14 A BIOMETRIC STUDY OF BASAL METABOLISM IN MAN. 



In metabolism work positive correlations are numerous. For 

 example, the correlation between body-weight and total heat-produc- 

 tion in the 136 men available for this investigation is +0.796, or about 

 80 per cent of perfect interdependence. Physiologists have, of course, 

 known of the existence of this relationship. The statistical method 

 has not been necessary to demonstrate its existence. What the statis- 

 tical formula has done is to measure on a quantitative scale a relationship 

 concerning which ideas were heretofore vague and qualitative only. 

 The positive sign shows that total heat-production increases with 

 body-weight. 



Age is the only character for which correlations have in this work 

 been found to be consistently negative in sign. The correlation between 

 age and total heat-production in these 136 men has been found to be 

 0.306. This shows that heat-production decreases as age increases 

 and measures, on the universally comparable scale of unity, the close- 

 ness of the interrelationship between these two variables. 



For purposes of comparison a measure of the interrelationship of 

 two variables on a universal scale is invaluable. Fortunately it is 

 possible, by proper statistical formulas, to pass from measures in terms 

 of correlation to measures of interdependence expressing in the con- 

 crete units of actual measurement the average change in the y character 

 associated with a unit variation in the x character, or vice versa. 



The formulas are 



(y-y) =r xu ^ (x-x) (x-x) =r xy ^ (y-y) 



ff x v 



or in a somewhat different form 



V = (y~r xv ^ x) +r xv ^x x = (x-r xv ^ y) +r xu - y 



<r x <r x ff v ff v 



All the symbols in these equations are familiar to the reader from the 

 immediately foregoing paragraphs. 



In statistical terminology such equations are called regression 

 equations. This term, which has an historical significance, is now well 

 established in the literature and we shall use it, or sometimes a perhaps 

 better term prediction equation, throughout this volume. In equations 

 like the first of the two above we speak of the regression of y on x, 

 which is equivalent to saying the prediction of y from x. In the case 

 of the second equation we speak of the regression of x on y, or of the 

 prediction of x from y. 



Such equations are easily reduced to numerical form by the sub- 

 stitution of the statistical constants. For example, the correlation 

 between body-weight and total heat-production in a group of athletes 

 has been shown above to be expressed by a coefficient of r vh = 0.958. 



