Encyclopedia of Sociology

CORRELATION AND REGRESSION ANALYSIS

b*y1.2 =

ry1 – ry2ry12

1 – r^212

( 14 )

b*y2.1 =

ry2 – ry1r 12

1 – r^212

( 15 )

where by1.2 = the standardized partial regression
coefficient of Y on X 1 , controlling for X 2 ; and
by2.1 = the standardized partial regression coeffi-
cient of Y on X 2 , controlling for X 1.

Standardized partial regression coefficients,
here symbolized by b* (read ‘‘b star’’), are fre-
quently symbolized by the Greek letter beta, and
they are commonly referred to as ‘‘betas,’’ ‘‘beta
coefficients,’’ or ‘‘beta weights.’’ While this is com-
mon usage, it violates the established practice of
using Greek letters to refer to population parame-
ters instead of sample statistics.

A comparison of equation 14, describing the
standardized partial regression coefficient, b*y1.2,
with equation 11, describing the partial correla-
tion coefficient, ry1.2, will make it evident that
these two coefficients are closely related. They
have identical numerators but different denomi-
nators. The similarity can be succinctly expressed by

r^2 y1.2 = b*y1.2 b*1y.2 ( 16 )

If any one of the quantities in equation 16 is 0, all
are 0, and if the partial correlation is 1.0 in abso-
lute value, both of the standardized partial regres-
sion coefficients in equation 16 must also be 1.0 in
absolute value. For absolute values between 0 and
1.0, the partial correlation coefficient and the
standardized partial regression coefficient will have
somewhat different values, although the general
interpretation of two corresponding coefficients is
the same in the sense that both coefficients repre-
sent the relationship between two variables, with
one or more other variables held constant. The
difference between them is rather subtle and rare-
ly of major substantive import. Briefly stated, the
partial correlation coefficient—e.g., ry1.2—is the
regression of one standardized residual on anoth-
er standardized residual. The corresponding stand-
ardized partial regression coefficient, b*y1.2, is the
regression of one residual on another, but the
residuals are standard measure discrepancies from
standard measure predictions, rather than the

residuals themselves having been expressed in the

form of standard deviates.

A standardized partial regression coefficient

can be transformed into an unstandardized partial

regression coefficient by

by1.2 = b*y1.2

sy

s 1 (^17 )

by2.1 = b*y2.1

sy

s 2 (^18 )

where by1.2 = the unstandardized partial regres-

sion coefficient of Y on X 1 , controlling for X 2 ; by2.1

= the unstandardized partial regression coefficient

of Y on X 2 , controlling for X 1 ; b*y2.1 and b*y2.1 are

standardized partial regression coefficients, as de-

fined above; sy = the standard deviation of Y; s 1 =

the standard deviation of X 1 ; and s 2 = the standard

deviation of X 2.

Under all but exceptional circumstances, stand-

ardized partial regression coefficients fall between

−1.0 and +1.0. The relative magnitude of the stand-

ardized coefficients in a given regression equation

indicates the relative magnitude of the relation-

ship between the criterion variable and the predic-

tor in question, after holding constant all the other

predictors in that regression equation. Hence, the

standardized partial regression coefficients in a

given equation can be compared to infer which

predictor has the strongest relationship to the

criterion, after holding all other variables in that

equation constant. The comparison of unstandardized

partial regression coefficients for different predic-

tors in the same equation does not ordinarily yield

useful information because these coefficients are

affected by the units of measure. On the other

hand, it is frequently useful to compare unstandardized

partial regression coefficients across equations.

For example, in separate regression equations pre-

dicting income from education and work experi-

ence for the United States and Great Britain, if the

unstandardized regression coefficient for educa-

tion in the equation for Great Britain is greater

than the unstandardized regression coefficient for

education in the equation for the United States,

the implication is that education has a greater

influence on income in Great Britain than in the

United States. It would be hazardous to draw any

such conclusion from the comparison of standard-

ized coefficients for Great Britain and the United