¢
The general formula may now be evaluated on the bases of the two
references, Inserting first the constants for the Minnesota standard
man and the values e, == 0.16, and ¢, = 0.14 proposed by Keys and
Brozek (1958), the fat estimating equation is
(32)
f = 0.696 e — 1.620 w + 1.041
With the fat-free body as the reference, f, 1, e, =w,— f,=—0,
and the fat estimating equation becomes
(33)
the residual dispersion in body density of humans after adjustment
to the same proportion of fat as the reference body that was selected.
The magnitude of the uncertainty in d, is based here on the dispersion
in normal total body water of o,, = +. 2% body weight together with
a dispersion of +: 0.1 in the mineral-protein ratio. The resultant uncertainty in d, is then derived as follows, assuming a == ap:
The reference body density may be expressed as
f=A= 0.619 e---1618 w41
a
the inherent uncertainty of this method, consequently, the choice of
reference, adipose tissue composition, or other assumptions that may
be introduced, are relatively unimportant. Conversely, the method
cannot be expected to give a very reliable estimate of body composition.
The introduction of extracellular space merely compounds the difficulties by adding greater uncertainties than those associated with
estimating body composition solely from total body water. However,
the most important conclusion is this: in the presence of edema, the
method ia subject to serious systematic error, and for normally
hydrated persons, an extracellular-total body water method does not
in fact exist. The latter conclusion may be demonstrated by formulating the method for conditions of normal hydration, in which case
either the extracellular fluid apace or the total body water cancela
out of the formulation. One or the other measurement is redundant,
the appropriate rule for calculating the cumulative uncertainty in f.
For simplicity the formula is expressed below in terms of variances
(standard deviations squared, oa“):
z
°
=
6F
\2
24
bi?
24
&Fo- \?
._
3%
(ya t Cy Jeet Ge dat
where 6f'/5a) is the partial derivative of the function with respect
to quantity a, and o, is the standard deviation in a.
Appendix 2
As explained in the text, the standard deviation of +0.01 gm/ce in
the value of the reference body dengity is intended as a measure of
241
fe
Ww + Qa. ~ fo — we) (dn + ont ,.)
dy
(1+ alan
Applying the Law of Propagation of Errors, with the condition that
f. is constant, the variance is then
o,= a(t -
de t od, Joa at (bt — fo — we){d, — fn)! :
(+ adywl.,
(Ll + a)
= 0.164 06+ 0.0042 &2
Witho, = + 0.02 ando, = + 0.1, the standard deviation in d, be-
comes oy== + 0.01 gm/cc,
Appendix 3
A.
Variance in densitometric estimate of fat.
2
dido(fi— fod?
_- dy
2o
" ( d(da— ¢y)+) [: + (Race
=a75) %
Appendix 1
If a quantity f is related by a function F (a, b,c,....} to the quantitiesa, b,e..... , each of which is subject to an uncertainty expressed
as standard deviation, o, the Law of Propagation of Errors provides
=
dd;
Estimates of fat on the basia of the two reference bodies never differ
by more than 1.5% of body weight. This difference is far smaller than
5+ menmeonzemn:
LAOTMAY I
di —
do
4
d -- dy
ids
+ (ioc) % + (iGefy) ont duh si)7]
The corresponding variance in the differential fat estimate, Af, is
also given by the equation above if /, is set equal to zero and the fourth
term in the bracketis omitted.
B. Variance in fat estimated from total body water.
fi- fo Nf:
wih?
s
wo — wy? s
gpa (LEB
ot Gwom
=mot+
(YY ot,
Wa — Wy
Up > Uy
Wa wy
.
aC
Wo Up
fi a)
4
“ye
(?-wy
The corresponding variance in the differential fat estimate, Af, may
also be calculated by deleting the last term and f,.
C. Variance in fat from combined density and total body water.
242
ISSWTD WSTTOUV LEAK NOLLRLLIN
O61 YAGOLIO YAMIWALdAs % ON 6 “1IOA NOLLIL
AN Opt
eae
Basses.