CLIN. CHEM. 22/8.
1383-1389
(1976)
End-Point Parameter Adjustment on a Small
Desk-Top Programmable Calculator for Logit-Log
Analysis of Radloimmunoassay Data
Kenneth F. Hatch, Eric Coles, Hugh Busey, and Samuel C. Goldman
We describe an improved method of logit-log curve
fitting,
by adjusting end-point parameters in radioimmunoassay studies,foruse with a small desk-top programmable calculator. Straight logit-log analyses are often
deficient because of their high sensitivity to small errors
in the end-point parameters B0 and NSB (the actual measured activity in the tubes). The literature suggests techniques for adjusting these end-point parameters, but they
require too much computing time and programming space
to be used with a desk-top programmable calculator. The
extension to the logit-log model presented here is easily
handled by the programmable calculator and provides a
good estimate of the change requiredin B0 and NSB to
obtain a better fit. The program requires 1.5 mm to run on
our desk-top programmable calculator, and has resulted
in improved
data analysis
for all of the 11 types of ra-
dioimmunoassay studied.
The main feature of this work is the mathematical
formulation
of the end-point adjustment,
which is designed to be readily programmed
on a desk-top programmable
calculator
and obtained
in a reasonable
time.
Materials and Methods
The mathematical
procedures
were developed
with
the aid of the Tektronix 31 desk-top programmable
calculator.
In addition,the fmal analysisprograms were
written for the Tek 31, and these programs were used
to produce the examples presented in this paper. An
equipment
manufactured
by Picker Corporation,
marketed as the PACE-i system, was used to gather and
analyze data.
Results
AdditIonal
technique
Keyphrases: PALL (Progranl Adjusted Loglt-Log)
#{149}hormones
Computer analysis of radioimmunoassay
(RIA) data
has become quite widespread,
as evidenced by the
amount of literature being published and the proliferation of commercial data-processing
systems. A need
has arisen for reliable automatic reduction of RIA data
techniques amenable to low-cost desk-top programmable calculators.
This work was undertaken in an attempt to choose a practical, workable mathematical
analysis that will yield good results for a wide range of
RIA
and competitive protein binding procedures.
Most analysis procedures in current use are either too
involved for a desk-top calculator or fail to yield satisfactory results in special cases. This leads to very long
calculation times or essential
inability to perform
adequate analysis. The procedure presented in this paper
follows the basic logit-log approach described by Rodbard et al. (1), but with the addition of end-point
adjustment as suggested by Healy (2), and used or implied
by others (3-6). End-point adjustment
generalizes the
procedure so that satisfactory
results are obtained for
special cases that otherwise
would require a more
complicated
model.
Clinical
Laboratory
Department, Picker Corp., Northford,
06472.
ReceivedApril14,1976;
accepted June 4, 1976.
Conn.
Curve
Analyses
Many linearizationtechniques are discussed in the
literature,
such as reciprocal bound (3, 7,8), logit-log
(1),
probit (9),arcsin (9, 10), and splinefunctions (11). The
probit,arcsin,and spline functions are empirical attempts to fit data, and their complexitylimits their use
in small-memory, relatively slow computers. On the
other hand, reciprocal bound and logit-log can be related to direct physical/chemical
phenomena (7, and
Appendix I), and their behavior is readily understood.
Thus we narrow the choice to one of these two methods.
Because of their simplicity, reciprocal bound plots are
preferred by some authors (3, 7), but for several reasons
we chose to use the logit-log formulation.
First, it has
been our experience
that the logit-log plots yield
straightlinesfor a broader selection of cases. Figure 1
shows two examples
of plots that are linear using
logit-log, but not for reciprocal
bound. Furthermore
Appendix!!
shows that any data that result in a linear
reciprocal bound plot will also yield a linear logit-log
plot if the appropriate
end-point parameters are used.
