33.1 Bad Controls

33.1.1 M-bias

Traditional textbooks (G. W. Imbens and Rubin 2015; Angrist and Pischke 2009) consider \(Z\) as a good control because it’s a pre-treatment variable, where it correlates with the treatment and the outcome.

This is most prevalent in Matching Methods, where we are recommended to include all “pre-treatment” variables.

However, it is a bad control because it opens the back-door path \(Z \leftarrow U_1 \to Z \leftarrow U_2 \to Y\)

# cleans workspace
rm(list = ls())

# DAG

## specify edges
model <- dagitty("dag{x->y; u1->x; u1->z; u2->z; u2->y}")

# set u as latent
latents(model) <- c("u1", "u2")

## coordinates for plotting
coordinates(model) <-  list(x = c(
    x = 1,
    u1 = 1,
    z = 2,
    u2 = 3,
    y = 3
),
y = c(
    x = 1,
    u1 = 2,
    z = 1.5,
    u2 = 2,
    y = 1
))

## ggplot
ggdag(model) + theme_dag()

Even though \(Z\) can correlate with both \(X\) and \(Y\) very well, it’s not a confounder.

Controlling for \(Z\) can bias the \(X \to Y\) estimate, because it opens the colliding path \(X \leftarrow U_1 \rightarrow Z \leftarrow U_2 \leftarrow Y\)

n <- 1e4
u1 <- rnorm(n)
u2 <- rnorm(n)
z <- u1 + u2 + rnorm(n)
x <- u1 + rnorm(n)
causal_coef <- 2
y <- causal_coef * x - 4*u2 + rnorm(n)


jtools::export_summs(lm(y ~ x), lm(y ~ x + z))
Table 33.1:
Model 1Model 2
(Intercept)-0.04    -0.03    
(0.04)   (0.03)   
x1.96 ***2.76 ***
(0.03)   (0.02)   
z       -1.57 ***
       (0.02)   
N10000       10000       
R20.32    0.57    
*** p < 0.001; ** p < 0.01; * p < 0.05.

Another worse variation is

# cleans workspace
rm(list = ls())

# DAG

## specify edges
model <- dagitty("dag{x->y; u1->x; u1->z; u2->z; u2->y; z->y}")

# set u as latent
latents(model) <- c("u1", "u2")

## coordinates for plotting
coordinates(model) <-  list(
  x = c(x=1, u1=1, z=2, u2=3, y=3),
  y = c(x=1, u1=2, z=1.5, u2=2, y=1))

## ggplot
ggdag(model) + theme_dag()

You can’t do much in this case.

  • If you don’t control for \(Z\), then you have an open back-door path \(X \leftarrow U_1 \to Z \to Y\), and the unadjusted estimate is biased

  • If you control for \(Z\), then you open backdoor path \(X \leftarrow U_1 \to Z \leftarrow U_2 \to Y\), and the adjusted estimate is also biased

Hence, we cannot identify the causal effect in this case.

We can do sensitivity analyses to examine (Cinelli et al. 2019; Cinelli and Hazlett 2020)

  1. the plausible bounds on the strength of the direct effect of \(Z \to Y\)
  2. the strength of the effects of the latent variables

33.1.2 Bias Amplification

# cleans workspace
rm(list = ls())

# DAG

## specify edges
model <- dagitty("dag{x->y; u->x; u->y; z->x}")

# set u as latent
latents(model) <- c("u")

## coordinates for plotting
coordinates(model) <-  list(
  x = c(z=1, x=2, u=3, y=4),
  y = c(z=1, x=1, u=2, y=1))

## ggplot
ggdag(model) + theme_dag()

Controlling for Z amplifies the omitted variable bias

n <- 1e4
z <- rnorm(n)
u <- rnorm(n)
x <- 2*z + u + rnorm(n)
y <- x + 2*u + rnorm(n)

jtools::export_summs(lm(y ~ x), lm(y ~ x + z))
Table 33.2:
Model 1Model 2
(Intercept)-0.02    -0.02    
(0.02)   (0.02)   
x1.35 ***2.00 ***
(0.01)   (0.01)   
z       -1.97 ***
       (0.03)   
N10000       10000       
R20.72    0.80    
*** p < 0.001; ** p < 0.01; * p < 0.05.

