Directed Acyclic Graphs课件.ppt

Causal Sufficiency We want to measure the effect of y on z (write this as ?y ) and we have x, y and z in our study, but we leave another variable, w, out of the study. The world is generated by the graph: w x ?w y 0 ?y z 精品文档 Causal Sufficiency Continued If we fail to include w in our sample we will end up with the following graph (let by and bx represent our measured effects based on x,y and z): w x y E( bx ) ? 0 E (by ) ? ?y z The key to causal sufficiency is that we don’t have to have every variable that causes z in our study. But we do need all variables that cause two or more variables in our study. (Here E (by ) is the expected value of or measure of the effect of y on z). 精品文档 Faithfulness Here we assume that if we measure the correlation between two variables, say x and y, as zero, it is zero because there is no edge between x and y in the “true” model. It is not zero because of cancellation of deeper parameters. 精品文档 Faithfulness Continued Say we have the following true model: ?xy x y ?xz ?yz z The”true” parameters connecting these variables are given by the betas (?xy ?xz ?yz ). 精品文档 Faithfulness Continued If it so happens that in the real world: ?xz = - ?xy ?yz then the correlation between x and z will equal zero. PC algorithm will remove the edge between x and z, even though the true model has such an edge. 精品文档 Example: Traffic Fatalities Variables and Data taken from Peltzman Journal of Political Science 1977 Eight variables for the U.S. 1947 - 1974 data: number of traffic fatalitie