How to convert quadratic logarithmic equation

[#1]

Running an ELISA test I got the best fit for the standard curve using fit special and log/log transformation as a quadratic function: Ln(OD) = -4.139548 + 1.301394*Ln(Concentration of α-CD3 [ng/ml]) - 0.0934906*Ln(Concentration of α-CD3 [ng/ml])^2. Is the square operator applicable for concentration only or for its logarithmic derivative. How to convert this equation to be able to calculate concentration from the measured OD? I prefer an interactive explanation. Thanks.

Bivariate(
 Y( :OD ),
 X( :Name( "Concentration of α-CD3 [ng/ml]" ) ),
 Fit Special(
 xTran( "Log" ),
 yTran( "Log" ),
 Degree( 2 ),
 {Line Color( {142, 176, 40} )}
 ),
 SendToReport(
 Dispatch(
 {},
 "1",
 ScaleBox,
 {Scale( "Log" ), Min( 5 ), Max( 400 ), Inc( 1 ), Minor Ticks( 1 )}
 ),
 Dispatch(
 {},
 "2",
 ScaleBox,
 {Scale( "Log" ), Min( 0.1 ), Max( 2 ), Inc( 1 ), Minor Ticks( 1 )}
 )
 )
)

[#2]

You can also achieve the same fit using Analyze > Fit Model.

• Right-click the concentration column and select the log transform.
• Select this new column.
• Click the Macros button and select Polynomial to Degree (2 is default).
• Right-click the OD column and select the log transformation.
• Select the new column and click Y.
• Click Run.
• Click the red triangle and select Estimates > Inverse Prediction.
• Enter your OD values and select your confidence, then click OK.

The benefit of Fit Least Squares over Bivariate is that you can use Estimates > Inverse Prediction to estimate the concentration from the OD.

It is typical with dose-response standard curves to use log base 10 for interpretation. It won't change the goodness of fit.

[#3]

Thank you Mark,
The solution is very nice. Unfortunately I am getting an alert: "Inverse Prediction does not support squared terms in the variable to predict".
Is there a way how to bypass issue, please?
Thanks,
Peter

[#4]

Sorry about that issue. No, there is no way to bypass the issue using my suggestion.

Is this model especially important or useful? Did you try to fit the data using one of the models available in Analyze > Specialized Modeling > Fit Curve?

[#5]

I haven't tried Modelling, I went directly to Bivariate Fit - Special. We usualy perform log transformation of both OD and concentration and linear fit works fine. This is a new test and the linear fit is not good, so I tried a higher polynomial and quadratic was good. The manufacturer recommends a 4-parameter logistic fit, but also let the user explore some other alternatives. So the model is not critical. I just want to find the best model from the standard curve and then apply it for my samples. That's why I need an inversion function to calculate concentrations from measured OD.

[#6]

I understand.

A quadratic polynomial generally has two values of Log Conc that equate to the same Log OD. JMP doesn't know that it should only use the half of the parabola. The Fit Curve has the Logistic models and Custom Inverse Prediction. These curves are monotonic, unlike a quadratic polynomial.

[#7]

I realized when I had got a quadratic fit (equation) that the SW would have a problem to handle a dual solution for a single OD. My only solution seems to be using a more narrow range of concentrations where I can apply a linear fit. Thank you, Mark, for your help.

[#8 · Accepted solution]

I would try the logistic 4P as an alternative to shrinking your range. 

[#9 · Accepted solution]

Additionally, you might try modeling the OD directly with the 4PLC as well. This model accounts for a non-linear response. The log transform is an attempt to linearize it.

[#10]

Thank you, Mark, I will try that