A TRANSFORMATIONIWEIGHfING MODEL FOR
ESTIMATING MlaIAELIS-MENTEN PARAMErERS
Raymond J. Carro 11
Noel Cressie
David Ruppert
Raymond J.
Statistics,
27514.
Carroll
and
David
are
with
the
Department
of
the University of North Carolina at Chapel Hill, Chapel Hill, NC
Noel Cressie is with the
University,
Ruppert
Ames,
Iowa
50011.
Force Office of Scientific
Department
of
Statistics
at
Iowa
State
Carroll's research was supported by the Air
Research
Contract
AFOSR
F
49620
85
C 0144.
Ruppert's research was supported by the National Science Foundation Grant MCS
8100748.
Key
The authors thank Perry Haaland for helpful conversations.
Words
and
Transformations,
Phrases
Weighted
Maximum
Least
Spawner-Recruit Fisheries models.
likelihood,
Squares,
Michaelis-Menten
Hormone-Receptor
models,
assays,
ABSTRAcr
There
has been considerable disagreement about how best to estimate the
parameters in Michaelis-Menten models.
methods
are
based
on
different
We
stochastic
point
models.
squares estimates after appropriate transformation.
model
which
can
choice of weights.
be
used
to
out
We
that
many
fitting
being weighted least
propose
a
flexible
help determine the proper transformation and
The method is illustrated by examples.
- 1 -
SECfION 1
Introduction
The Michaelis-Menten relation between a response y and a predictor x can
be written as
(1.1)
y
where a
O
variety
= 1/V
of
and a
1
= Vx/(K+x)
= KIV.
biological
and
The model applies at least approximately
biochemical situations. see Cressie
(1981). Currie (1982). Johansen (1984) and Ruppert
fisheries
& Carroll
in
a
& Keightley
(1985).
In
research. (1.1) is called the Beverton-Holt (1957) spawner-recruit
model.
There is considerable debate
parameters
in model (1.1).
as
to
how
one
can
best
estimate
the
The simplest method is to linearize (1.1) by the
double reciprocal or Lineweaver-Burke transformation :
(1.2)
and then regress l/y on l/x using least squares to estimate
(aO.a1 ).
While
the Lineweaver-Burke model is often used. in some areas it is thought to be a
particularly poor fitting technique.
Currie (1982) states that the estimates
from this technique are "more prone to be biased" than are the estimates from
"any
other
commonly
used
technique".
Other authors have come to the same
conclusion, based in part on a study by Dowd & Riggs (1965).
A second common method is the Woolf
linearizing
multiplies through by x in (1.2), leading to the model
transformation.
which
- 2 -
(1.3)
Cressie
& Keightley (1981)
find
that
for hormone-receptor assays, least
squares applied after the Woolf transformation is
points
than
is
least
squares
after
more
robust
to
outlying
the Lineweaver-Burke transformation,
although when using a robust fit the latter was marginally better.
and
other reasons (Keightley, Fisher
For
this
& Cressie, 1983), they prefer the Woolf
transformation for analyZing hormone-receptor assays.
Currie (1982) reconunends
squares.
fitting
directly
(1.1)
by
nonlinear
least
He also states that "The Woolf transformation, the best among the
linearizing transformations, ... provides unreliable parameter estimates
(linearizing)
transformations
should" not
be
used",
although he adds the
qualifying statement "except1 in cases where they stabilize the error".
It seems to us that the proper fitting method necessarily depends on the
particular subject area, experiment and underlying distribution.
No
general
conclusions about a fitting method can-or should be made without taking these
considerations
into
account.
squares should be seen as
models
for
the
data.
a
If
Fitting
consequence
~
(1.1)
of
- (1.3) by unweighted least
fitting
different
stochastic
has mean zero and variance one, fitting (1.1) -
(1.3) is a consequence of assuming, respectively
(1.4)
Nonlinear (Currie)
y
(1.5)
Lineweaver-Burke
y
(1.6)
Woolf
y
= {aO +
= {aO +
= {aO +
a /x}
1
-1
+
a~
-1
a /x +
1
a~}
a /x +
1
a~/x}
-1
That these may be very different stochastic models for data can
letting
a be small and taking a simple Taylor series expansion.
