Basic Statistics

182 REGRESSION AND CORRELATION

Y

CY

xu 1 X(2) 43)

Figure 12.6 Graph depicting assumptions for the fixed-X model.

I

xu 1 X(2)

Figure 12.6 Graph depicting assumptions for the fixed-X model.

We lose the ability to estimate px and oz because X values were deliberately

chosen. For example, if we chose values of X that are far apart, we will get a larger

variance and standard deviation than we would if we choose them close together.

Since we are assuming that the Y values depend on X, they are also affected and

we cannot estimate py or og. The correlation T is also affected by the choice of X

values, so we cannot estimate p.

One advantage of the fixed-X model is that when individuals are randomly as-

signed to, say, three treatment levels as given in the leg circumference example, we

can make causal inferences. In the case of the mortality rates for the most recent

years, direct causality cannot be established. If we find changes in the rates over

time, we do not know what caused them.

The assumptions made in using the fixed-X model should first be checked graph-

ically by looking at a scatter diagram with a line drawn on it. The points should fall

along a straight line with the residuals from the line approximately the same size for

different values of X (equal variance assumption). If there are multiple Y values at

a limited number of X values, the means of the Y values at each X value should fall

close to the line and the range of the Y values for each X value should be similar.

Indications of not meeting the assumptions would be to have the ranges of the Y

values at each X value increase with increasing levels of the means of the Y variable

or to have the means of Y for various values of X not fall close to a straight line.