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410 CHAPTER 9 SIMPLE LINEAR REGRESSION AND CORRELATION
9.1 INTRODUCTION
In analyzing data for the health sciences disciplines, we find that it is frequently desirable
to learn something about the relationship between two numeric variables. We may,
for example, be interested in studying the relationship between blood pressure and age,
height and weight, the concentration of an injected drug and heart rate, the consumption
level of some nutrient and weight gain, the intensity of a stimulus and reaction time, or
total family income and medical care expenditures. The nature and strength of the relationships
between variables such as these may be examined by regression and correlation
analysis, two statistical techniques that, although related, serve different purposes.
Regression Regression analysis is helpful in assessing specific forms of the relationship
between variables, and the ultimate objective when this method of analysis is
employed usually is to predict or estimate the value of one variable corresponding to a
given value of another variable. The ideas of regression were first elucidated by the English
scientist Sir Francis Galton (1822–1911) in reports of his research on heredity—first
in sweet peas and later in human stature. He described a tendency of adult offspring, having
either short or tall parents, to revert back toward the average height of the general population.
He first used the word reversion, and later regression, to refer to this phenomenon.
Correlation Correlation analysis, on the other hand, is concerned with measuring the
strength of the relationship between variables. When we compute measures of correlation
from a set of data, we are interested in the degree of the correlation between variables.
Again, the concepts and terminology of correlation analysis originated with Galton, who
first used the word correlation in 1888.
In this chapter our discussion is limited to the exploration of the linear relationship
between two variables. The concepts and methods of regression are covered first,
beginning in the next section. In Section 9.6 the ideas and techniques of correlation are
introduced. In the next chapter we consider the case where there is an interest in the
relationships among three or more variables.
Regression and correlation analysis are areas in which the speed and accuracy of
a computer are most appreciated. The data for the exercises of this chapter, therefore,
are presented in a way that makes them suitable for computer processing. As is always
the case, the input requirements and output features of the particular programs and
software packages to be used should be studied carefully.
9.2 THE REGRESSION MODEL
In the typical regression problem, as in most problems in applied statistics, researchers
have available for analysis a sample of observations from some real or hypothetical
population. Based on the results of their analysis of the sample data, they are interested
in reaching decisions about the population from which the sample is presumed to have
been drawn. It is important, therefore, that the researchers understand the nature of the
population in which they are interested. They should know enough about the population