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 asymptote
   
   In 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 slope
   
   The 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 point
   
   The 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

4. D = maximum asymptote
   
   In 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 equal to 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 a curve-fitting software with the 4PL model equation and also does weighting, then try out the Free 14-Day Trial of MasterPlex ReaderFit (fully-functional) or our online free version at ReaderFit.com (light version).

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

1. Use Weighting to Offset Heteroscedasticity

Heteroscedasticity is a situation that arises in almost all fields, including chemical- and immuno-assays, in which the variance of the dependent variable varies across the data. When dealing with RLU and concentration values, the concentrations usually increase as the RLU increases. When dealing with the high end of the standard curve, it is natural for the concentration values to have a greater variance when compared to the small concentration values on the low end of the standard curve. Many of the regression analyses used in analyzing Luminex data, such as the popular 5PL, assume equal variance.

Heteroscedasticity - Nonconstant Variance in Immunoassay Data

For analyzing ELISA, the 1/Y^2 (preferred) and 1/Y weighting algorithms are recommended in addition to using the 5-PL model equation.

• 1/Y^2 – Minimizes residuals (errors) based on relative RLU values
• 1/Y – This algorithm is useful if you know the errors follow a Poisson distribution

Here is an actual case where weighting was able to resurrect data that would otherwise have been useless. The platform and assay is slightly different than ELISA but the underlying concepts of data analysis are identical.

MasterPlex ReaderFit offers these weighting algorithms in addition to a couple of other options.

Weighting Algoritms available in MasterPlex ReaderFit

2. Use the 5 Parameter Logistic (5PL) Nonlinear Regression Model

The 5 Parameter Logistic or 5PL nonlinear regression model is an asymmetric function that is ideal for analyzing ELISA data. Learn more about the 5PL model equation.

MasterPlex ReaderFit offers the 5PL curve-fit regression model in addition to a number of other curve-fitting model equations.

Model Equations available in MasterPlex ReaderFit

3. Run your samples in replicates

Having replicate samples will bring good karma in addition to the following:

• Backups – If one sample well is accidentally prepared, you will have backups that you can rely on.
• Statistics – Naturally, you will obtain more reliable results when dealing with a larger pool of data. You will also be able to obtain statistics such as %CV and standard deviation that will tell how much variation or dispersion you have amongst data points of the same replicate group.

MasterPlex ReaderFit allows you to group your replicate samples and automatically generates the mean, %CV, and standard deviation statistics for the MFI and concentration values.

MasterPlex ReaderFit's Powerful Reporting Engine

4. Knock out those standard outliers

One of the simplest ways to identify outliers in your standards is by analyzing the Residuals and % Recovery.

• Residual = Calculated Concentration – Expected Concentration
• % Recovery = ( Calculated Concentration / Expected Concentration ) * 100

% Recovery is a better metric to use because it is a relative metric. Obviously, the further the % Recovery deviates from 100%, the higher the probability that the data point is an outlier.

Although there is no strict rule or standard on what the limits are for considering outliers, I tend to mark any standard that has < 50% recovery or > 150% recovery as outliers and that has generated pretty good results.

MasterPlex ReaderFit has a user-friendly interface that allows one to view the Residuals, % Recovery, and the Standard Curve all at the same time while being able to mark outliers.

Viewing Standard Data and Marking Outliers in MasterPlex ReaderFit

5. Use controls

This should go without saying but it is quite alarming for me to see that most of the analysis data files that I see are missing control groups. It should be as important as your background groups. Not only will they tell you if your assay worked but knowing this information is invaluable when it comes time to have to troubleshoot when something goes wrong. The workflow for setting up and running an Elisa assay can become quite complicated and there are many opportunities for things to go awry. Having control groups will keep you sane.

6. Use a sufficient standard curve range for your unknowns

Extrapolation is the process of inferring or estimating the concentrations for points that are within calculable limits but outside of the standard curve range. Extrapolations are often less meaningful especially when the values lie on the flatter parts of the curve. Please read this blog post for more on the dangers of extrapolating data.

