Statistical Methods for Psychology

otherwise stated. (We do occasionally fit straight lines to curvilinear data, but we do so on

the assumption that the line will be sufficiently accurate for our purpose—although the

standard error of prediction might be poorly estimated. There are other forms of regression

besides linear regression, but we will not discuss them here.)

As mentioned earlier, whether or not we make various assumptions depends on what

we wish to do. If our purpose is simply to describe data, no assumptions are necessary. The

regression line and rbest describe the data at hand, without the necessity of any assump-

tions about the population from which the data were sampled.

If our purpose is to assess the degree to which variance in Yis linearly attributable to

variance in X, we again need make no assumptions. This is true because and are

both unbiased estimators of their corresponding parameters, independent of any underly-

ing assumptions, and

is algebraically equivalent to.

If we want to set confidence limits on bor Y, or if we want to test hypotheses about ,

we will need to make the conditional assumptions of homogeneity of variance and normality

in arrays of Y. The assumption of homogeneity of variance is necessary to ensure that is

representative of the variance of each array, and the assumption of normality is necessary

because we use the standard normal distribution.

If we want to use rto test the hypothesis that r50, or if we wish to establish confi-

dence limits on r, we will have to assume that the (X, Y) pairs are a random sample from a

bivariate-normal distribution, but keep in mind that for many studies the significance of r

is not particularly an issue, nor do we often want to set confidence limits on r.

9.14 Factors That Affect the Correlation

The correlation coefficient can be substantially affected by characteristics of the sample.

Two such characteristics are the restriction of the range (or variance) of Xand/or Yand the

use of heterogeneous subsamples.

The Effect of Range Restrictions

A common problem concerns restrictions on the range over which Xand Yvary. The effect

of such range restrictionsis to alter the correlation between Xand Yfrom what it would

have been if the range had not been so restricted. Depending on the nature of the data, the

correlation may either rise or fall as a result of such restriction, although most commonly r

is reduced.

With the exception of very unusual circumstances, restricting the range of Xwill in-

crease ronly when the restriction results in eliminating some curvilinear relationship. For

example, if we correlated reading ability with age, where age ran from 0 to 70 years, the

data would be decidedly curvilinear (flat to about age 4, rising to about 17 years of age and

then leveling off) and the correlation, which measures linearrelationships, would be rela-

tively low. If, however, we restricted the range of ages to 5 to 17 years, the correlation

would be quite high, since we would have eliminated those values of Ythat were not vary-

ing linearly as a function of X.

The more usual effect of restricting the range of Xor Yis to reduce the correlation. This prob-

lem is especially pertinent in the area of test construction, since here criterion measures (Y) may

be available for only the higher values of X. Consider the hypothetical data in Figure 9.8. This

s^2 Y#X

b*

r^2

SSY 2 SSresidual

SSY

s^2 Y s^2 Y#X

Section 9.14 Factors That Affect the Correlation 281

range
restrictions