Dear Allstat
I would like to thank everyone who responded to my query, your help and
advice was very useful. My original posting was :
I have been asked to reproduce a 4 parameter curve fit for analyte
signal vs concentration. The curve has already been defined by the
machine that produced the signals and so I have the parameter values
that define the curve but do not know the function that relates signal
to concentration.
The output from the machine reports that a "semi-log sigmoid fit with
tails" was used. I have looked through the internet and deduced that
semi log indicates that one of the variables is logged. This seems to
tie in with a plot I saw (of different data but same assay) where the
concentration was plotted in the logarithmic scale (X axis).
My understanding in that a sigmoid curve is just one that looks like an
'S' and is a special case of the logistic function. Is it possible that
the term sigmoid is being used as a general term and may include the
logistic function?
In SAS JMP I found a 4 parameter logistic function which was S-shaped,
although when I used it to fit the data I am getting nothing meaningful
and so am currently messing around with various parameter values to try
and get a reasonable fit. There is also a growth curve which looks as
though it may be a possible candidate, but I haven't tried it yet.
I suppose there are few questions that I have:
Does anyone know what the function would be for a "semi-log sigmoid fit
with tails" to model signal as a function of log concentration?
Is it reasonable to use an alternative 4 parameter equation even though
it is not the same as the one already evaluated?
Is there only one form of the logistic 4 parameter curve fit?
The responses I received are given below:
There are several types of sigmoidal curves (many CDFs will do!), one of
which is the logistic. It sounds like you are using the 4-parm logistic
curve using the log scale. Please realise that what one person calls
the 'semi-log sigmoid fit with tails' may be what another person calls
the 'LL4' model (as I do); what might be wiser is to look at the assumed
model function just to be sure. Also, please be mindful that there is
nothing sacred about the log scale - the square root scale may work
better for your data, so you may want to play around with the proper
scale as well.
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It is likely that a four-parameter logistic curve is being fit the
analyte signal versus log(concentration). However, there are many
different ways of parameterizing the four-parameter logistic (see
Ratkowsky's "A Handbook of Nonlinear Regression Models" (Dekker, 1990)).
So you will need to know both the parameter values and the form of the
model that has been fit.
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I think there is more than one 4-parameter logistic. Usually, of
course, it has only 3 parameters, (upper) asymptote, slope, and EC50.
The fourth
parameter is usually some sort of asymmetry parameter (eg a power on the
concentration), but alternatively it could be the lower asymptote if
this is non-zero.
As you say, there are other forms of sigmoid curve, perhaps the
best-known
is the Gompertz. There is also one which I derived in my Ph.D thesis
and
has at least once been described in print as the 'Chanter growth curve',
although sad to say it never seems to have caught the public
imagination.
It's just a generalization of which the logistic and Gompertz curves are
special cases, but if memory serves me correctly it has more than four
parameters.
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If you are using a reputable program,
then the functional form should be
clear in its documentation. If not,
or there's no documentation, it may
be difficult to resolve the issue.
More positively, sigmoid does indeed
mean just S-shaped. In my experience
it is often used broadly and need
_not_ imply logistic in either narrow
or broad senses.
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I would agree that semi-log means one of the variables had had logs
taken. I would take the term sigmoid to mean any S-shaped curve. A
logistic curve is one form of sigmoid, with either 2 or 3 parameters. A
more general one with 4 parameters is the Gumbel.
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A '4 parameter logistic' curve is a good bet: e.g. as
in
http://www.ats.ucla.edu/stat/sas/library/analystelisa.pdf
It is described there in terms of dose(x) but could
equally well be given in terms of ln(x).
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I am not sure whether this will help, but when I was involved with
calibration curve analysis for assays, the function most often used was
called the 4 parameter log logistic. Its form was
Response = A + (B-A)/(1+Z) where Z=exp(C+DlogX). A and B are the
asymptotes.
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a) "Semi-log" does usually indicate that one of the variables is
logged, in my experience, usually the "y" variable.
b) In the microbiological community there is a presumption that
semi-log growth and especially death curves are "naturally"
straight. There is no real justification for this assumption,
although appeal is sometimes made to first order reaction
kinetics. In practice both type of curve are often sigmoidal.
For death, microbiologists refer to "shoulders" and "tailing",
for growth they refer to "lag phase", "linear phase", and
"stationary phase".
c) A number of different functional forms are used to fit the
sigmoidal curves. Usually there are four parameters, because
there are four distinct characteristics of a sigmoidal curve
i) the left hand asymptote
ii) the x position of increased/maximum slope
iii) the value of maximum slope
iv) the right hand asymptote
d) The different functional forms include, but are not limited to,
i) three straight lines
ii) logistic
iii) Gompertz,
iv) Baranyi
e) There are 3 and 4 parameter versions of the logistic and Gompertz,
and I think the Baranyi has up to 7 parameters. In addition, there
are different parameterisations with the same number of parameters.
f) The sigmoidal functions are generally non-linear, often multiple
exponentials. Fitting them to data can be difficult! I've never
used SAS, I use S-Plus which has a number of different sigmoidal
functional forms (generally implemented as "self-starting" functions)
that can be fitted using "nls".
g) So yes, "the term sigmoid is being used as a general term and
may include the logistic function" but no, "a sigmoid curve
is ... NOT ... a special case of the logistic function". On
the contrary, the logistic function is a special case of the
class of sigmoid curves.
h) Whether it is "reasonable to use an alternative 4 parameter
equation even though it is not the same as the one already
evaluated" depends on the circumstances.
What was the basis for choosing the original equation?
Why are you fitting another equation?
How close is the fit between the data and the original?
How close does your fit need to be?
Can't you get any information from the machine manufacturers?
In my experience (microbiological data) the deviations of the data
from the fitted curve are large compared to the deviations of
different functional forms from each other, so choosing different
equations doesn't' make much practical difference.
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Yes, sigmoid is often used as a description for the 4 parameter logistic
curve.
There isn't a unique function, but a reasonable one to use is:
Y=Bottom + (Top-Bottom)/(1+10^((LogEC50-X)*HillSlope))
;X is the logarithm of concentration. Y is the response
;Y starts at Bottom and goes to Top with a sigmoid shape.
This one relies on log to base 10; you could use logs to base e. You may
sometimes find other versions which re-write the equation in minor ways.
Unfortunately, it is sometimes hard to get meaningful solutions for this
curve, especially if the plateaus at the left and right hand side of the
curve are not very well-defined. If this happens, you will see large
standard errors for the parameters of the curve.
An excellent resource for curve-fitting is a free book at
www.curvefit.com <http://www.curvefit.com/> , which is a site run by the
makers of (in my opinion) the best curve-fitting software, GraphPad
Prism. You can get a demo copy of the software for free at
www.graphpad.com <http://www.graphpad.com/> .
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