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

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

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

We recently had a customer that was interested in MasterPlex QT because his current analysis software for his Bio-Plex instrument was reporting a lot of “OOR <” or out of range concentration values (below the lower asymptote in this case) for points on the lower end of curve. This is what you would normally expect to see for values that fall below the minimum asymptote BUT the software did not have the capability to use weighting in calculating the lower asymptote which can greatly affect points on the lower part of the curve.
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There seem to be Top 10 lists for everything these days so why not have one as well for Luminex/Bio-Plex® Data Analysis.
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The 5-PL or 5 Parameter Logistic is a nonlinear regression model used for prediction of the probability of occurrence of an event by fitting data to a logistic curve. It differs from the 4-PL or 4 Parameter Logistic model in that it is an asymmetric function which is a better fit for immunoassay or bioassay data.  As the name suggests, there are 5 parameters in the 5-PL model equation:

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