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.
# Multicollinearity: A good example from my work is that if a pharmaceutical company wants to evaluate the risk of getting a certain disease based on some factors such as age, sex, height, weight, diet, etc. This design of study will lead to multicollinearity and false inference and statistical analysis result because obviously some of the supposed independent varibles are actually high correlated such as weight and diet.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. X and Y are independent, conditional on Z. In both a fork and a pipe, conditioning of the middle variable blocks the path.
# Collider: X->Z<-Y. Different from the other two types of relationship, there is no association between X and Y unless condition on Z. Conditioning on Z, information flows between X and Y.
# Descendent: Condition on a descendent of Z in the pipe.6E4. How is a biased sample like conditioning on a collider? Think of the example at the open of the chapter.
# As we discussed in the trustworthiness/newsworthiness example, a biased sample either overrepresentes or underrepresentes a portion of the population in a study, which will creates a selection-distortion effect and inccur dependencies between variables. In the example, there will be a negative association between newsworthy and trustworthy when conditioning on the final scores, which is due to biased sample, i.e. the selected proposalare will have either high newsworthiness or high trust worthiness.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?
# There are four paths from X to Y:
# 1. X<-U<-A->C<-V->Y, this path contains a fork U<-A->C, and a collider C<-V->Y
# 2. X<-U->B<-C<-V->Y, this path contains a collider X<-U->B
# 3. X <- U <- A -> C -> Y
# 4. X <- U -> B <- C -> Y, this path contains a collider in the middle, X<-U->B, and B<-C<-V
#3 should be closed and condition on A to close it.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)
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 1.0000000# Yes, there is multicollinearity between X and Z as X and Z are strongly correlated.The main difference is that in this example, Z will provide extra information about but the length of two legs are not providing same information in legs example.