Biostatistics

510 CHAPTER 10 MULTIPLE REGRESSION AND CORRELATION
10.5.5 Refer to Exercise 10.3.5 and let x 1j ¼ 90 and x 2j ¼ 80:
10.5.6 Refer to Exercise 10.3.6 and let x 1j ¼ 50; x 2j ¼ 95:0; x 3j ¼ 2:00; x 4j ¼ 6:00; x 5j ¼ 75, and
x 6j ¼ 70:
10.6 THE MULTIPLE CORRELATION MODEL
We pointed out in the preceding chapter that while regression analysis is concerned with
the form of the relationship between variables, the objective of correlation analysis is to
gain insight into the strength of the relationship. This is also true in the multivariable case,
and in this section we investigate methods for measuring the strength of the relationship
among several variables. First, however, let us define the model and assumptions on which
our analysis rests.
The Model Equation
We may write the correlation model as
y j ¼ b 0 þ b 1 x 1j þ b 2 x 2j þþb k x kj þ e j (10.6.1)
where y j is a typical value from the population of values of the variable Y, the b’s are the
regression coefficients defined in Section 10.2, and the x ij are particular (known) values of
the random variables X i . This model is similar to the multiple regression model, but there is
one important distinction. In the multiple regression model, given in Equation 10.2.1, the X i
are nonrandom variables, but in the multiple correlation model the X i are random variables.
In other words, in the correlation model there is a joint distribution of Y and the X i that we
call a multivariate distribution. Under this model, the variables are no longer thought of as
being dependent or independent, since logically they are interchangeable and either of the
X i may play the role of Y.
Typically, random samples of units of association are drawn from a population of
interest, and measurements of Y and the X i are made.
A least-squares plane or hyperplane is fitted to the sample data by methods described
in Section 10.3, and the same uses may be made of the resulting equation. Inferences may
be made about the population from which the sample was drawn if it can be assumed that
the underlying distribution is normal, that is, if it can be assumed that the joint distribution
of Yand X i is a multivariate normal distribution. In addition, sample measures of the degree
of the relationship among the variables may be computed and, under the assumption that
sampling is from a multivariate normal distribution, the corresponding parameters may be
estimated by means of confidence intervals, and hypothesis tests may be carried out.
Specifically, we may compute an estimate of the multiple correlation coefficient that
measures the dependence between Y and the X i . This is a straightforward extension of the
concept of correlation between two variables that we discuss in Chapter 9. We may also
compute partial correlation coefficients that measure the intensity of the relationship
between any two variables when the influence of all other variables has been removed.
The Multiple Correlation Coefficient As a first step in analyzing the
relationships among the variables, we look at the multiple correlation coefficient.