Causal Estimands

The difference

Estimand

@tigga_mac TikTok

Estimator



https://www.bluey.tv/make/bluey-celebration-cake/

Estimate


Coles Mudcake Hacks Facebook

Home baker attempts to make a Bluey birthday cake using Woolies mudcakes - but is left in hysterics over her ‘abomination’ of a dessert

The difference

Estimand

\[E[Y(1) - Y(0)]\]

Estimator

\[\sum_{i=1}^N\frac{Y_i\times X_i}{N_A}-\frac{Y_i \times(1 - X_i)}{N_B}\]

Estimate

library(tidyverse)
set.seed(928)
ab <- tibble(
  x = rbinom(100, 1, 0.5),
  y = x + rnorm(100)
)

ab |>
  summarise(
    n_a = sum(x),
    n_b = sum(1 - x),
    estimate = sum(
      (y * x) / n_a -
        y * (1 - x) / n_b
    )
  )
# A tibble: 1 × 3
    n_a   n_b estimate
  <int> <dbl>    <dbl>
1    54    46     1.15

Weighted estimator

\[\frac{\sum_{i=1}^NY_i\times X_i\times w_i}{\sum_{i=1}^NX_i\times w_i}-\frac{\sum_{i=1}^NY_i\times(1-X_i)\times w_i}{\sum_{i=1}^N(1-X_i)\times w_i}\]

Example

Photo by Anna CC-BY-SA-4.0

Historically, guests who stayed in a Walt Disney World resort hotel were able to access the park during “Extra Magic Hours” during which the park was closed to all other guests.

These extra hours could be in the morning or evening.

The Seven Dwarfs Mine Train is a ride at Walt Disney World’s Magic Kingdom. Typically, each day Magic Kingdom may or may not be selected to have these “Extra Magic Hours”.

We are interested in examining the relationship between whether there were “Extra Magic Hours” in the morning and the average wait time for the Seven Dwarfs Mine Train the same day between 9am and 10am.

Average treatment effect

Question

Should we make every day an extra magic morning (or not)?

Estimand

\(E[Y(1)-Y(0)]\)

Weight

\[w_{ATE} = \frac{X}{p} + \frac{(1 - X)}{1 - p}\]

ATE weighted table

Characteristic No Magic Hours
N = 3541		 Extra Magic Hours
N = 3571		 Overall
N = 7111
park_ticket_season


    peak 78 (22%) 81 (23%) 160 (22%)
    regular 193 (54%) 187 (52%) 380 (53%)
    value 83 (23%) 89 (25%) 172 (24%)
park_close


    16:30:00 2 (0.4%) 0 (0%) 2 (0.2%)
    18:00:00 50 (14%) 72 (20%) 122 (17%)
    20:00:00 20 (5.6%) 19 (5.3%) 39 (5.5%)
    21:00:00 31 (8.7%) 0 (0%) 31 (4.3%)
    22:00:00 108 (30%) 86 (24%) 193 (27%)
    23:00:00 95 (27%) 81 (23%) 176 (25%)
    24:00:00 48 (14%) 94 (26%) 142 (20%)
    25:00:00 1 (0.3%) 6 (1.6%) 7 (1.0%)
park_temperature_high 84 (78, 89) 83 (78, 87) 84 (78, 88)
1 n (%); Median (Q1, Q3)

ATE weighted histogram

Average treatment effect among the treated

Question

Should we stop extra magic hours on days that have them?

Estimand

\(E[Y(1)-Y(0) | X = 1]\)

Weight

\[w_{ATT} = X + \frac{(1 - X)p}{1 - p}\]

ATT weighted table

Characteristic No Magic Hours
N = 601		 Extra Magic Hours
N = 601		 Overall
N = 1201
park_ticket_season


    peak 18 (30%) 18 (30%) 36 (30%)
    regular 35 (58%) 35 (58%) 70 (58%)
    value 7 (12%) 7 (12%) 14 (12%)
park_close


    16:30:00 1 (0.9%) 0 (0%) 1 (0.4%)
    18:00:00 13 (21%) 18 (30%) 31 (26%)
    20:00:00 2 (3.3%) 2 (3.3%) 4 (3.3%)
    21:00:00 3 (4.6%) 0 (0%) 3 (2.3%)
    22:00:00 17 (28%) 11 (18%) 28 (23%)
    23:00:00 17 (28%) 11 (18%) 28 (23%)
    24:00:00 8 (14%) 17 (28%) 25 (21%)
    25:00:00 0 (0.3%) 1 (1.7%) 1 (1.0%)
park_temperature_high 83 (74, 88) 83 (75, 87) 83 (75, 88)
1 n (%); Median (Q1, Q3)

ATT weighted histogram