Outcome Modeling

Because the causal effect of A on Y is identified by adjusting for the confounders L1 and L2, we can estimate by outcome modeling.

  1. Model \(E(Y\mid A, L_1, L_2)\), the conditional mean of \(Y\) given the treatment and confounders
  2. Predict potential outcomes
    • set A = 1 for every unit. Predict \(Y^1\)
    • set A = 0 for every unit. Predict \(Y^0\)
  3. Aggregate to the average causal effect

The code below assumes you have generated data as on the data page.

1) Model

The code below uses Ordinary Least Squares to estimate an outcome model.

model <- lm(Y ~ A*(L1 + L2), data = data)

Call:
lm(formula = Y ~ A * (L1 + L2), data = data)

Residuals:
    Min      1Q  Median      3Q     Max 
-4.1448 -0.7105  0.0097  0.6998  3.1743 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  0.01606    0.05699   0.282  0.77827    
A            1.11555    0.18021   6.190 1.26e-09 ***
L1           1.06333    0.05938  17.907  < 2e-16 ***
L2           1.11199    0.05951  18.685  < 2e-16 ***
A:L1        -0.39475    0.14279  -2.765  0.00591 ** 
A:L2        -0.28935    0.13940  -2.076  0.03844 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.111 on 494 degrees of freedom
Multiple R-squared:  0.6732,    Adjusted R-squared:  0.6699 
F-statistic: 203.6 on 5 and 494 DF,  p-value: < 2.2e-16

We chose a model where treatment A is interacted with an additive function of confounders L1 + L2. This is also known as a t-learner (Kunzel et al. 2019) because it is equivalent to estimating two separate regression models of outcome on confounders, one among those for whom A == 1 and among those for whom A == 0.

2) Predict

The code below predicts the conditional average potential outcome under treatment and control at the confounder values of each observation.

First, we create data with A set to the value 1.

data_1 <- data |>
  mutate(A = 1)
# A tibble: 500 × 7
        L1     L2      Y0    Y1 propensity_score     A       Y
     <dbl>  <dbl>   <dbl> <dbl>            <dbl> <dbl>   <dbl>
1  0.00304  1.03   0.677   1.59          0.276       1  0.677 
2 -2.35    -1.66  -4.09   -3.53          0.00244     1 -4.09  
3  0.104   -0.912  0.0659  1.31          0.0569      1  0.0659
# ℹ 497 more rows

Then, we create data with A set to the value 0.

data_0 <- data |>
  mutate(A = 0)
# A tibble: 500 × 7
        L1     L2      Y0    Y1 propensity_score     A       Y
     <dbl>  <dbl>   <dbl> <dbl>            <dbl> <dbl>   <dbl>
1  0.00304  1.03   0.677   1.59          0.276       0  0.677 
2 -2.35    -1.66  -4.09   -3.53          0.00244     0 -4.09  
3  0.104   -0.912  0.0659  1.31          0.0569      0  0.0659
# ℹ 497 more rows

We use our outcome model to predict the conditional mean of the potential outcome under each scenario.

predicted <- data |>
  mutate(
    Y1_predicted = predict(model, newdata = data_1),
    Y0_predicted = predict(model, newdata = data_0),
    effect_predicted = Y1_predicted - Y0_predicted
  )
# A tibble: 500 × 10
        L1     L2      Y0    Y1 propensity_score     A       Y Y1_predicted
     <dbl>  <dbl>   <dbl> <dbl>            <dbl> <int>   <dbl>        <dbl>
1  0.00304  1.03   0.677   1.59          0.276       0  0.677         1.98 
2 -2.35    -1.66  -4.09   -3.53          0.00244     0 -4.09         -1.81 
3  0.104   -0.912  0.0659  1.31          0.0569      0  0.0659        0.451
# ℹ 497 more rows
# ℹ 2 more variables: Y0_predicted <dbl>, effect_predicted <dbl>

3) Aggregate

The final step is to aggregate to an average causal effect estimate.

aggregated <- predicted |>
  summarize(average_effect_estimate = mean(effect_predicted))
# A tibble: 1 × 1
  average_effect_estimate
                    <dbl>
1                    1.13

Closing thoughts

Outcome modeling is a powerful strategy because it bridges nonparametric causal identification to longstanding strategies where outcomes are modeled by parametric regression.

Here are a few things you could try next:

  • replace step (1) with another approach to estimate conditional mean outcomes, such as a different functional form or a machine learning method
  • evaluate performance over many repeated simulations
  • evaluate performance at different simulated sample sizes