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.
# selection bias
# missing cofounding variables
# colinearity6E2. For one of the mechanisms in the previous problem, provide an example of your choice, perhaps from your own research.
# For example, in a study of relationship for hours of study and final test result. people who are more willing to partcipate in this kinds of study might tends to be more willing to spend time in studying.
# Thus, these people might be more efficient in terms of the learning. Thus, the outcome of the study might be not be able to generatlized for casual relationship.
# 6E3. List the four elemental confounds. Can you explain the conditional dependencies of each?
#The Descendant- Variable X causes Z and variable Z causes A and Y. Condition on Z is equivalent to condition on A
#The Fork- Variable Z is a common cause of variables X and Y. X and Y are independent and condition on Z
#The Pipe- Variable X causes Z and Z further causes variable Y. X and Y are independent and condition on Z
#The Collider- Both variables X and Y cause variable Z. X and Y are dependent when condition on Z6E4. How is a biased sample like conditioning on a collider? Think of the example at the open of the chapter.
# Both electricity and swtich determine if the light is on. If the switch is controlled for by study design (selection bias), a distorted association will occur betwen electricity and light would occur. "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?
# 4 paths between X to Y:
# (1) X<-U<-A->C<-V->Y: this path contains a fork U<-A->C, then a pipe X<-U, and a collider C<-V->Y. This is open at the left side, X←U←A. If no conditioning on V, there is no relation to the right hand side, C<-V->Y. To close this path, we need to condition on A.
# (2) 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?
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)## Loading required package: parallel## rethinking (Version 2.13)##
## Attaching package: 'rethinking'## The following object is masked from 'package:stats':
##
## rstudent#Samples
N <- 100
set.seed(10)
x <- rnorm(N)
z <- rnorm(N, x* 0.99)
y <- rnorm(N, z*0.90)
#Data frame
df <- data.frame(x, y, z)
cor(df)## x y z
## x 1.0000000 0.5476033 0.6708713
## y 0.5476033 1.0000000 0.7451832
## z 0.6708713 0.7451832 1.0000000##
## There are multicollilnearity. Given that we assign beta between x and z to be 0.99. However, it is very suprising to see that in corr table we are able seeing 0.99 level of correlation.