Math 300Z: In-class group activity

Simple causal paths

Mediator

A A B B A->B C C B->C

  1. To see the full path from A to C, should you include B as a covariate?
  2. To observe the relationship A \(\rightarrow\) B, should you include C as a covariate?
  3. To observe the relationship B \(\rightarrow\) C, should you include A as a covariate?
mediator <- dag_make(
  A ~ exo(),
  B ~ 2.5*A + exo(),
  C ~ -1.2*B + exo()
)
dag_draw(mediator)

Samp <- sample(mediator, size=1000)
# Question 1
lm(C ~ A, data=Samp) |> conf_interval()
lm(C ~ A + B, data=Samp) |> conf_interval()
# Question 2
lm(B ~ A, data=Samp) |> conf_interval()
lm(B ~ A + C, data=Samp) |> conf_interval()
# Question 3
lm(C ~ B, data=Samp) |> conf_interval()
lm(C ~ B + A, data=Samp) |> conf_interval()

Common cause

B B C C B->C A A A->C 

common_cause <- dag_make(
  A ~ exo(),
  B ~ exo(),
  C ~ 5*B - 2*A + exo()
)
dag_draw(common_cause)

Samp <- sample(common_cause, size=1000)
  1. To see the direct relationship between A and C, should you include B as a covariate?
  2. To see the relationship between B and A, should you include C as a covariate?
# Question 1
lm(C ~ A, data=Samp) |> conf_interval()
lm(C ~ A + B, data=Samp) |> conf_interval()
# Question 2
lm(A ~ B, data=Samp) |> conf_interval()
lm(A ~ B + C, data=Samp) |> conf_interval()

Contributing causes

A A C C A->C B B B->C 

contrib_causes <- dag_make(
  B ~ exo(),
  A ~ exo(),
  C ~ -2*B + 3*A + exo()
)
Samp <- sample(contrib_causes, size=1000)
  1. To see the effect of B on C, do you need to include A as a covariate?
# A tibble: 2 × 4
  term           .lwr  .coef   .upr
  <chr>         <dbl>  <dbl>  <dbl>
1 (Intercept) -0.0387  0.158  0.355
2 B           -2.02   -1.81  -1.60 
# A tibble: 3 × 4
  term           .lwr   .coef    .upr
  <chr>         <dbl>   <dbl>   <dbl>
1 (Intercept) -0.0449  0.0172  0.0793
2 B           -2.01   -1.94   -1.88  
3 A            2.95    3.01    3.07
  1. Which model has the better confidence interval on B?

Collider

Same network as contributing causes, but focussing on relationship between A & B.

  1. To see the correct (lack of) relationship between A and B, should you include C as a covariate?
# A tibble: 2 × 4
  term           .lwr  .coef  .upr
  <chr>         <dbl>  <dbl> <dbl>
1 (Intercept) -0.0152 0.0468 0.109
2 B           -0.0216 0.0436 0.109
# A tibble: 3 × 4
  term           .lwr     .coef   .upr
  <chr>         <dbl>     <dbl>  <dbl>
1 (Intercept) -0.0201 -0.000492 0.0191
2 B            0.562   0.586    0.609 
3 C            0.293   0.299    0.306

Confounder

A A C C A->C B B A->B B->C 

confounder <- dag_make(
  A ~ exo(),
  B ~ 4*A + exo(),
  C ~ 2*A  + 0.5*B + exo()
)
Samp <- sample(confounder, size=1000)
  1. To see the correct direct connection between B and C, should you include A as a confounder?
lm(C ~ B, data=Samp) |> conf_interval()
lm(C ~ B + A, data=Samp) |> conf_interval()