From the examples cited in Section 1.2, the reader will notice that in regression analysis we are concerned with what is known as the statistical, not functional or deterministic, dependence among variables, such as those of classical physics. In statistical relationships among variables we essentially deal with random or stochastic4 variables, that is, variables that have probability distributions. In functional or deterministic dependency, on the other hand, we also deal with variables, but these variables are not random or stochastic.
The dependence of crop yield on temperature, rainfall, sunshine, and fertilizer, for example, is statistical in nature in the sense that the explanatory variables, although certainly important, will not enable the agronomist to predict crop yield exactly because of errors involved in measuring these variables as well as a host of other factors (variables) that collectively affect the yield but may be difficult to identify individually. Thus, there is bound to be some "intrinsic" or random variability in the dependent-variable crop yield that cannot be fully explained no matter how many explanatory variables we consider.
In deterministic phenomena, on the other hand, we deal with relationships of the type, say, exhibited by Newton's law of gravity, which states: Every particle in the universe attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them. Symbolically, F = k(m1m2/r2), where F = force, m1 and m2 are the masses of the two particles, r = distance, and k = constant of proportionality. Another example is Ohm's law, which states: For metallic conductors over a limited range of temperature the current C is proportional to the voltage V; that is, C = (%)V where 1 is the constant of proportionality. Other examples of such deterministic relationships are Boyle's gas law, Kirchhoff's law of electricity, and Newton's law of motion.
In this text we are not concerned with such deterministic relationships. Of course, if there are errors of measurement, say, in the k of Newton's law of gravity, the otherwise deterministic relationship becomes a statistical relationship. In this situation, force can be predicted only approximately from the given value of k (and m1, m2, and r), which contains errors. The variable F in this case becomes a random variable.
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