The Book of Why: The New Science of Cause and Effect

by Judea Pearl

Although the miasma theory has by now been discredited, poverty was undoubtedly a confounder, as was location. But even without measuring these (because Snow’s door-to-door detective work only went so far), we can still use instrumental variables to determine how many lives would have been saved by purifying the water supply.

Thus we have two equations: ab = rZY and a = rZX. If we divide the first equation by the second, we get the causal effect of X on Y: b = rZY/rZX. In this way, instrumental variables allow us to perform the same kind of magic trick that we did with front-door adjustment: we have found the effect of X on Y even without being able to control for, or collect data on, the confounder, U.

Also notice that we have gotten information on the second rung of the Ladder of Causation (b) from information about the first rung (the correlations, rZY and rZX). We were able to do this because the assumptions embodied in the path diagram are causal in nature, especially the crucial assumption that there is no arrow between U and Z. If the causal diagram were different—for example, if Z were a confounder of X and Y—the formula b = rZY/rZX would not correctly estimate the causal effect of X on Y. In fact, these two models cannot be told apart by any statistical method, regardless of how big the data.

He was interested in predicting how the output of a commodity would change if a tariff were imposed, which would raise the price and therefore, in theory, encourage production. In economic terms, he wanted to know the elasticity of supply.

Philip Wright borrowed the idea of path coefficients from his son. No economist had ever before insisted on the distinction between causal coefficients and regression coefficients; they were all in the Karl Pearson–Henry Niles camp that causation is nothing more than a limiting case of correlation. Also, no one before Sewall Wright had ever given a recipe for computing regression coefficients in terms of path coefficients, then reversing the process to get the causal coefficients from the regression. This was Sewall’s exclusive invention.

It shows that Philip took the trouble to understand his son’s theory and articulate it in his own language.

Simpson’s paradox refers to a trend that seems to go one direction in each layer of a population (women are accepted at a higher rate in each department) but in the opposite direction for the whole population (men are accepted at a higher rate in the university as a whole).