1.3 STATISTICAL VERSUS DETERMINISTI

1.3 STATISTICAL VERSUS DETERMINISTIC RELATIONSHIPS
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/r 2), 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 = ( 1
k )V where 1
k 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.