Transition Over Variables

1. Transitions over variables: assigning categories wobbles, humps and sudden jumps

2. Assigning categories (1 of 2) Indicator variable is often a frequency of occurrence or an intensity How much of an indicator variable should there be in order for a category “to be there”? • How many prepositions must an infant use so that we can say that the preposition category is present? • How hyperactive must a child be in order to count as “hyperactive”? • How stable must the child’s sociometric choices be in order to count as stable? • Examples • Does the child “have” prepositions, object concept, .. Or not • ADHD or not, PDD-Nos or not, … • Is the child’s sociometric rating stable or not..? • Standard Assumptions • Category is a determinate property • Mutually exclusive categories • Observer disagreement is error • Assignment often based on • Continuous indicator variables • A number of such indicators wobbles, humps and sudden jumps - transitions over variables

3. Assigning categories (2 of 2) present Value of indicator variable conventional boundary or measurement error not present wobbles, humps and sudden jumps - transitions over variables

4. Categories as a transition problem (1 of 3) Method • Define stable rating as any rating that is based on stronger rating constraints than those of your null-hypothesis model • Define a null-hypothesis model. E.g. • No rating constraints • No stronger rating constraints than gender preference • Run null-hypothesis model many times • Determine probability that null model produces agreement between ratings that are as good as or better than observed agreements between ratings • The probability specifies a transition function • Example: “when” are the social preferences stable? • actual preference = determinate property; preference “in general” = indeterminate property • Preference in general can be expressed in terms of constraints on choices • If those constraints are obeyed, then the rating is stable • Ratings are stable if they cannot be produced by a rating where such constraints are absent wobbles, humps and sudden jumps - transitions over variables

5. Categories as a transition problem (3 of 3) stable transition variable Results • Transition curve depends on the properties of the null-model (i.e. a model with no constraints, or not enough constraints) and on the number/nature of ratings given by a child • The form of the transition curve makes it easy to divide the %-of-agreement dimension in three categories • Stable raters (more constraints than null-model) • Variable raters (consistent variation) • Transition group: ambiguous state, or both • The child’s preference is either an actual choice (determinate property) or an indeterminate property (range); the child occupies a range on the stability dimension (how stable is the child’s stability?) wobbles, humps and sudden jumps - transitions over variables

6. Categories as a transition problem (3 of 3) characteristicness The stability of a child is also represented by a range wobbles, humps and sudden jumps - transitions over variables

7. Transition curves and fuzzy categories Degree-of-membership • Example: prepositions • When is a “preposition” a preposition? • “How many prepositions should the child produce before we can say that it has the preposition category?” • In language development, categories, such as prepositions, are constructed • The construction process can be described by a transition curve • During the process, the child will go through a phase of ambiguity wobbles, humps and sudden jumps - transitions over variables

8. A note on interobserver (dis)agreement • Disagreement in categorization of behaviors seen as error • We correct for chance agreement (e.g. Kappa) • The reasoning behind the chance correction is strange • Disagreement contains information: If categories are ambiguous, disagreement among competent observers reflects the ambiguity of the observed category • Transform disagreement into a measure of ambiguity or a measure of the fuzziness of the categories wobbles, humps and sudden jumps - transitions over variables

9. Recapitulation wobbles, humps and sudden jumps

10. Recapitulation (1 of 3) • I started from a dynamic and contextual view on developing behavior • And applied the distinction between determinate and indeterminate properties to measuring actual behavior versus making a statement about capacities, abilities generalizing over actual contexts • Measured properties are indeterminate • And can be represented as specific constraints on the degrees of freedom • Which amounts to specifying ranges instead of true scores or levels wobbles, humps and sudden jumps

11. Recapitulation (2 of 3) • I used this view to study developmental transitions and claimed such transitions are intimately linked to variability • I made a distinction between transitions over time and transitions over descriptive variables (categorization) • A distinction was made between continuous and discontinuous transitions • A common property is the occurrence of anomalies • I demonstrated techniques for describing and testing those anomalies wobbles, humps and sudden jumps

12. Recapitulation (3 of 3) • I showed how categorization can be viewed a s a form of transition over a descriptive category • And demonstrated a method to make a distinction between “stable” and “unstable” sociometric ratings • I argued that categorization is intimately linked to ambiguity and fuzziness • Ambiguity and fuzziness can be quantified in terms of transition functions • Disagreement among (trained and competent) observers may provide information about ambiguity But basically I have argued that a dynamic and contextual view on development requires a rethinking of measurement and a renewed interest in the study of phenomena that have suffered from too much suspicion, namely intra-individual variability, extremes, anomalies, ambiguity, fuzziness … and yellow socks. wobbles, humps and sudden jumps