Assignment #6

Questions

6-1. Use the Waffle House data, data(WaffleDivorce), to find the total causal influence of number of Waffle Houses on divorce rate. Justify your model or models with a causal graph (draw a DAG).

data(WaffleDivorce)
data <- WaffleDivorce

DAG <- dagitty("dag {
    A -> D
    A -> M -> D
    A <- S -> M
    S -> W -> D
  }")

coordinates(DAG) <- list(
  x = c(A = 1, S = 1, M = 2, W = 3, D = 3),
  y = c(A = 1, S = 3, M = 2, W = 3, D = 1)
)
drawdag(DAG)

adjustmentSets(DAG, exposure = "W", outcome = "D")

## { A, M }
## { S }

model <- quap(
  alist(
    Divorce ~ dnorm(mu, sigma),
    mu <- a + bS * South + bW * WaffleHouses,
    a ~ dnorm(0, 0.2),
    c(bS, bW) ~ dnorm(0, 0.5),
    sigma ~ dexp(1)
  ),
  data = data
)
precis(model)

##             mean         sd        5.5%      94.5%
## a     0.24107433 0.20212236 -0.08195624 0.56410490
## bS    0.21518881 0.49545902 -0.57665040 1.00702801
## bW    0.06218672 0.01549949  0.03741555 0.08695789
## sigma 7.82233826 0.72280963  6.66714888 8.97752765

6-2. Build a series of models to test the implied conditional independencies of the causal graph you used in the previous problem. If any of the tests fail, how do you think the graph needs to be amended? Does the graph need more or fewer arrows? Feel free to nominate variables that aren’t in the data.

impliedConditionalIndependencies(DAG)

## A _||_ W | S
## D _||_ S | A, M, W
## M _||_ W | S

# Check A _||_ W | S
model1 <- quap(
alist(
    MedianAgeMarriage ~ dnorm(mu, sigma),
    mu <- a + bS * South + bW * WaffleHouses,
    a ~ dnorm(0, 0.2),
    c(bS, bW) ~ dnorm(0, 0.5),
    sigma ~ dexp(1))
  ,
  data = data
)

precis(model1)

##              mean         sd        5.5%      94.5%
## a      0.10712613 0.20030749 -0.21300392  0.4272562
## bS     0.08254169 0.49920381 -0.71528241  0.8803658
## bW     0.15553093 0.03841526  0.09413593  0.2169259
## sigma 19.75045318 1.57124236 17.23930442 22.2616019

# Check D _||_ S | A, M, W
model2 <- quap(
  alist(
    Divorce ~ dnorm(mu, sigma),
    mu <- a + bA * MedianAgeMarriage + bS * South + bM * Marriage + bW * WaffleHouses,
    a ~ dnorm(0, 0.2),
    c(bA, bS, bM, bW) ~ dnorm(0, 0.5),
    sigma ~ dexp(1)
  ),
  data = data
)

precis(model2)

##              mean         sd        5.5%      94.5%
## a     0.019664997 0.19996091 -0.29991115 0.33924115
## bA    0.168845872 0.04072965  0.10375202 0.23393972
## bS    0.406129109 0.41291916 -0.25379546 1.06605368
## bM    0.246454650 0.05151472  0.16412418 0.32878512
## bW    0.005314709 0.00405780 -0.00117044 0.01179986
## sigma 1.639864521 0.16147716  1.38179284 1.89793620

# Check M _||_ W | S
model3 <- quap(
  alist(
    Marriage ~ dnorm(mu, sigma),
    mu <- a + bS * South + bW * WaffleHouses,
    a ~ dnorm(0, 0.2),
    c(bS, bW) ~ dnorm(0, 0.5),
    sigma ~ dexp(1)
  ),
  data = data
)

precis(model3)

##             mean         sd        5.5%      94.5%
## a      0.1267937 0.20045205 -0.19356743  0.4471547
## bS     0.1059600 0.49881614 -0.69124453  0.9031645
## bW     0.1224513 0.03098106  0.07293764  0.1719651
## sigma 15.8854610 1.31519414 13.78352671 17.9873952

6-3. Consider your own research question. Draw a DAG to represent it. What are the testable implications of your DAG? Are there any variables you could condition on to close all backdoor paths? Are there unobserved variables that you have omitted? Would a reasonable colleague imagine additional threats to causal inference that you have ignored?

# Research question: How the higher demand for houses impact house prices.
# Implication: There is a direct relationship between the housing demand and house prices; higher housing demand increases house prices.
# In the long term, factors such as increase in the number of millennials who have high savings, increase in wealthy immigrants would continue to increase the demand for houses. Constructions company will catch up this trend and construct more houses Therefore, there is a backdoor channel that the increase in housing demand will also make construction companies expand their construction plans and sell more houses.The increasing supply of houses may impact house prices. 

DAG2 <- dagitty("dag {
    Demand -> Price
    Demand -> Construction -> Price

  }")

drawdag(DAG2)

adjustmentSets(DAG2, exposure = "Construction", outcome = "Price")

## { Demand }

6-4. For the DAG you made in the previous problem, can you write a data generating simulation for it? Can you design one or more statistical models to produce causal estimates? If so, try to calculate interesting counterfactuals. If not, use the simulation to estimate the size of the bias you might expect. Under what conditions would you, for example, infer the opposite of a true causal effect?

N <-50000
set.seed(123)
demand <-rnorm(N,10,3) 
construction <-demand*rnorm(N,0,2)+rnorm(N,10,3)
price <-demand - construction + rnorm(N,0,2)
data2 <-data.frame(demand,construction,price)

model4 <- quap(
alist(
    price ~ dnorm(mu, sigma),
    mu <- a + bD * demand + bS * construction,
    a ~ dnorm(0, 1),
    c(bD, bS) ~ dnorm(0, 2),
    sigma ~ dexp(1))
  ,
  data = data2
)

precis(model4)

##               mean          sd        5.5%       94.5%
## a      0.001632909 0.031492266 -0.04869782  0.05196363
## bD     1.002050653 0.002989124  0.99727346  1.00682785
## bS    -1.000808625 0.000425442 -1.00148856 -1.00012869
## sigma  2.006203306 0.006343963  1.99606443  2.01634218

Assignment #6

Hien Nguyen

2022-08-01

Week 6 - Good & Bad Controls

Questions