There are several ways to avoid extrapolating data:

1. If your unknown RLU is above the standard curve range, you can dilute the unknown sample enough to bring it back within range and just take the dilution factor into account when calculating your concentrations. MasterPlex ReaderFit allows you to input the dilution factors of your unknown samples and it will automatically take them into account when calculating the concentrations.
   

   Setting the dilution factors in MasterPlex ReaderFit

2. If your response value is below the standard curve range, you can create one or more standard groups on the lower end by extending your serial dilutions.

7. Individual Standard Points vs. Mean of each Standard Replicate Group

Generally speaking, the more standard points you have (using individual standard points for the curve fit), the better the curve fit because you will have a greater number of degrees of freedom. This method will be more sensitive to outliers though so it is recommended that you first knock out any outliers that you might have and proceed with the standard calculations.

Using the mean of each standard replicate group will lead to less data points and that in turn will lead to a worse curve fit. This method is less susceptible to outliers though if you do not intend to mark them.

MasterPlex ReaderFit let’s the user choose between both these methods.

Individual and Average Standard Point Options in MasterPlex ReaderFit

8. Keep it in the Linear Range

In an ideal situation, you will want to be interpolating your unknown concentrations from the dynamic or linear portion of the standard curve. You definitely do not want to be interpolating or extrapolating any concentration values near the top of the curve where it is flat. Minute changes in RLU changes in this part of the curve can lead to huge differences in concentration so any errors that may have occurred will be multiplied by a very large factor.

9. Use the 4PL or 5PL for easy EC50/IC50 Determination

Dose response curves are best described using nonlinear regression models such as the 4PL and 5PL. Using these model equations, one can easily determine that the EC50/IC50 values.

Do you have anything additional tips that you would like to share? If so, please comment below.

Download the FREE fully-functional trial version of MasterPlex ReaderFit today!

You may also be interested in reading our blog post on:

• 4PL nonlinear regression model
• 5PL nonlinear regression model
• The importance of weighting with the 4PL and 5PL
• Dose response analysis and EC₅₀/IC₅₀ determination

Quick Demo of MasterPlex ReaderFit (1:43 min.)

The microplate analysis demo includes the following simple steps:

1. Importing an Excel file with the RLU values
2. Marking the background, standard, and unknown samples
3. Filling in the standard concentrations with the AutoFill Feature
4. Pressing the Best Fit button
5. Reviewing standard curve, concentrations, and charts

Save time, stop copying & pasting, and download the FREE MasterPlex ReaderFit trial and see for yourself how easy it can be!

• File Import Wizard – Although MasterPlex ReaderFit can automatically import some of the known file formats from common plate readers that are out there in the market, it does not handle all 100% of them. Unknown file formats can now be handled by the File Import Wizard in 1 of 2 ways:
  1. Plate Format

     

  2. List

     

  More details on using the File Import Wizard for MasterPlex ReaderFit

• Best Fit Options – The Best Fit feature now offers 3 parameters for which you can optimize for to determine the best curve fit:
  1. Root Mean Square Error (RMSE) – RMSE is a good measure of accuracy or goodness of fit which aggregates the individual differences or residuals between your expected and calculated values into a single measure of predictive power. This is the default option as it is the one that is recommended.
  2. Standard Deviation of % Recovery – This option will provide the model equation and weighting algorithm combination with the lowest standard deviation of percent recovery. The percent recovery is basically the calculated concentration – the expected concentration * 100.
  3. R-squared – This option is the only Best Fit option in previous builds of MasterPlex ReaderFit. It is not as sensitive to the goodness of fit as the RMSE option.

  

• Competitive Binding and Inhibition Assays – This build provides improved support for competitive binding or inhibition assays.

  

• More Intuitive Auto Fill – The Auto Fill feature helps to automate the entry of the standard concentration values for serial dilutions. This feature has been simplified with more intuitive dilution direction buttons and being able to preview the standard values.

  

Try it out and download the free trial now!