate, and interpretable machine learning method for computing probability of failure

We introduce a novel machine learning method called the Penalized Profile Support Vector Machine based on the Gabriel edited set for the computation of the probability of failure for a complex system as determined by a threshold condition on a computer model of system behavior. The method is designed to minimize the number of evaluations of the computer model while preserving the geometry of the decision boundary that determines the probability. It employs an adaptive sampling strategy designed to strategically allocate points near the boundary determining failure and builds a locally linear surrogate boundary that remains consistent with its geometry by strategic clustering of training points. We prove two convergence results and we compare the performance of the method against a number of state of the art classification methods on four test problems. We also apply the method to determine the probability of survival using the Lotka--Volterra model for competing species.