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566 CHAPTER 11 REGRESSION ANALYSIS: SOME ADDITIONAL TECHNIQUES
is motivated by a desire to describe, understand, and make use of the relationship
between independent variables and a dependent (or outcome) variable that is discrete.
Particularly plentiful are circumstances in which the outcome variable is dichotomous.
A dichotomous variable, we recall, is a variable that can assume only one of two
mutually exclusive values. These values are usually coded Y = 1 for a success and
Y = 0 for a nonsuccess, or failure. Dichotomous variables include those whose two
possible values are such categories as died–did not die; cured–not cured; disease
occurred–disease did not occur; and smoker–nonsmoker. The health sciences professional
who either engages in research or needs to understand the results of research
conducted by others will find it advantageous to have, at least, a basic understanding
of logistic regression, the type of regression analysis that is usually employed when
the dependent variable is dichotomous. The purpose of the present discussion is to
provide the reader with this level of understanding. We shall limit our presentation to
the case in which there is only one independent variable that may be either continuous
or dichotomous.
The Logistic Regression Model Recall that in Chapter 9 we referred to
regression analysis involving only two variables as simple linear regression analysis. The
simple linear regression model was expressed by the equation
y = b 0 + b 1 x +P
(11.4.1)
in which y is an arbitrary observed value of the continuous dependent variable. When
the observed value of Y is m yƒx , the mean of a subpopulation of Y values for a given value
of X, the quantity P, the difference between the observed Y and the regression line (see
Figure 9.2.1) is zero, and we may write Equation 11.4.1 as
m y ƒ x = b 0 + b 1 x
(11.4.2)
which may also be written as
E 1yƒx2 = b 0 + b 1 x
(11.4.3)
Generally the right-hand side of Equations 11.4.1 through 11.4.3 may assume any value
between minus infinity and plus infinity.
Even though only two variables are involved, the simple linear regression model
is not appropriate when Y is a dichotomous variable because the expected value (or mean)
of Y is the probability that Y = 1 and, therefore, is limited to the range 0 through 1,
inclusive. Equations 11.4.1 through 11.4.3, then, are incompatible with the reality of the
situation.
If we let p = P1Y = 12, then the ratio p>11 - p2 can take on values between 0 and
plus infinity. Furthermore, the natural logarithm (ln) of p>11 - p2 can take on values