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?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.2Simulation 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.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.3A 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-16# 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