User:Altsalt/Mathematical Theorizing

Approaches to Modeling
not a mere reexpression of data

not purely statistical "models" like MR or FA

"a mathematical model is a representation of a situated theory" (160)

"yields greater consistency and precision of statement" (161)

"rarely taken to be a substitute for a verbal theory" (161)

"minimum necessary requirement was precision" (163)

"primary duty is to represent, mathematically, conceptually understandable processes which are candidates for explaining data" (164)

"attempting to develop a useful model for a theory is a way of critically examining it and improving it" (164)

Explanatory Modeling
"representation of a conceptual theory in mathematical terms" (165)

Values of Representation (166)

 * 1) contain propositions which have testable consequences relative to meaningful alternative theories and which are required or justified by some central assumptions of the theory
 * 2) description should clearly distinguish between representations, interpretations, and inclusions for convention or convenience.
 * 3) take as axioms the central assumptions and as deductable theorems the propositions of the theory.
 * 4) refer back from propositions to theoretical assumptions, tests should question whether the propositions are true for the reasons given.

Stages of Construction (167)

 * 1) Instantiation of the theory - specifying unit of analysis, a context, and set of variables
 * 2) Expressing theoretic propositions in mathematical form - as multiple forms can correspond to any single proposition, this stage minimizes the generality and/or arbitrariness of the model
 * 3) Interpretation of parameters - can be made of of variables from another theory or perhaps grouped by rate of change.
 * 4) Analytical-critical deductions and test - use math to deduce necessary consequences and implications as well as linking to data or established empirical knowledge.
 * 5) Model extension - often due to abstraction or simplification, modeler may extend theory to a series of models.

Theory Building (172)

 * "Strong theoretic implications out of weak data"
 * "Specific theoretic implications out of complex data"
 * Examples: catastrophe theory, blockmodels, and self-organization and dissipative structure

Issues (179)

 * Simplification distorts reality
 * Identification of realistic assumptions
 * Comparison testing of multiple distinct models

Descriptive Modeling
"to discover or explicate a justification for the mathematical choice is often to transform descriptive modeling into explanatory modeling" (180)

Danger is that it it becomes the unquestioned basis for further research.

Markov process is most often used due to tradition of empirical success and the goal of explaining state change.

"Without theoretical constraints, the descriptive modeler can play fast and loose with success."

Difficult to discern when descriptive models become explanatory ones.

Models of Data
"relationship among indicators" (182) primary difference from explanatory is that models consist primarily of direct links between observed indicators and unobserved "factors" with little attention given to direct links among indicators or links outside of epistemic relations of measure to construct measured

Issues
1: "should a data model also include theoretical relations among unobserved variables and relations of those variables to their indicators?" "any unobserved variable is linked both to its own indicators and to those of other variables with which it is causally related" "if these links have conflicting implications, a resultant bias could arise in data model parameter estimates"

2: "systematic theory of indicators is both possible and necessary" "such arguments are in clear conflict with the practice of path analysts and others who use various means to generate items and purely empirical considerations to select items and develop a measurement model" lack of a clear theory of measurement could invalidate data models

Simulation Modeling
"use of computers to derive the long-term trajectories based on system of equations which predict short-term change" (183) fall between explanatory and descriptive models

Motivations

 * 1) produce practically useful forecasts
 * 2) test a relatively complex system of theoretic relationships
 * 3) explore interrelations and holes in body of knowledge

Issues

 * "can we justifiably choose functional form s and parameter values"
 * "how are we to assess surprising results"

Dangers of Modeling

 * possible to setup so it cannot be rejected or can be saved in face of negative evidence
 * deviant cases are not studied
 * introduce restrictive assumptions into more-flexible theories
 * add assumptions of dubious value
 * serve nothing more than window dressing