Lecture 05 - Elemental Confounds

Author

Isabella C. Richmond

Published

February 22, 2024

Rose / Thorn

Rose: completely changed my approach to science

Thorn: why set M and not A in intervention?

Correlation

  • correlation is common in nature, causation is sparse
  • correlation over time series especially common
  • scientific question/estimand -> recipe/estimator -> result/estimate

Association & Causation

  • we must defend against confounding in our stats
  • confounds mislead us - feature of the sample + how we use it
  • four elemental confounds
    • causes of confounds can be extremely diverse but they are all at their core made up of relationships between 3 variables

The Fork

  • X and Y are associated \(Y \not\!\perp\!\!\!\perp X\) (not independent)
  • Share a common cause Z
  • Once stratified by Z, no association \(Y \perp\!\!\!\perp X | Z\)
dag <- dagify(
    X ~ Z,
    Y ~ Z
)

ggdag(dag) +
    theme_dag()

  • simulation:
n <- 1000
Z <- rbern(n, 0.5)
X <- rbern(n, (1-Z)*0.01 + Z*0.9)
Y <- rbern(n, (1-Z)*0.01 + Z*0.9)

# correlated
cor(X, Y)
[1] 0.7962541
# no longer correlated when we pull out Z
cor(X[Z==0], Y[Z==0])
[1] -0.00895323
cor(X[Z==1], Y[Z==1])
[1] -0.08114448
cols <- c(4,2)
n <- 300
Z <- rbern(n, 0.5)
X <- rnorm(n, (1-Z)*0.01 + Z*0.9)
Y <- rnorm(n, (1-Z)*0.01 + Z*0.9)

plot(X, Y, col=cols[Z+1]) + 
  abline(lm(Y[Z==1] ~ X[Z==1]), col = 2) + 
  abline(lm(Y[Z==0] ~X[Z==0]), col = 4) + 
  abline(lm(Y~ X))

integer(0)

  • example:

    • why do regions of the USA with higher rates of marriage also have higher rates of divorce?

    • estimand: causal effect of marriage rate on divorce rate

dag <- dagify(
    D ~ A,
    M ~ A,
    D ~ M
)

ggdag(dag) +
    theme_dag()

  • causes are not in the data

  • is effect of marriage rates on divorce rates just a symptom of common cause age?

  • need to break the fork to test direct effect – stratify by A

  • continuous variable stratification means that we are adding to that variable essentially to the intercept

    • stratifying means for every level of the stratified value, what is the association between the other two variables?

    • what does this mean for continuous variables?

    • every value of A produces a different relationship between D and M

    • \[ \mu_i = (\alpha + \beta_A*A) + B_D*D \]

  • standardizing variables is almost always helpful in linear regression

    • transform measurement scales to the scale we want, as long as we remember how we changed them
    • make the mean zero and divide by sd
    • alpha prior is the value of average divorce rate (when standardized = 0)
    • beta is within 1 and -1 for standardized variables because outside of those values the variable would be explaining all of the variation

  • to stratify by A, include as a term in the linear model:

    • \(D _i \sim Normal(\mu _i, \sigma)\)

    • \(\mu _i = \alpha + \beta_MM_i + \beta_AA_i\)

    • \(\alpha \sim Normal(0, 0.2)\)

    • \(\beta_M \sim Normal(0,0.5)\)

    • \(\beta_A \sim Normal(0, 0.5)\)

    • \(\sigma \sim Exponential(1)\)

library(rethinking)
data(WaffleDivorce)
d <- WaffleDivorce

# model
dat <- list(
    D = standardize(d$Divorce),
    M = standardize(d$Marriage),
    A = standardize(d$MedianAgeMarriage)
)

m_DMA <- quap(
    alist(
        D ~ dnorm(mu,sigma),
        mu <- a + bM*M + bA*A,
        a ~ dnorm(0,0.2),
        bM ~ dnorm(0,0.5),
        bA ~ dnorm(0,0.5),
        sigma ~ dexp(1)
    ) , data=dat )

plot(precis(m_DMA))

  • a causal effect is a manipulation of the generative model, an intervention

  • distribution of D when we intervene (“do”) M \(p(D|do(M))\)

    • implies deleting all arrows into M and simulating D
    • set values of M while holding A constant
post <- extract.samples(m_DMA)

# sample A from data
n <- 1e3
As <- sample(dat$A, size = n, replace = T)

# simulate D for M=0 (sample mean)
DM0 <- with(post, rnorm(n, a + bM*0 + bA*As, sigma))

# simulate D for M = 1 (+1 standard deviation)
# use the same A values 
DM1 <- with(post, rnorm(n, a + bM*1 + bA*As, sigma))

