Statistical Methods for Psychology

Standardized Regression Coefficients

Although we rarely work with standardized data (data that have been transformed so as to

have a mean of zero and a standard deviation of one on each variable), it is worth consider-

ing what bwould represent if the data for each variable were standardized separately. In

that case, a difference of one unit in Xor Ywould represent a difference of one standard

deviation. Thus, if the slope were 0.75, for standardized data, we would be able to say that

a one standard deviation increase in Xwill be reflected in three-quarters of a standard devi-

ation increase in. When speaking of the slope coefficient for standardized data, we often

refer to the standardized regression coefficientasb(beta)to differentiate it from the co-

efficient for nonstandardized data (b). We will return to the idea of standardized variables

when we discuss multiple regression in Chapter 15. (What would the intercept be if the

variables were standardized?)

Correlation and Beta

What we have just seen with respect to the slope for standardized variables is directly

applicable to the correlation coefficient. Recall that ris defined as , whereas bis

defined as. If the data are standardized, and the slope and the

correlation coefficient will be equal. Thus, one interpretation of the correlation coefficient

is that it is equal to what the slope would be if the variables were standardized. That sug-

gests that a derivative interpretation of r 5 .80, for example, is that one standard deviation

difference in Xis associated on the averagewith an eight-tenths of a standard deviation dif-

ference in Y. In some situations such an interpretation can be meaningfully applied.

A Note of Caution

What has just been said about the interpretation of band rmust be tempered with a bit of

caution. To say that a one-unit difference in family income is associated with 0.75 units dif-

ference in academic achievement is not to be interpreted to mean that raising family income

for Mary Smith will automatically raise her academic achievement. In other words, we are

not speaking about cause and effect. We can say that people who score higher on the income

variable also score higher on the achievement variable without in any way implying causa-

tion or suggesting what would happen to a given individual if her family income were to in-

crease. Family income is associated (in a correlational sense) with a host of other variables

(e.g., attitudes toward education, number of books in the home, access to a variety of envi-

ronments) and there is no reason to expect all of these to change merely because income

changes. Those who argue that eradicating poverty will lead to a wide variety of changes in

people’s lives often fall into such a cause-and-effect trap. Eradicating poverty is certainly a

worthwhile and important goal, one which I strongly support, but the correlation between

income and educational achievement maybe totally irrelevant to the issue.

9.6 Other Ways of Fitting a Line to Data

While it is common to fit straight lines to data in a scatter plot, and while that is a very use-

ful way to try to understand what is going on, there are other alternatives. Suppose that the

relationship is somewhat curvilinear—perhaps it increases nicely for a while and then lev-

els off. In this situation a curved line might best fit the data. There are a number of ways of

fitting lines to data and many of them fall under the heading of scatterplot smoothers.The

different smoothing techniques are often found under headings like splinesand loess,and

covXY>sX^2 sX=sY=s^2 X= 1

covXY>sXsY

YN

Section 9.6 Other Ways of Fitting a Line to Data 257

standardized
regression
coefficient
b(beta)

scatterplot
smoothers

splines

loess