Second, as seen in Figures 1 A and C, reciprocal bound
plots exhibit bunching of data for low concentrations,
owing to the typically
logarithmic
arrangement
of
standard concentrations. Thus one has difficultyin
using that part of the graph. Third, the additional
CLINICAL CHEMISTRY,
Vol. 22, No. 8, 1976
1383
S
A
AI.DOSTERONE
3-
2_
0
0
/
-2
I
I
I
2
3
4
5
6
LN CONCENTRATION
‘N
ALDOSTERONE
Fig. 1. Comparison of reciprocal bound (T/) and Ioglt-log for
aldosterone and thyrotropin
(A) total/bound for aldosterone, (B) logit-log for aldosterone, (C) total/bound
for thyrotropin No. 1, (D) Ioglt-Iog for thyrotropin No. 1
B
3
2
-I
-2
I
I
I
I
I
2
3
4
5
6
LN CONCENTRATION
Fig.3.SensItivity
of loglt-iog
toB0 errors
(A) Linear logit-log plot for aldosterone. B,, = 3291 cpm. (B) Nonlinearity Introduced by 3.5% decrease inB,, to 3176 Cpm. Oashed line represents equation
10
The logit-log linearityisextremely sensitive to small
variations in these end-point parameters, as is evident
from Figure 3. Here an aldosterone study that is linear
for the measured value of B0 (Figure 3A) becomes quite
nonlinear when B0 is reduced 3.5% (Figure 3B). The
3.5% reduction in B0 corresponds to the 95% confidence
limit from counting statistics
for a single 1-mm
Fig. 2. Logit-log plots exhibiting nonlinearity
(A) Corticotropin, (B)
computation
grammable
somatotropin.
of logit-log
(C) ttiyrotropin No. 2, (D) testosterone
is easily
handled
by the pro-
calculator we wish to use. Therefore,
to
avoid bunching of data and to obtain a more general
formulation, we chose the logit-logtechnique for our
program.
Unfortunately
the simple logit-log technique does not
always yield adequate results. Figure 2 illustrates four
sets of data from actual radioimmunoassays in which
the logit-log plots are nonlinear. There are some physical/chemical reasons for nonlinearity, such as foreign
reagent binding (but not incomplete reaction where the
reaction coefficient is small, i.e., some antibody is unbound, see Appendix I). Nevertheless the nonlinearities
are primarily due to inadequate determinations of the
two end-point parameters, B0 and NSB. Occasionally
an end-point parameter is omitted by the RIA kit.
These quantities are essential to the logit-log fQrmulation, and they are subject
1384
CLINICAL CHEMISTRY,
to many statistical
Vol. 22, No. 8, 1976
errors.
count.
To improve the logit-log linearity, better values of B0
and nonspecific
binding (NSB) are needed. Some authors (2, 6, 12) have suggested allowing a computer to
adjust these values to obtain the best fit. Even if the
nonlinearity
is not due to statistical
or measurement
errors of end-point parameters, but rather to a physical
or chemical phenomenon, adjustment of the end-points
will still result in a better fit.
By deliberately
choosing values of B0 and NSB that
optimize linearity, the curves of Figure 4 are obtained
for the same studies of Figure 2. The straight-line
fits
are remarkably improved.
End-point
adjustment
is especially helpful in cases
where free antigen is counted. To apply logit-log techniques, an average total count must first be determined.
Next, all measurements
are subtracted
from the total
count to obtain the corresponding bound counts. Counts
The logit function
isdefined as:
Logity
ln
=
(2)
to
1-y
2
where
_B-NSB
B0-NSB
(3)
Combining equations 3 and 2:
Logit y
=
ln
B-NSB
(4)
B0-B
The logit-log
formulation is an attempt to fit a
straightlineto the logittransformed data vs. log con‘N cWIOVflMTN*
Fig. 4.Program adjustedlogit-log
plotsfor same studies as in
Figure 2
centration by using a least-squares curve fitting technique. The resulting slope and intercept define a calibration curve. The governing expression is:
(A) Cortlcotropln, (B) somatotropin, (C) thyrotropin No. 2, (D)testosterone
ln
B-NSB
(5)
=mlnp+b
B0-B
for bound radioactivity are finally applied to the logitlog analysis in the normal manner. Obtaining data for
bound analyte in thisway compounds the errors because
of the dependence on a total measurement. Errors in
total will shift all of the net data for bound analyte
higher or lower, creating nonlinear logit-log plots.
where m = slope, b = intercept,and p = unlabeled antigen concentration.
