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The Nature of Scientific EvidenceStatistical, Philosophical, and Empirical Considerations$
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Mark L. Taper and Subhash R. Lele

Print publication date: 2004

Print ISBN-13: 9780226789552

Published to Chicago Scholarship Online: February 2013

DOI: 10.7208/chicago/9780226789583.001.0001

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Constraints on Negative Relationships: Mathematical Causes and Ecological Consequences

Constraints on Negative Relationships: Mathematical Causes and Ecological Consequences

(p.298) 10 Constraints on Negative Relationships: Mathematical Causes and Ecological Consequences
The Nature of Scientific Evidence

James H. Brown

Edward J. Bedrick

S. K. Morgan Ernest

Jean-Luc E. Cartron

Jeffrey F. Kelly

University of Chicago Press

One problem in inferring process from pattern is that offsetting and indirect effects, nonlinearities, and other difficulties in complex systems may prevent a clear pattern from being generated even though the hypothesized process is operating. One such complication is the constraint on the magnitudes of correlations and covariances when some of the relationships are negative. The positive semidefinite (PSD) condition requires that each of the eigenvalues and each of the principal minors of a correlation matrix be nonnegative, thereby placing limits on the possible values of the correlation coefficient. Using simple ecological examples, this chapter illustrates how processes that can generate strongly negative correlations in a simple two-variable system can be constrained from generating such clear patterns when there are multiple variables. The PSD condition is more than a statistical curiosity; it reflects real constraints on the magnitudes and patterns of interactions in ecological networks and other complex systems.

Keywords:   PSD condition, constraints, strongly negative correlations, patterns, ecological networks, complex systems, variables

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