Welcome to Part II of Causal Inference Book

11.1 Data cannot speak for themselves

• A: anti-retroviral therapy
• Y: CD4 cell count at the end of the study

• N: 16 individuals

• Estimator: \(\hat{E}[Y|A=a]\) (a function of the data) used to estimate the unknown populational parameter

• Consistent estimator: \(\hat{E}[Y|A=a]\) satisfies the criterion with the increased sample size the estimate is closer to the populational value \(E[Y|A=a]\)
• Possible estimators:
  • sample average of Y among those receiving \(A=a\) (a consistent estimator)
  • the Y value of the first observation in the dataset with \(A=a\) (not a consistent estimator)

11.1 Data cannot speak for themselves

11.1 Data cannot speak for themselves

library(tidyverse)
library(magrittr)
# Sample averages by treatment level
# Data for Figure 11.1
A <- c(1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0)
Y <- c(200, 150, 220, 110, 50, 180, 90, 170, 170, 30, 70, 110, 80, 50, 10, 20)
data <- tibble(A, Y) %>% 
  mutate(A = factor(A, levels = c("0", "1"), labels = c("Untreated", "Treated")))
p <- data %>% ggplot(aes(x = A, y = Y, color = A, fill = A)) +
  geom_point() +
  geom_boxplot(alpha = 0.3) +
  theme_minimal() +
  theme(legend.position = "none") +
  scale_color_manual(values = wesanderson::wes_palette(name = "Darjeeling2", n = 2)) +
  scale_fill_manual(values = wesanderson::wes_palette(name = "Darjeeling2", n = 2))

data %>% group_by(A) %>% summarise(mean = mean(Y)) %>% kableExtra::kable()

11.1 Data cannot speak for themselves

11.1 Data cannot speak for themselves

# Sample averages by treatment level
# Data for Figure 11.2
A <- c(1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4)
Y <- c(110, 80, 50, 40, 170, 30, 70, 50, 110, 50, 180, 130, 200, 150, 220, 210)
data <- tibble(A, Y) %>% 
  mutate(A = factor(A))
p <- data %>% ggplot(aes(x = A, y = Y, color = A, fill = A)) +
  geom_point() +
  geom_boxplot(alpha = 0.3) +
  theme_minimal() +
  theme(legend.position = "none") +
  scale_color_manual(values = wesanderson::wes_palette(name = "Darjeeling1", n = 4)) +
  scale_fill_manual(values = wesanderson::wes_palette(name = "Darjeeling1", n = 4))

data %>% group_by(A) %>% summarise(mean = mean(Y)) %>% kableExtra::kable()

11.1 Data cannot speak for themselves

11.1 Data cannot speak for themselves

# 2-parameter linear model
# Data for Figures 11.3
A <- c(3, 11, 17, 23, 29, 37, 41, 53, 67, 79, 83, 97, 60, 71, 15, 45)
Y <- c(21, 54, 33, 101, 85, 65, 157, 120, 111, 200, 140, 220, 230, 217, 11, 190)
data <- tibble(A, Y)
rm(A, Y)
res_lm <- lm(Y ~ A, data = data) %>% 
  broom::tidy(., conf.int = T) %>% 
  select(1, 2, 6, 7)
p <- data %>% 
  ggplot(aes(x = A, y = Y)) +
  geom_point() +
  theme_minimal()

## # A tibble: 2 x 4
##   term        estimate conf.low conf.high
##   <chr>          <dbl>    <dbl>     <dbl>
## 1 (Intercept)    24.5    -21.2      70.3 
## 2 A               2.14     1.28      2.99

11.2 Parametric estimators of the conditional mean

• Aim: to estimate mean of Y among individuals with treatment level A = 90, or \(E[Y|A=90]\)
• Y ~ Normal(A, \(\epsilon\)) (Y is a function of A with some error term)
• The mean of Y changes from some value \(\theta_0\) by \(\theta_1\) units per unit of treatment A: \(E[Y|A]=\theta_0 + \theta_1A\)
• The shape of conditional mean \(E[Y|A]\) is determined by this equation - linear mean model
• \(\theta_0\) and \(\theta_1\) are parameters of the model
• If model describes the expectation with a finite number of parameters, the model is parametric

11.2 Parametric estimators of the conditional mean

• A model restricts the joint distribution of the data
• Parametric models come with the assumptions
• The inferences are valid only when the model is correctly specified
• Assumption of no model misspecification for the model-based causal inference

11.3 Nonparametric estimators of the conditional mean

For dichotomous treatment A:

• \(E[Y|A]=\theta_0 + \theta_1A\)
• \(E[Y|A=1] = E[Y|A=0] + \theta_1\)
• Saturated model
• "Model is saturated whenever the number of parameters in a conditional mean model is equal to the number of unknown conditional means in the population"
• "When a model has only a few parameters but it is used to estimate many population quantities, it is parsimonious"

11.4 Smoothing

• Linear model with quadratic term \(A^2\) (or other polynomials: \(...\), \(A^{15}\))
• \(E[Y|A]=\theta_0 + \theta_1A + \theta_2A^2\)
• The more parameters the model has, the less smooth the curve is

11.4 Smoothing

data %<>% mutate(A_sq = A*A)
lm(Y ~ A + A_sq, data = data) %>% 
  broom::tidy(., conf.int = T) %>% 
  select(1, 2, 6, 7)

## # A tibble: 3 x 4
##   term        estimate conf.low conf.high
##   <chr>          <dbl>    <dbl>     <dbl>
## 1 (Intercept)  -7.41   -76.0      61.2   
## 2 A             4.11     0.800     7.41  
## 3 A_sq         -0.0204  -0.0535    0.0127

# predict by hand
-7.40687745 + 4.10722663*90 -0.02038477*90^2

## [1] 197.1269

# 3 parameters
p <- data %>% 
  ggplot(aes(x = A, y = Y)) +
  geom_point() +
  theme_minimal() + 
  stat_smooth(method = "glm", formula = y ~ poly(x, 2), color = "#00868B")
# 7 parameters
p2 <- data %>% 
  ggplot(aes(x = A, y = Y)) +
  geom_point() +
  theme_minimal() + 
  stat_smooth(method = "glm", formula = y ~ poly(x, 6), color = "#00868B")

11.5 The bias-variance trade-off

• Under 2-parameter model the prediction for CD4 cell count given A=90 was 216.9 and under 3-parameter model it was 197.1
• 3-parameter model is correctly specified under both straight line and curvelinear scenarios
• More parameters, less restrictions model implies
• Less smooth models provide less biased, but more imprecise result (estimate with a larger variance)

References

Hernán MA, Robins JM (2020). Causal Inference: What If. Boca Raton: Chapman & Hall/CRC (v. 31mar21)