The slope and intercept are obtained by performing
a linear regression on the data. Four sums are required:
formed on a low cost programmable calculator and executed in reasonable time. A program called PALL
(Program Adjusted Logit-Log) was written for our
calculator and used to derive the plots of Figure 4. A
brief review of logit-log mathematics and a mathematical description of PALL follows.
Logit-Log
2. lnp
3. ln2p
4.
For the remaining
Upon completing these sums, the slope and intercept
can be found from:
m
work, the end-point
TOTAL/BOUND
=
T-NSB
B - NSB
=
n
parameters
in an arbitrarystandard or unknown. These definitions
imply that each measurement must be reduced by the
background count to obtain the true activityof that
item. Additionally, the nonspecific binding should
generallybe subtracted from each quantity.However,
background counts and nonspecific binding can be
combined in the NSB term. Thus the TOTAL/BOUND
plots herein were determined from:
(1)
ln p1 Logit y1
where p1 = concentration of the ith standard tube, y1
= NSB
corrected B/Bo ratio(equation 3) for ith measured count rate, and n = total number of standards.
Formulation
are defined as follows:
TOTAL or T = actual measured activity in the total
tubes.
NSB = actual measured activity in the NSB tubes.
B0 = actual measured activity in the B0 tubes.
Also, the term B refersto the actualmeasured activity
Logit y1
1.
End-point
adjustment
will remove this error.
Finally, end-point adjusting is useful in cases where
no B0 or NSB data are provided, or where free antigen
is counted and no total count is obtained. In these cases
an estimate can be made of the appropriate
end-points,
and the values adjusted until the best fit is obtained.
The purpose of this work is to develop a mathematical
procedure for adjusting B0 and NSB that can be per-
b =
1n2p,
lnp1 Logity1
in2 Pt
n
Logit y, n ln2p
-
(
-
lnp
lnp1)2
Logity1
ln p1 Logit y1
-
(6)
ln p1
(lnp)2
(7)
where the sum limits, i-i and n, are dropped from the
sum symbols, , for simplicity.
The mathematical
manipulations
can be represented
by a matrix equation (see Appendix III):
r
ln p1
in r-1
inp1
n
1 [m 1 [
J L.b] 1
=
in p1 Logit
Logit y1
(8)
The solutionto equation 8 provides a slope and in5 very nearly repre-
tercept m and b so that equation
sents the measured data of the standard tubes. The
accuracy of the result will depend on how well the logit
CLINICAL CHEMISTRY,
Vol. 22. No. 8, 1976
1385
data,Logit y1, falls intoa straightline when
plottedvs.logconcentrationInp1.It isthisstraightline
fitthat we wish to improve.
transformed
Program
Adjusted
Logit-Log (PALL)
B0.
As mentioned earlier,the major reason that the
logit-logformulation will fail to fit a straightlineisits
sensitivityto B0 and NSB. To improve the fitwe start
with the measured B0 and NSB and compute small
corrections
for these parameters, which can be found
by using linear three parameter models that are easily
handled by the calculator.
The following
numerical
example will provide an
indication
of the three parameter model to use. Refer
to the plots of Figure 3 in which a slight error was introduced into the value of B0. The equation:
in
=-0.851np+3.77
3176-B
B
=
-0.85 in p + 3.77 + 3176-
B
(10)
The resulting plot of equation 10 resembles the
modified data quite well (dashed line,Figure 3B).
The added term behaves in a similarmanner to small
changes in B0 because it isrelatedto the partialderivative of the logit function with respect to B0. Differentiating the logit function one obtains:
B0-B
(11)
Thus if i..B0 represents a small change to B0, the logit
transform
can be approximated
by:
Logit y
in
rB-NSB1
B0’
B
-
I
-
The resulting matrix expression for the three
rameters is (see Appendix III):
in2pj
-B0
B0’ - B
1
n
Llnp
___
in p1
______
1
(B0-B)J
B-NSB
ln B0’-B
b
____
I
1
(16)
lAl
I
Lj
0Lo1t)J
m, b, and A, four additional
1
B0-B’
1
Logity1
Therefore
determine
sums
must betocomputed:
.
