8g Journal of Agricultural Research voi. xvi, no. 3 



tween the percentage of butter fat and age is slightly over 2.7 times its 

 probable error when the doubtful observation is included in the data. 

 The difference and its probable error for this is 0.1066 ±0.0400, or 2.7 

 times its probable error when the doubtful observation is included in the 

 data. The difference and its probable error for this is 0.1066 ±0.400, 

 or the difference is 2.7 times its probable error. When this doubtful value 

 is not included in the calculations, the difference and its probable error 

 becomes 0.1645 ±0.0395, ^^ 4-2 times the probable error, a value which 

 certainly represents a greater effect of age on the solids-not-fat content 

 of cow's milk than of age on the butter-fat content of the same milk. 



Much the same statement holds for the relation of the percentage of 

 solids-not-fat and weight of milk produced and percentage of the solids- 

 not-fat and age at test. The difference is of the same magnitude as is 

 the difference between percentage of solids-not-fat and percentage of 

 butter fat and age — that is, the difference is only slightly significant if 

 we consider the correlation found in the presence of the doubtful value, 

 and is markedly significant when this value is thrown out of the table. 



LINEARITY OF REGRESSION 



The analytical constants necessary to test the linearity of regression 

 are given in Table VIII. In every case the correlation ratio is a somewhat 

 larger numerical quantity than the correlation coefficient for the same 

 table. These differences are sho\\Ti to be of little significance in view of 

 the fact that S^ and rf — r^ are substantially zero. The difference be- 

 tween the correlation ratios and the correlation coefficients are probably 

 not significant. In only one case is the difference tf' — r'^ greater than 

 three times the probable error (j) , and in this case the difference is only 

 3.3 times the probable error. For this one case the difference is in all 

 probability not significant. For the other correlations the difference is 

 certainly not significant. It may be concluded, therefore, that the re- 

 gressions are linear and that the correlation coefficient represents the true 

 correlation. 



The following conclusion may be drawn from the above analysis con- 

 cerning the relations between the constituents of cow's milk and the varia- 

 bles, age at beginning of the year test and amount of milk produced dur- 

 ing this year test. 



1. As the amount of milk given by the cows in this test increases, the 

 percentage composition of the butter fat in this milk decreases. The 

 amount of this decrease is statistically significant. Considered practically, 

 this fall in butter-fat content could not be easily detected in the small 

 samples usually handled. 



2. There is a slightly significant fall of the percentage of butter fat con- 

 tained in the milk as age advances. This slight fall may, however, be 

 accounted for by the rise in milk production which occurs coincident with 