33.1.3 Overcontrol bias

Sometimes, this is similar to controlling for variables that are proxy of the dependent variable.

# cleans workspace
rm(list = ls())

# DAG

## specify edges
model <- dagitty("dag{x->z; z->y}")


## coordinates for plotting
coordinates(model) <-  list(
  x = c(x=1, z=2, y=3),
  y = c(x=1, z=1, y=1))

## ggplot
ggdag(model) + theme_dag()

If X is a proxy for Z (i.e., a mediator between Z and Y), controlling for Z is bad

n <- 1e4
x <- rnorm(n)
z <- x + rnorm(n)
y <- z + rnorm(n)

jtools::export_summs(lm(y ~ x), lm(y ~ x + z))
Table 33.3:
Model 1Model 2
(Intercept)0.01    0.02    
(0.01)   (0.01)   
x1.00 ***0.00    
(0.01)   (0.01)   
z       0.99 ***
       (0.01)   
N10000       10000       
R20.33    0.67    
*** p < 0.001; ** p < 0.01; * p < 0.05.

Now you see that \(Z\) is significant, which is technically true, but we are interested in the causal coefficient of \(X\) on \(Y\).

Another setting for overcontrol bias is

# cleans workspace
rm(list = ls())

# DAG

## specify edges
model <- dagitty("dag{x->m; m->z; m->y}")


## coordinates for plotting
coordinates(model) <-  list(
  x = c(x=1, m=2, z=2, y=3),
  y = c(x=2, m=2, z=1, y=2))

## ggplot
ggdag(model) + theme_dag()

n <- 1e4
x <- rnorm(n)
m <- x + rnorm(n)
z <- m + rnorm(n)
y <- m + rnorm(n)


jtools::export_summs(lm(y ~ x), lm(y ~ x + z))
Table 33.4:
Model 1Model 2
(Intercept)0.03 *  0.02    
(0.01)   (0.01)   
x0.98 ***0.50 ***
(0.01)   (0.01)   
z       0.48 ***
       (0.01)   
N10000       10000       
R20.32    0.49    
*** p < 0.001; ** p < 0.01; * p < 0.05.

Another setting for this bias is

# cleans workspace
rm(list = ls())

# DAG

## specify edges
model <- dagitty("dag{x->z; z->y; u->z; u->y}")

# set u as latent
latents(model) <- "u"

## coordinates for plotting
coordinates(model) <-  list(
  x = c(x=1, z=2, u=3, y=4),
  y = c(x=1, z=1, u=2, y=1))

## ggplot
ggdag(model) + theme_dag()

set.seed(1)
n <- 1e4
x <- rnorm(n)
u <- rnorm(n)
z <- x + u + rnorm(n)
y <- z + u + rnorm(n)

jtools::export_summs(lm(y ~ x), lm(y ~ x + z))
Table 33.5:
Model 1Model 2
(Intercept)-0.01    -0.01    
(0.02)   (0.01)   
x1.01 ***-0.47 ***
(0.02)   (0.01)   
z       1.48 ***
       (0.01)   
N10000       10000       
R20.15    0.78    
*** p < 0.001; ** p < 0.01; * p < 0.05.

The total effect of \(X\) on \(Y\) is not biased (i.e., \(1.01 \approx 1.48 - 0.47\)).

