What is the general mathematical description of necessary conditions?

What is the mathematical description of a single necessary condition?

What is the mathematical description of multiple necessary conditions?

What is the general mathematical description of necessary conditions?

The general mathematical description of necessary conditions is Y ≤ f_i(X_i), where Y is the outcome, X_i is the condition, and f_i(X_i) is the ceiling line for X_i. NCA assumes that the ceiling line is non-decreasing. This allows making statements like "X_i ≥ X_ic is necessary for Y = Y_c", where C is a point (X_ic ,Y_c) on the ceiling line.  

What is the mathematical description of a single necessary condition?

With one condition, NCA puts a line on the data and this line is the ceiling line f₁(X₁), such that X₁ ≥ X_1c is necessary for Y = Y_c. The maximum possible Y = y_c for a given value X₁ = x₁,  is  y_c = f₁(x₁).

What is the mathematical description of multiple necessary conditions?

With two conditions (X₁ and X₂), there is a three dimensional space (X₁,X₂,Y) and  NCA puts a blanket on the data (ceiling surface). The projection of the ceiling surface on the X₁,Y plane is the ceiling line f₁(X₁) for X₁. The projection of the ceiling surface on the X₂,Y plane is the ceiling line f₂(X₂) for X₂.  Both X₁ ≥ X_1c  AND X₂ ≥ X_2c are necessary for Y = Y_c. The maximum possible Y = y_c for given values X₁ = x₁,  and X₂= x₂ is y_c = min {f₁(x₁), f₂(x₂)}.

With more than two conditions there is a multidimensional ceiling with projections f_i(X_i). The mathematical descriptions of a necessary AND configuration with several conditions are: X_i ≥ X_ic for Y = Y_c and the maximum possible  Y = y_c for given values of X_i = x_i  is  y_c =  min { f_i(X_i) }, where Y is the outcome, X_i  is the i-th condition, f_i(X_i)  is the i-th ceiling line and C is a point on the ceiling line.