Adjusting for the wrong confounders is one of the ways to create spurious correlations. For example, consider the causal system X -> Z <- Y in which X and

Adjusting for the wrong confounders is one of the ways to create spurious correlations. For example, consider the causal system X -> Z <- Y in which X and Y are random coin tosses (heads=0, tails=1) and Z is their sum. Even though X and Y independent, they will be correlated if you control for Z. If you want to see it yourself, let's simulate this system in R: # Simulate 1000 trials in which X and Y are random coin tosses (T or F) # and Z is their sum. > X <- rnorm(1000) < 0.5 > Y <- rnorm(1000) < 0.5 > Z <- X + Y # As expected, X and Y are not correlated: > summary(lm(Y ~ X)) Call: lm(formula = Y ~ X) Residuals: Min 1Q Median 3Q Max -0.6947 -0.6893 0.3053 0.3108 0.3108 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.694704 0.025816 26.910 <2e-16 *** XTRUE -0.005455 0.031330 -0.174 0.862 <<< No correlation. --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.4625 on 998 degrees of freedom Multiple R-squared: 3.038e-05, Adjusted R-squared: -0.0009716 F-statistic: 0.03032 on 1 and 998 DF, p-value: 0.8618 # However, if we include Z as a possible confounder, we create a # spurious correlation between X and Y: > summary(lm(Y ~ X + Z)) Call: lm(formula = Y ~ X + Z) Residuals: Min 1Q Median 3Q Max -2.45e-14 -8.40e-18 2.04e-17 2.04e-17 3.59e-15 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -3.370e-16 6.292e-17 -5.356e+00 1.05e-07 *** XTRUE -1.000e+00 8.241e-17 -1.213e+16 < 2e-16 *** <<< Oops! Z 1.000e+00 5.873e-17 1.703e+16 < 2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 8.582e-16 on 997 degrees of freedom Multiple R-squared: 1, Adjusted R-squared: 1 F-statistic: 1.449e+32 on 2 and 997 DF, p-value: < 2.2e-16