1
B0-B’
(B0-B1)2’
B0
-
B,
Logit y
The resulting expressions for rn, b, and A are too
complicated
to write out here, but they are easily obtained from the matrix equation 16.The computations
are readilydone on a programmable calculator.
Once A is determined, a new value of B0 is computed
from equation 15, and the process is repeated. The
program PALL iterates five times to find the best value
of B0.
After finding a new B0, one may proceed similarly to
findan improvenlent
in the other end-point parameter,
NSB. The procedure isalmost identical,
and isgoverned
by the followingexpressions:
B-NSB
(12)
C
rninp+b+BNSB
(17)
and
5 can be written:
rnlnp+
rL inp1 Logit y1
rrnl
lnp
1
________
B0-B
B0-B1
in B0-B
where B0 = B0’ + .B0.
Using equation 12, equation
pa-
in p1
B0 - B1
ln pj
(9)
represents
the solid line shown, which fits the original
data (Figure 3A), but does not agree well with the new
B0 (Figure 3B). Suppose one adds a term A/(Bo - B) to
equation 9, where A = 90, B0 = 3176. Thus:
ln 3176-
but since statistical variations suggest that B0 as measured is not necessarily the best value to use, the further
step of adjusting B0 is recommended. This procedure
can then be repeated several times to find an optimum
d NSB
(13)
NSB1
{lnB-
=
B -NSB
(18)
A least-squarescurve fitisapplied to equation 17,and
Thus a bettercurve fitcan be obtained by including
a third term, A/(B0 - B), and relatingthe coefficient
A to a small correctionin B0. To do this,three parameters-rn,
b, and A-are
determined
such that a leastsquares curve fit is obtained for the relation:
Logityln
B-NSB
=rnlnp+b+
B0-B
A
a new value of NSB is determined
NSB (new)
=
NSB
inp1
B0-B
in
in2p
lflj
A new value of B0 willthen be determined from:
=
B0 (old)+ A
CLINICAL CHEMISTRY,
Vol. 22, No. 8, 1976
B
NSB
-
B1-NSB
(15)
Equation 14 itselfcan be used to represent the data,
1386
(old)+ C
(19)
The resulting matrix equation to findm, b, and C is (see
Appendix III):
(14)
B0 (new)
from:
in p1
B
-
NSB
B -NSB
(B1
-
NSB)2
[
m
x [b]
lnp1Logity1
Logit
=
Again, PALL iterates
y
(20)
-NSB
five times to find the best value
of NSB. Upon completing the five iterations for B0,
PALL tests to see if B0 is converging. If not, the order
of adjusting B0 and NSB is reversed, and NSB is adjusted first. (It can happen that if the first estimate of
NSB, or measured value, is particularly
bad, it will
prevent B0 from converging.) If the data are extremely
poor, it may happen that neither B0 nor NSB will converge. In that case, PALL aborts the technique altogether
and defaults to a simple logit-log by using the measured
B0 and NSB.
Discussion
of Endpoints
Estimation
There are cases in which B0 and (or) NSB are not
measured at all. The ordinary logit-log analysis cannot
be applied to these cases. However, with PALL, one can
arbitrarily set B0 and NSB to reasonable values, and the
program will find the correct values, or those that give
the best curve fit.
The procedure used in the PALL program arbitrarily
sets B0 to ten percent higher than the largest data point
when either none is given, or when the measured value
is illogical (less than the data, e.g.). NSB is correspondingly set to 10 percent less than the smallest data
if none is given, or if the measured value is illogical.
When free antigen is counted and no total is available,
the total is set to 10 percent higher than the largest data
point. Any errors introduced by these arbitrary estimates are oorrected by the iterative procedure of
PALL.
