Abstract
Epistemic network analysis (ENA) has been used in more than 300 published studies to date. However, there is no work in publication that describes the transformations that constitute ENA in formal mathematical terms. This paper provides such a description, focusing on the mathematical formulations that lead to two key affordances of ENA that are not present in other network analysis tools or multivariate analyses: (1) summary statistics that can be used to compare the differences in the content rather than the structure of networks and (2) network visualizations that provide information that is mathematically consistent with those statistics. Specifically, we describe the mathematical transformations by which ENA constructs matrix representations of discourse data, uses those representations to generate networks for units of analysis, places those networks into a metric space, identifies meaningful dimensions in the space, and positions the nodes of network graphs within that space so as to enable interpretation of those dimensions in terms of the content of the networks. We conclude with a discussion of how the mathematical formalisms of ENA can be used to model networks more generally.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
SVD is related to other commonly used dimension reduction/visualization techniques, such as principal component analysis (PCA) and factor analysis. SVD decomposes the original (non-symmetric) data matrix, Z, without centering, while PCA standardizes the columns of the original matrix and decomposes the data into a set of ordered, orthogonal components using the eigenvalues of the symmetric sample covariance matrix, thus centering the data prior to decomposition. Factor analysis derives a set of influential factors by decomposing the sample covariance matrix of a set of centered variables into two additive components, one of which is attributed to the set of common factors and the other is specific to each observation.
- 2.
Briefly, we prove orthogonality of V and \(\varvec{\mathbf {\mu }}\) by \(V^\prime \varvec{\mathbf {\mu }} = 0\). By Eqs. 3.17 and 3.18, and because \(\varvec{\mathbf {\mu }}\) is a unit vector \(\varvec{\mathbf {\mu }}^\prime \varvec{\mathbf {\mu }} = 1\),
$$\begin{aligned} V^\prime \varvec{\mathbf {\mu }}&= D^{-1}U^\prime (\mathbf{N}-\mathbf{N} \varvec{\mathbf {\mu }}\varvec{\mathbf {\mu }}^\prime )\varvec{\mathbf {\mu }}\\&= D^{-1}U^\prime \mathbf{N}\varvec{\mathbf {\mu }}-D^{-1}U^\prime \mathbf{N} \varvec{\mathbf {\mu }}\varvec{\mathbf {\mu }}^\prime \varvec{\mathbf {\mu }}\\&= D^{-1}U^\prime \mathbf{N}\varvec{\mathbf {\mu }}-D^{-1}U^\prime \mathbf{N} \varvec{\mathbf {\mu }}\\&= 0 \end{aligned}$$.
- 3.
We use \(\frac{1}{2}\) of the normalized weights because for any pair of codes \(a_i\) and \(a_j\), one half of the normalized weight of their connection is associated with each code. Thus the total normalized weights are preserved and \(\sum _i \tilde{\mathbf{w}}_i^k = \sum _i \mathbf{N}_i^k\).
- 4.
There is no intercept term included in Eq. 3.23. This is because the columns of \(\mathbf {\tilde{w_i}}\) (which function as the independent variables in the regression) are not linearly independent. In particular, by Eq. 3.20, \(\forall k, \sum _j \mathbf {\tilde{w}_{ij}^k} = 1\). Thus, adding an intercept term would make the equation ill-defined—that is, without a unique solution.
