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There are many consensus measures that can be computed using Likert data. Although these measures should work with any number

The significance of reaching consensus in a group or among groups can easily be appreciated by anyone who has ever been involved in a group decision-making process. Indeed, some researchers believe that coming to consensus plays a key role in group decision-making [

The importance of consensus can be further appreciated when we consider the vast number of fields, other than just group decision-making, that use consensus. For example, politics, economics, social choice, and science all use the idea of consensus [

Since consensus is an important idea in many major areas, there is a demand for an accurate way to measure consensus. The central idea of many researchers working in this field of study concerns how to build consensus (or diminish the disagreement) among all people in one group or in more than one group. Many of these measures have an iterative process to try to come to agreement or build consensus [

There are several different meanings or definitions in the literature for the term “consensus” or “disagreement.” However, these two concepts are always the compliment of each other. In this paper, we use the term consensus to mean an opinion or belief reached by a group of persons who can agree on Likert scale items, while disagreement refers to a difference of opinion or perception.

The research to find a mathematical or statistical measure of consensus (or disagreement) started with simple measures and has progressed over time to more sophisticated techniques. The simplest and most widely used measures of consensus and disagreement are the percent agreement measure and the variance. The percent agreement measure has been used in different cases and has been applied to small group consensus. This measure gives only a percentage of team members who accept a particular opinion [

Another common approach to calculating the consensus is

There are many other more advanced methods presented to measure the consensus within a group of individuals. The method of Beliakov et al. [

Based on what is presented above, there are several ways to define consensus and various ways to measure consensus within and among groups. However, none of these approaches use computational geometry concepts in

Consequently, Abdal Rahem and Darrah in [

The rest of the paper is organized as follows. Section

The main theoretical foundations of the consensus measurement are introduced in Akiyama et al. [

In the second paper, Abdal Rahem and Darrah [

Let us start by introducing the notation for the two main variables we will work with. Suppose

Other common notations used throughout are the probabilities

As utilized in [

Using the notion

where

or equivalently

In order to simplify (

where

In order to find minimum

For purposes of finding the area

And in

We set

where

Note that all

Now that we have presented all the equations required for restricting the target area, the next step is to compute these areas in any dimension.

Although mathematicians may prefer an analytical process to get an exact solution, in many cases it is extremely hard or even impossible to find one. The way of finding the area

To examine what it means to determine these areas or volumes in

Example of the area

By adding one more probability, the volume is then in three dimensions. In this case, the equations

Example of the area

Consequently and since the region has such a strange shape in general, calculating its area or volume proves to be very difficult with analytical methods, especially as we go to higher dimensions, but calculating the area of rectangles or cubes is simple. We will use this method to simplify our calculations by subdividing the region into small rectangles or cubes, as is a common method for approximating an area or volume. One of the popular approaches to finding the area under a curve numerically is using the

Before presenting the steps of the algorithms, recall that the area for any

Divide the interval

For

For

For

If

If

Note that in this approach of finding the area, the larger the

Once you get

In order to ensure that the methods described above produce acceptable results, they are applied to different cases. For

For

which means that the system of inequities in (

For determining the volume restricted by two planes and the

Now we can look at some examples to see if the method produces the desired results. First, we consider some special cases of sets of probabilities. We use the probabilities to determine the mean and the variance. We apply the algorithms above to get the consensus values for these special cases to show that they make sense.

The first case we consider is the case when all the respondents have chosen the same response. In other words, when one of the probabilities, say

The second case is when the responses are evenly distributed at opposite ends of the scale. For example, probabilities are

Before looking at the different values for the mean and variance, we will look at one more special case. Now, suppose all probabilities are equal. Symbolically,

For selected values of

The Index of disagreement

Mean | Variance ( | | |
---|---|---|---|

1.1 | 0.09 | 0.0000 | 1.0000 |

| |||

0.1 | 0.0238 | 0.9762 | |

| |||

0.2 | 0.3562 | 0.6435 | |

| |||

0.3 | 0.755 | 0.245 | |

| |||

0.39 | 0.9595 | 0.0405 | |

| |||

0.49 | 0.0000 | 1.0000 | |

| |||

1.2 | 0.16 | 0.0000 | 1.0000 |

| |||

0.2 | 0.0134 | 0.9866 | |

| |||

0.3 | 0.0874 | 0.9126 | |

| |||

0.4 | 0.2609 | 0.7391 | |

| |||

0.5 | 0.5275 | 0.4725 | |

| |||

0.6 | 0.7736 | 0.2264 | |

| |||

0.7 | 0.9374 | 0.0626 | |

| |||

0.76 | 0.9899 | 0.0101 | |

| |||

0.96 | 1.0000 | 0.0000 |

The results of all examples above are reasonable and acceptable especially if we compare the results with the same cases for two dimensions,

A new approach for generalizing the consensus measure (or index of disagreement) presented in Akiyama et al. [

Another distinguishing feature of this new method is that it is easy to understand and apply. In fact, we can summarize this work as easy as A, B, C. (

In future studies, we plan to develop the same ideas to compute the consensus measure for continuous scales. We also plan to continue looking for easier and faster methods to find the areas and volumes described in the paper by using advanced computation geometry and high-dimensional statistics concepts [

The data used are available upon request.

The authors declare that they have no conflicts of interest.