A program has been written for a desk-top calculator
and has been used to evaluate the data for a number of
studies. Results of these studies are tabulated in Table
1, which gives the original and adjusted B0 and NSB as
well as the correlation coefficient and maximum error
for PALL vs. straight logit-log.
Table 1. Comparison
Test
For the studies presented in Table 1, the correlation
coefficient for PALL is always equal to or better than for
straight logit-log. Furthermore for thyrotropin, renin,
B12, corticotropin, and testosterone, there is a considerable improvement in the maximum error. In one case,
somatotropin, the maximum error is slightly worse,
which occurs because a better fit to the remaining data
is achieved at the expense of this one bad point. Note
that the correlation coefficient is better.
The PALL technique appears to be quite general, and
has succeeded in fitting the data for all the cases studied
in this work. A typical data analysis takes about 1.5 mm
with the desk-top calculator. It is particularly suited to
on-line automated analysis, where several studies of
different types may be loaded into the system and run
together.
We thank all of those who contributed
to the development
of this
work, especially
Dr. John Crigler of Children’s
Hospital,
Boston,
Mass., and Drs. Dos Remedios and Paul Weber of Kaiser Permanente
Hospital,
Oakland, Calif., for providing some of the data we used in
our analysis. We also thank the Clinical Laboratory
Equipment
group
at Picker Corp. for their contribution
of equipment
and technical
assistance
during this project, and Dr. Nib Herrara of Danbury
Hospital
for helping
to evaluate
the results of our analyses
programs.
Ed. note: Names used for peptide hormones
in this paper may be
unfamiliar
to some readers. The nomenclature
is that recently
recommended
by IUPAC/IUB
[J. Biol. Chem. 250,3215 (1975)].
References
1. Rodbard,
D., Rayford,
P. L., Cooper, J. A., and Ross, G. T., Statistical quality control of radioimmunoassays.
J. Clin. Endocrinol.
28, 1412 (1968).
2. Healy, M. J. R., Statistical analysis of radioimmunoassay
data.
Biochem. J. 130, 207 (1972).
3. Leclercq, R., Taljedal, I. B., and Wold, S., Evaluation of radioisotope data in steroid assays based on competitive
protein binding. Clin.
Chim. Acta 36, 257 (1971).
4. Burger,
H. G., Lee, V. W. K., and Rennie,
of the Resultsof PALL vs. Straight Logit-Log
NSBINSB adj
Corr. coeff.
(PALL/straight)
Thyrotropin
Bo/Bo adj
3921/5038
Thyroxine,
38546/38850
0/268 1
.9993/.9880
.9995/.9993
15290/15266
0/-7633
1.0000/.9882
(CPB)
Corticotropin
Testosterone
Vitamin B,,
Renin
Aldosterone
Triiodothyronine
(RIA)
Insulin
Somatotropin
Digoxin
524/482
7505/8829
0/239
15466/15708
6238/5421
0/-167
0/69
3291/3418
10065/9841
0/25
0/289
5018/7447
0/-151
0/113
0/46
5886/5942
10438/10374
G. C., A generalized
computer program for the treatment of data from competitive protein-binding assays including radioimmunoassays.
J. Lab. Clin. Med.
80,302 (1972).
5. Arigucci, A., Forti, G., Fiorelli, G., et aL, The Endocrine Function
of the Human Testis, V. H. T. James, L. Martini, and M. Serio, Eds.,
Academic Press, New York, N.Y., 1972.
6. Rodbard, D., and Hutt, D. J., Statistical analysis of radioimmu-
.99781.9965
.99771.9969
.9997/.9985
.9994/.9987
.9968/.9967
Max. % error
(PALL/straight)
11/35
5/7
1/23
13/19
5/11
3/7
10/12
6/7
.9928/.99 12
30/35
.9974/.9969
.9999/.9999
15/14
CLINICAL CHEMISTRY,
2/2
Vol. 22. No. 8, 1976
1387
noassays and immunoradiometric
SM-177/208,
labelled antibody
assays. IAEA-
165 (1973).
7. Hales, C. N., and Randle, P. J., Immunoassay
of insulin
sulin-antibody
precipitate.
Biochem. J. 88, 137 (1963).
