Biostatistics

514 CHAPTER 10 MULTIPLE REGRESSION AND CORRELATION
For example, the partial sample correlation coefficient r y:12 is a measure of the correlation
between Y and X 1 after controlling for the effect of X 2 .
The partial correlation coefficients may be computed from the simple correlation
coefficients. The simple correlation coefficients measure the correlation between two
variables when no effort has been made to control other variables. In other words, they are
the coefficients for any pair of variables that would be obtained by the methods of simple
correlation discussed in Chapter 9.
Suppose we have three variables, Y, X 1 , and X 2 . The sample partial correlation
coefficient measuring the correlation between Y and X 1 after controlling for X 2 , for
example, is written r y1:2 . In the subscript, the symbol to the right of the decimal point
indicates the variable whose effect is being controlled, while the two symbols to the left of
the decimal point indicate which variables are being correlated. For the three-variable case,
there are two other sample partial correlation coefficients that we may compute. They are
r y2:1 and r 12:y .
The Coefficient of Partial Determination The square of the partial
correlation coefficient is called the coefficient of partial determination. It provides useful
information about the interrelationships among variables. Consider r y1:2 , for example. Its
square, r 2 y1:2 tells us what proportion of the remaining variability in Y is explained by X 1
after X 2 has explained as much of the total variability in Y as it can.
Calculating the Partial Correlation Coefficients
the following simple correlation coefficients may be calculated:
For three variables
r y1 , the simple correlation between Y and X 1
r y2 , the simple correlation between Y and X 2
r 12 , the simple correlation between X 1 and X 2
The MINITAB correlation procedure may be used to compute these simple correlation
coefficients as shown in Figure 10.6.2. As noted earlier, the sample observations are
stored in Columns 1 through 3. From the output in Figure 10.6.2 we see that
r 12 ¼ :08; r y1 ¼ :043, and r y2 ¼ :535.
The sample partial correlation coefficients that may be computed from the simple
correlation coefficients in the three-variable case are:
1. The partial correlation between Y and X 1 after controlling for the effect of X 2 :
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r y1:2 ¼ r y1 r y2 r 12 = ð1 r 2 y2Þð1 r 2 12Þ
(10.6.4)
2. The partial correlation between Y and X 2 after controlling for the effect of X 1 :
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r y2:1 ¼ r y2 r y1 r 12 = ð1 r 2 y1Þð1 r 2 12Þ
(10.6.5)
3. The partial correlation between X 1 and X 2 after controlling for the effect of Y:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r 12:y ¼ r 12 r y1 r y2 = ð1 r 2 y1Þð1 r 2 y2Þ
(10.6.6)