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This book builds a much-needed bridge between biostatistics and organismal biology by linking the arithmetic of statistical studies of organismal form to the biological inferences that may follow from it. It incorporates a cascade of new explanations of regression, correlation, covariance analysis, and principal components analysis, before applying these techniques to an increasingly common data resource: the description of organismal forms by sets of landmark point configurations. For each data set, multiple analyses are interpreted and compared for insight into the relation between the arithmetic of the measurements and the rhetoric of the subsequent biological explanations. The text includes examples that range broadly over growth, evolution, and disease. For graduate students and researchers alike, this book offers a unique consideration of the scientific context surrounding the analysis of form in today's biosciences.
Cimpian & Salomon (C&S) appear to characterize the inherence heuristic and essentialism as unwise or childish aspects of human reasoning. But actually, these cognitive modes lie at the core of statistical analysis across all of the quantitative sciences, including the developmental cognitive psychology in which the argument here is couched. Their whole argument is as much an example of its topic as an analysis of it.
To find numerical structures that are both surprising and explanatorily useful in complex organized systems, we need a general-purpose numerical pattern engine. This chapter, building on the foundation in Chapter 5, introduces a range of variations on one theme, the singular-value decomposition (SVD). Section 6.1 is about an elegant geometric diagram, the doubly ruled hyperbolic paraboloid. Section 6.2 describes the algebra of the singular-value decomposition, whose diagram this is. Sections 6.3, 6.4, and 6.5 explicate three important application contexts of this tool – principal components analysis, Partial Least Squares, and principal coordinates analysis – with variants and worked examples. Chapter 7 is occupied mainly with demonstrations of the craft by which all this may be interwoven in one context, morphometrics, that very nicely matches the theme of organized systems on which Part III concentrates. What makes morphometrics so suitable an example is its systematic attention to the parameterization of the variables that go into the SVD pattern engine, a parameterization that the SVD is specifically designed to inspect and indeed to try and simplify.
The Hyperbolic Paraboloid
How can geometry help us apprehend numerical explanations – help us turn arithmetic into understanding – in more complex systems?
Most readers who have ever taken a statistics course have seen straight lines used as if they were explanations.
This last chapter summarizes the implications of all that has preceded for the praxis of statistical science in the near future. It is divided into three sections. Section 8.1 introduces one final example: another study of fetal alcohol exposure, this time based on ultrasound brain images of infants. With its aid I review the notions of abduction and consilience from a point of view emphasizing the psychosocial structures of science rather than the logic of these forms of inference per se. If a scientific fact is “a socially imposed constraint on speculative thought” (Ludwik Fleck's main theme), then abduction and consilience work in somewhat contrasting ways to effect that constraint depending on whether the scientific context is one of simple measurement or the probing of a complex system. Section 8.2 shows how all these procedures depend on prior consensus as to what constitutes agreement or disagreement between a numerical representation of some pattern and an expectation about a scientific regularity. In the final section I step back to examine the whole protocol by which forms of numerical reasoning are communicated across the academic and technological generations. The chapter concludes with some recommendations for major changes in the way we teach statistics. If this praxis of abduction cum consilience toward the understanding of complex systems is to keep up with the requirements for science and for the public understanding of science over the next few decades, special attention must be paid to the curricula by which all these strategies are taught to the next generation of our colleagues.
Chapter 2 reviewed diverse aspects of consilience, the general principle by which agreement of numerical values or matching of numerical patterns is turned into evidence for scientific assertions. But except for one example in each of the preceding two chapters we have not yet dealt with the step even earlier at which potential agreements were made the focus of our attention: the step at which some “surprising fact C is observed” from which a novel hypothesis will drain the surprise. The present chapter closes the logical circle by a detailed examination of this process, the launch of an abduction. It will prove to rely as much on human psychology as on any formal properties of numbers and the symbolic reasoning they convey.
As in earlier chapters, the first section reviews one specific example, here the demonstration by the Intergovernmental Panel on Climate Change (IPCC, 2007) that the rise of average global temperature beginning around 1960 is very likely the result of human action. I pay some attention to intellectual history, to publication details, and to several associated ironies. Section 3.2 introduces Charles Sanders Peirce, 19th-century American philosopher, and then, intentionally climbing higher on the scale of abstraction, presents in some detail the general syllogism he discovered, of which the episode in the introductory section serves as a particularly shining, if anachronistic, example.