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,collider bias6E2. For one of the mechanisms in the previous problem, provide an example of your choice, perhaps from your own research.
#I want to test the living standard with a person's education and income. Because education and income is highly correlated, I will have multicollinearity in my model. 6E3. List the four elemental confounds. Can you explain the conditional dependencies of each?
# 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.
# Use the example of the editor. The editor want to choose both high newsworthiness and trustworthiness scores proposal to fund, but the relationship between newsworthiness and trustworthiness scores is negative. Therefore if the editor choose one proposal with high trustworthiness scores, it will have low trustworthiness scores.6M1. 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?
# X<-U<-A->C<-V->Y
#this path conatains a fork U<-A->C, then a pipe X<-U, and a collider C<-V->Y. And it is open at the left side, X←U←A. If no conditioning on V, there is no assocation at the right hand side, C<-V->Y. To close this path, we need to condition on A.
# X<-U->B<-C<-V->Y
#This path contains a collider in the middle, X<-U->B, and B<-C<-V. And a second collider at the right hand side, C<-V->Y. This path is closed if no conditioning on B or V.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?
n<- 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)
m1 <- data.frame(x,y,z)
cor(m1)## x y z
## x 1.0000000 0.4562717 0.6924074
## y 0.4562717 1.0000000 0.6351279
## z 0.6924074 0.6351279 1.0000000# There is multicollinearity. The difference is that in the legs example, bad priors were used.