Comparing the strengths of two causes of an effect − zeroing-out, adding-in, and the need for baselines

Abstract Suppose 50-year-old Sue now has lung cancer, due to the fact that C = c & A = g (meaning that Sue smoked c cigarettes and inhaled g grams of asbestos over the previous 30 years), and that neither cause caused the other. Given this, a retrospective question arises – did one of those actual causes have a stronger influence than the other on her getting lung cancer? We propose a “zeroing-out” criterion for making sense of this question; it says that C = c was a stronger causal influence than A = g precisely when Pr(lung cancer | C = c & A = g) – Pr(lung cancer | C = 0 & A = g) > Pr(lung cancer | C = c & A = g) – Pr(lung cancer | C = c & A = 0). Zeroing-out uses Pr(lung cancer | C = c & A = g) as a baseline and relates that baseline to two counterfactual probabilities. We discuss how zeroing-out applies to three evolutionary examples − the influences of selection and drift on the fixation of an allele, the influences of group and individual selection on the evolution of altruism, and the influences of stabilizing selection and ancestral influence (aka “phylogenetic inertia”) on the evolution of tetrapody in land vertebrates. Zeroing-out differs from an “adding-in” criterion, which uses Sue’s probability of having lung cancer at age 50, given her actual state at age 20 (at which time she was cancer free and C = 0 & A = 0) as a baseline and asks whether her risk of having lung cancer at age 50 would be greater if C = c were true of the 30 years in between than it would be if A = g were true of those years. Zeroing-out and adding-in generate identical criteria for comparing causal influences in this example because the probabilities are related “monotonically” (a concept we define). We then describe examples in which monotonicity fails and the two criteria differ. We prove theorems that describe when the two criteria disagree and when they do not. We then consider how zeroing-out and adding-in are related to six quantitative measures of causal strength that have been proposed. Inter alia, we discuss how our framework is related to interventionism.