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).
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6E1. List three mechanisms by which multiple regression can produce false inferences about causal effects.
# multicollinearity
# conditioning on post-treatment variables
# collider bias6E2. For one of the mechanisms in the previous problem, provide an example of your choice, perhaps from your own research.
# Multicollinearity: usually means a very strong association between two or more predictor variables.
# one example: education level and annual income, the higher the education level, and the higher the annual income.6E3. List the four elemental confounds. Can you explain the conditional dependencies of each?
# 1. Fork: X<-Z->Y. Y ⊥⊥ X|Z, X and Y are independent, conditional on Z.
# 2. Pipe: X->Z->Y. Y ⊥⊥ X|Z, X and Y are independent, conditional on Z.
# 3. Collider: X->Z<-Y. Y⊥̸⊥ X|Z no association between X and Y unless condition on Z. Conditioning on Z, information flows between X and Y.
# 4. Descendant: a variable influenced by another. when condition on a descendant, weakly condition on its parent, and so what happens depends on the parent's place in the graph.6E4. How is a biased sample like conditioning on a collider? Think of the example at the open of the chapter.
# scientific studies' newsworthy and trustworthy as an example.
# When selecting proposals to be funded, editors will rate newsworthiness and trustworthiness equally, then pick the top 10% of the proposals putting newsworthy score and trustworthy score together.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 five paths to connect X and Y
# 1. X -> Y (the causal path)
# 2. X <- U -> B <- C -> Y
# 3. X <- U -> B <- C <- V -> Y
# 4. X <- U <- A -> C -> Y
# 5. X <- U <- A -> C <- V -> Y
# we want to let path 1 open and make sure all of the others are closed, as they are non- causal paths that confound inference.
# before, all the paths through B are closed, as B is collider, so we don't condition on B.
# new paths through A and V are closed, as C is a collider on the paths. conditon on A to close all non-causal paths.
# in the chapter, V is absent, it's also ok to condition on C. it isn't now6M2. 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
X <- rnorm(N)
Z <- rnorm(N,X,0.1)
Y <- rnorm(N,Z)
cor(X,Z)## [1] 0.9947315it’s high correlation, x and z are nearly same variable. regress:
library(rethinking)## Loading required package: rstan## Warning: package 'rstan' was built under R version 4.0.3## Loading required package: StanHeaders## Loading required package: ggplot2## Warning: package 'ggplot2' was built under R version 4.0.3## 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## Loading required package: dagitty## Warning: package 'dagitty' was built under R version 4.0.3## rethinking (Version 2.01)##
## Attaching package: 'rethinking'## The following object is masked from 'package:stats':
##
## rstudentm_6M2 <- quap(
alist(
Y ~ dnorm( mu , sigma ),
mu <- a + bX*X + bZ*Z,
c(a,bX,bZ) ~ dnorm(0,1),
sigma ~ dexp(1)
) , data=list(X=X,Y=Y,Z=Z) )
precis( m_6M2 )## mean sd 5.5% 94.5%
## a 0.019572463 0.03171272 -0.03111059 0.07025551
## bX 0.006784403 0.28750350 -0.45270171 0.46627052
## bZ 0.964994192 0.28646915 0.50716116 1.42282723
## sigma 1.002871111 0.02240962 0.96705621 1.03868601The SD are larger than we can expect, for such a large sample. This is an example of conditioning on a post-treatment variable, it knocks out X as Z mediates X’s effect. We can’t explain multicollinearity outside a specific model. It isn’t only a property of the data