Basic Statistics

LINEAR REGRESSION ASSUMING THE FIXED-X MODEL 181

12.4.1

In the fixed-X model, the values of X are selected by the investigator or occur because

of the nature of the situation. For example, in studying the effects of pressure bandages

on leg circumference, nurses were randomly assigned to wear a pressure bandage

for either 30, 60, or 120min. The outcome or Y variable is the difference in leg

circumference before and that after applying the pressure bandage. In this experiment,

the three time periods are the X variable. The values of X are considered to be fixed

and known without error. This example would be a case where the values of X are

selected by the investigator. This model is useful in analyzing experiments when

the X values (the variable whose values are set by the investigator) are continuous

(interval or ratio data).

Alternatively, in studies of change over time, data are obtained on some outcome

variable such as mortality rates or expected length of life by year. The year becomes

the fixed-X variable. Usually, the investigator does not take a random sample of years

but instead looks at the most recent 20 years or some other time period of interest.

Here, the nature of the situation dictates how the sample is taken. Again, the X

variable will be assumed to be fixed and known without error.

To make inferences to the population from confidence intervals or tests of hypothe-

ses, different assumptions are made in this model than in the single-sample case. For

each value of X being considered, a population of Y values exists and the following

three assumptions must hold:

Model Underlying the Fixed-X Linear Regression

1. The Y values at each X are normally distributed.


2. Their means lie on a straight line.


3. Their variances are equal to each other.

Figure 12.6 presents a hypothetical example where measurements have been taken
at three values of X: that is, X(1), X(2), and X(3). The Y values for each X value
are normally distributed with the same variance, and the mean of each set of Y values
falls on a straight line.

12.4.2 Linear Regression Using the Fixed-X Model

Computation of the regression line Y = a + bX proceeds in exactly the same way for

the fixed-X model as for the bivariate normal model. The same formulas are used for

a, b, s~.~, se(a), and se(b). The confidence intervals for the population intercept and

slope, CI: and b, are precisely the same. Tests of hypotheses concerning the population

CI: and ,!3 are also the same as those given for the single-sample bivariate normal mode.

With the fixed-X model we cannot estimate all the parameters that we could for the

bivariate normal model. For the fixed-X model we cannot estimate the parameters

px, py, cx, o9, cxy, or p. The sample statistics for these parameters do not provide

valid estimates of the population parameters. Note that computer programs will print

out X, Y, s,, sy, and r, but these should be ignored if this model is assumed.

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