tl + d.f.

and d.f. = (n - 5) in this case (one a and four ps).

Partial correlations are very important in deciding whether or not to include more explanatory variables. For instance, suppose that we have two explanatory variables x, and x,, and rjj is very high, say 0.95, but , is very low, say 0.01. What this means is that if Xj alone is used to explain y, it can do a good job. But after x, is included, Xj does not help any more explaining y; that is, x, has done the job of X2. In this case there is no use including X2. In fact, we can have a situation where, for instance,

= 0.95 and /2 = 0.96

/,2 = 0.1 and , = 0.1

In this case each variable is highly correlated with but the partial correlations are both very low. This is called multicoUinearity and we will discuss this problem later in Chapter 7. In this example we can use x, only or Xj only or some combination of the two as an explanatory variable. For instance, suppose that X] is the amount of skilled labor, Xj the amount of unskilled labor, and the output. What the partial correlation coefficients suggest is that the separation of total labor into two components-skilled and unskilled-does not help us much in explaining output. So we might as well use x, + x or total labor as the explanatory variable.

4.6 Relationships Among Simple, Partial, and Multiple Correlation Coefficients

To study the relationships among the different types of correlation coefficients, we need to make use of the relationship that at each stage the residual sum of squares RSS = Syy{\ - R). Suppose that we have two explanatory variables X, and X2. Then

5vv(l - Rl 12) = residual sum of squares after both x, and X2 are included Syy{\ - ijt) = residual sum of squares after the inclusion of x, only

Now 1 measures the proportion of this residual sum of squares explained by X2. Hence the unexplained residual sum of squares after X2 is also included is

(1 - /•;2 ,)5 1 - rb)

in tiie multiple regression equation are all variables that are mentioned in the partial correlation coefficient. For instance, if we want to compute ; 237, we have to run a regression of on x, xi, x,, and x-,. Let 4 = P4/SE(P4), where P4 is the coefficient of x. Then

?4 237 =

Two Illustrative Examples

The following examples illustrate simple r, partial r, and R. The first example illustrates some problems in interpreting multiple regression coefficients when the explanatory variables are proportions. The second example illustrates the use of interaction terms.

Example 1: Hospital Costs

Consider the analysis of hospital costs by case mix by Feldstein. The data are for 177 hospitals. The explanatory variables are proportions of total patients treated in each category. There are nine categories: M = medical, P = pediatrics, S = general surgery, E = ENT, T = traumatic and orthopedic surgery, OS = other surgery, G = gynecology, Ob = Obstetrics. Other = miscellaneous others. The regression coefficients, their standard errors, r-values, partial rs, simple rs, and average cost per case are given in Table 4.2.

The entries in Table 4.2 need some explanation. The /-values are just the coefficients divided by the respective standard errors. The partial rs are obtained by the formula

/2 -I- d.f.

M. S.Feldstein, Economic Analysis for the Health Service Industry (Amsterdam: North-Holland, 1967), Chap. 1.

But this is equal to 5,1 - R] ,2). Hence we get the resuh that

1 - /?2,2 = (1 - f.Kl - 21)

If we have three explanatory variables we get

(1 - 2.,2 ) = (1 - rl,Kl - ,2.)(1 - .2) (4.12)

The subscripts, 1, 2, 3 can be interchanged in any required order. Thus if we need to consider them in order 3, 1, 2, we have

(1 - / ?,2 ) = (1 - ridO - /, )(1 - ,)

Note that the partial can be greater or smaller than the simpler r. For instance, the variable X2 might explain only 20% of the variance of y, but after Xi is included, it can explain 50% of the residual variance. In this case the simple ,-2 is 0.2 and the partial is 0.5. We are thus talking of the proportions explained of different variances.

On the other hand, the simple and the partial can never be greater than R (the square of the multiple correlation coefficient). This is clear from the formulas we have stated.

Table 4.2 Hospital Cost Regressions