.
be
seen
by
If E(y) and
- 3 -
SD(y} denote the mean and standard deviation of
y,
we
have
the
following
approximations :
Surely
Nonlinear:
SD(y}
~a
Lineweaver-Burke
SD(y}
~a
{E(y}}2
Woolf
SD(y}
~a
{E(y}}
it
2
/ x
should come as no surprise that if one runs a simulation in which
the errors in (1.1) are additive or nearly so, then ordinary nonlinear
squares
should
be a clear winner over linearizing transformations, with the
Woolf transformation second and Lineweaver-Burke a poor third.
essentially
least
what
Currie did in his
study.We~e
Yet
this
is
little doubt that had he
based his simulation on the underlying,vdistribution implied by say the
Woolf
model, then he would have found that linearizing is the method of choice.
Storer,
Darlison & Cornish-Bowden (1975)
~resent
real-life examples for
which the standard deviation of y appears to be proportional to the mean or a
power of the mean.
This evidence with real data
suggests
that
one
should
take care to study the underlying distributional properties of an experiment,
and not assume that the Nonlinear model holds in all situations.
The
three
models
(1.4)
(1.6)
generalization to the Transform Both Sides
are
special
approach
of
cases
of
Carroll
a
simple
& Ruppert
(1984, 1987a, 1987b), see also Bates, Wolf & Watts (1985) and Snee (1986) for
nice applications of this technique.
Define the usual Box - Cox (1964) power
transformations :
veAl
veAl
= (vA - I)
= log(v}
/ A for A # 0
for A
=0
- 4 -
The Extended Transform Both Sides Michaelis-Menten Model is
(1.7)
y
(A)
= { (aO +
aI/x)
-1
}
(A)
+ a x
a ~.
See Carroll & Ruppert (1987a) for an introduction to this model.
A =
1 and
a
= O. we obtain the nonlinear model (1.4).
model (1.5) sets A
= -1. a = O.
Transform Both Sides Model (1.7).
An
alternative
illustrate
the
use
of
= a = -1.
the
Extended
In section 2. we discuss properties of the
In section 3. we present examples.
method of) estimation
is discussed by Dalgaard & Johansen l (t986).
because
The Lineweaver-Burke
while the Woolf model (1.6) sets A
The purpose of this paper is to
model and methods of fitting it.
By choosing
due to Cornish-Bowden & Eisenthal
We
do
not
study
this
method
as they state "it ... appears dangerous to us to use these estimates
without making a careful investigation of the error distributions. and if one
can do that. it would appear more reasonable to apply the model based maximum
likelihood estimator".
- 5 -
SECTION 2
The Extended Transform Both Sides Model
We consider a slight generalization of model (1.7).
one
which
permits
any nonlinear regression model and a wider variety of models for the standard
deviation:
(2.1)
Here
f(x.~)
is called the regression function. g(x.8) is called the standard
deviation function and
estimated.
(A.~.8)
are
unknown
parameters
which are
to
be
In accordance with tradition. c is assumed to be approximately
normally distributed with mean zero andrVaI;!ance
deviations could also depend
on
th~
o~e.
mean.bu~·
we
In (2.1). the standard
do
not
discuss
this
refinement here.
To make the identifications with (1.7). we have
f{x.~)
Assuming
only
symmetry
of
g{x.8)
c.
independent of the values of A and 8.
heavily
under
Snee
this
=x8.
model y has median
(1986)
has
argued
might
that
for
skewed data in nonlinear regression. it is more natural to model the
median rather than the mean. with the transformation taking care of
and some
f(x.~)
of the heteroscedasticity in the data.
skewness
Alternatively to (2.1). one
choose to model the mean-variance relationship directly. i.e .• build a
As
an
is small. the two approaches are closely related.
In
heteroscedastic nonlinear regression model without
approximation
when
a
transformation.
this instance. the mean and median nearly coincide. skewness is not
a
great
- 6 -
issue
and
y
has approximate standard deviation og(x,9}f(x,p}1-A.
natural alternative to
the
Extended
Transform
Both Sides
Hence, a
Model
is
the
heteroscedastic model
E(y} = f(x,P}
(2.2)
SD(y} = og(x,9}f(x,P}
Methods
for
fitting
I-A
.
model (2.2) are discussed by Davidian & Carroll (1986)
and Carroll & Ruppert (1987a, Chapters 2 and 3).
9
offers some insight when g(x,9} = x.
The approximate model (2.2)
In many of these problems f(x,P} given
by (1.1) is approximately proportional to a power of x, so that 9 and A will
be
difficult
to
distinguish.
