Biostatistics

9.3 THE SAMPLE REGRESSION EQUATION
9.3 THE SAMPLE REGRESSION EQUATION 417
In simple linear regression the object of the researcher’s interest is the population
regression equation—the equation that describes the true relationship between the
dependent variable Y and the independent variable X. The variable designated by Y is
sometimes called the response variable and X is sometimes called the predictor variable.
In an effort to reach a decision regarding the likely form of this relationship, the
researcher draws a sample from the population of interest and using the resulting data,
computes a sample regression equation that forms the basis for reaching conclusions
regarding the unknown population regression equation.
Steps in Regression Analysis In the absence of extensive information
regarding the nature of the variables of interest, a frequently employed strategy is to
assume initially that they are linearly related. Subsequent analysis, then, involves the
following steps.
1. Determine whether or not the assumptions underlying a linear relationship are met in
the data available for analysis.
2. Obtain the equation for the line that best fits the sample data.
3. Evaluate the equation to obtain some idea of the strength of the relationship and the
usefulness of the equation for predicting and estimating.
4. If the data appear to conform satisfactorily to the linear model, use the equation
obtained from the sample data to predict and to estimate.
When we use the regression equation to predict, we will be predicting the value Y is
likely to have when X has a given value. When we use the equation to estimate, we will be
estimating the mean of the subpopulation of Y values assumed to exist at a given value of X.
Note that the sample data used to obtain the regression equation consist of known values of
both X and Y. When the equation is used to predict and to estimate Y, only the corresponding
values of X will be known. We illustrate the steps involved in simple linear regression
analysis by means of the following example.
EXAMPLE 9.3.1
Despres et al. (A-1) point out that the topography of adipose tissue (AT) is associated with
metabolic complications considered as risk factors for cardiovascular disease. It is
important, they state, to measure the amount of intraabdominal AT as part of the evaluation
of the cardiovascular-disease risk of an individual. Computed tomography (CT), the only
available technique that precisely and reliably measures the amount of deep abdominal AT,
however, is costly and requires irradiation of the subject. In addition, the technique is not
available to many physicians. Despres and his colleagues conducted a study to develop
equations to predict the amount of deep abdominal AT from simple anthropometric
measurements. Their subjects were men between the ages of 18 and 42 years who
were free from metabolic disease that would require treatment. Among the measurements
taken on each subject were deep abdominal AT obtained by CT and waist circumference as