METHODS OF STATISTICAL ANALYSIS. 17 



In some cases we have found it necessary to use regression equations 

 in which the value of one variable, z, is predicted from those of two 

 others, x and y, or from that of three others, w, x and y. Formulas 

 for these will be given when used. 



Throughout the following pages we shall have frequent occasion 

 to use partial correlation formulas. Total heat-production is correlated 

 with stature and with body- weight; but stature and body- weight are 

 also correlated, taller individuals being on the average heavier than 

 shorter ones. The problem now arises: May not the correlation 

 between stature and total heat-production be merely the resultant of 

 the correlation between body-weight and heat-production on the one 

 hand and body-weight and stature on the other? To solve this problem 

 we have to correct the correlation between stature and total heat- 

 production for the influence of body-weight. Or, in statistical termin- 

 ology, we must determine the partial correlation between stature, s, and 

 heat-production, h, for constant body-weight, w. This is done by the 

 use of the formula 



' sh ' ws 'wh 



v/i _ r 2 \/i r ., 2 



' WS Wh, 



Here w r sh is to be read "the correlation between stature and heat for 

 constant body- weight." The technical expression "for constant body- 

 weight" means merely "with the influence of body- weight eliminated." 

 If the correlation between stature and total heat-production were 

 merely the resultant of the correlation between weight and heat- 

 production and weight and stature, w r sh should be sensibly zero. For 

 example, for the 136 men, using the constants as given on pages 59 and 

 96, we have: 



r th = +0.6149 



r m = +0.5725 1 -r w , =0.6722 Vl^v7 = 0.8199 



r wh = +0.7960 1 -r wh * =0.3663 VT^J =0.6052 



0.6149 -0.5725 X0.7960 0.1592 



M _ . _ _ II -\ ) 1 



0.8199X0.6052 0.4962 



1 - w r sh * = 0.8969 Ejr,* = 0.6745 ~j = 0.0519 



V N 



Thus the partial correlation between stature and heat-production 

 for constant body-weight is only about half the magnitude of the 

 uncorrected value. It is clear, therefore, that the greater heat-produc- 

 tion of tall individuals is due largely to their greater weight. The fact 

 that the partial correlation has a material and statistically significant 

 positive value indicates that the observed relationship between stature 

 and metabolism is not merely the resultant of the correlations between 

 stature and weight and between weight and metabolism. 



