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 Due to extreme nonlinearities, a causal variable can induce a totally different kind of result at low concentration than at high concentration. An example is that nitroglycerin is a common medication for heart problems, yet the patient never explodes! Low concentrations of some causal variables can have surprisingly large effects, through development of a feedback cycle. Such a cycle, for example, is thought to account for the mechanism by which minute oscillations in the earth’s orbit cause enormous fluctuations in global climate known as ice ages and interglacial stages. Extreme nonlinearities are the researcher’s bane.

Correlation Conclusions
change.
 * Correlation can describe a relationship, but it cannot establish causality.
 * Many variables have secular trends, but the correlation with time is indirect: secular change in a possibly unidentified causal variable causes the measured dependent variable to exhibit secular
 * Crossplots are the most robust and reliable way to look for a relation between variables.
 * Statistical correlation techniques assume independent measurements, so they must be used with caution when measurements are not independent (e.g., time series or grouped data).
 * Interpolation between independent measurements is safe, but interpolation between nonindependent measurements is risky.
 * Extrapolation beyond the range of previous measurements is usually risky.
 * Linear regression and the correlation coefficient R assume a linear relationship between variables.
 * Examination of regression residuals is needed, to detect systematic mismatches.
 * Nonlinearity can complicate relationships among variables enormously.

Perspectives on Causality
“Felix qui potuit rerum cognoscere causas.”

(Happy is he who has been able to learn the causes of things) [Virgil, 70-19 B.C.] Causality is a foundation of science, but it is not a firm foundation. Our concept of causality has been transformed more than once and it continues to evolve.

During the classical Greek period, to seek causes meant to seek the underlying purposes of phenomena. This concept of causality as purpose is identified with Aristotle, but Aristotle was an advocate rather than an initiator of this focus. The search for underlying purpose is also a religious concern, and the overlap between science and religion was correspondingly greater in ancient Greece than in modern times. Perhaps the religious connotation partly explains the shift away from Aristotelian causality during the last few centuries, but I suspect that the decisive factor was the growing scientific emphasis on verifiability. Greek science felt free to brainstorm and speculate about causes, but modern science demands tests of speculations. Testing purposes is much less feasible than testing modern associative causality. Modern scientific concern about purpose is confined primarily to some aspects of biology and social science. Even most of these questions (e.g.,