For Interpretable Treatment Effect Estimation

 Marco Morucci Duke University

marco.morucci@duke.edu

## Treatment Effects and High-Stakes Decisions

Causal Estimates are increasingly used to justify decisions in all sectors...

Goal: We want to provide decision-makers with trustworthy methods that produce justifiable estimates.

### Treatment effect estimation

#### Setting: Observational Causal Inference

• We have a set of $N$ units
• They have potential outcomes $Y_i(1), Y_i(0)$
• They are associated with a set of contextual covariates, $\mathbf{X}_i$
• They are assigned either treatment $T_i=1$ or $T_i=0$
• We observe $Y_i = T_iY(1) + (1-T_i)Y_i(0)$
• We would like to know: $$CATE(\mathbf{x}_i) = \mathbb{E}[Y_i(1) - Y_i(0)|\mathbf{X_i = \mathbf{x}_i}]$$
• Assume SUTVA, Overlap, ignorability

All methods presented here can also be applied to experimental data, for example, to analyze treatment effect heterogeneity.

### Treatment effect estimation

How do we estimate a quantity that's half unobserved?

We can under two main assumptions

• Conditional Ignorability $(Y_i(1), Y_i(0))$ is independent of $T_i$ conditional on $\mathbf{X}_i$
• SUTVA: Unit $i$ is exclusively affected by its own treatment

## Matching in a nutshell

 Q:What would happen if A didn't receive the treatment? Idea:Compare A to B, its identical twin who didn't receive the treatment But...Not everyone has an identical twin! Sometimes similar is good enough.

### Matching in a nutshell

Matching allows us to take advantage of CI and SUTVA to estimate the CATE in an interpretable way.

• For each unit, $i$ find several other units that are similar in the covariate values.
• Form a Matched Group (MG) with all the units matched to $i$
• Estimate the CATE for $i$ with $$\widehat{CATE}_i = \frac{1}{N^t_{MG}}\sum_{i \in MG}Y_iT_i - \frac{1}{N^c_{MG}}\sum_{i \in MG}Y_i(1-T_i)$$

### Why matching?

We choose matching to estimate CATEs because matching is interpretable and interpretability is important

• Interpretability permits greater accountability of decision-makers in high-stake settings
• Interpretability allows researchers to debug their models, datasets and results
• Interpretability makes results from Machine Learning tools more trustworthy to human observers

### Matching is interpretable because estimates are case-based

"We came up with that estimates because we looked at these similar cases."

### Matching with Continuous Covariates

Problem: do we find know who to match to whom when covariates are continuous?

Solution: Coarsening (binning) variables and then forming matched groups based on bins seems natural

More problems: ...but how do we know how much to coarsen each variable?

### Carefully Coarsening Covariates

Consider the problem of having to bin a covariate, $x$

### Takeaways

• Relationship between $x$ and $y$ should be considered when binning
• Even so, optimal-but-fixed width bins are not enough, because...
• Different parts of the covariate space should be binned according to variance of $y$

### Takeaways

• Relationship between $x$ and $y$ should be considered when binning
• Even so, optimal-but-fixed width bins are not enough, because...
• Different parts of the covariate space should be binned according to variance of $y$

[Morucci, Orlandi, Roy, Rudin, Volfovsky (UAI 2020)]

An algorithm to adaptively learn optimal coarsenings of the data that satisfy these criteria.

### The AHB Algorithm

Step 1: On a separate training set, fit a flexible ML model to predict both potential outcomes for each unit in the matching set.

### The AHB Algorithm

Step 2: Construct a p-dimensional box around each unit, such that, in the box:

 There are many units Predicted outcomes are similar Predictions from outcome averages are similar to ML predictions

### The AHB Algorithm

Step 2: Construct a p-dimensional box around each unit, such that, in the box:

 There are many units Predicted outcomes are similar Predictions from outcome averages are similar to ML predictions

### The AHB Algorithm

Step 2: Construct a p-dimensional box around each unit, such that, in the box:

 There are many units Predicted outcomes are similar Predictions from outcome averages are similar to ML predictions

### The AHB Algorithm

Step 2: Construct a p-dimensional box around each unit, such that, in the box:

 There are many units Predicted outcomes are similar Predictions from outcome averages are similar to ML predictions

### The AHB Algorithm

Step 3: Match together units that are in the same box

That's it!

### AHB Formally

$\min_{\mathbf{H}} Err(\mathbf{H}, \mathbf{y}) + Var(\mathbf{H}, \mathbf{y}) - n(\mathbf{H})$

where:

• $\mathbf{H}$ is the hyper-box we want to find
• $Err(\mathbf{H}, \mathbf{y})$ is the outcome prediction error for all points in $\mathbf{H}$
• $Var(\mathbf{H}, \mathbf{y})$ is the outcome variance between points in $\mathbf{H}$
• $n(\mathbf{H})$ is the number of points in $\mathbf{H}$

### AHB Formally

In practice, we solve, for every unit, $i$:

$\min_{\mathbf{H}_i} \sum_{j=1}^n \sum_{t=0}^1 w_{ij}|\hat{f}(\mathbf{x}_i, t) - \hat{f}(\mathbf{x}_j, t)| - \gamma\sum_{j=1}^n w_{ij}$

• $w_ij$ denotes whether unit $j$ is contained in box $i$
• $\hat{f}(\mathbf{x}, t)$ is the predicted outcome

This function controls all terms previously introduced.

### Two solution methods for AHB

#### An exact solution via a MILP formulation

• Can be solved by any popular MIP solver (CPLEX, Gurobi, GLPK...)
• Simple preprocessing steps make solution relatively fast

### Two solution methods for AHB

#### A fast approximate algorithm

1. Determine which unit is closest in absolute distance along each covariate
2. Expand the box in the direction with the lowest predicted outcome variance
3. Repeat until a desired threshold of matches or loss is met

### AHB Performance

Test case: Lalonde data

• Effect of training program on income
• Several samples, two observational (PSID, CPS), one experimental (NSW)
• Idea: match experimental treated units (N=185) to observational controls (N>2000) to recover experimental estimates.
• Matching covariates: Age, Education, Prior income, Race, Marital status

### AHB Performance

Test case: Lalonde data

Who gets closest to experimental?

Method CPS sample PSID sample
Experimental 1794$1794$
AHB1720$1762$
Naive-7729-14797
Full Matching708816
Prognostic13192224
CEM3744-2293
Mahalanobis1181-804

### AHB Performance

 Recall: ideally, adaptive bins would look like this: AHB outputs these bins on the Lalonde data:

Pretty close to the theoretical ideal!

### Are our Matches interpretable?

AME/AHB matches vs. Pscore matches

 Age Education Black Married HS Degree Pre income Post Income (outcome) Treated unit we want to match 22 9 no no no 0 3595.89 Matched controls: prognostic score 44 9 yes no no 0 9722 22 12 yes no yes 532 1333.44 18 10 no no no 0 1859.16 Matched controls: AHB 22 9 no no no 0 1245 20 8 no no no 0 5862.22 24 10 yes no no 0 4123.85

### Are our Matches interpretable?

AHB Interactive output

### Software

https://almost-matching-exactly.github.io/

### Conclusion

Interpretability of causal estimates is important!

• We propose matching methods that produce interpretable and accurate CATE estimates
• Our methods take advantage of machine learning backends to boost accuracy
• Our methods explain both estimates in terms of cases, and why cases were selected in terms of covariate similarity.