title: Does the model of hierarchical complexity produce significant gaps between orders and are the orders equally spaced?
abstract: The model of hierarchical complexity (MHC) provides an analytic a priori measurement of the difficulty of tasks. As part of the theory of measurement in mathematical psychology, the model of hierarchical complexity (Commons and Pekker, 2008) defines a new kind of scale. It is important to note that the orders of hierarchical complexity of tasks are postulated to form an ordinal scale. A formal definition of the model of hierarchical complexity is presented along with the descriptions of its five axioms that help determine how the model of hierarchical complexity orders actions to form a hierarchy. The fourth and the fifth axioms are of particular importance in establishing that the orders of hierarchical complexity form an equally spaced ordinal scale. Previously, it was shown that Rasch-scaled items followed the same sequence as their orders of hierarchical complexity. Here, it is shown that the gaps between the highest Rasch scaled item scores at a lower order and the lowest scores at the next higher order exist. We found there was no overlap between the Rasch-scaled item scores at one order of complexity, and those of the adjoining orders. There are 'gaps' between the stages of performance on those items. Second, we tested for equal spacing between the orders of hierarchical complexity. We found that the orders of hierarchical complexity were equally spaced. To deviate significantly from the data, the orders had to deviate from linearity by over .25 of an order. This would appear to be an empirical and mathematical confirmation for the equally spaced stages of development.
authors.affiliation: Harvard Medical School, 234 Huron Avenue, Cambridge, MA 02138, USA, commons@tiac.net.
authors.forename: Michael Lamport
authors.initials: ML
authors.lastname: Commons
authors.forename: Eva Lujia
authors.initials: EL
authors.lastname: Li
authors.forename: Andrew Michael
authors.initials: AM
authors.lastname: Richardson
authors.forename: Robin
authors.initials: R
authors.lastname: Gane-McCalla
authors.forename: Cory David
authors.initials: CD
authors.lastname: Barker
authors.forename: Charu Tara
authors.initials: CT
authors.lastname: Tuladhar
mesh_terms: D000293:Adolescent; D000328:Adult; D000368:Aged; D016477:Artifacts; D003198:Computer Simulation; D003627:Data Interpretation, Statistical; D056344:Executive Function / Q000145:classification*; D005260:Female; D006801:Humans; D008297:Male; D008875:Middle Aged; D015233:Models, Statistical; D011594:Psychometrics / Q000379:methods* / Q000706:statistics & numerical data*; D015203:Reproducibility of Results; D012680:Sensitivity and Specificity; D013647:Task Performance and Analysis; D055815:Young Adult