Logic of PCI2

Concepts of consensus and disagreement / conflict are common in economics, political science, psychology, sociology, and natural resources. A lack of consensus arises because people do not necessarily share similar value orientations, attitudes or norms regarding what behaviors are acceptable.

PCI₂ for Bipolar Scales with a Neutral Value

In responding to survey questions about cognitions (e.g., norms, attitudes), people may form their evaluations relative to where they perceive others are on the topic. The rating of person (x) relative to that of person (y) can be thought of as a function of the distance between their responses (d_x,y = f(r_x,r_y)). However, there are alternative ways to formulate d_x,y. For example, d_x,y could be defined as the absolute value of x’s response (r_x) minus y’s response (r_y) (i.e., d_x,y = |r_x – r_y|).

Logic, however, suggests two issues with this formulation.

• Two people with responses of -3 and -2 are not necessarily in conflict; they both find the situation unacceptable and differ only slightly in the degree to which their views are held.
• People with negative or positive responses may perceive no conflict with a person who is neutral on the topic. Thus, a d_x,y > 0 may only exist between any negative response and any positive response.

D₁ does not include “neutral” or “neither” responses in the calculation of distance. By subtracting 1, the distance from a person who has a negative evaluation to a person who has a positive evaluation is calculated as if there was no neutral category (e.g., distance from -2 to +1 is 2, not the algebraic difference of 3).

Alternatively, if circumstances associated with given research support believing that neutral ratings should affect distance, a second distance formulation, D₂, for a bipolar scale with a neutral valueis defined by:

D₂ = d_x,y = |r_x – r_y|   if sign(r_x) ≠ sign(r_y) otherwise d_x,y = 0

D₂ includes “neutral” responses in the calculation of distance. When using D₂, the distance from -1 to +1 is 2 and the distance from -2 to +1 is 3.

PCI₂ for Bipolar Scales without a Neutral Value

Not all researchers include a neutral category in bipolar scales. For example, a 4-point scale might be -2, -1, 1, 2, where -2 = highly unacceptable, and 2 = highly acceptable. PCI2 can be computed for 2, 4, 6 and 8-point bipolar scales using D₁ as the distance function.

PCI₂ for Unipolar Scales

Other researchers use unipolar scales such as “not at all important” to “extremely important” or “not at all crowded” to “extremely crowded.” To accommodate these types of scales, a third distance function, D₃, was constructed:
D₃ = |r_x – r_y|^p

where:

r_x and r_y = ratings by person x and person y
p = a power (p > 0 and < 5)

Powers of Distance Function

D₁, D₂, and D₃ need not be linear functions of responses. Powers of differences or some other non-linear function of distance should also be considered. For example, someone with +1 may not see someone responding with -1 as being much in conflict. Someone responding -3, however, may be seen as threatening to what the +1 person wants because the -3 person may push strongly for change.

To reflect these non-linear perceptions, the difference scores can be raised to some power. If the initial difference scores were 1, 2, and 3 (i.e., power = 1), squaring the differences (i.e., power = 2) results in distances of 1, 4 and 9. A power of 2 gives more weight to larger differences between individuals. The greatest difference occurs between individuals who express the most extreme values on a scale (e.g., for a 7-point scale for D₁, -3 and +3 differ by 36). The PCI₂ estimation allows for alternative powers (e.g., 1, 1.5, 2) greater than 0 and less than 5.

The general PCI₂ distance expression for distances with a power is:

Dp_x,y= d_x,y = (|r_x – r_y| – (m – 1))^p if sign(r_x) ≠ sign(r_y) for p > 0
otherwise d_x,y = 0

where:

Dp_x,y = Distance raised to some power
m = D₁ or D₂
p = power