8. Bliss, C. I., Dose-response
with in-
curves for radioimmunoassays.
In StaJ. W. McArthur, and J. Colton, Eds. M.I.T.
tistics in Endocrinology,
Press, Cambridge, Mass., 1970, p 431.
9. Finney, D. J., Statistical Method in Biological
Charles Griffin & Co. Ltd., London, 1964.
Assay,
2nd ad.,
Appendix I. Logit-Log and Reciprocal
Bound from Mass-Action Laws
K (q
=
(Al)
Cb)
(q
-)
+ ln T
B
B1-B
-
-
B2 approaches T. The last term becomes the log of unity
form of the logit-log equation,
term In (T B)/(B2
B). This
noninfinite values of K. Indeed,
infinity,
B1 approaches B0 and
and drops out, yielding the typical linear logit-log
equation. Thus nonlinearities in logit-log from incomplete reaction (K
cc) are embodied in this last term.
However, one finds that the last term is quite constant
for normal ranges of B. For large K, B2 is close to T and
the effect of the last log term is small. For small K, the
range of B turns out to be small compared to values of
T such that the last term is nearly constant.
Thus, for incomplete binding, K finite, the major
effect on the logit-log plot is a shift in the intercept. It
remains quite linear with a slope of minus one.
(A4)
into equation A3, and rearranging, one obtains
iT
+q- T
KT-B
Appendix ii. Equivalence of Adjusted
Logit-Log and Reciprocal Bound
(AS)
B
Note that this equation is linear in (TIE) when K =
Thus reciprocal bound plots will be linear if the binding
affinity is high. When the binding reaction coefficient
is not high, the second term in equation A5 becomes
important, and reciprocal bound plots become nonlinear. However the logit-log plot remains quite linear, as
is indicated in the following derivation. Suppose one
solves equation AS for the values of B that result in zero
concentration
- B
(A7)
B2-B
where B, and B2 are the roots as given by equation
A6.
ln
=
(A3)
Substituting
R =B/F=B/(T-B)
+ lnp*
This is exactly the
except for an additional
last term is a result of
when Kp* approaches
where R is the ratio of bound over free, B/F, K is the
binding reaction coefficient, q is the total concentration
of antibody binding sites, Cb is the concentration of
bound ligand, p is the unknown ligand concentration,
p * is the radioactive labeled ligand concentration, and
B, F, and T are bound, free, and total radioactivity, respectively. A one-to-one combining ratio between ligand
and binding sites has been assumed.
pp*_.
A5 can be put into the
equation
(A2)
Chase’ derives the following relationship
_p*+
F., Scriba, P. C., Calculation.of the racurve by “spbine function”. IAEA-SM-
12. Ekins, R. P., Automation of radioimmunoassay
and other saturation assay procedures.
IAEA-SM-2771206,
106 (1973).
-lnp
-
Cb=
=
I., Erhardt,
form
and the relation
p
11. Marschner,
dioimmunoassay standard
177/72, 111 (1973).
With some algebra,
From the Scatchard equation
R
10. Vivian, S. R., and LaBeila, F. S., Classic bioassay statistical procedures applied to radioisnmunoassay
of bovine thyrotropin,
growth
hormone and prolactin. J. Clin. Endocrinol. 33, 255 (1971).
p. There are two solutions,
Suppose
a plot
TOTAL/BOUND,
of reciprocal
bound,
specifically
is linear. Then one may write:
(AS)
T/B=mp+b
where p is unknown concentration; rn and b are slope
and intercept.
Performing algebraic manipulations:
=
(m/b)p
-
1
+ 1
(A9)
which we will
call roots of equation AS. These can be obtained from
the simple quadratic formula, and are:
T/b
B
=
(m/b)p
(AlO)
=
(rn/b)p
(All)
B1,2 =(T/2)(1+-+)
±\/(T/2)2(l
where B0
Grafton
1
B02
+B0T
(A6)
TIb
ln TIb-
=
B
B
=
=
-lnp
(b/m)p
+ ln (b/rn)
(A12)
(Al3)
p*
D. Chase,
from an unpublished
1388
+-_-)
Philadelphia
paper
entitled
CLINICAL CHEMISTRY.
