Nonadditive

Nonadditive is a term used to describe a function that relates one set of scores (e.g., scores on a variable, Y) to two or more other sets of scores (e.g., scores on variables X1 and X2). The relationship between Y and the other variables is said to be additive if Y can be expressed as a sum of those other variables. The sum can be a weighted sum and has the general expression

where a and the various ws are constants.

In some applications, restrictions are placed on the constants. For example, if a = 0 and each weight is set to 1.0, then Y is literally the sum of the X variables. Another type of restriction that a researcher might apply is that the weights must sum to 1.0 in accordance with the formulation

where cj is an absolute weight for variable Xj. The summation is across the k weights. As an example, suppose that Y is an additive function of X1 and X2, and that a = 0,c1 = 1, and c2 = 1. Then, w1 = 1/(1 + 1) = 0.50, and w2 = 1/(1 + 1) = 0.50, yielding

Note that this expression is equivalent to

so that Y is the average of X1 and X2. This example shows that expressing Y as an average of a set of variables is a type of additive function.

A nonadditive function is one where Y cannot be expressed as a (weighted) sum of the other variables. Nonadditive functions can take many forms—indeed, there is an infinite number of such functions. An example of a nonadditive function is a MULTIPLICATIVE function, where Y is said to equal the product of two variables, X1 and X2.

Traditional multiple regression analysis is based on an additive model,

where α is the intercept and the various βs are the REGRESSION COEFFICIENTS.