In most of the examples we have studied, we
have found confidence regions for (A,9) to be fairly broad, but not so
as
to
broad
include all three of the Nonlinear, Woolf and Lineweaver-Burke models
simultaneously.
Assuming
the
transformation achieves
approximate
normality,
the
loglikelihood for model (2.1) is given by
l(A,9,P,o} =
~
[ (A-I) 10g(Yi} - log{o g(x ,9}} ]
i
i=l
For
given values
of
(A,9), p and
0
can be estimated by weighted nonlinear
2
least squares regression of Yi(A} on f(xi,P}(A} with weights 1/g (X ,9}.
If
i
A2
the weighted mean squared error is denoted by 0 (A,9), we thus obtain the
- 7 -
maximized loglikelihood
(2.3)
The maximum likelihood estimates of X and 9 can
(2.3).
over
loglikelihood
surface,
Nonlinear
model
a
grid
of
This
values.
enables
us
by
maximizing
X,9
~
to
1, and
~
plot
the
and as a simple by-product make tests for the fit of
(X=1,
9=0),
Lineweaver-Burke model (X=-1, 9=0).
done.
computed
In the Michaelis-Menten model, we usually restrict -1
compute (2.3)
the
be
the
Woolf
model
(X=9=-1)
and
the
Such computation is usually very qUickly
In the examples of the next section, using a grid size of 0.05 and the
matrix
programming language GAUSS on an IBM PClAT, the loglikelihood surface
took less than 5 minutes to compute.
For more complex regression and standard deviation
maximize
the
likelihood
general maximum likelihood
programs.
respectively.
Let
Ym
and
in
all
program
functions,
one
can
the parameters simultaneously either by a
or
by
using
nonlinear
least
squares
gm(9) be the geometric means of {Yi} and {g(x i ,9)}
Define
Then the maximum likelihood estimates of
(X,9,~)
minimize the sum of squares
To use a nonlinear least squares program to make this
computation,
identify
- 8 -
D (A. 0.13)
i
as
the
zero; see Carroll
"regression
function" and identify all "responses" to be
& Ruppert (1987a. Chapter 5).
Professor D. Bates (personal communication) has
useful
to
plot
suggested
that
it
is
the individual components of 13 as a function of (A.O). as a
check on the sensitivity of the parameter estimates
to
the
choice
of
the
model.
As argued in Carroll & Ruppert (1984). the effect on the distribution of
,..
13
due
to
estimating
found it useful to
Weisberg
(1982).
model. see Carroll
(A.O) is often small.
adjust
For
standard
robust
In practice. we have sometimes
residuals
estimation
for
leverage.
Cook
&
and diagnostics in this type of
& Ruppert (1987a. Chapter 6) and (1987b).
"
see
- 9 -
SECfION 3
Examples
In this section. we analyze two data sets.
The analyses are meant to be
illustrative numerical examples rather than complete investigations.
Skeena River Sockeye Salmon
EXAMPLE 3.1
The data in Table 1. taken from
relationship between x
recruits
for
1940 - 1967.
= number
& Smith (1975).
Ricker
of spawners and y
Skeena River Sockeye Salmon.
= total
concern
the
return or number of
The data are given for the years
Since our intention is to prOVide numerical illustrations only.
we have deleted 1951 which was affected by a rockslide and 1955 because other
analyses indicate that it is rather extreme.
consider is that of dependencies in
including
the
A possibility which we will not
data.
For
alternative
analyses
robust methods and influence diagnostics and different models. see
Carroll & Ruppert (1987a. 1987b).
The studentized residuals from the Woolf model, while not pictured
to
conserve
space,
heteroscedasticity.
studentized
suggest
The
residuals
a
systematic
Spearman
and
the
rank
values
fit
between
is
studentized
and/or
the
residuals
model
Figure 1.
Some idea of the severity of the problem can be gleaned
that
the
level of 0.007.
severe
absolute
0.60 with a formally
Nonlinear
fact
suggest
The
of
correlation
predicted
computed significance level of 0.001.
lack
here
from
the
a clear pattern of classic heteroscedasticity; see
from
the
Spearman rank correlation is 0.50 with a formal significance
The Lineweaver-Burke
studentized
residuals
show
no
such
pattern of obvious heteroscedasticty, but they do seem skewed; see Figure 2.
Model (1.7) was fit to the data and the maximum likelihood estimates for
-10-
(X.9)
are (0.34. 0.77).
x2
The likelihood ratio
test statistic based on two
2
2
degrees of freedom is X = 10.7 for the Nonlinear model (X=1. 9=0). X = 18.5
2
for the Woolf model (X=9=-1) and X = 8.43
(X=-1.
