Intro to causal inference

Good and bad controls

Researchers have long understood that adding variables (“controls”) to a regression model is necessary to eliminate “omitted variable bias”. Scientists exposed only to this view may get the impression that adding “more controls” to a regression model is always better. However some variables when added to the regression equation can produce unintended discrepancies between the regression coefficient and the effect that the coefficient is expected to represent. Such variables have become known as “bad controls”. This tutorial describes graphical tools for understanding, visualizing, and resolving the problem of selecting variables for inclusion in a regression model through a series of illustrative simulations in R. An applied example will be provided on the role of intergenerational transfer to determine whether grandparents directly affect the wealth, education, status of their grandchildren.

Intro

I am Richard Morris.

www.sydney.edu.au/medicine-health/about/our-people/academic-staff/richard-morris.html

 

…

Background

x x y y x->y

This is a directed acyclic graph (DAG)



\[ \begin{align} y =& \ bx \ + \ e \\ b =& \ cov(x, y) / cov(x, x) \\ \end{align} \]

\(b\) is unbiased when:

\[ \begin{align} cov(x, e) = 0 \\ \end{align} \]

Background

x x y y x->y

DAGS can represent causal networks

# Simulating data:
N <- 200              # number of obs./subjects
X <- rnorm(N)            
Y <- rnorm(N, 1 - X)  # Y is dependent on X

summary(lm(Y ~ X))     

Call:
lm(formula = Y ~ X)

Residuals:
    Min      1Q  Median      3Q     Max 
-2.5466 -0.7045 -0.0519  0.6637  3.2969 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  0.97786    0.07065   13.84   <2e-16 ***
X           -0.99604    0.07318  -13.61   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.9988 on 198 degrees of freedom
Multiple R-squared:  0.4834,    Adjusted R-squared:  0.4808 
F-statistic: 185.3 on 1 and 198 DF,  p-value: < 2.2e-16

Forks (confounds)

x x y y x->y z z z->x z->y

z causes both x and y



\[ \begin{align} y =& \ b_1x \ + \ b_2z \ + \ e \\ b_1 =& \ cov(x, y | z) / var(x | z) \end{align} \]

Forks (confounds)

x x y y z z z->x z->y

z causes both x and y

N <- 200                     # number of obs./subjects
Z <- rnorm(N)
X <- rnorm(N, 1 + Z)         # X depends on Z
Y <- rnorm(N, 1 + 0*X + 2*Z) # Y is dependent on Z, not X

summary(lm(Y ~ X))           # estimate of X should be 0

Call:
lm(formula = Y ~ X)

Residuals:
    Min      1Q  Median      3Q     Max 
-4.1195 -0.9933 -0.0924  1.1107  5.4105 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) -0.09482    0.15379  -0.617    0.538    
X            0.99664    0.09470  10.524   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.768 on 198 degrees of freedom
Multiple R-squared:  0.3587,    Adjusted R-squared:  0.3555 
F-statistic: 110.8 on 1 and 198 DF,  p-value: < 2.2e-16

Forks (confounds)

x x y y z z z->x z->y

z causes both x and y

# 
# 
# 
# Including Z as a predictor

summary(lm(Y ~ X + Z))    # estimate of X should be 0

Call:
lm(formula = Y ~ X + Z)

Residuals:
     Min       1Q   Median       3Q      Max 
-2.46314 -0.64511  0.03695  0.60648  2.08276 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  1.01843    0.09422  10.809   <2e-16 ***
X           -0.03631    0.06715  -0.541    0.589    
Z            1.99847    0.08742  22.862   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.9276 on 197 degrees of freedom
Multiple R-squared:  0.8245,    Adjusted R-squared:  0.8227 
F-statistic: 462.6 on 2 and 197 DF,  p-value: < 2.2e-16

Problem: Unmeasured confounds

x x y y x->y u u u->x u->y

u is an unobserved cause of x and y

N <- 200                 # number of obs./subjects
U <- rnorm(N)
X <- rnorm(N, 1*U)       # X depends on U
Y <- rnorm(N, 1*X + 2*U) # Y depends on X and U

summary(lm(Y ~ X))       # estimate of X should be 1

Call:
lm(formula = Y ~ X)

Residuals:
    Min      1Q  Median      3Q     Max 
-3.4101 -1.2809 -0.0926  1.1634  5.2391 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  0.01629    0.12133   0.134    0.893    
X            2.01297    0.08348  24.112   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.715 on 198 degrees of freedom
Multiple R-squared:  0.746, Adjusted R-squared:  0.7447 
F-statistic: 581.4 on 1 and 198 DF,  p-value: < 2.2e-16

Colliders

x x z z x->z y y y->z

z is a collider

X <- rnorm(N)            
Y <- rnorm(N)            # Y is independent of X
Z <- rnorm(N, 2*X + 1*Y) # Z depends on X and Y

summary(lm(Y ~ X + Z))   # estimate of X should be 0

Call:
lm(formula = Y ~ X + Z)