College of Pharmacy and Science;
“Reduction
of RIA data”.
Vol. 22. No. 8. 1976
Equation A13 is exactly of the logit-log form as given by
equation S in the main text. However NSB is zero,
having not been introduced into equation AS. Thus the
logit-log must be linear if equation A8 is true, provided
the proper value of B0 can be found (Bo = T/b).
The
iterative program should find this value.
Note that the slope is exactly minus one. Therefore
other linear plots can occur for logit-log, where the slope
is not minus one, which are not linear as reciprocal
bound plots. A more complete comparison between
logit-log and TOTAL/BOUND is evidenced by equations
A14 and A15, which are entirely equivalent to each
a11x + ai2y
=
a2lx + a22y
=
[a,1
b1
b2
a121
x [xl
a21 a22]
yi
#{149}
where
T
-
NSB
B
-
NSB
B0
=
=mlnp+b
(A14)
[
in2 p1
in p11
in p,
n
fi
(AlS)
1/3)
(A16)
ap -m +
Tfl3 + NSB(1
-
]
1m 1
Lbi
r
X
I
________
in
I
B1 - NSB
NSB
B0-B,
p1 ln
B
=
I
-
L
n B0-B,
which is equation 8 in the main text.
Similarly,
=
(A22)
we may write
other.
B-NSB
in
B0 - B
Fbi]
Lb2
=
14 of the main text can be solved
equation
for values
of m,The
b, and
minimize
square
errors.
sum Aofthat
square
errors the
is: sum of the
(
E2-
inBNSB
B0
B
-
in p
m
-
b
-
A
B0
-
-
2
B1)
and
(A24)
a
(Al7)
13e’
=
Equations Al6 and A17 define the necessary relationships between the parameters.
In summary, logit-log and TOTAL/BOUND are exactly
equivalent
if the values of B0, slopes, intercepts,
power terms are properly interrelated.
and
setting
t9E2
am
8E2
=
>
(in B1
NSB
-
B0
Appendix Ill. Derivation of Matrix Equations
From the logit-log expression, equation 5 in the main
text, we wish to determine rn and b such that the sum
of the square errors is minimized. Thus we wish to
minimize the function:
‘
________
- m In p
b)2
-
B0-B,
m in pj
-
B1
-
=
0,
3E2
=
(in B
-
B0
(Al8)
0
(A19)
(A25)
b
A
-
NSB
B1
BI-NSB
B0 - B
(in
-
m in p
B0
m in p1
=
0
B1)
-
=
A
b
-
B0
-
xl
coiiectin
o1)
A
B0
-
-
B1) (-in
-
-
This is done by differentiating
with respect to rn and b
and equating both partial differentials to zero. Thus:
aE2
0
=
-
B1-NSB
/
in
=
we obtain
-
=
0
B1)
-
-1
“
\B0
=0
(A26)
B1!
-
g terms
in p1
for which
m
(in B-
(
NSB
0-B1
in
-
B-NSB
B0-B1
m lnp
-
-mlnp
b) (-inp)
=
in2p, + b
inp + A
B0
o
inp
=
1B
______
-
B0
-
NSB
B,
____
1_b)(_1)=0
m1np1+b(n)+A
B0
1
-
(A20)
B1
=in
Collecting terms and rearranging:
B
-
_________
NSB
,.
D#{216}
El
mln2p1+blnp1=lnp1ln
minp1+b(n)=>Jln
B,
-
-
NSB
B0-B1
B1-NSB
______
B0
-
B1
m
_____
iflP
B0-B1
_____
+bB1B+A(B’B)2
_____
____
(A21)
B’Blfl
______
B1 - NSB
B0-B
_____
(A27)
where n is the number of terms in the sum.
Using the general matrix representation for simui-
from which the matrix form, equation 16 of the main
text, follows immediately. The derivation of equation
taneous
20 is similar.
equations:
CLINICAL CHEMISTRY,
Vol. 22, No. 8, 1976
1389