9=0).
respectively.
that
none
these
having
for
the
Lineweaver-Burke
model
significance levels of 0.005. 0.0001. and 0.015
The likelihood ratio tests agree with our graphs in concluding
of
the
three
standard
models
have
constant
variance
with
approximately normally distributed errors.
no
course.
Of
choice
heteroscedasticty in these data.
=
1
(x
2
= 2.24)
of
(X.9)
perfectly
explains
However. X = 9 = .5. (X
both seem adequate.
2
skewness
and
= 1.14) and X = 9
The choice X = 9 = 1. for which the
residual plot is given in Figure 3. has the feature that it
is
a
model
of
heteroscedasticity without data transformation. which some will find to be an
advantage.
In
this
model.
the standard deviation is proportional to the
number of spawners. which changes, by a
data.
f~ctor
of about 4
over
the
observed
Alternatively. one might model the yariance as proportional to a power
of the mean. see Carroll & Ruppert (1987a. Chapter 3 and example 6.4.1).
- 11 -
EXAMPLE 3.2
A Hormone Receptor Assay
In Table
we
2
list
the results of a small hormone-receptor assay as
taken from Cressie & Keightley (1979. Table
different
3);
the
ordering
is
slightly
to reflect the fact that there were duplicates at six levels.
Our
purpose is to illustrate the application of Extended Transform Both Sides.
but
not
to
give
a
complete analysis.
Indeed. the data set is very small
(only 12 observations). and it would be unwise to
read
too
much
into
the
analysis.
{x.a}
The estimates of
-1
x.a
~
~
1.
The
were constrained to lie in the unit square. i.e ..
maximum
A
Lineweaver-Burke model, being X
for
likelihood
= -1.0
and
the Lineweaver-Burke model (X = -1.0.
= -1.0. a = -1.0)
model (X
Nonlinear
model
a
1.0.
=
(X
i t is
estimate
a = -0.1.
a = O)
)(-'J.~ 8.29. Wh'fle
=~.).
occurs very near the
A
is
')(?
The
test
)(2 = 0.78.
i t is)(
2
=
statistic
for the Woolf
for
4.54
the
Even though the confidence region is
fairly large. the Extended Transform Both Sides model suggests that the Woolf
and (to a lesser extent) the Nonlinear models might provide a worse fit
than
does Lineweaver-Burke.
In
Figures
4
-
6
we
plot the studentized residuals from the Woolf.
Nonlinear and Lineweaver-Burke fits against the logarithm
values.
the
predicted
We have also looked at plots of the absolute studentized residuals.
To our eye. as suggested by the
model
of
seems
likelihood
analysis.
First.
the
Spearman
rank
of absolute studentized residuals against their predicted values
is .44 •. 17 and -.32 for the Woolf.
respectively.
Lineweaver-Burke
to do the best job of accounting for heterogeneity of variance.
There are additional factors of some interest.
correlation
the
This
might
suggest
Nonlinear
that
the
and
Lineweaver-Burke
Lineweaver-Burke
models
model has
- 12 -
overcorrected for heterogeneity of variance.
Note too that the plots suggest
that there may be both within and between components of
variance.
which
we
have not been attempting to model.
There is also the issue of robustness and leverage.
fit
to
One can do a robust
the data here. but in the interest of space we only do the following
analysis.
Since the data are essentially collected in pairs. it is
interest
to
see
In Table 3 we have recomputed
the likelihood after deleting succesive pairs of data points.
this
here
pair
is
some
if there is a single pair of observations which has a very
large effect on the final parameter estimates.
striking
of
What
is
most
is the effect of deleting the first pair of observations.
If
deleted.
of
Lineweaver-Burke.
Woolf
then
in
terms
of
likelihood.
all
three
and the Nonlinear model are essentially equivalent.
Either using the full data or keeping the first pair and deleting any of
the
others
for
leads
to
Lineweaver-Burke.
one
a
strong
preference
(in
terms
of
likelihood)
This example suggests that especially for small data
sets
will need to examine the data carefully to ascertain whether the answers
are being driven by a small group of observations.
As a postscript. there is another linearizing transformation (y/x
= aO +
a y) yielding the Scatchard model. which has been used heaVily in the context
1
of hormone-receptor assays.
(1.7).
