w= (1l—--A) w, + Aw, ps (1—A) p, + AM roa (L—A)} m, + Am, One may now choose whatever composition seems appropriate for adipose tissue and for the reference body. . In the following sections, the general formulation for each method will also be evaluated for two extremes in reference body composition. The first is based on the Minnesota ‘‘reference man” (1952), characterized by d,-= 1.062 gm/ce, f,=—= 0.14, #2, —=0.61, p= 0.19, m., == 0.06, together with Keys and Brozek’s estimate of the composition of ‘‘adipose tissue’; d, — 0.948 gm/cc, f, == 0,62, w, = 0.31, py = 0,07, and mi, == 0.00. The second example is evaluated on the basis of the fat-free body, assuming the ratios between water, protein, and mineral are constant for all adult humans, and by identifying ‘adipose tissue’ with pure fat. Under these conditions f, == 1.0, w, =p, == m™, = 0, and the remaining quantities have approximately the following values: d,—= 1.1 gm/cc, f, == 0.0, mw, == 0.72, p,== 0.21, m, = 0.07. These two standards of reference are used primarily because they illustrate opposite extremes in concepts of reference bodies. It will be apparent in analyses of most methods that the choice of reference yt tool HAWOLOO ddd ldds & ON 6 HON NOLLRLLIN body may have less material effect. on the estimate of fat or of protein and mineral than do the underlying uncertainties in the method. In view of the insensitivity of most methods and the consequent uncertainty associated with them, the characteristic values indicated above appear to be justifiable, even where there may be disagreement on the precise values of the proportions of constituents, Technical Errors and Biological Uncertainty It would be a misleading simplification to asaume that the accuracy with which body composition can be estimated is dependent solely upon the accuracy with which corporeal density or the fluid spacer can be measured. Even if experimental errors were non-existent, there would still remain in most methods for estimating body composi- tion a substantial residual uncertainty (standard deviation), esti- mated at about +4% of body weight. Each method contains, whether explicitly or implicitly, a fixed reference body (or its equivalent) which incorporates a set of assumptions inter-relating constituents that cannot be measured directly. Thus, for example, all methods assume that mineral constitutes a fixed fraction of the fat-free body, or that it has a fixed ratio to protein, or that it conforms to some alternative empirical relationship. Since it can hardly be expected 227 that all individuals will conform exactly to the same numerical conatants in such relationships, individual deviations from the “‘standard” constitute an irreducible biological variability. The empirical constants in fat estimating formulas may at best represent an average for a selected population. Furthermore, they are correct in only a limited segment of the obesity-emaciation range. The variability in each constituent therefore contributes its share to the uncertainty in an estimate of fat, protein, or mineral. Biological variability sets the limit of confidence one may have in estimates of body composition by methods now available, and it also sets a useful limit of accuracy that is desirable in measuring density and fluid spaces. This latter consideration is particularly significant from a practical standpoint. On the one hand, it may save the expenditure of great effort put into improving the accuracy of a measuring technique that would in reality produce no significant improvement in the estimate of fat, and on the other hand, would avoid interpreting an already precise measurement of density or fluid space as a comparably accurate determination of fat and body composition generally. The over-all uncertainty in an estimate of fat must consequently include both biological variability and experimental error. Since the various methods can be formulated explicitly in terms of the biological variables, an estimate of this uncertainty expressed as standard deviation can be found by applying the Law of Propagation of Errors to the general formulas (See Appendix 1). This will also yield an estimate of optimum experimental accuracy that seems justified in applying a specific method. The formulas for calculating the variance in the fat estimates are expressed in terms of the biological variables and their variance, experimental and bivlogical. Obviously, the biological uncertainties must be the same in every method for estimating body compvsition from density and fluid spaces, although their cumulative effect may vary with the method used. The standard deviations listed below are intended primarily to illustrate, when substituted into the appropriate formulas, the approximate magnitude of the uncertainty asseciated with each method, Nevertheless, their values are believed to be justified by the available data on body composition. The quantities to which they refer are indicated by subscripts. Experimental: Biological : oo, == :+. 0.0025 gem/‘cec T, == 7 0.02 oo ==01 o, = + 0.02 Oy, == + O01 oy, = + 0.01 body weight reference body weight gm/cc gm/ce 228 : 1 rw f= (1—A) fo + Afi (4) a i