128 MCEWEN AND MICHAEL. 



pendent variable in one group equals the first value in the succeeding 

 group. This is the case, for example, in the illustrative wheat prob- 

 lem where the highest temperature in group 1 (see table 5) is 64.2° F. 

 which is also the lowest temperature in group 2. In such cases it 

 might seem better to vary the number of entries in the groups so as 

 to bring the point of division between different values of the indepen- 

 dent variable, c. g., in the particular case cited, to increase the number 

 of entries in group 1 and decrease that in group 2 by 1 so that the 

 highest temperature in group 1 would be 64.2° F. and the lowest in 

 group 2 would be 64.4° F. However, this is only an apparent advan- 

 tage, for, in the case when variability within the group is. neglected 

 the error introduced depends primarily upon the range in value of the 

 independent variable within the group and not upon the precise point 

 of dixdsion. The latter is comparatively insignificant, and the little 

 effect it does have in the end results decreases as the number of entries 

 per group increases and as the range in value of the independent vari- 

 able decreases. Finally, in the case when regressions are used the 

 effect of a change in the point of division is almost if not quite elimi- 

 nated. * 

 Basis of mathemalical reasoning. 



As stated on page 100, the special form of expression, upon which 

 the mathematical demonstration of the method of successive approxi- 

 mation is based, implies that the change in the dependent variable, w, 

 corresponding to a given change in any independent variable is 

 negligibly affected by the magnitude of the constant values to which 

 the remaining independent variables are reduced. Stated in the 

 concrete terms of the wheat problem, the assumption is that a change 

 of say two degrees in temperature increases the wheat yield by essen- 

 tially the same amount whether the precipitation is 3.5 or 11.6 inches. 

 It is evident that this involves a second assumption ; that the change 

 in w is approximately independent of the magnitude of iv, or, expressed 

 in terms of the wheat problem, that an increase in temperature, say 

 from 62.9° F. to 65.3° F., increases the wheat yield by approximately 

 0.11 bushels per acre (see table 7), irrespective of whether the yield 

 corresponding to 62.9° F. is 4 or 17 bushels per acre. Although both 

 of these assumptions are likewise inherent in the method of multiple 

 correlation, which, as a matter of fact, is but a special case of this 

 more general method of successive approximation, the question at 

 once arises: to what extent are these assumptions valid, and how 

 may one proceed in any particular case when they are known not to 

 be valid? 



