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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

Updated 12/10/2013: For those of you who are looking for the back-calculations to solve for x, here is the formula.
x = C*(((A-D)/(F(x)-D))-1)^(1/B) Special thanks to Pawel S. for providing this formula!

Not surprisingly, the 4PL model equation comprises of 4 parameters:


4 Parameter Logistic Nonlinear Regression Model
  1. 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.
  2. 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.
  3. 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.

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
Sign Up for Free Account

ReaderFit Desktop - Robust curve-fitting, quality control and reporting desktop software
Download Free Trial

MasterPlex QT - Robust curve-fitting, quality control and reporting desktop software for multiplex ELISA data (Luminex, Bio-Plex, Meso Scale Discovery and Applied BioCode platforms)
Download Free Trial

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