Practical methods for working multivariable linear correlation and regression problems
Here we will take a new illustration of a multivariable correlation problem, and work through the calculations of the net regression coefficients, the coefficients of multiple and partial correlation, and the standard errors for these statistics. The meaning of such standard errors is discussed in more detail in Chapters 17 and 19.
For this exercise, we will use data on the per capita consumption of beef in the United States, and on three variables logically related to it--retail price of beef, deflated for changes in price levels; disposable income per capita, similarly deflated; and the consumption of pork per capita. The hypothesis assumed in selecting these data is that consumption of beef per person will be affected by real prices of beef, real income available for expenditure, and consumption of the other main meat, pork. The figures, thus adjusted, are given in the first five columns of Table 13. 1.
The final column in the table, headed check sum, is calculated by making a total of all the other values for each observation, from X1, through X4.
The next step is to add all the columns, including the Σ0, and to divide by n, the number of observations (20 in this case), and enter the average. The results are shown at the foot of the table. All the calculations to this point are now verified by the two equations:
X1 + ΣX2 + ΣX3 + ΣX4 = Σ(Σ0) (13.1)
M1 + M2 + M3 + M4 = M0 (13.2)
That is, in each of these last two lines, the sum of the entries in the four columns from X1, to X4 will equal the sum of the Σ0 column.