To make the results easier to interpret, we define a descriptive word scale for the consensus range from zero to unity. For this we analyzed the consensus within 140 hierarchy nodes (a set of criteria or sub-criteria within a decision hierarchy) of 35 AHP group decision projects. It could be shown (Fig. 1) that the consensus SAHP is normal distributed with a mean value of 64 % ± 3 %. With a 99.5% probability the consensus of all projects lies between 28 % and 99 %. Therefore we divided the range of the scale in four equal segments from 50 % to 100 % (going from ‘low’ to ‘very high’), and defined the consensus for values below 50 % as ‘very low’.
Table 1 Qualitative wording scale for AHP consensus indicator
Consensus SAHP
< 0.5
0.5 – 0.625
0.625 – 0.75
0. 75 – 0.87.5
> 0.875
Word Scale
Very low
low
moderate
high
Very high
Switching from the consensus indicator SAHP to the relative homogeneity S shifts the mean value from 64 % to 70 %, which can be explained by the fact that in AHP we have a limited 1 to 9 scale and Hα,min is a function of the maximum scale value.
Figure 1 Distribution of consensus SAHP in blue actual values, in red normal distribution
Since April 2022 a new feature of AHP-OS, Group Consensus Cluster Analysis is available. It can be reached from the AHP-OS main page.
The idea of the program is to cluster a group of decision makers into smaller subgroups with higher consensus. For each pair of decision makers the similarity of priorities is calculated, using Shannon alpha and beta entropy. The result is arranged in a similarity matrix and sorted into clusters of higher similarity based on a consensus threshold.
In order to use the program, you first need to load a priority json file, exported from the AHP-OS Group result menu, containing the priorities of all participants:
Group Result Menu – Export priorities using Priorities (json).
Once downloaded to your computer, you can import this file via the Group Consensus Menu:
Group Consensus Menu
Click on Browse… to select the file; then click Analyze.The result is structured in
Input data
Threshold table
Result for selected node and a
Similarity Matrix
Input Data
Project session code, selected node (default: pTot), number of categories, number of participants and scale are shown. pTot stands for the global priorities of a hierarchy.
Threshold Table
The program calculates the number of clusters and number of unclustered participants based on a similarity threshold in the range between 70% and 97.5% in steps of 2.5%. For each step the values are displayed in the threshold table.
Consensus Threshold Table
Automatically the optimal threshold is determined.
In this case as 0.85 with 2 clusters and no unclustered members. If you want to change, for example the number of clusters to 3, you can enter 0.9 as new threshold in the AHP Group Consensus Menu manually.
Manual Threshold input field in the Group Consensus Menu
In the menu you also find a drop-down selection list for all nodes of the project. With Load new data another json file can be loaded.
Result for selected Node
First the AHP group consensus S* or relative homogeneity S for the whole group is shown, followed by the number of clusters. Next, for each cluster (subgroup) S* or S of the subgroup and the number of members in this cluster are displayed. Individual members are shown with a number and their name. The participants number corresponds to the number displayed on the project result page (Project Participants), so it is easy to select or deselect them by their number on the AHP-OS result page based on the result of the cluster analysis.
Similarity Matrix
The similarity matrix is a visualization of the clusters. Each cell (i,j) contains the AHP consensus S* or relative Homogeneity S for the pair of decision makers i and j in percent. Darker green color means higher values as show in the scale above the matrix. Clusters are always rectangles along the diagonal of the matrix, and are framed by borders.
Similarity Matrix
As you can see in the figure above, the program found two clusters with members 1,3,6,7,10,11,12 respectively 2,4,5,8,9, and one unclustered member 13. In this example the group consensus without clustering is 52.4% (low), the consensus for subgroup 1 is 80.5% (high) and subgroup 2 80.7% (high). This means that within the group there are two individual parties in higher agreement. You can easily go back to the project’s group result page to analyze the consolidated priorities for each group by selecting the individual participants.
Once the number of participants exceeds 40, the similarity matrix is shown without values in order to better fit on the output page.
Example of the similarity matrix with 72 participants. You can clearly identify three clusters.
References
Goepel, K.D. (2022). Group Consensus Cluster Analysis using Shannon Alpha- and Beta Entropy. Submitted for publication.Preprint
Goepel, K.D. (2018). Implementation of an Online Software Tool for the Analytic Hierarchy Process (AHP-OS). International Journal of the Analytic Hierarchy Process, Vol. 10 Issue 3 2018, pp 469-487, https://doi.org/10.13033/ijahp.v10i3.590
The video explains partitioning of Shannon diversity into two independent components: alpha (within group) and beta (in between groups) diversity. It helps to understand beta diversity as a measure of variation between different samples of data distributions. Some practical applications in the field of business analysis are shown.
The diversity calculator is an excel template that allows you to calculate alpha-, beta- and gamma diversity for a set samples (input data), and to analyze similarities between the samples based on partitioning diversity in alpha and beta diversity.
The template works under Windows OS and Excel 2010 (xlsx extension). No macros or links to external workbooks are necessary. The workbook consists of an input worksheet for a set of data samples, a calculation worksheet, where all necessary calculations are done, and a result worksheet “beta” displaying the results.
Applications
The template may be used to partition data distributions into alpha and beta diversity, it can be applied in many areas, for example
Bio diversity – local (alpha) and regional (beta) diversity
AHP group consensus – identify sub-goups of decision makers with similar priorities
Marketing – cluster analysis of similarities in markets
Business diversification over time periods
and many more.
Let me know your application! If you just need to calculate a set of diversity indices, you can use my online diversity calculator.
Calculations and results
Following data will be calculated and displayed:
Shannon Entropy H (natural logarithm) alpha-, beta- and gamma, and corresponding Hill numbers (true diversity of order one) for all samples
Homogeneity measure
Mac Arthur homogeneity indicator M
Relative homogeneity S
AHP group consensus S* (for AHP priority distributions)
Table 1: Shannon alpha-entropy, Equitability, Simpson Dominance, Gini-Simpson index and Hill numbers for each data sample
Table 2: Top 24 pairs of most similar samples
Page 2: Matrix of pairs of data samples
Diagram 1: Gini-Simpson index and Shannon Equitability
Diagram 2: Average proportional distribution for all classes/categories
Diagram 3: Proportional distribution sorted from largest to smallest proportion (relative abundance)
The work is licensed under the Creative Commons Attribution-Noncommercial 3.0 Singapore License. For terms of use please see our user agreement and privacy policy.
As this version is the first release, please feedback any bugs or problems you might encounter.
The video explains partitioning of Shannon diversity into two independent components: alpha (within group) and beta (in between groups) diversity. It helps to understand beta diversity as a measure of variation between different samples of data distributions. Some practical applications in the field of business analysis are shown.
The diversity calculator is an excel template that allows you to calculate alpha-, beta- and gamma diversity for a set samples (input data), and to analyze similarities between the samples based on partitioning diversity in alpha and beta diversity.
