44 BULLETIN 1277, TJ. S. DEPARTMENT OF AGRICULTURE 
roughly the relation of the corn fed to the pork produced. Com- 
paring this chart with the two others, the reason for the difference 
in the slope of the regression lines is apparent ; the regression line 
for the heavier feeding tended to be parallel to the upper part of the 
curve, and the one for lighter feeding to the lower part of the curve. 
The slope of a regression line is therefore a poor measure of the effect 
of input upon output, as it will change, depending upon the part of the 
curve where the bulk of the records happen to fall in any given sample. 
The relation shown by the curve in Figure 16 is of much more 
value than that shown by the regression lines in either of the other 
charts. The curve gives an approach to that exact measurement of 
diminishing returns which has been shown to be essential to any 
detailed input-output analysis. 
A method for determining the exact curve from the group aver- 
ages has been evolved by Dr. W. J. Spillman. This method gives a 
rather satisfactory way of determining the curve in cases of simple 
two-variable relationship such as this. A measure of the accuracy 
of the results, comparable with the coefficient of correlation, can like- 
wise be obtained for such curvilinear relations. 22 
MULTIPLE CURVILINEAR CORRELATION 
The curve in Figure 16 showing the relation between corn fed and 
pork produced is still subject, however, to one limitation that was 
stated with regard to the straight line in the case of simple correla- 
tion. It tells, it is true, how much gain in weight per pig was found 
for the average of the droves receiving each stated input of corn; but 
it does not tell how much of this gain was due to the corn alone, and 
how much was due to the other feeds the pigs received. " Other 
things " are still not constant. 
An extension of the method of multiple correlation- for linear re- 
lations is applicable to< the solution of this problem. The use of 
the multiple curvilinear analysis enables one to determine approxi- 
mately the curve of diminishing returns for each variable separately, 
and supplies a means of making estimates of what the gains will be 
for any given combination of inputs, taking into account the location 
of each input on its curve of diminishing returns. The reliability 
of the curves for each variable and the accuracy with which the 
output may be estimated, can also be determined by this method. 23 
Even the method of multiple curvilinear correlation will not solve 
all the difficulties met in input-output analysis. Many problems 
involve not merely determining the net effect of individual inputs, 
but determining the effect of one input in the presence or absence 
of others, or with specified variations in others. How to handle these 
more complex relations lias yet to be worked out. As the simpler 
phases of input-output analysis are attacked and conquered, the more 
difficult problems will be disposed of with increasing facility. 
22 References on simple curvilinear correlation: 
Spillman, \Y. .1. Application of the Law of Diminishing Returns to Some Fertilizer 
and Feed Data, Journal <>;' Farm Economics, Vol. V, No. 1, pp. 36-52, Jan lary, L923. 
Spillman, \\\ J., ami Lang, El The I aw of Diminishing Return.-. 1924, pp. 12-17, 
70 I. - ,. 
Ezekiel, Mordecai. A Method of Handling Curvilinear Correlation for any Number of 
Variables. (Pari I!, Journal of the American Statistical Association, Vol. XIX, New 
L924. 
^Reference <>a curvilinear multiple correlation: Ezekiel, Mordecai. A Method of 
Handling Curvilinear Correlation foi any Number of Variables. (Part II), Journal of 
the American Statistical Association, Vol. XIX, Now Series, No. 148, 1924. 
o 
