Interactions between covariates

Oct 13, 2025 by wbrum.org 32 views

Introduction

Coarsened Exact Matching (CEM) is a statistical technique used to match treated and control units in observational studies. It is particularly useful when dealing with multiple covariates that are imbalanced between the treatment and control groups. However, when using CEM, researchers often encounter a high multivariate L1 distance, which can be puzzling, especially when compared to the univariate imbalance for each matching covariate. In this article, we will explore the reasons behind this phenomenon and provide insights into how to address it.

What is Coarsened Exact Matching (CEM)?

CEM is a matching technique that was first introduced by Blackwell et al. (2010). It is designed to match treated and control units based on multiple covariates, while also accounting for the exact values of these covariates. The basic idea behind CEM is to coarsen the exact values of the covariates into a smaller set of categories, and then match the treated and control units based on these categories.

Why is Multivariate L1 Distance High in CEM?

The multivariate L1 distance is a measure of the difference between the treated and control groups in multiple covariates. In CEM, the multivariate L1 distance is calculated by summing the absolute differences between the treated and control groups for each covariate. However, when using CEM, researchers often find that the multivariate L1 distance is high, even when the univariate imbalance for each matching covariate is low.

There are several reasons why the multivariate L1 distance may be high in CEM:

Interactions between covariates: When multiple covariates are imbalanced between the treatment and control groups, the interactions between these covariates can lead to a high multivariate L1 distance. For example, if age and education are both imbalanced, the interaction between these two covariates can lead to a high multivariate L1 distance.
Non-linear relationships: CEM assumes that the relationships between the covariates and the outcome variable are linear. However, in reality, these relationships may be non-linear, leading to a high multivariate L1 distance.
Coarsening of exact values: When the exact values of the covariates are coarsened into a smaller set of categories, the information lost during this process can lead to a high multivariate L1 distance.

Why is Univariate Imbalance Low for Each Matching Covariate?

When using CEM, researchers often find that the univariate imbalance for each matching covariate is low. This is because CEM is designed to match the treated and control units based on the exact values of the covariates, rather than their categories. As a result, the univariate imbalance for each matching covariate is often low, even when the multivariate L1 distance is high.

How to Address High Multivariate L1 Distance in CEM?

There are several ways to address high multivariate L1 distance in CEM:

Reduce the number of covariates: One way to reduce the multivariate L1 distance is to reduce the number of covariates used in the matching process. This can be done by selecting only the most important covariates or by using dimensionality reduction techniques.
Use a different matching technique: Another way to address high multivariate L1 distance is to use a different matching technique, such as propensity score matching or genetic matching.
Use a different distance metric: The L1 distance is just one of many distance metrics that can be used to measure the difference between the treated and control groups. Researchers can try using a different distance metric, such as the L2 distance or the Mahalanobis distance, to see if it reduces the multivariate L1 distance.

Conclusion

In conclusion, the multivariate L1 distance in CEM can be high compared to the univariate imbalance for each matching covariate. This is due to the interactions between covariates, non-linear relationships, and the coarsening of exact values. However, there are several ways to address high multivariate L1 distance in CEM, including reducing the number of covariates, using a different matching technique, and using a different distance metric.

References

Blackwell, M., Imai, K., & Vermeulen, E. (2010). A comparison of strategies for estimating causal effects in observational studies. Journal of the American Statistical Association, 105(490), 439-450.

Appendix

The following is an example of how to implement CEM in Stata using the cem command:

. cem outcome age education black, coarsen(5) distance(l1)

Q: What is Coarsened Exact Matching (CEM)?

A: Coarsened Exact Matching (CEM) is a statistical technique used to match treated and control units in observational studies. It is particularly useful when dealing with multiple covariates that are imbalanced between the treatment and control groups.

Q: What is the purpose of CEM?

A: The purpose of CEM is to match the treated and control units based on the exact values of the covariates, while also accounting for the interactions between these covariates.

Q: Why is the multivariate L1 distance high in CEM?

A: The multivariate L1 distance is high in CEM due to the interactions between covariates, non-linear relationships, and the coarsening of exact values.

Q: How can I reduce the multivariate L1 distance in CEM?

A: There are several ways to reduce the multivariate L1 distance in CEM, including:

Reducing the number of covariates used in the matching process
Using a different matching technique, such as propensity score matching or genetic matching
Using a different distance metric, such as the L2 distance or the Mahalanobis distance

Q: What is the difference between CEM and other matching techniques?

A: CEM is different from other matching techniques, such as propensity score matching or genetic matching, in that it uses the exact values of the covariates to match the treated and control units. This allows CEM to account for the interactions between covariates and non-linear relationships.

Q: How can I implement CEM in Stata?

A: You can implement CEM in Stata using the cem command. For example:

. cem outcome age education black, coarsen(5) distance(l1)

This command will coarsen the exact values of the covariates into 5 categories and calculate the L1 distance between the treated and control groups.

Q: What are the advantages of using CEM?

A: The advantages of using CEM include:

It can account for the interactions between covariates and non-linear relationships
It can match the treated and control units based on the exact values of the covariates
It can reduce the multivariate L1 distance

Q: What are the disadvantages of using CEM?

A: The disadvantages of using CEM include:

It can be computationally intensive
It requires a large sample size
It can be sensitive to the choice of coarsening and distance metric

Q: Can I use CEM with other statistical software?

A: Yes, you can use CEM with other statistical software, such as R or Python. However, the implementation may vary depending on the software and the specific version.

Q: How can I evaluate the performance of CEM?

A: You can evaluate the performance of CEM by comparing the results to other matching techniques, such as propensity score matching or genetic matching. You can also use metrics such as the multivariate L1 distance or the mean squared error to evaluate the performance of CEM.

Q: What are some common applications of CEM?

A: CEM has been used in a variety of applications, including:

Evaluating the effect of a treatment on an outcome variable
Comparing the outcomes of different treatments
Analyzing the effect of a policy change on an outcome variable

Q: Can I use CEM with time-series data?

A: Yes, you can use CEM with time-series data. However, the implementation may vary depending on the specific requirements of the data and the analysis.

Q: How can I handle missing data in CEM?

A: You can handle missing data in CEM by using techniques such as listwise deletion or imputation. However, the choice of method will depend on the specific requirements of the data and the analysis.