15.2 Average Marginal Effect Algorithm

For one-sided derivative \(\frac{\partial p(\mathbf{X},\beta)}{\partial X}\) in the probability scale

1. Estimate your model
2. For each observation \(i\)

   1. Calculate \(\hat{Y}_{i0}\) which is the prediction in the probability scale using observed values

   2. Increase \(X\) (variable of interest) by a “small” amount \(h\) (\(X_{new} = X + h\))

      • When \(X\) is continuous, \(h = (|\bar{X}| + 0.001) \times 0.001\) where \(\bar{X}\) is the mean value of \(X\)

      • When \(X\) is discrete, \(h = 1\)

   3. Calculate \(\hat{Y}_{i1}\) which is the prediction in the probability scale using new \(X\) and other variables’ observed vales.

   4. Calculate the difference between the two predictions as fraction of \(h\): \(\frac{\bar{Y}_{i1} - \bar{Y}_{i0}}{h}\)

3. Average numerical derivative is \(E[\frac{\bar{Y}_{i1} - \bar{Y}_{i0}}{h}] \approx \frac{\partial p (Y|\mathbf{X}, \beta)}{\partial X}\)

Two-sided derivatives

1. Estimate your model
2. For each observation \(i\)

   1. Calculate \(\hat{Y}_{i0}\) which is the prediction in the probability scale using observed values

   2. Increase \(X\) (variable of interest) by a “small” amount \(h\) (\(X_{1} = X + h\)) and decrease \(X\) (variable of interest) by a “small” amount \(h\) (\(X_{2} = X - h\))

      • When \(X\) is continuous, \(h = (|\bar{X}| + 0.001) \times 0.001\) where \(\bar{X}\) is the mean value of \(X\)

      • When \(X\) is discrete, \(h = 1\)

   3. Calculate \(\hat{Y}_{i1}\) which is the prediction in the probability scale using new \(X_1\) and other variables’ observed vales.

   4. Calculate \(\hat{Y}_{i2}\) which is the prediction in the probability scale using new \(X_2\) and other variables’ observed vales.

   5. Calculate the difference between the two predictions as fraction of \(h\): \(\frac{\bar{Y}_{i1} - \bar{Y}_{i2}}{2h}\)

3. Average numerical derivative is \(E[\frac{\bar{Y}_{i1} - \bar{Y}_{i2}}{2h}] \approx \frac{\partial p (Y|\mathbf{X}, \beta)}{\partial X}\)

library(margins)
library(tidyverse)

data("mtcars")
mod <- lm(mpg ~ cyl * disp * hp, data = mtcars)
margins::margins(mod) %>% summary()
#>  factor     AME     SE       z      p    lower   upper
#>     cyl -4.0592 3.7614 -1.0792 0.2805 -11.4313  3.3130
#>    disp -0.0350 0.0132 -2.6473 0.0081  -0.0610 -0.0091
#>      hp -0.0284 0.0185 -1.5348 0.1248  -0.0647  0.0079

# function for variable
get_mae <- function(mod, var = "disp") {
    data = mod$model
    
    pred <- predict(mod)
    
    if (class(mod$model[[{
        {
            var
        }
    }]]) == "numeric") {
        h = (abs(mean(mod$model[[var]])) + 0.01) * 0.01
    } else {
        h = 1
    }
    
    data[[{
        {
            var
        }
    }]] <- data[[{
{
var
}
}]] - h

    pred_new <- predict(mod, newdata = data)

    mean(pred - pred_new) / h
}

get_mae(mod, var = "disp")
#> [1] -0.03504546