The 2 x 2 x 2 case in causality, of an effect, a cause and a confounder. A cross-over’s guide to the 2 x 2 x 2 contingency table

Basic causality is that a cause is present or absent and that the effect follows with a success or not. This happy state of affairs becomes opaque when there is a third variable that can be present or absent and that might be a seeming cause. The 2 x 2 x 2 layout deserves the standard name of the ETC contingency table, with variables Effect, Truth and Confounding and values {S, -S}, {C, -C}, {F, -F}. Assuming the truth we can find the impact of the cause from when the confounder is absent. The 8 cells in the crosstable can be fully parameterized and the conditions for a proper cause can be formulated, with the parameters interpretable as regression coefficients. Requiring conditional independence would be too strong since it neglects some causal processes. The Simpson paradox will not occur if logical consistency is required rather than conditional independence. The paper gives a taxonomy of issues of confounding, a parameterization by risk or safety, and develops the various cases of dependence and (conditional) independence. The paper is supported by software that allows variations. The paper has been written by an econometrician used to structural equations models but visiting epidemiology hoping to use those techniques in experimental economics.