Controlling for Z will fail to identify the direct effect of \(X\) on \(Y\) and opens the biasing path \(X \rightarrow Z \leftarrow U \rightarrow Y\)

33.1.4 Selection Bias

Also known as “collider stratification bias”

rm(list = ls())

# DAG

## specify edges
model <- dagitty("dag{x->y; x->z; u->z;u->y}")

# set u as latent
latents(model) <- "u"

## coordinates for plotting
coordinates(model) <-  list(
  x = c(x=1, z=2, u=2, y=3),
  y = c(x=3, z=2, u=4, y=3))

## ggplot
ggdag(model) + theme_dag()

Adjusting \(Z\) opens the colliding path \(X \to Z \leftarrow U \to Y\)

n <- 1e4
x <- rnorm(n)
u <- rnorm(n)
z <- x + u +  rnorm(n)
y <- x + 2*u + rnorm(n)

jtools::export_summs(lm(y ~ x), lm(y ~ x + z))
Table 33.6:
Model 1Model 2
(Intercept)0.00    0.02    
(0.02)   (0.02)   
x0.98 ***-0.03    
(0.02)   (0.02)   
z       1.00 ***
       (0.01)   
N10000       10000       
R20.17    0.49    
*** p < 0.001; ** p < 0.01; * p < 0.05.

Another setting is

rm(list = ls())

# DAG

## specify edges
model <- dagitty("dag{x->y; x->z; y->z}")

## coordinates for plotting
coordinates(model) <-  list(
  x = c(x=1, z=2, y=3),
  y = c(x=2, z=1, y=2))

## ggplot
ggdag(model) + theme_dag()

Controlling \(Z\) opens the colliding path \(X \to Z \leftarrow Y\)

n <- 1e4
x <- rnorm(n)
y <- x + rnorm(n)
z <- x + y + rnorm(n)

jtools::export_summs(lm(y ~ x), lm(y ~ x + z))
Table 33.7:
Model 1Model 2
(Intercept)-0.00    -0.01    
(0.01)   (0.01)   
x1.03 ***-0.01    
(0.01)   (0.01)   
z       0.51 ***
       (0.00)   
N10000       10000       
R20.51    0.76    
*** p < 0.001; ** p < 0.01; * p < 0.05.

33.1.5 Case-control Bias

rm(list = ls())

# DAG

## specify edges
model <- dagitty("dag{x->y; y->z}")

## coordinates for plotting
coordinates(model) <-  list(
  x = c(x=1, z=2, y=3),
  y = c(x=2, z=1, y=2))

## ggplot
ggdag(model) + theme_dag()

Controlling \(Z\) opens a virtual collider (a descendant of a collider).

However, if \(X\) truly has no causal effect on \(Y\). Then, controlling for \(Z\) is valid for testing whether the effect of \(X\) on \(Y\) is 0 because X is d-separated from \(Y\) regardless of adjusting for \(Z\)

n <- 1e4
x <- rnorm(n)
y <- x + rnorm(n)
z <- y + rnorm(n)

jtools::export_summs(lm(y ~ x), lm(y ~ x + z))
Table 33.8:
Model 1Model 2
(Intercept)0.00    -0.00    
(0.01)   (0.01)   
x1.00 ***0.50 ***
(0.01)   (0.01)   
z       0.50 ***
       (0.00)   
N10000       10000       
R20.50    0.75    
*** p < 0.001; ** p < 0.01; * p < 0.05.

References

Angrist, Joshua D, and Jörn-Steffen Pischke. 2009. Mostly Harmless Econometrics: An Empiricist’s Companion. Princeton university press.
Cinelli, Carlos, and Chad Hazlett. 2020. “Making Sense of Sensitivity: Extending Omitted Variable Bias.” Journal of the Royal Statistical Society Series B: Statistical Methodology 82 (1): 39–67.
Cinelli, Carlos, Daniel Kumor, Bryant Chen, Judea Pearl, and Elias Bareinboim. 2019. “Sensitivity Analysis of Linear Structural Causal Models.” In International Conference on Machine Learning, 1252–61. PMLR.
Imbens, Guido W, and Donald B Rubin. 2015. Causal Inference in Statistics, Social, and Biomedical Sciences. Cambridge University Press.