## ABSTRACT

In this chapter we propose a three-stage procedure for analyzing APC data. Stage 1 (Analysis without assumptions) consists of three steps. First, the linear and nonlinear effects should be separated. Second, the nonlinear effects should be graphed and their existence assessed. Third, the canonical solution line consisting of the different combinations of the linear effects that are consistent with the data should be represented in a 2D-APC graph.

In the second stage (Bounds on overall effects), researchers should carry out a set of bounds analysis involving monotonicity constraints. In the first step, using a 2D-APC graph, researchers should examine the consequences of their theoretical assumptions. In the second step, they should combine the resulting bounds with the nonlinear effects to produce bounds for the combination of the linear and nonlinear effects.

The third stage (Mechanism-based models of APC effects) also has two steps. Assuming there is data on mechanisms, in the first step one fits one or more mechanism based models and tests whether any remaining nonlinear effects are zero. In the second step, researchers should carry out a sensitivity analysis to assess the possible effects of any unobserved causal pathways.

We demonstrate this procedure using data from the GSS on religiosity, showing how parameter bounds may be nearly as informative as point estimates.