Finding archetypal patterns for binary questionnaires

Abstract: Archetypal analysis is an exploratory tool that explains a set of observations as mixtures of pure (extreme) patterns. If the patterns are actual observations of the sample, we refer to them as archetypoids. For the first time, we propose to use archetypoid analysis for binary observations. This tool can contribute to the understanding of a binary data set, as in the multivariate case. We illustrate the advantages of the proposed methodology in a simulation study and two applications, one exploring objects (rows) and the other exploring items (columns). One is related to determining student skill set profiles and the other to describing item response functions.

• The paper introduces ADA, an unsupervised method for binary data that selects actual data points as archetypoids for enhanced interpretability.
• It employs mixed-integer optimization to decompose binary observations as convex combinations of extreme profiles, reducing misclassification error.
• Empirical results demonstrate ADA’s superiority over AA and PAA in recovering clear, interpretable archetypal patterns in noisy survey and test datasets.

Archetypoid Analysis for Binary Questionnaires: Theory and Applications

Problem Statement and Methodological Contributions

The paper "Finding archetypal patterns for binary questionnaires" (2003.00043) rigorously formulates the use of Archetypoid Analysis (ADA) for binary data. ADA extends archetypal analysis (AA), an unsupervised matrix factorization technique, by requiring the archetypal patterns (archetypoids) to be actual data points from the observed sample, thereby preserving interpretability—a key property for exploratory data analysis (EDA) in the social sciences. The paper motivates ADA as particularly suitable for binary matrices ($n \times m$, with entries in $\{0,1\}$), often arising in dichotomous item tests, surveys, and behavioral diagnostics.

ADA is positioned as a superior alternative to both AA and probabilistic AA (PAA) when interpretable binary patterns are required. Whereas AA and PAA generate mixtures in a continuous parameter space and can return infeasible (non-binary) patterns, ADA leverages a mixed-integer optimization so that all archetypoids are valid sample members.

The core claim is that ADA enables a geometric, non-parametric decomposition of binary matrices, approximating each observation as a convex combination of $k$ extreme binary samples, facilitating robust profile extraction and visualization.

Theoretical Framework

The ADA procedure is defined as follows for $n \times m$ binary data $\mathbf{X}$:

• Each sample $\mathbf{x}_i$ is approximated as a convex combination of $k$ archetypoids $\mathbf{z}_j$, themselves chosen to be rows from $\mathbf{X}$ ($\mathbf{z}_j = \mathbf{x}_{l_j}$ for some $l_j$).
• The mixture coefficients $\alpha_{ij}$ are nonnegative and sum to 1 for each observation.
• The optimal solution minimizes the residual sum of squares (RSS):

$$\text{RSS} = \sum_{i=1}^n \|\mathbf{x}_i - \sum_{j=1}^k \alpha_{ij} \mathbf{z}_j \|^2$$

Subject to the constraints outlined above and requiring that archetypoids correspond to actual sample rows.

For comparison, AA and PAA relax the constraint and allow archetypal mixtures that typically do not assume values in $\{0,1\}^m$, compromising interpretability. The paper includes an algorithmic discussion of practical estimation, with ADA implemented through BUILD and SWAP steps to seek archetypoids minimizing RSS, and selection of $k$ guided by the 'elbow' heuristic.

Empirical Analysis: Simulation Study

The simulation study precisely quantifies the appropriateness of ADA for binary data relative to AA and PAA. Six archetypes are generated with ten binary variables, and one hundred datasets are synthesized through noisy convex combinations.

Key findings include:

• ADA achieves the lowest mean misclassification error (mean: 3.19, SD: 1.88) relative to AA (mean: 3.59, SD: 1.99) and PAA (mean: 4.20, SD: 1.86) in restoring ground-truth archetypes.
• ADA archetypoids show less bias and better recovery of binary structure under noise.
• Binarization of AA and PAA outputs does not close the gap to ADA in recovering interpretably extreme patterns.

These results substantiate the claim that ADA provides more reliable and interpretable extraction of archetypal patterns for binary matrices, even under high noise and complex mixture scenarios.

Application 1: Student Skill Set Profiling

ADA is applied to a 690 $\times$ 21 binary matrix of mathematics test results for first-year university students, with each row encoding a student's binary response vector. Comparative analyses include PAM, $k$-means, LCA, AA, and PAA.

Highlights include:

• ADA recovers three archetypoids: a student with very poor skills, and two students with complementary profiles in mastery of specific mathematical topics (e.g., one excels in nonlinear systems and linear functions, the other in calculus and algebraic interpretation).
• Archetypoids have high percentile scores and significant Hamming distances, demonstrating extreme and diverse mastery.
• Cluster-based methods (PAM, $k$-means, LCA) yield representative profiles centered closer to the bulk mass, with less interpretability and complementarity.
• ADA provides richer composition information via $\alpha$-mixture weights per sample, whereas conventional methods yield only hard assignments.

Statistical interpretation is enhanced by the ability to express each student's skill set as a combination of ADA archetypoids, supporting adaptive educational interventions, targeted instruction, and nuanced group formation.

Application 2: Item Response Functions in ACT Matrices

The methodology is adapted to functional binary data associated with an ACT mathematics test, examining 0/1 responses across 2115 male students and 60 items.

Key outcomes:

• Functional Archetypoid Analysis (FADA) identifies extreme IRFs (item response functions) among the test items relating $\theta$ (latent ability) to $P_i(\theta)$ (probability of success).
• FADA archetypoid items (e.g., items 2, 18, 28, 60) correspond to extreme, interpretable question profiles, with distinct slopes, difficulty levels, and discriminative power not recoverable via FPCA.
• FPCA captures principal axes of variance but fails to identify human-readable archetypal representatives, corroborating prior theoretical critique.
• FADA quantifies the composition of each item's IRF as a convex combination of archetypoid IRFs, supporting nuanced psychometric analyses.

The technical implication is that FADA enables effective pattern extraction and visualization for large-scale testing scenarios where item-level characteristics are critical for test improvement and student outcome analysis.

Implications and Prospects

The application of ADA and FADA to binary and functional binary datasets provides a robust, interpretable alternative to classic EDA methods, especially in scenarios where raw clustering and PCA fail to capture extreme or complementary profiles. Practically, ADA supports the design of adaptive surveys, diagnostics, and assessments, while theoretically, it motivates further research into mixed, nominal, and ordinal data generalizations and scalable computational approaches.

The methodology encourages viewing data sets as compositional mixtures of archetypal patterns, facilitating "human-readable" summaries valuable even beyond expert analysis—although the approach remains fundamentally geometric and distribution-free.

For future research, directions include weighted variable importance, adaptation to mixed and missing data, nominal and ordinal generalization, and optimized large-scale algorithms for ADA in truly big data regimes.

Conclusion

Archetypoid Analysis, as introduced and formalized in this work, is demonstrably suitable for mining binary questionnaires, surpassing conventional unsupervised methods in interpretability, complementary profile extraction, and compositional data representation. The empirical and theoretical evidence supports ADA and FADA as valuable tools for EDA, item response theory, and functional data analysis in survey-driven domains, broadly enhancing both practical application and methodological discourse.