Its
This model is not a special case
of
the
model
performance on hormone-receptor assay data is shown by Cressie
and Keightley (1981) to be markedly inferior to that of either the
the Lineweaver-Burke models.
Woolf
or
- 13 -
SECTION 4
In
Discussion
fitting a Michaelis-Menten model to data. the three standard fitting
models are a consequence of different distributions for the responses.
It is
clear that no single fixed distribution will fit all situations. so it should
also
be
clear
appropriate.
that
no
single
fitting
technique
will
be
universally
The proper model and its associated fitting techniques should
be allowed to vary among subject areas and even within a subject area.
We view Extended Transform Both Sides
effective
easily
as
a
reasonable.
method for helping to decide upon a model for data.
computed
Naturally.
as
and
enables
indicated
in
Michaelis-Menten relationships.
standard
section
model
2.
the
checks
model
to
is
flexible
and
The method is
be
performed.
not restricted to
We do not advertise Extended Transform
Sides as a panacea. nor do we advocate that it be used as a black box.
Both
While
the technique is widely applicable. no simple model will fit all data sets.
Inferential
problems
concerning
the
parameters
(1.1) has not been our concern in this article.
confidence
validity
of
Wald-type
statements depend on the design and parameterization. see Bates
Watts (1980).
2. 4 and 5).
The
V and K in the model
For further discussion. see Carroll
&
& Ruppert (1987a. Chapters
- 14 -
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Notes
in
Biomathematics 61. M. Mangel editor.Springer Verlag. New York.
Snee. R. D. ( 1986) . An al ternat i ve .', 'approach to
reexpression
of
the
response 'is useful.
Technology 18. 211-225.
fitting models when
Journal of Quality
Storer. A. C.• Darlison. M. G. & Cornish-Bowden, A. (1975).
experimental error in enzyme kinetic measurements.
Journal 151. 361-367.
.,::'
The nature of
Biochemistry
TABLE 1
Skeena River Sockeye Salmon data. Years 1951 and 1955 have been deleted
the illustrative example of section 3. Data are in millions of fish.
YEAR
SPAWNERS
1940
1941
1942
1943
1944
1945
1946
1947
1948
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
0.963
0.572
0.305
0.272
0.824
0.940
0.486
0.307
1.066
0.480
0.393
0.176
0.237
0.700
0.511
0.087
0.370
0.448
0.819
0.799
0.273
0.936
0.558
0.597
0.848
0.619
0.397
0.616
in
RECRUITS
2.215
1.334
0.800
0.438
3.071
0.957
0.934
0.971
2.257
1.451
0.686
0.127
0.700
1.381
1.393
0.363
0.668
2.067
0.644
1.747
0.744
1.087
1.335
1.981
0.627
1.099
1.532
2.086
e
TABLE 2
Results of a Hormone
fmol/mg cytosolprotein.
Receptor Study.
~
X
0.358
1.43
0.358
1.46
0.782
2.23
0.771
2.36
1.725
2.88
1.703
3.15
2.672
3.34
2.680
3.43
- ',?')
3.653
3.58
3.629
' 3.81
f-:
4.630
3.75
4.697
3.57
Data are in units of
TABLE 3
Hormone-Receptor Study. The values of the log likelihood
listed after successively deleting pairs of points.
for
(x.e) are
Woolf
Lineweaver
Burke
Nonlinear
(1.2)
19.44
20.42
20.55
(3.4)
19.85
23.57
21.02
(5.6)
20.20
24.70
22.43
(7.8)
18.91
23.23
20.66
(9.10)
22.09
24.05
21.86
(11, 12)
20.92
24.05
21.86
Deleted
Observations
e
Figure 1
The studentized residuals plotted against the logarithm of the
predicted values for the Nonlinear model in the Skeena river data set.
Figure 2
predicted
set.
The studentized residuals plotted against the logarithm of the
values for the Lineweaver-Burke model in the Skeena river data
Figure 3
The studentized residuals plotted against the logarithm of the
predicted values for the untransformed heteroscedastic model in the Skeena
river data set.
Figure 4
The studentized residuals plotted against the logarithm of the
predicted values for the Woolf model in the Hormone Receptor Study.
Figure 5
•
The studentized residuals plotted against the logarithm of the
predicted values for the Nonlinear model in the Hormone Receptor Study.
Figure 6
predicted
Study.
The studentized residuals plotted against the logarithm of the
values for the Lineweaver-Burke model in the Hormone Receptor
e
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Skeena River Data
Nonlinear Model
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