A new development in the hierarchical clustering of repertory grid data
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A New Development in the Hierarchical
Clustering of Repertory Grid Data
Mark Heckmann & Richard C. Bell University of Bremen, Germany, University of Melbourne, Australia
ICPCP, Sydney, July 19, 2013
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The Context: Tight & Loose Construct Systems
• The importance of the 9ghtness – looseness construct – Fragmented vs Monolithic construing dimension – Involved in Kelly’s Crea9vity Cycle. Therapy involves a series of Crea9vity Cycles, each of which • Starts with loosened construc9on • Ends with 9ghtened construc9on
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Measuring 9ghtness and looseness
• Using the Repertory Grid • Overall Grid 9ghtness & looseness of construing – Cogni9ve Complexity measures such as • Bannister’s intensity (Average correla9on) • PVAFF (Percentage of Variance Accounted for by the First Factor) • Number of components
• Finding subsystems of 9ght and loose construing
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Measuring 9ghtness and looseness
• Using the Repertory Grid to find subsystems of 9ght and loose construing
• Requires representa9on of rela9onships between constructs that are differen9ated in terms of “closeness”. – Spa9al representa9ons (principal components) – Tree representa9ons (clustering)
• Neither readily permits objec9ve iden9fica9on of 9ght and loose rela9onships
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Hierarchical Clustering of Grid Data
• Appears to have originated with Thomas & Mendoza in 1974 at Brunel University but
• Made famous by Thomas & Shaw in 1976 as the FOCUS program – Never en9rely clear which cluster method was used – either McQuiby or Single Linkage
– Nor was the measure made clear – probably city-‐block (Manhaban) distances
• More of an impact in industrial seengs than clinical
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Hierarchical Clustering of Grid Data
• Advantage – Shows grouping clearly
• Disadvantages – Representa9on (dendrogram) depends on method of clustering and measure of similarity (between constructs)
– Can’t tell whether clusters are significant (but also true of other representa9ons such as principal components)
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Iden9fying Significant Clusters • Recent advances in compu9ng have enabled us to assess significance without resor9ng to tradi9onal theore9cal distribu9ons such as t, F, or z.
• Such methods involve mul9ple samples and include – Jackknife (crea9ng new samples using all cases except (a different) one each 9me)
– Monte Carlo (random data generated by model) – Bootstrap (crea9ng new samples by sampling with replacement)
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Sarah‘s dataset rearranged
Sarah‘s dataset
Sarah‘s grid
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Prelude: A lot of grid sta9s9cs are derived from similarity measures
Complexity (RMS)
Conflic9ng triads
Implica9ve Dilemma
Cluster analysis
Usually these sta9s9cs are interpreted ‚as-‐are‘
Correla9ons
Distances
...
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Standard hierarchical cluster analysis
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Sarah‘s dataset rearranged
Sarah‘s grid
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Some more reliability observa9ons
1. Appr. 70% of constructs remain the same1
2. Ra9ngs of same grids will vary2
t1 t2
We get a glimpse but not the whole picture à sampling from a universe of constructs / elements
1) Hunt 1951, Fjeld & Landfield 1961 2) Bell 1990
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Descriptive Inference
r = 0.35 r ∈ [0.3;0.4]
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Descriptive Inference
r ∈ [0.3;0.4]
cc
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r = 0.35
cc
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r = .30Not feel guilty -‐ Feel guilty
Powerful-‐ Powerless
Element child self ommibed
r = .61Not feel guilty -‐ Feel guilty
Powerful-‐ Powerless
Correla9ons vary with the element set
All elements
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Element partner ommibed
r = .39
à the similarity measure also is a random variable
Not feel guilty -‐ Feel guilty
Powerful-‐ Powerless
Idea: Thinking of the set of elements and constructs as realisia9ons of random variables
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How much does a correla9on vary?
Similarity measures may vary if a different (sub)set of elements is used
Safe to detect e.g. implica9ve dilemmas at r=0.35 no maber what?
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What about dendrograms?
No indica9on of associa9on
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Element „Child self“ omibed
Dendrograms are based on similari9es and will be affected by element selec9on
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Assessing the stability of cluster solu9ons
• How can we assess which parts of the cluster structure are stable?
• Similar problem in phylogene9c research
• Felstenstein (1985): Suggests Bootstrapping
• Idea: Resampling from the data we have and assess which structures remain stable
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① ②
③
Dendrogram
① AB|CDEF ② ABCD|EF ③ ABC|DEF
Corresponding Par33ons
A B C D E F
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A B C D E F A B D C F E A B C E D F
AB|CDEF ABC|DEF ABCD|EF
AB|CDEF ABD|CEF ABCD|EF
BC|ADEF ABC|DEF ABCE|DF
Bootstrap Replicates
Corresponding Par33ons
AB|CDEF ABC|DEF ABCD|EF
AB|CDEF ABD|CEF ABCD|EF
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Par$$on f BC|ADEF 1 ABC|DEF 2 ABCD|EF 2 AB|CDEF 2 ABCE|DF 1 ABD|CEF 1
h BP .33 33 .67 67 .67 67 .67 67 .33 33 .33 33
A B C D E F Par$$on f BC|ADEF 1 ABC|DEF 2 ABCD|EF 2 AB|CDEF 2 ABCE|DF 1 ABD|CEF 1
67 67
67
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Bootstrap Probabili9es
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Approximately Unbiased
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AU and BP
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Possible measures of interest: 1. Number of (TOP-‐LEVEL) significant clusters 2. Propor9on of ALL constructs in significant clusters 3. Propor9on of UNIQUE constructs in significant clusters
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What can we make of it?
• Do significant clusters indicate 9ghtly knibed parts of the construct system?
• Does it have any meaning at all? Currently lack of a valida9on criterion!
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Some similarity measures and cluster methods
• Manhaban distance • Euclidean distance • Correla9ons • ...
• Ward • Single linkage • Complete linkage • Average • McQuiby • Median • Centroid • …
PCP: FOCUS procedure =
Manhaban distances plus Single linkage. But
why?
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Manhaban Single linkage
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Manhaban Complete linkage
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Euclidean Single linkage
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Euclidean Complete linkage
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Conclusions
• Developments in other fields offer chances for transfer
• Adop9ng an inference view • No substan9al associa9ons with global measures of complexity
• Meaning of significant clusters: subject to further research, valida9on or invalida9on
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www.onair.openrepgrid.org
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Thanks !
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Bell, R. (1990). Repertory Grid as Mental tests: Implica9ons of test theories for grids. Journal of Construc6vist Psychology, 3(1), 91-‐103.
Feixas, G., Saúl, L. A., & Sanchez, V. (2000). Detec9on and analysis of implica9ve dilemmas: implica9ons for the therapeu9c process. In J. W. Scheer (Ed.), The Person in Society: Challenges to a Construc6vist Theory. Giessen: Psychosozial-‐Verlag.
Felsenstein, J. (1985). Confidence Limits on Phylogenies: An Approach Using the Bootstrap. Evolu6on, 39(4).
Krauthauser, H., Bassler, M., & Potratz, B. (1994). A new approach to the iden9fica9on of cogni9ve conflicts in the repertory grid: A nomothe9c study. Journal of Construc6vist Psychology, 7(4), 283–299.
Slade, P. D., & Sheehan, M. J. (1979). The measurement of “conflict” in repertory grids. Bri6sh Journal of Psychology, 70(4), 519–524.
Suzuki, R., & Shimodaira, H. (2006). Pvclust: an R package for assessing the uncertainty in hierarchical clustering. Bioinforma6cs (Oxford, England), 22(12), 1540–1542.
References