4/20/2010 36-402/608 ADA-II H. SeltmanBreakout #23: Mediation 1Simulation of an experimentx = rnorm(n=100, mean=5, sd=1)x2 = rnorm(n=100, mean=5, sd=1)y = rnorm(n=100, mean=15+3*x+4*x2, sd=2.5)summary(lm(y ~ x))# Estimate Std. Error t value Pr(>|t|)# (Intercept) 39.1052 2.7368 14.289 < 2e-16# x 2.1867 0.5406 4.045 0.000104summary(lm(y ~ x2))# Estimate Std. Error t value Pr(>|t|)# (Intercept) 32.5712 1.6107 20.22 <2e-16# x2 3.4515 0.3109 11.10 <2e-16summary(lm(y ~ x + x2))# Estimate Std. Error t value Pr(>|t|)# (Intercept) 16.8382 1.8540 9.082 1.29e-14# x 2.8418 0.2690 10.563 < 2e-16# x2 3.7677 0.2152 17.506 < 2e-16Question 1: Draw a “directed acyclic graph” (DAG) in the form of a simplediagram of the variables x, x2, and y connected with arrows showing causality,i.e. A→B means changes in A cause changes in B. Compare the estimated(causal) effects to the true effects. What happens when x and x2 are corre-lated?x → y ← x2The x coefficients (2.1867 and 2.8418) are estimates of the true causal effect of x on y(when x goes up by 1, y goes up by 3). The x2 coefficients similarly estimate the true x2causal effect of 4.Here is an example with correlated x’s:library(MASS)# 0.9 * 1 * 1 = 0.9 # covariance for cor=0.9, vars=1x34 = mvrnorm(30, mu=c(3,4), Sigma=matrix(c(1,0.9,0.9,1),2))x3 = x34[,1]x4 = x34[,2]cor(x3,x4) # 0.89y34 = rnorm(30, mean=15+3*x3+4*x4, sd=7)summary(lm(y34~x3))# Estimate Std. Error t value Pr(>|t|)# (Intercept) 23.816 3.585 6.642 3.31e-07# x3 5.686 1.075 5.291 1.25e-05summary(lm(y34~x4))# Estimate Std. Error t value Pr(>|t|)# (Intercept) 15.968 4.924 3.243 0.00305# x4 6.108 1.133 5.390 9.55e-06summary(lm(y34~x3+x4))# Estimate Std. Error t value Pr(>|t|)# (Intercept) 18.357 5.302 3.462 0.0018# x3 2.765 2.367 1.168 0.2529# x4 3.475 2.519 1.379 0.1791If x and x2 are correlated, then either or both may be “nonsignificant” in the combinedmodel. This is because with sufficient “shared” information between the x’s, neither addsinformation about y beyond what is provided by the other.Simulation of an observational studyz = rnorm(n=100, mean=5, sd=1)x = rnorm(n=100, mean=20+2*z, sd=2)y = rnorm(n=100, mean=15+3*z, sd=1.5)summary(lm(y ~ x))# Estimate Std. Error t value Pr(>|t|)# (Intercept) 7.35008 2.95870 2.484 0.0147# x 0.76111 0.09902 7.687 1.18e-11Question 2: Draw the DAG. Explain why this shows that observational studiescan’t be used to claim causal relationships.x ← z → yEven though x and z are highly correlated it would be a mistake to conclude that x causesy. In fact z cause x and y, and if we could/would manipulate x, that would have no effecton y. Variable z is a confounder (lurking variable). One or more confounding z’s is always2possible (and not unlikely) in any observational study. In a randomized experiment theaverage of z (and therefore the average causal effect of z on y) is the same for for eachlevel of x, so we can attributed any observed change in y to the manipulation of x.Simulation of a mediator (causal) modelx = rnorm(n=100, mean=20, sd=2)m = rnorm(n=100, mean=10+3*x, sd=1.5)y = rnorm(n=100, mean=15+2*m, sd=1)summary(lm(m ~ x))# Estimate Std. Error t value Pr(>|t|)# (Intercept) 10.97590 1.85094 5.93 4.55e-08# x 2.94580 0.09072 32.47 < 2e-16summary(lm(y ~ m))# Estimate Std. Error t value Pr(>|t|)# (Intercept) 15.74659 1.18391 13.3 <2e-16# m 1.99179 0.01666 119.5 <2e-16summary(lm(y ~ x))# Estimate Std. Error t value Pr(>|t|)# (Intercept) 37.431 3.775 9.915 <2e-16# x 5.876 0.185 31.758 <2e-16summary(lm(y ~ m + x))# Estimate Std. Error t value Pr(>|t|)# (Intercept) 15.91940 1.22443 13.002 <2e-16# m 1.95986 0.05733 34.188 <2e-16# x 0.10280 0.17654 0.582 0.562Question 3: Draw the DAG. Interpret each regression with respect to theDAG. The effects of X on M, M on Y, and X on Y ignoring M (with M not inthe model) are called “direct” effects. Relate the X on M and M on Y directestimates to the simulated (causal) values. The “indirect” effect of X on Yis defined as the product of the two direct effects. How does it relate to thedirect effect of X on Y? Explain what happened to the X coefficient in thefinal model.x → m → yThis is “complete” mediation when x has no effect on y except through its effect on m.According to the simulation, when x goes up by 1, m goes up by 3 on average. And when3m goes up by 3, y goes up by 6 on average. So when x goes up by 1, y goes up by 6 onaverage. In general the indirect mediated effect of x on y is the product of the X on Meffect (usually designated “a”) and the M on Y effect (“b”) which equals ab.The x coefficient becomes non-signficant and falls to near zero when it is in a regressionmodel with y because a change in x while holding m constant has no effect no y, whilea change in m while holding x constant would change y. This is another way of statingthat m mediates the effect of x on y.Question 4: Construct a simple set of non-quantitative rules that are basedon high (>0.05) vs. low (<=0.05) p-values and that could be used to assessmediated causation.A common set of rules is:1. the regression of y on x should have a significant (slope) coefficient2. the regression of m on x should have a significant coefficient3. the regression of y on m should have a significant coefficient4. the coefficient of x in the regression of y on m and x should drop to near zero, andits p-value should become non-significant.A partial mediation modelx = rnorm(n=100, mean=20, sd=2)m = rnorm(n=100, mean=10+3*x, sd=1.5)y = rnorm(n=100, mean=15+1.5*x+2*m, sd=1)summary(lm(m ~ x))# Estimate Std. Error t value Pr(>|t|) f# (Intercept) 11.85906 1.51144 7.846 5.39e-12# x 2.90992 0.07541 38.588 < 2e-16summary(lm(y ~ m))# Estimate Std. Error t value Pr(>|t|)# (Intercept) 10.30802 1.39136 7.409 4.53e-11# m 2.49497 0.01983 125.796 < 2e-16summary(lm(y ~ x))# Estimate Std. Error t value Pr(>|t|)# (Intercept) 38.4438 3.3605 11.44 <2e-164# x 7.3329 0.1677 43.74 <2e-16summary(lm(y ~ m + x))# Estimate Std. Error t value Pr(>|t|)# (Intercept) 13.36256 1.32948 10.051 < 2e-16# m 2.11494 0.06963 30.372 < 2e-16# x 1.17863 0.20919 5.634 1.72e-07Question 5: How would you modify the rules to accommodate partial media-tion?In the more common partial mediation (as opposed to complete mediation), the fourthrule becomes “the coefficient of x in the regression of y on m and x should drop, and itsp-value should rise.This additional example shows that use of mediation analysis does not protect against falsecausal conclusions in