# contrast
M10_contrast <- DM1 - DM0
dens(M10_contrast, lwd=4, col=2, xlab="effect of increase in M")

The Pipe

  • X and Y are associated \(Y \not\!\perp\!\!\!\perp X\)

  • Influence of X on Y transmitted through Z

  • Once stratified by Z, no association \(Y \perp\!\!\!\perp X | Z\)

dag <- dagify(
    Y ~ Z,
    Z ~ X
)

ggdag(dag) +
    theme_dag()

n <- 1000
X <- rbern(n, 0.5)
Z <- rbern(n, (1-X)*0.01 + X*0.9)
Y <- rbern(n, (1-Z)*0.01 + Z*0.9)
cor(X,Y)
[1] 0.7876689

  • everything that Y knows about X, is already known by Z

    • once you learn Z, there is nothing more to learn about the association
cols <- c(4,2)
n <- 300
X <- rbern(n)
Z <- rbern(n, inv_logit(X))
Y <- rbern(n, (2*Z-1))
Warning in rbinom(n, size = 1, prob = prob): NAs produced
# plot
  • including the mediator at the wrong time can lead to incorrect inference
dag <- dagify(
   H1 ~ H0 + Treat,
   H1 ~ Fungus,
   Fungus ~ Treat
   
)

ggdag(dag) +
    theme_dag()

  • what is the total causal effect of treatment?

  • Treat -> Fungus -> H1 is a pipe, should not stratify by F

  • post-treatment bias

    • if you stratify by a consquence of the treatment, it can induce post-treatment bias - gives you a misleading estimate of what you’re after

    • consequences of treatment should not usually be included in the estimator

The Collider

  • X and Y are not associated (share no causes) \(Y \perp\!\!\!\perp X\)

  • X and Y both influence Z

  • Once stratified by Z, X and Y are associated \(Y \not\!\perp\!\!\!\perp X | Z\)

dag <- dagify(
    Z ~ X + Y
)

ggdag(dag) +
    theme_dag()

n <- 1000
X <- rbern(n, 0.5)
Y <- rbern(n, 0.5)
Z <- rbern(n, ifelse(X+Y>0, 0.9, 0.2))
  • strong correlations caused by the collider could be read as correlation but causes are not in the data
cols <- c(4,2)
N <- 300
X <- rnorm(N)
Y <- rnorm(N)
Z <- rbern(N, inv_logit(2*X+2*Y-2))

plot(X,Y, cols = cols[Z+1]) + 
  abline(lm(Y[Z==1] ~ X[Z==1]), col = 2) + 
  abline(lm(Y[Z==0] ~X[Z==0]), col = 4) + 
  abline(lm(Y~ X))
Warning in plot.window(...): "cols" is not a graphical parameter
Warning in plot.xy(xy, type, ...): "cols" is not a graphical parameter
Warning in axis(side = side, at = at, labels = labels, ...): "cols" is not a
graphical parameter

Warning in axis(side = side, at = at, labels = labels, ...): "cols" is not a
graphical parameter
Warning in box(...): "cols" is not a graphical parameter
Warning in title(...): "cols" is not a graphical parameter

integer(0)

  • sometimes samples come already stratified by collider

  • associations among the things you have measured post-selection is dangerous because the selection is often collider bias

  • endogenous colliders

    • if you include a collider in your estimator you can induce a spurious correlation

    • happens within your analysis

  • example: age and happiness

    • estimand: influence of age on happiness

    • possible confound: marital status

    • suppose age has zero influence on happiness, but that both age and happiness influence marital status

dag <- dagify(
   M ~ A + H
)

ggdag(dag) +
    theme_dag()

  • stratified by marital status, negative association between age and happiness even though they are not related -> age/happiness are unrelated

The Descendant

  • how it behaves depends on what it is attached to

  • A is the descendant, contains information of its “parent”

  • X and Y are causally associated through Z \(Y \not\!\perp\!\!\!\perp X\)

  • A holds information about Z

  • Once stratified by A, X and Y less associated (if strong enough) \(Y \perp\!\!\!\perp X | A\)

dag <- dagify(
   Z ~ X,
   Y ~ Z,
   A ~ Z
)

ggdag(dag) +
    theme_dag()

n <- 1000
x <- rbern(n, 0.5)
z <- rbern(n, (1-x)*0.1 + x*0.9)
y <- rbern(n, (1-z)*0.1 + z*0.9)
a <- rbern(n, (1-z)*0.1 + z*0.9)
  • descendants are everywhere - proxies