1. Hierarchical Clustering and Topology
for Psychometric Validation
By Colleen M. Farrelly

2. Creating a New Survey: Psychometrics
 Many types of surveys/tests exist for assessing academic achievement,
psychological traits, or sociological constructs; the field that studies the
construction and functioning of tests is called psychometrics.
 Sometimes, a new survey must be created to either improve upon a
previous/discontinued one or assess a new idea/context for a given behavior
or trait.
 These new surveys pose several statistical challenges:
 Consistency within a survey (measuring what a survey is thought to measure)
 Crohnbach’s alpha, differential item functioning…
 Validation across samples (measures the same thing across populations/time)
 Exploratory factor analysis followed by confirmatory factor analysis
 Subscales for easier computation and interpretation of results (need to figure out
what items on the survey function similarly)
 Statistical frameworks exist for assessing these challenges, but they typically
require large sample sizes and assume certain structures underlie the survey
design.

3. Example Survey
a) Red:Rainbow::July:____ (Month, Year, Hot, Cloud)
b) Soothing:Anodyne::____:Esoteric (Eccentric, School, Abstruse, Calming)
c) Pyrrhic:Victory::Potemkin:____ (Village, Battle, Hollow, Achilles)
d) Stegasaurus:Jurassic::Trilobite:____ (Triassic, Dinosaur, Mesozoic, Cambrian)
e) Mice:Men::Cabbages:____ (Women, Lettuce, Salad, Kings)
f) Fill in the following series: 1, 1/8, 1/27, 1/64, ___
g) Fill in the following series: ___, 25, 168, 1229, 9592
h) Fill in the following series: 3, ___,4,1,5

4. Factor Analysis
 Creation of new surveys requires internal and external validation, typically
done through factor analysis.
 Exploratory factor analysis is used to cluster items measuring similar underlying
processes.
 Confirmatory factor analysis can then be applied to validate those clusters, or
subscales, that were found in the exploratory analysis.
 Crohnbach’s alpha establishes internal consistency.
Verbal
Math
f
g
h
a
b c d
e

5. Potential Pitfalls in Psychometric
Validation with Factor Analysis
 Two major problems challenge the assumptions of these methods and
necessitate the development of a new way to analyze and validate the
measure.
 Time-wise or context-wise measurement can introduce non-independent, non-
hierarchical components into the model.
 Study habits across terms (longitudinal effects on measurement), identity across social
spheres (student perception of intellectual ability when with friends, work, and school)
 Factor analysis can be broadened to Bayesian networks and structural equation models, but
this method comes with its own assumptions on the underlying geometry and sample size.
 Small sample size can create numerical instability in traditional algorithms for both
factor analysis and structural equation models (suggest 5-10 participants per item).
 If there are 90 items, at least 450 students would be needed to discover subscales, and
another 450 would be needed to validate these findings.
 Cost and population size can be prohibitive to the study.
 Ex. Bridging constructs, or loosely connected concepts without a defined hierarchy,
typically run into both limitations and require a new method to validate their
surveys.
 Many of these issues arise from the dependence on linear mapping from the
survey response space to a lower-dimensional space.

6. Moving from Euclidean-Based Statistics
to Topologically-Based Statistics
Loss of information with
each projection to a lower-
dimensional space (errors)
Topological methods work by
partitioning existing space
into homogenous components
(no maps, no error)
2D example

7. Algebraic Topology and Topological Spaces
 Spaces, such as the one formed by survey response data, can be defined
topologically and decomposed using algebraic topology/geometry.
 Data follows discrete versions of many theoretical results in this area of math.
 Topology is rubber sheet geometry, with areas analogous to gluing together children’s building
blocks, examining connections on shapes, or hunting for mountain/valley water flows.
 Examining how the pieces fit together in a given space allows one to study the topological
space’s defining characteristics and the behavior of functions in that space.
 One can define connections between pieces of this space via algebra and examine
structural properties computationally:
 Homotopy (shrinking connected paths to a point)
 Homology (hole-counting to define topological classification of structure)
1 2 3Homotopy/
Homology Basins of Attraction (Morse Theory)
Hodge Theory

