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. Make sure to include plots if the question requests them. 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.
# Post-treatment bias; collider bias; multicollinearity.6E2. For one of the mechanisms in the previous problem, provide an example of your choice, perhaps from your own research.
# Post-treatment bias: Control for race and voting intentions on the Vote.6E3. List the four elemental confounds. Can you explain the conditional dependencies of each?
# Fork: X<-Z->Y. X and Y are independent, conditional on Z.
# Pipe: X->Z->Y. Conditioning of the middle variable blocks the path
# Collider: X->Z<-Y. There is no association between X and Y until condition on Z.
# Descendant: X->Z->Y & Z->A Conditioning on A is like conditioning on Z.6E4. How is a biased sample like conditioning on a collider? Think of the example at the open of the chapter.
#The concept for collider is the collider, when being conditioned on, creates statistical but not necessarily causal associations for its cause. If the switch of a light is controlled in a study, then if a light can be turned on might be associated with electricity, even though in reality both electricity and switch matter.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. Draw the DAG. How many paths connect X to Y? Which must be closed? Which variables should you condition on now?
# (1) X<-U<-A->C<-V->Y: This path is open. Without conditioning on V, there is no relation to the right hand side, C<-V->Y. To close this path, condition on A.
# (2) X<-U->B<-C<-V->Y: This path is closed, since it contains colliders (X<-U->B, and B<-C<-V; 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?
library(rethinking)
N <- 1000
set.seed(77)
x <- rnorm(N)
z <- rnorm(N, x* 0.99)
y <- rnorm(N, z*0.90)
df <- data.frame(x, y, z)
cor(df)## x y z
## x 1.0000000 0.5392626 0.7030183
## y 0.5392626 1.0000000 0.7740856
## z 0.7030183 0.7740856 1.0000000# There is multicollinearity because the correlation is >0.5