Continuous exposures and g-computation

Normal regression estimates associations. But we want causal estimates: what would happen if everyone in the study were exposed to x vs if no one was exposed.

G-Computation/G-Formula

Fit a model for y ~ x + z where z is all covariates
Create a duplicate of your data set for each level of x
Set the value of x to a single value for each cloned data set (e.g x = 1 for one, x = 0 for the other)

G-Computation/G-Formula

Make predictions using the model on the cloned data sets
Calculate the estimate you want, e.g. mean(x_1) - mean(x_0)

Advantages of the parametric G-formula

Often more statistically precise than propensity-based methods

Incredibly flexible

Basis of other important causal models, e.g. causal survival analysis and TMLE

Greek Pantheon data (`greek_data`)

The name of a Greek god	A prognostic factor	The treatment, a heart transplant	The outcome, death
Rheia	0	0	0
Kronos	0	0	1
Demeter	0	0	0
Hades	0	0	0
Hestia	0	1	0
Poseidon	0	1	0
Hera	0	1	0
Zeus	0	1	1
Artemis	1	0	1
Apollo	1	0	1

+ 10 more rows

1. Fit a model for `y ~ a + l`

greek_model <- lm(y ~ a + l, data = greek_data)

2. Create a duplicate of your data set for each level of `a`

The name of a Greek god	A prognostic factor	The treatment, a heart transplant	The outcome, death
Rheia	0	0	0
Kronos	0	0	1
Demeter	0	0	0
Hades	0	0	0
Hestia	0	1	0
Poseidon	0	1	0
Hera	0	1	0
Zeus	0	1	1
Artemis	1	0	1
Apollo	1	0	1

2. Create a duplicate of your data set for each level of `a`

The name of a Greek god	A prognostic factor	The treatment, a heart transplant	The outcome, death
Rheia	0	0	0
Kronos	0	0	1
Demeter	0	0	0
Hades	0	0	0
Hestia	0	1	0
Poseidon	0	1	0
Hera	0	1	0
Zeus	0	1	1
Artemis	1	0	1
Apollo	1	0	1

The name of a Greek god	A prognostic factor	The treatment, a heart transplant	The outcome, death
Rheia	0	0	0
Kronos	0	0	1
Demeter	0	0	0
Hades	0	0	0
Hestia	0	1	0
Poseidon	0	1	0
Hera	0	1	0
Zeus	0	1	1
Artemis	1	0	1
Apollo	1	0	1

3. Set the value of `a` to a single value for each cloned data set

The name of a Greek god	A prognostic factor	The outcome, death
Rheia	0	0
Kronos	0	1
Demeter	0	0
Hades	0	0
Hestia	0	0
Poseidon	0	0
Hera	0	0
Zeus	0	1
Artemis	1	1
Apollo	1	1

The name of a Greek god	A prognostic factor	a	The outcome, death
Rheia	0	1	0
Kronos	0	1	1
Demeter	0	1	0
Hades	0	1	0
Hestia	0	1	0
Poseidon	0	1	0
Hera	0	1	0
Zeus	0	1	1
Artemis	1	1	1
Apollo	1	1	1

3. Set the value of `a` to a single value for each cloned data set

#  set all participants to have a = 0
untreated_data <- greek_data |>
  mutate(a = 0) 

#  set all participants to have a = 1
treated_data <- greek_data |>
  mutate(a = 1) #  set all participants to have a = 0
untreated_data <- greek_data |>
  mutate(a = 0) 

#  set all participants to have a = 1
treated_data <- greek_data |>
  mutate(a = 1)

4. Make predictions using the model on the cloned data sets

#  predict under the data where everyone is untreated
predicted_untreated <- greek_model |>
  augment(newdata = untreated_data) |>
  select(untreated = .fitted)

#  predict under the data where everyone is treated
predicted_treated <- greek_model |>
  augment(newdata = treated_data) |>
  select(treated = .fitted)

predictions <- bind_cols(
  predicted_untreated, 
  predicted_treated
) #  predict under the data where everyone is untreated
predicted_untreated <- greek_model |>
  augment(newdata = untreated_data) |>
  select(untreated = .fitted)

#  predict under the data where everyone is treated
predicted_treated <- greek_model |>
  augment(newdata = treated_data) |>
  select(treated = .fitted)

predictions <- bind_cols(
  predicted_untreated, 
  predicted_treated
)

5. Calculate the estimate you want

predictions |>
  summarise(
    mean_treated = mean(treated),
    mean_untreated = mean(untreated),
    difference = mean_treated - mean_untreated
  )predictions |>
  summarise(
    mean_treated = mean(treated),
    mean_untreated = mean(untreated),
    difference = mean_treated - mean_untreated
  )

# A tibble: 1 × 3
  mean_treated mean_untreated difference
         <dbl>          <dbl>      <dbl>
1          0.5            0.5          0

Continuous exposures

We recommend g-computation over propensity scores for continuous exposures because of stability issues

Do posted wait times at 8 am affect actual wait times at 9 am?

Your Turn

Work through Your Turns 1-3 in `10-continuous-g-computation-exercises.qmd`

10:00