8. Applied Homology: Filtrations and
Persistence
 Filtration
 This is an iterative changing of lens with which
to examine data (height, neighbors…).
 Topological features appear and disappear as
the lens changes.
 This creates a nested sequence of features with
underlying algebraic objects, called a homology
sequence:
 Hom1⊂Hom2⊂Hom3⊂Hom4
 Persistence is the length of feature existence in
a homology sequence, which can be visualized.
 This information maps back to the data
space’s topology (shape).
 The first level of algebraic objects
corresponds to connectedness of the space
(0th Betti numbers), and this is directly
related to a type of clustering analysis.
0 2 4 6 8 10
time
Connected
space
Vertices
Hole in
middle

9. Solution: Use Machine Learning to Exploit
Underlying Topology of Survey Data
 Single-linkage hierarchical clustering partitions data space according to
connected components (0th Betti numbers) across filtration levels (i.e. a series
of distance filtrations).
 This method has been successfully applied to neuroimaging studies focused on
patterns of brain activity across diseases, neuropsychological tests, and drug states.
 This provides a nuanced scanning of topologically-based features within the datasets at
different correlation/similarity thresholds.
 These can be summarized in feature plots, called persistence diagrams, that track the birth
and death of a given feature across thresholds, and can be compared through existing
statistical tests, such as a nonparametric Wasserstein metric test.
 It has also been used to track gene expression pattern changes across time and/or
disease states in microarray studies.
 These studies particularly emphasize the visualization of hierarchical clustering through
dendrograms (tree diagrams of relationships at different filtration levels) and heat maps
(color-coded expression-similarity plots among genes in the microarray).
 These visualizations provide a user-friendly way to understand and communicate key
findings of this statistical method.
 This method can handle data with fewer observations than predictors (p>>n), and,
thus, does not require large sample sizes.
 Internal correlations do not pose issues; in fact, the method excels at separating
data within and across dependencies.

10. Hierarchical Clustering: Example Survey
Math Verbal
Heatmap
Very distinct separation of items (noted
by sharp color contrast of heatmap and
long height bars on dendrogram)

11. Validation: Dendrograms and Topology
 Dendrograms are a special type of graph,
called a tree.
 Because graphs have a defined topological
space and dendrograms are a type of
graph, they can be studied or measured
through the tools of topology and metric
geometry.
 Hausdorff distance allows two objects of
the same dimension to be compared by a
defined metric.
 This examines the greatest distance
between close points, allowing for a
nearness-of-match type of metric on two
objects (top left).
 Within a graph framework, it allows one to
calculate worst best match between two
graphs (as shown at bottom left).
 This allows for the development of a
distance-based nonparametric test to test
for dendrogram structural differences in a
statistical framework.
Hausdorff
Distance

12. Steps in Exploration and Validation of
Surveys with Hierarchical Clustering
1) Partition sample into training and validation sets/draw a small number of
bootstrap samples from the original dataset.
2) Calculate distance metrics in each sample.
3) Run a single-linkage hierarchical clustering algorithm on the training set to
obtain exploratory clusters of similar survey items (pvclust R package
statistically tests internal survey structure like the Crohnbach alpha metric).
Create heat map and dendrogram.
4) Repeat (3) on validation sets to obtain a set of dendrograms.
5) Calculate Hausdorff distance (a topological metric) between dendrograms to
estimate differences in results (validation step).
6) Obtain p-value through permuting the extant dendrograms or generating
random dendrograms.
7) If p-value is larger than 0.05/n (Bonferroni correction) for dendrograms in (5),
no statistically significant differences exist in dendrogram structure, meaning
that the survey clusters are consistent and valid.

13. Example Measure: Bridging Constructs
 Identity expression across life contexts (ILLCQ Survey):
 There are many components to identity in leading theories of identity.
 Example: religious identity in school, family, and friends contexts
 It was unknown whether identity type or social context plays a greater role in the
expression of identity within an individual.
 Identity type as more influential would suggest that identity is a fairly static trait.
 Context as more influential would suggest that identity is fluid.
 Sample size and survey size
 406 participants (FIU students) and 91 distinct survey items.
 5 draws of 130 participants each for validation and consistency checks.
 Results suggest certain aspects of identity are fluid and others are fixed.
 Political and racial/ethnic identity are fairly fixed.
 Other types, such as athletic or gender, are fairly fluid.
 Bootstrapped samples suggest consistency of measure and validate findings.
 Subscales hold over different samples (tests of difference, all p>0.05).
 This validates the measure and allows for inference into the psychology of identity.

