Multiple regression is no oracle, but only a golem. It is logical, but the relationships it describes are conditional associations, not causal influences. Therefore additional information, from outside the model, is needed to make sense of it. This chapter presented introductory examples of some common frustrations: multicollinearity, post-treatment bias, and collider bias. Solutions to these frustrations can be organized under a coherent framework in which hypothetical causal relations among variables are analyzed to cope with confounding. In all cases, causal models exist outside the statistical model and can be difficult to test. However, it is possible to reach valid causal inferences in the absence of experiments. This is good news, because we often cannot perform experiments, both for practical and ethical reasons.
Place each answer inside the code chunk (grey box). The code chunks should contain a text response or a code that completes/answers the question or activity requested. Problems are labeled Easy (E), Medium (M), and Hard(H).
Finally, upon completion, name your final output .html file as: YourName_ANLY505-Year-Semester.html and publish the assignment to your R Pubs account and submit the link to Canvas. Each question is worth 5 points.
6E1. List three mechanisms by which multiple regression can produce false inferences about causal effects.
#Multicollinearity, Post-treatment bias, and Collider bias..6E2. For one of the mechanisms in the previous problem, provide an example of your choice, perhaps from your own research.
# Multicollinearity
#Example: divorce rate and marriage rate; Education level and income6E3. List the four elemental confounds. Can you explain the conditional dependencies of each?
# Four elemental confounds: The Fork, The Pipe, The Collider, and The Dexcendant.
# The Confounding Fork: Z is a common cause of X and Y
# The Perplexing Pipe: X causes Z causes Y
# The Explosive Collider: X and Y jointly cause Z
# The Descendant: conditioning on A is like conditioning on Z6E4. How is a biased sample like conditioning on a collider? Think of the example at the open of the chapter.
# A biased sample like conditioning on a collider is not always relational.
# Example: Newsworthiness and Trustworthiness scores6M1. Modify the DAG on page 186 to include the variable V, an unobserved cause of C and Y: C ← V → Y. Reanalyze the DAG. How many paths connect X to Y? Which must be closed? Which variables should you condition on now?
# There are four paths in total:
# 1. X ← U ← A → C → Y
# 2. X ← U ← A → C → V → Y
# 3. X ← U → B ← C → Y
# 4. X ← U → B ← C → V → Y
#
#
# Path 1: condition on A to close
# Path 4: condition on B to open 6M2. Sometimes, in order to avoid multicollinearity, people inspect pairwise correlations among predictors before including them in a model. This is a bad procedure, because what matters is the conditional association, not the association before the variables are included in the model. To highlight this, consider the DAG X → Z → Y. Simulate data from this DAG so that the correlation between X and Z is very large. Then include both in a model prediction Y. Do you observe any multicollinearity? Why or why not? What is different from the legs example in the chapter?
#There is observed multicollinearity between X and Z because they are highly correlated.
library(rethinking)## Loading required package: rstan## Loading required package: StanHeaders## Loading required package: ggplot2## rstan (Version 2.21.2, GitRev: 2e1f913d3ca3)## For execution on a local, multicore CPU with excess RAM we recommend calling
## options(mc.cores = parallel::detectCores()).
## To avoid recompilation of unchanged Stan programs, we recommend calling
## rstan_options(auto_write = TRUE)## Do not specify '-march=native' in 'LOCAL_CPPFLAGS' or a Makevars file## Loading required package: parallel## rethinking (Version 2.12)##
## Attaching package: 'rethinking'## The following object is masked from 'package:stats':
##
## rstudentn<- 1000
b_xz<- 0.9
b_zy<- 0.7
set.seed(100)
x<- rnorm(n)
z<- rnorm(n,x*b_xz)
y<- rnorm(n,z*b_zy)
d <- data.frame(x,y,z)
cor(d)## x y z
## x 1.0000000 0.4562717 0.6924074
## y 0.4562717 1.0000000 0.6351279
## z 0.6924074 0.6351279 1.0000000m6m2<- quap( alist(
y ~ dnorm( mu , sigma ),
mu <- a + b_xz*x + b_zy*z,
a ~ dnorm( 0 , 100 ),
c(b_xz,b_zy) ~ dnorm( 0 , 100 ),
sigma ~ dexp( 1 ) ),
data=d )## Caution, model may not have converged.## Code 1: Maximum iterations reached.precis(m6m2)## mean sd 5.5% 94.5%
## a -0.008135237 0.03332760 -0.06139918 0.04512871
## b_xz 0.041459808 0.04483542 -0.03019585 0.11311547
## b_zy 0.619099380 0.03398903 0.56477834 0.67342042
## sigma 1.053730409 0.02383294 1.01564077 1.09182005