180 REGRESSION AND CORRELATION
By substituting in the values from the sample of 10 adult males, we have
(^357) - .857
t=
J(1 - .8572)/(10 - 2) - dm
or
= 4.70
357
t=-
.1822
If we compare the computed t value with a t value of 2.306 with n - 2 = 8 d.f. from
TableA.3, we would reject the HO : p = 0 at the Q = .05 significance level. Note that
the test of p = 0 is equivalent to the test that the population slope /3 = 0. If we cannot
say that the population slope is not 0, we cannot say that the population correlation
is not 0. This test of HO : p = 0 is commonly included in statistical programs, but
the test that p is some value other than 0 is not included.
12.3.6 Interpreting the Correlation Coefficient
An advantage held by the correlation coefficient over the slope of the regression line is
that the correlation coefficient is unaffected by changes in the units of X or Y. In the
example, if weight had been in kilograms, we would still obtain the same correlation
coefficient, .857. In fact, r is unaffected by any linear change in X and Y. We can
add or subtract constants from X or Y or multiply X or Y by a constant and the value
of T remains the same.
The correlation r has a high magnitude when the ellipse depicted in Figure 12.3
is long and thin. All the points lie close to a straight line. A value of T close to 0
results if the points in a scatter diagram fall in a circle or the plot is nonlinear. Scatter
diagrams are a great help in interpreting the numerical value of r.
When a high degree of correlation has been established between two variables, one
is sometimes tempted to conclude that “a causal relation between the two variables
has been statistically proved.” This is simply not true, however. There may or may
not be a cause-and-effect relationship; all that has been shown is the existence of a
straight-line relationship.
An explanation of a high correlation must always be sought very carefully. If the
correlation between X and Y is positive and high, then possibly a large X value may
tend to make an individual’s Y value large or perhaps both X and Y are affected by
some other variables. The interpretation must be based on knowledge of the problem,
and various possible interpretations must be considered.
12.4 LINEAR REGRESSION ASSUMING THE FIXED-X MODEL
Up to this point in Chapter 12, we have been discussing the single-sample model.
The single-sample model is suitable in analyzing survey data or when examining data
from one of two or more groups. We will now take up the fixed-X model. This is a
briefer section since the formulas given for the single-sample model remain the same
and the fixed-X model is not used as much as the random-X model.