14. Identity by Context Survey HeatmapILLCa_school_success_family
ILLCa_school_success_school
ILLCa_gender_dating
ILLCa_age_dating
ILLCa_age_freetime
ILLCa_sexual_or_dating
ILLCa_beauty_dating
ILLCa_sport_dating
ILLCa_sport_freetime
ILLCa_sport_religion
ILLCa_religion_freetime
ILLCa_religion_family
ILLCa_religion_school
ILLCa_religion_neighborhood
ILLCa_politics_dating
ILLCa_religion_group
ILLCa_sexual_or_religion
ILLCa_gender_religion
ILLCa_age_religion
ILLCa_politics_religion
ILLCa_politics_family
ILLCa_politics_neighborhood
ILLCa_politics_group
ILLCa_politics_school
ILLCa_politics_freetime
ILLCa_tribe_dating
ILLCa_tribe_group
ILLCa_tribe_freetime
ILLCa_tribe_family
ILLCa_tribe_school
ILLCa_tribe_neighborhood
ILLCa_tribe_religion
ILLCa_beauty_neighborhood
ILLCa_look_neighborhood
ILLCa_school_success_religion
ILLCa_look_religion
ILLCa_music_neighborhood
ILLCa_race_religion
ILLCa_status_religion
ILLCa_beauty_religion
ILLCa_religion_religion
ILLCa_religion_dating
ILLCa_race_school
ILLCa_race_freetime
ILLCa_sexual_or_school
ILLCa_beauty_family
ILLCa_beauty_freetime
ILLCa_beauty_school
ILLCa_beauty_group
ILLCa_look_freetime
ILLCa_look_family
ILLCa_look_school
ILLCa_status_dating
ILLCa_status_group
ILLCa_race_group
ILLCa_race_dating
ILLCa_sexual_or_group
ILLCa_sexual_or_freetime
ILLCa_gender_freetime
ILLCa_gender_family
ILLCa_gender_school
ILLCa_age_family
ILLCa_age_school
ILLCa_school_success_neighborhood
ILLCa_race_neighborhood
ILLCa_sexual_or_neighborhood
ILLCa_status_neighborhood
ILLCa_gender_neighborhood
ILLCa_age_neighborhood
ILLCa_sport_school
ILLCa_sport_family
ILLCa_sport_group
ILLCa_music_freetime
ILLCa_music_religion
ILLCa_music_dating
ILLCa_sport_neighborhood
ILLCa_school_success_dating
ILLCa_school_success_group
ILLCa_school_success_freetime
ILLCa_music_school
ILLCa_music_family
ILLCa_music_group
ILLCa_gender_group
ILLCa_age_group
ILLCa_look_group
ILLCa_look_dating
ILLCa_race_family
ILLCa_sexual_or_family
ILLCa_status_freetime
ILLCa_status_family
ILLCa_status_school
-0.2
0
0.2
0.4
0.6
0.8
1

15. Conclusion
 This method offers a robust way to create survey subscales and validate
measures without needing a large sample or a pre-defined measure structure.
 Flexible
 Deeply routed in mathematics
 Statistically testable
 Internal validity by pvclust’s statistical test of cluster hierarchy for cut-points
 External validity by Hausdorff nonparametric test on bootstrapped samples
 It has been successfully applied to a bridging concept survey (factorial design),
as well as more traditional survey designs.
 This offers a general way to extend traditional areas of statistics to a more
general framework through the use of topological theory and tools.
 Likely to be useful as data becomes more complex in industry and academia.
 May be able to circumvent other problems in modern statistics.
 Item response theory (how people in different groups perform on test items)
 Network comparison (social networks, covariance networks…) between groups or over time
 Structural equation modeling when data does not meet method assumptions

16. Co-authors
 The Analysis of bridging constructs with hierarchical clustering methods: An
application to identity (under review Journal of Research in Personality)
 Seth Schwartz, University of Miami
 Anna Lisa Amodeo, University of Naples
 Daniel Feaster, University of Miami
 Douglas Steinley, University of Missouri
 Alan Meca, University of Miami
 Simona Picariello, University of Naples