same uncertainties affect both Af andf. Third, the reference body cannot be formulated from densitometric analysia alone without danger of introducing a large systematic error. variance o/, in difference in fat between subject and reference body are given in explicit form in Appendix 8. Numerical evaluation of ao, and o,, requires only approximate value of d., dy, far fi to be generally valid. Using the values proposed by Keys and Brozek given above, and a subject of density 1.050 gm/cc, the variances become (11) conatituents other than fat introduce a comparable indeterminate error in fat estimate. ‘The method is obviously invalid, for example, oh, = 24.2864 + 18.6604,+ 0 Bey+ 0.0107, The standard deviation a, represents solely the error in measuring the subject's density and for the present. purpose is taken as +0.0025 gm/cc. The remaining standard deviations reflect primarily biological variability; thus, variations in the mineral-protein ratio in total body water introduce a dispersion into d,, even though the reference body may be a true average for the population and its composition known precisely. The estimated values, which ure discussed in the Section on Technical Errors and Riological Uncertainty, are Ts, = + 0.01 gm/ce, og, = + 0.01 gm/cc, o, = 2+ 0.02 reference body weight, and o;, = + 0.05 unit adipose tissue. The standard deviation in fat estimated by the densitometric method becomes ap = $4.00 body weight as, = 4.0% body weight. Several conclusions may be drawn from the foregoing analysis of the densitometric method. First, it is evident that little is gained, espe- cially in view of the increased technical difficulties, in attempting to measure body density more accurately than about +0.005 gm/cc. If there were no error whatever in measuring density, the uncertainty in fat estimate would still remain +3.8% body weight primarily because of normal variability in body constituents, and also because of the uncertainty in attempting to establish the compositions of adipose 231 in the presence of abnormal hydration. Fifth, the nature of tissue gained or lost during weight change cannot be deduced from densitometry alone if other lissues in addition to adipose tissue nxre involved. It is conceivable, for example, that the apparent density of tissue Jost could be less than that of pure fat, i.e, 0.9 emcee, if there occurred a gain in muscle mass concurrently with a loss of adipose tissue. Total Body Water Method Investigations of the gross composition of amall animals by direct analysis reveal for the most part a relatively constant fraction of water in the fat-free body and a high inverse correlation between ether-extractable fat and total water. This has been demonstrated most extensively in the guinea pig (Pacé and Rathbun, 1945; Pitts, 1956), suggesting that, at least in a limited range of fatness, such animals consist of a basic lean atructure to which pure fat is appended without greatly altering the relative proportions between water, protein, and mineral. If this conclusion is accepted, the proportion of fat is given on the avernge by the widely used formula (13) fay! w ‘ where wis the meacured total body water and w’ the proportion of water in the fat-free body, which has been variously estimated from 67 to 74%. 232 TIN: N\ og) = WNSGoa + ME 1Ren + 0.230%, + O10,7 + 0.010; oN! 66. WAHOLIO WAGAALAAS © 6 ON 6 ‘TOA NOLLALLIN ¢ or grenter than the uncertainty in the estimate of total fat. While this result is not intuitively evident, it follows from the fact that the This error does not atem from lack of precision in measuring density, but from the impossibility of establishing body composition solely by measuring one quantity such as density or total body water. As a corollary to this, it may be noted that even if the densities of both subject and reference were determined with great accuracy, the uncertainty in the estimate of fat would still be 3.844. body weight. Fourth, significant differences from the average in any of the gross fat, The over-all uncertainty, expreased aa the variance oi in fat and (12) SRP Second, the uncertainty in the estimate of difference in fat, Af, or in adipose tissue, A, between subject and reference is the same ty ca aa X. Ibe Pag on Technical Errors and Riclogical Uncertainty, muat include not only the error of measurement in d, but also the biological variability associated with the assumptions made in formulating the methed. The standard deviation in the estimated value of fat may be derived from the general Eqs. (6) and (7) by applying the Law of, Propagation of Errors, recognizing that there will be dispersion in d,, di, f., and f, due mainly to the variability in total body water and in the mineral-protein ratio among individuals with the same weight and tissue and reference man that are true averages for the category of subjecta measured. YSsk de Weal evi A true estimate of the uncertainty associated with the determina- tion of fat by the densitometric method, as pointed out in the Section