The 4 Parameter Logistic or 4PL nonlinear regression model is commonly used for curve-fitting analysis in bioassays or immunoassays such as ELISAs or dose-response curves.
The following is the 4PL model equation where x is the concentration (in the case of ELISA analysis) or the independent value and F(x) would be the response value (e.g. absorbance, OD, response value) or dependent value.
F(x) = ((A-D)/(1+((x/C)^B))) + D
Not surprisingly, the 4PL model equation comprises of 4 parameters:
4 Parameter Logistic Nonlinear Regression Model
- A = minimum asymptoteIn an ELISA assay where you have a standard curve, this can be thought of as the response value at 0 standard concentration.
- B = Hill slopeThe Hill Slope or slope factor refers to the steepness of the curve. It could either be positive or negative. As the absolute value of the Hill slope increases, so does the steepness of the curve.
- C = inflection pointThe inflection point is defined as the point on the curve where the curvature changes direction or signs. This can be better explained if you can imagine the concavity of a sigmoidal curve. The inflection point is where the curve changes from being concave upwards to concave downwards (see picture below).
Inflection point and change in curvature or concavity
- D = maximum asymptoteIn an ELISA assay where you have a standard curve, this can be thought of as the response value for infinite standard concentration.
The following are some key characteristics of the 4PL curve-fit model:
- Symmetry- There is perfect symmetry for the sigmoidal curve around the inflection point for 4PL curve fits.
Symmetry around inflection point for 4PL
- Monotonic- A monotonic function is either always increasing or decreasing for all values of x.
An example of a monotonic increasing function
An example of a monotonic decreasing function
- Assumptions made by the 4PL model equation
- It assumes that the standard deviation of the scatter is the same for all values of x (homoscedastic data). In the example of a standard curve, this is saying that the standard deviation for all the replicates of a low standard is equalto the standard deviation of the replicates for your high standard (see example curve below).
Homoscedastic Data
Of course, this is rarely the case when dealing with bioassays or immunoassays (ELISAs) where the data is heteroscedastic. We normally see something like this where the standard deviation increases as x increases:
Heteroscedastic Data
Applying weighting algorithms for 4PL and 5PL curve fitting is something that can be done to offset the assumption that data is homoscedastic.
- The 4PL model equation also assumes that the scatters a normal (or Gaussian) distribution.
- It assumes that the standard deviation of the scatter is the same for all values of x (homoscedastic data). In the example of a standard curve, this is saying that the standard deviation for all the replicates of a low standard is equalto the standard deviation of the replicates for your high standard (see example curve below).
If you are looking for curve-fitting software with the 4PL model equation that also does weighting we have a couple options:
ReaderFit.com – Free online curve-fitting application
ReaderFit Desktop – Robust curve-fitting, quality control and reporting desktop software
You may also be interested in reading our blog post on:
- Dummies Guide to Curve-Fitting for Quantitative Analysis
- 5PL nonlinear regression model
- Tips for ELISA analysis
- The importance of weighting with the 4PL and 5PL
- Dose response analysis and EC50/IC50 determination
I need this software for my OPN elisa reading graph. I am a Ph.D student. Please send me free version of the software so that I can use it.
Reply
aliu reply on February 3rd, 2011 11:10 am:
Hi. We have a free fully functional 14-day trial of MasterPlex ReaderFit (for ELISA analysis). Please let me know if you have any other questions or you can email me directly at aliu at miraibio dot com.
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Hy,
i used excel with the 4PL Formular and the ABCD values from the Demo software / your screens here and estimate that i can calc the same values for the curve via excel, as the software, but they are diffrent, why?.
I also tried to plot a curve on excel by changing the numbers from 0.01 till 3 ABS but it don’t looks sigmuid.
I am not sure if i understand 4PL right.
Regards
PhD Student JH
Reply
aliu reply on June 9th, 2011 10:42 am:
Hi JH,
Even though the 4PL model equation can be identical in 2 different software, the actual curve-fitting algorithms can vary quite a bit leading to different parameters (i.e. A, B, C and D). Here is a post that goes through curve fitting at a very high level. As you can imagine, there can be many ways an algorithm can come up with these parameters. This does not even include weighting (which you should be using) which will have an effect on the parameters as well. Here is a good read on the 4PL and there is a part that talks about why weighting is necessary.
Please let me know if you have any questions.
Best Regards,
Allen Liu
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when we draw a standard curve manually using four known standard concentrations with their response values, we are getting almost accurately the unknown sample concentration if we interpolate the response value in the graph. Where as when I am using 4-or 5-pl nonlinear regression model by using your software the concentrations for unknown samples are not accurate. We are taking 4 standard concentrations and 6-8 unknown samples. can you explain why?
Reply
aliu reply on June 9th, 2011 10:36 am:
Hi Dr. Prasad,
I’d be more than happy to help you out with your question.
Is it possible for you to give me the MasterPlex ReaderFit project file and the standard curve & results that you did manually so that I can compare. By “manually,” are you using a model equation or are you simply connecting the standard points?
Best Regards,
Allen Liu
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Dear sir
I am a research scholar doing my Ph D from . I am working on antihypertensives and want to determine IC50 values for my newly synthesised compounds .I further want to know how following soft ware of yours can help me for this .pl suggest what data I need to provide .
Dose Response Curves with EC50 & IC50 Determination using MasterPlex ReaderFit
The MiraiBio Group Blog.
Regards
Reply
Hi Shefali,
In order to use MasterPlex ReaderFit for IC50 determination, you will need the following data:
– Measured response values for your samples from your immunoassay – This value is typically something like OD, absorbance, RLU, fluorescence intensity, etc.
– Known concentrations or dilutions of the synthesized compound used in each corresponding sample. In ReaderFit, this would be your “Independent Value”
Here is a post that goes into more detail on the procedure for determining IC50:
http://www.miraibio.com/blog/2010/03/dose-response-curves-ec50-ic50-determination-masterplex-readerfit/
We also have some tutorial videos on how you can actually use it:
http://www.miraibio.com/masterplex-readerfit/online-demo.html
If you would like, you can also send me your data and I would be more than happy to look at your data for you: aliu@miraibio.com
Thanks for your interest!
Allen
Reply
I’m looking for a program that can compute the predction interval for nonlinear regression (the 4PL, for now).
Reply
Allen Liu reply on September 19th, 2011 11:52 am:
Hi Joe,
I apologize for the delay in my response.
We provide several solutions for nonlinear regression curve fitting with the 4PL. If you have a Windows computer and you are looking for desktop software, then you should take a look at MasterPlex ReaderFit. There is a free 14-day demo that you can download and begin using right away.
If you are looking for a web application, we do have an online solution at ReaderFit.com. This is a light-weight solution compared to our desktop version but all that will change in the near future when we launch with some huge updates!
Please let me know if you have any questions.
Best Regards,
Allen Liu
Reply
many thanks for you assistance
sorry I got disconnected before I could read the response when I clicked on the readerfit.com link.
Kind Regards,
Pia
Reply
Robert Lynde reply on October 18th, 2011 11:20 am:
Pia,
Glad I could help! I hope readerfit.com was able to give the curve fit you needed.
Regards,
Robert
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