Residuals:
     Min       1Q   Median       3Q      Max 
-1.81241 -0.42887 -0.01173  0.42913  2.26573 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  0.02104    0.04885   0.431    0.667    
X           -0.98516    0.08510 -11.577   <2e-16 ***
Z            0.48411    0.03492  13.864   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.6895 on 197 degrees of freedom
Multiple R-squared:  0.494, Adjusted R-squared:  0.4889 
F-statistic: 96.18 on 2 and 197 DF,  p-value: < 2.2e-16

Problem: Measured confounds

x x y y x->y z z x->z y->z

z is a collider


X <- rnorm(N)            
Y <- rnorm(N)            # Y is independent of X
Z <- rnorm(N, 2*X + 1*Y) # Z depends on X and Y

compare(
  lm(Y ~ X + Z),
  lm(Y ~ X)
  )   
Coefficients:
         Estimate Std. Error  t value Pr(>|t|)    
X (+ Z) -0.972343   0.087097 -11.1639   <2e-16 ***
X       -0.107300   0.075666  -1.4181   0.1577    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Prevailing wisdom

  1. Condition on all pre-treatment predictors of the treatment assignment (e.g., propensity score logic)
  2. All post-treatment variables are “bad controls”

Mediators

x x z z x->z y y z->y

z is a post-treatment variable (mediator)

X <- rnorm(N)            
Z <- rnorm(N, 1*X)      # Z depends on X
Y <- rnorm(N, 1 + 1*Z)  # Y depends on Z

compare(
  lm(Y ~ X + Z),     # Conditional effect of X = 0   
  lm(Y ~ X)          # Total effect of X = 1 (via Z)
)
Coefficients:
        Estimate Std. Error t value Pr(>|t|)    
X (+ Z) 0.083811   0.109865  0.7629   0.4465    
X       1.149480   0.099836 11.5137   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Mediators II

x x m m x->m y y z z z->m m->y

z is a post-treatment variable, m is a mediator

set.seed(123)

X <- rnorm(N)      
Z <- rnorm(N)
M <- rnorm(N, .5*X + 3*Z)  # M depends on X and Z
Y <- rnorm(N, 1 + 2*M)     # Y depends on M

compare(
  lm(Y ~ X + Z),           # Z improves precision
  lm(Y ~ X)
)
Coefficients:
        Estimate Std. Error t value  Pr(>|t|)    
X (+ Z)  0.88714    0.16708  5.3098 2.947e-07 ***
X        0.71220    0.47851  1.4884    0.1382    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Bias Amplification

x x y y x->y z z z->x u u u->x u->y

z is a pre-treatment variable

U <- rnorm(N)
Z <- rnorm(N)
X <- rnorm(N, 1*Z + 2*U) # X depends on U and Z
Y <- rnorm(N, 0*X + 2*U) # Y depends on U, not X

compare(
  lm(Y ~ X + Z),         # True effect of X = 0
  lm(Y ~ X)
)   
Coefficients:
        Estimate Std. Error t value  Pr(>|t|)    
X (+ Z) 0.778428   0.037786  20.601 < 2.2e-16 ***
X       0.659921   0.040536  16.280 < 2.2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Instruments

x x y y x->y z z z->x u u u->x u->y

z is an instrument of x

x <- predict(lm(X ~ Z)) # get predicted values of X

summary(lm(Y ~ x))      # matches true effect of X = 0 

Call:
lm(formula = Y ~ x)

Residuals:
    Min      1Q  Median      3Q     Max 
-5.7985 -1.3678 -0.1371  1.4928  6.6315 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)
(Intercept)  0.14341    0.16169   0.887    0.376
x           -0.05111    0.16397  -0.312    0.756

Residual standard error: 2.276 on 198 degrees of freedom
Multiple R-squared:  0.0004905, Adjusted R-squared:  -0.004557 
F-statistic: 0.09718 on 1 and 198 DF,  p-value: 0.7556

Example: Intergenerational transfer

https://academic.oup.com/esr/article/34/6/603/5094485

Breen 2018

Example: Intergenerational transfer

Parent Parent Child Child Parent->Child u u u->Parent u->Child Grandparent Grandparent Grandparent->Parent Grandparent->Child

Direct effects grandparent -> child

Example: Intergenerational transfer

z z y y z->y u u u->z u->y x x x->z x->y

Z is a mediator (X -> Y) and a collider (U + X)

U <- rnorm(N)                  # Shared parent-child influences
X <- rnorm(N)                  # Grandparents           
Z <- rnorm(N, 2*X + 2*U)       # Parents          
Y <- rnorm(N, 1 + X + Z + 2*U) # Direct effect of X = 1

compare(
  lm(Y ~ X + Z),         
  lm(Y ~ X)
)   
Coefficients:
        Estimate Std. Error t value  Pr(>|t|)    
X (+ Z) -0.66469    0.11966 -5.5548  8.93e-08 ***
X        3.15471    0.29039 10.8638 < 2.2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Conclusions

In general:

  • Parents of X, which are not necessary for identification (see Instruments), are harmful for precision
  • Parents of Y which do not spoil identification are beneficial for precision

https://www.dagitty.net/dags.html


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