References
Andrist, S., Ruis, A.R., Shaffer, D.W.: A network analytic approach to gaze coordination during a collaborative task. Comput. Hum. Behav. 89, 339–348 (2018)
Arastoopour, G., Shaffer, D.W., Swiecki, Z., Ruis, A.R., Chesler, N.C.: Teaching and assessing engineering design thinking with virtual internships and epistemic network analysis. Int. J. Eng. Educ. 32(3), 1492–1501 (2016)
D’Angelo, A.L.D., Ruis, A.R., Collier, W., Williamson Shaffer, D., Pugh, C.M.: Evaluating how residents talk and what it means for surgical performance in the simulation lab. Am. J. Surg. (2020, in Press)
DiSessa, A.A.: Knowledge in pieces. In: Forman, G., Pufall, P. (eds.) Constructivism in the Computer Age, pp. 47–70. Erlbaum, Hillsdale (1988)
Dorogovtsev, S.N., Mendes, J.F.F.: Evolution of Networks: From Biological Nets to the Internet and WWW. Oxford University Press, Oxford (2003)
Herman, I., Melancon, G., Marshall, M.: Graph visualization and navigation in information visualization: a survey. IEEE Trans. Vis. Comput. Graph. 6, 24–43 (2000)
Knight, S., Arastoopour, G., Shaffer, D.W., Shum, S.B., Littleton, K.: Epistemic networks for epistemic commitments (2014)
Krause, A.E., et al.: Adaptations in a hierarchical food web of southeastern lake Michigan. Ecol. Model. 220(22), 3147–3162 (2009)
Marquart, C.L., Swiecki, Z., Collier, W., Eagan, B.R., Woodward, R., Shaffer, D.W.: rENA: epistemic network analysis [r package]. https://cran.r-project.org/web/packages/rENA/index.html
Nash, P., Shaffer, D.W.: Epistemic trajectories: mentoring in a game design practicum. Instruc. Sci. 41(4), 745–771 (2013)
Quardokus Fisher, K., Hirshfield, L., Siebert-Evenstone, A.L., Arastoopour, G., Koretsky, M.: Network analysis of interactions between students and an instructor during design meetings. In: Proceedings of the American Society for Engineering Education, p. 17035. ASEE (2016)
Ruis, A.R.: ‘Trois empreintes d’un même cachet’: toward a historical definition of nutrition. In: Ewing, E.T., Randall, K. (eds.) Viral Networks: Connecting Digital Humanities and Medical History, pp. 179–212. VT Publishing, Blacksburg (2018)
Ruis, A.R., Rosser, A.A., Quandt-Walle, C., Nathwani, J.N., Shaffer, D.W., Pugh, C.M.: The hands and head of a surgeon: modeling operative competency with multimodal epistemic network analysis. Am. J. Surg. 216(5), 835–840 (2018)
Shaffer, D.W.: Models of situated action: computer games and the problem of transfer. In: Steinkuehler, C., Squire, K.D., Barab, S.A. (eds.) Games, Learning, and Society: Learning and Meaning in the Digital Age, pp. 403–431. Cambridge University Press, New York (2012)
Shaffer, D.W.: Quantitative Ethnography. Cathcart Press, Madison (2017)
Shaffer, D.W., Collier, W., Ruis, A.R.: A tutorial on epistemic network analysis: analyzing the structure of connections in cognitive, social, and interaction data. J. Learn. Analytics 3(3), 9–45 (2016)
Shaffer, D.W., et al.: Epistemic network analysis: a prototype for 21st century assessment of learning. Int. J. Learn. Media 1(1), 1–21 (2009)
Sullivan, S.A., et al.: Using epistemic network analysis to identify targets for educational interventions in trauma team communication. Surgery 163(4), 938–943 (2018)
Wooldridge, A.R., Carayon, P., Eagan, B.R., Shaffer, D.W.: Quantifying the qualitative with epistemic network analysis: a human factors case study of task-allocation communication in a primary care team. IISE Trans. Healthcare Syst. Eng. 8(1), 72–82 (2018)
Zörgő, S., Peters, G.-J.Y.: Epistemic network analysis for semi-structured interviews and other continuous narratives: challenges and insights. In: Eagan, B., Misfeldt, M., Siebert-Evenstone, A. (eds.) ICQE 2019. CCIS, vol. 1112, pp. 267–277. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-33232-7_23
Acknowledgements
This work was funded in part by the National Science Foundation (DRL-1661036, DRL-1713110), the Wisconsin Alumni Research Foundation, and the Office of the Vice Chancellor for Research and Graduate Education at the University of Wisconsin–Madison. The opinions, findings, and conclusions do not reflect the views of the funding agencies, cooperating institutions, or other individuals.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Bowman, D. et al. (2021). The Mathematical Foundations of Epistemic Network Analysis. In: Ruis, A.R., Lee, S.B. (eds) Advances in Quantitative Ethnography. ICQE 2021. Communications in Computer and Information Science, vol 1312. Springer, Cham. https://doi.org/10.1007/978-3-030-67788-6_7
Download citation
DOI: https://doi.org/10.1007/978-3-030-67788-6_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-67787-9
Online ISBN: 978-3-030-67788-6
eBook Packages: Computer ScienceComputer Science (R0)