Path Analysis

Examples

One well-known example relates income to occupation, education, father's education, father's occupation, and number of siblings. In another example, a professor wants to relate students' scores on the final exam to those on the two exams during the term. Exam 1 can be related to the final exam score either directly or indirectly via Exam 2. What are the sizes of these effects of Exam 1? If both are small, then the professor might give only a midterm and a final next semester. In a third example, Profit can be studied as a function of Sales and Assets. Assets support Sales, and hence indirectly contribute to Profit. But Assets can also contribute directly to Profit, via interest and rents. What are the relative sizes of these effects? (See Table 1.)

In considering the relationship between Profit, Sales, and Assets, it becomes apparent that Assets, by supporting Sales, contribute indirectly to Profit. Assets also contribute directly to Profit via dividends on investments, interest on savings, and rents on property. The prediction equation for Profit based on Sales and Assets, estimated by the statistical technique of multiple linear regression, is

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Table 1 Sales, Profits, and Assets of the 10 Largest U.S. Industrial Corporations (in millions of dollars)
Company	Sales	Profits	Assets
GM	126974	4224	173297
Ford	96933	3835	160893
Exxon	86656	3510	83219
IBM	63438	3758	77734
GE	55264	3939	128344
Mobil	50976	1809	39080
Philip Morris	39069	2946	38528
Chrysler	36156	359	51038
Du Pont	35209	2480	34715
Texaco	32416	2413	25636
Source: “Fortune 500,” Fortune, 121 (April 23, 1990), 346–367. © 1990 Time Inc. All rights reserved.

The fitting of such a relationship is called regressing Profit on Sales and Assets. The interpretation of the coefficient (multiplier) of Assets is that if Assets increase by one unit (1 $M) and Sales remain the same, Profit is predicted to increase by 0.010069. But if Assets increase by one unit, what are Sales expected to do? Would not higher Assets possibly mean higher Sales? We need also to regress Sales on Assets. This gives

So, in view of our understanding of Assets as contributing both directly and indirectly to Profit, we really need this system of two regression equations, one for Sales on Assets, the other for Profit on Sales and Assets. Now we can see that if Assets increase by 1 unit, Sales are expected to increase by 0.507, and the expected change in Profit is