Conditioning on W will open the path UWVY, allowing unwanted association to flow from U to Y. This target effect is equal to the unconditional association of U and Y, because there is no unconditional biasing path. For example, suppose our target effect is the net effect of U on Y in Figure 1 (associations transmitted via UXY, UWY, and UWXY ). Ĭonditioning can also produce bias due to closing target paths. Bias that results from such conditioning on a shared effect of the target variables is often called “Berksonian,” in honor of the discoverer of this type of bias, Joseph Berkson. In other words, conditioning on W transforms UWV into a biasing path which is not a confounding path, because it does not terminate with an arrow into V. Because there is no causal pathway from U to V, this effect is zero, or “null.” But conditioning on W will open the path UWV, allowing unwanted association to flow from U to V. For example, suppose our target effect is the net effect of U on V in Figure 1. Not all biasing paths opened by conditioning are confounding paths, however. Thus, conditioning on W in Figure 1 will “control” (remove) any confounding of the X effect on Y that was present in the original system, but at the same time may introduce new confounding by opening paths that were previously closed. In observational sciences, however, “control of a variable” is often used more broadly to include conditioning on a variable, whether it removes bias or creates bias. It should be noted that, in accord with ordinary language, experimental sciences use the term “control” to refer to a physical alteration of a system to remove sources of bias, such as randomization. At the same time, the open paths XUWY, XWVY, and XWY become closed and can no longer transmit associations, and thus are no longer confounding paths. For example, if we condition on W in Figure 1, the closed path XUWVY becomes open and now can transmit associations it thus becomes a confounding path for the X effect on Y. As a consequence, the status of paths may reverse. One key notion is that the open/closed status of a variable along a path is reversed by conditioning (stratifying) on the variable: A collider becomes open and noncollider becomes closed. Independencies in the conditional distribution Pr( g| c) implied by the graph may then be seen using just a few more concepts. Let G and C be disjoint subsets of variables in the graph, with g and c being sets of values for G and C. Sander Greenland, in Philosophy of Statistics, 2011 Conditioning and Control
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