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The partial correlations given in Eqs. (7.11.1) to (7.11.3) are called firstorder correlation coefficients. By order we mean the number of secondary

22Most computer programs for multiple regression analysis routinely compute the simple correlation coefficients; hence the partial correlation coefficients can be readily computed.

CHAPTER SEVEN: MULTIPLE REGRESSION ANALYSIS: THE PROBLEM OF ESTIMATION 231

subscripts. Thus r12.34 would be the correlation coefficient of order two, r12.345 would be the correlation coefficient of order three, and so on. As noted previously, r12, r13, and so on are called simple or zero-order correlations. The interpretation of, say, r12 34 is that it gives the coefficient of correlation between Y and X2, holding X3 and X4 constant.

Interpretation of Simple and Partial Correlation Coefficients

In the two-variable case, the simple r had a straightforward meaning: It measured the degree of (linear) association (and not causation) between the dependent variable Y and the single explanatory variable X. But once we go beyond the two-variable case, we need to pay careful attention to the interpretation of the simple correlation coefficient. From (7.11.1), for example, we observe the following:

1. Even if r12 = 0, r12 3 will not be zero unless r13 or r23 or both are zero.

2. If r12 = 0 and r13 and r23 are nonzero and are of the same sign, r12.3 will be negative, whereas if they are of the opposite signs, it will be positive. An example will make this point clear. Let Y = crop yield, X2 = rainfall, and X3 = temperature. Assume r12 = 0, that is, no association between crop yield and rainfall. Assume further that r13 is positive and r23 is negative. Then, as (7.11.1) shows, r12 3 will be positive; that is, holding temperature constant, there is a positive association between yield and rainfall. This seemingly paradoxical result, however, is not surprising. Since temperature X3 affects both yield Y and rainfall X2, in order to find out the net relationship between crop yield and rainfall, we need to remove the influence of the "nuisance" variable temperature. This example shows how one might be misled by the simple coefficient of correlation.

3. The terms r12 3 and r12 (and similar comparisons) need not have the same sign.

4. In the two-variable case we have seen that r2 lies between 0 and 1. The same property holds true of the squared partial correlation coefficients. Using this fact, the reader should verify that one can obtain the following expression from (7.11.1):

which gives the interrelationships among the three zero-order correlation coefficients. Similar expressions can be derived from Eqs. (7.9.3) and (7.9.4).

5. Suppose that r13 = r23 = 0. Does this mean that r12 is also zero? The answer is obvious from (7.11.4). The fact that Y and X3 and X2 and X3 are uncorrelated does not mean that Y and X2 are uncorrelated.

In passing, note that the expression r^2 3 may be called the coefficient of partial determination and may be interpreted as the proportion of the variation in Y not explained by the variable X3 that has been explained

232 PART ONE: SINGLE-EQUATION REGRESSION MODELS

by the inclusion of X2 into the model (see exercise 7.5). Conceptually it is similar to R2.

Before moving on, note the following relationships between R2, simple correlation coefficients, and partial correlation coefficients:

In concluding this section, consider the following: It was stated previously that R2 will not decrease if an additional explanatory variable is introduced into the model, which can be seen clearly from (7.11.6). This equation states that the proportion of the variation in Y explained by X2 and X3 jointly is the sum of two parts: the part explained by X2 alone (= r22) and the part not explained by X2 (= 1 — r22) times the proportion that is explained by X3 after holding the influence of X2 constant. Now R2 > rj2 so long as rj3 2 > 0. At worst, r23 2 will be zero, in which case R2 = r22.

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