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30/10/2024
This article focuses on the concept maps produced by the solution used at Polygon, rather than the classic method developed by Trochim (1989).
The ultimate goal of a concept mapping exercise is to represent the results of the analysis in the form of a map that allows the visualization of emerging themes. More specifically, this map combines the results of cluster analysis, or clustering, and dimensionality reduction.
The points on the map represent the analysed items (i.e. phrases, concepts) and the colours of the points indicate the groups to which they belong. Items within the same group therefore represent the emerging themes. Typically, a convex polygon is superimposed on the points of the same group to make the map easier to read.
The position of the points on the map is determined by the algorithm used to project the objects into a reduced dimensional space, usually 2D. Since the sorting exercise can generate non-metric distances, it is preferable to use methods adapted to this type of distance. In the solution developed by Polygon, the user can choose between three methods for visualizing partitions: Isomap (Balasubramanian and Schwartz, 2002) Laplacian Eigenmaps (Belkin and Niyogi, 2003) and UMAP (McInnes et al., 2020)1.
The projections resulting from these three methods are presented in a Euclidean space2, which makes the map easier to read. These projections estimate the dissimilarity between items, and consequently, the items closest in the projection are potentially more similar (i.e. they were more frequently grouped together by participants). Nevertheless, the projections resulting from these three methods will differ, and this can have a major impact on the interpretation of the results. In fact, these methods are based on different approaches to estimating the coordinates of the points in the projection space, especially when trying to preserve the global or local structure of the dissimilarity matrix3. Furthermore, the presence of outliers can also affect the results, and these points can significantly influence the projection. Therefore, it is best to compare different projection methods to better understand their limitations and specificities (Fig. 2).
The clusters identified by the cluster analysis can also be interpreted in terms of spatial proximity on the map. Denser clusters indicate items that were grouped more frequently by participants, while sparser clusters indicate items that were grouped less frequently. The former configuration would suggest a ‘stronger’ or more ‘consensual’ theme, while the latter would suggest a ‘weaker’ or more ‘divergent’ theme. Clusters can also show more complex configurations, such as both dense and sparse sections, or certain isolated items that don’t seem to belong to any particular group.
While 2D or 3D projection makes it easier to interpret clusters, it’s important to consider the bias introduced by projection methods. Therefore, comparing these results with other forms of analysis can help to avoid over-interpretation. The dendrogram is particularly useful when clustering is done using hierarchical methods.
Finally, the axes on the concept maps produced by the above methods have no particular significance and therefore cannot be interpreted as dimensions or latent factors4. As such, the maps can be rotated or mirrored without affecting the significance of the results.
For more information on concept mapping and Polygon’s CM* tool, see the following links: What is concept mapping? and CM*
Notes:
1. Traditionally, the projection is performed using non-metric multidimensional scaling (nMDS). However, the projections produced by this method are not easy to interpret when the clustering analysis is performed on the dissimilarity matrix.
2. Euclidean space is a space in which the distance between two points is defined by the Euclidean distance. When represented in 2D, Euclidean space is simply a Cartesian plane where points are represented by coordinates (x, y), as on a geographical map. As a result, the interpretation of this projection is relatively intuitive.
3. From a topological perspective, ISOMAP attempts to preserve the global structure of the dissimilarity matrix based on geodesic distances, whereas Laplacian Eigenmaps focus on preserving local structures by minimizing variance within the neighbourhoods of each point. UMAP is a more balanced method that attempts to preserve both the global and local structure of the dissimilarity matrix.
4. A similar principle applies to classic concept maps generated by non-metric multidimensional scaling (nMDS).
Bibliography:
Balasubramanian, M., & Schwartz, E. L. (2002). The isomap algorithm and topological stability. Science, 295(5552), 7–7. https://doi.org/10.1126/science.295.5552.7a
Belkin, M., & Niyogi, P. (2003). Laplacian eigenmaps for dimensionality reduction and data representation. Neural Computation, 15(6), 1373–1396. https://doi.org/10.1162/089976603321780317
McInnes, L., Healy, J., & Melville, J. (2020). UMAP: Uniform manifold approximation and projection for dimension reduction. ArXiv. https://arxiv.org/abs/1802.03426
Trochim, W. M. K. (1989). An introduction to concept mapping for planning and evaluation. Evaluation and Program Planning, 12(1), 1–16. https://doi.org/10.1016/0149-7189(89)90016-5
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Concept map interpretation
