[4PL fitting] Different inflection points using different units of concentration

[#1]

Hello the community,

We are facing some issue in our inflection point calculation for a nonlinear 4P Rodbard fitting. 

The model is basic:

X: concentration [nM]

Y: %Specific Killing 

By: samples

I usually run this on JMP, but other team members in our departement are using the service of ou bio-informatitian and she used the concentration in pM. 

At first sight, no issue, only a factor of 1000 should be the difference. However, we found that for some fitting, she do not have any issue but I observed different results on JMP.

Here is an example:

Concentration [nM]	Concentration [pM]	Specific Killing [%]
20	20000	103.2
2	2000	103.5
0.2	200	102.6
0.02	20	104.9
0.002	2	99.9
0.0002	0.2	-5.0
0.00002	0.02	13.3
0.000002	0.002	9.2
2E-07	0.0002	0.3
2E-08	0.00002	6.1

And here my model results

Using x as Concentration [nM]

Parameter	Estimate	Std Error	Wald ChiSquare	Prob > ChiSquare	Lower 95%	Upper 95%
Growth Rate	-5.39	282.61	0.00	0.98	-559.29	548.50
Inflection Point	0.001096	0.03	0.00	0.97	-0.07	0.07
Lower Asymptote	4.77	2.98	2.57	0.11	-1.07	10.61
Upper Asymptote	103.55	2.98	1209.02	<.0001	97.71	109.39

Using x as Concentration [pM]

Parameter	Estimate	Std Error	Wald ChiSquare	Prob > ChiSquare	Lower 95%	Upper 95%
Growth Rate	-6.76	6445.81	0.00	1.00	-12640.32	12626.80
Inflection Point	1.2354	567.78	0.00	1.00	-1111.60	1114.07
Lower Asymptote	4.77	2.98	2.57	0.11	-1.06	10.61
Upper Asymptote	103.54	2.98	1209.97	<.0001	97.71	109.38

As you can see, there is not just a 1000 fold differences in the inflection point (and other differences for example the growth rate).

Do you know why we have such difference ? For almost all of our samples there is no differences it is only few of them, 

We also see different value if are running a 4PL with variable slope on Prism on with R using our bio-info sript (when is it usually similar).

Can you help us ?

Thanks in advance for your help

[#2]

You are confusing mathematics and computing. Yes, the change in units means that the levels of Concentration are different by a factor of 1000. But the parameters are estimated from the data. You are pushing the numerical accuracy of the estimation. Furthermore, the designated model is (1) a poor choice in this case or (2) is not supported well by the data.

I used JMP: File > Internet Open > Web Page to create a JMP data table from your example.

Here is the fit using Analyze > Specialized Modeling > Fit Curve and the Rodbard model with Concentration in nM.

Here is the same analysis but using Concentration as pM:

You can see the problem right away. The sigmoidal shape is not visible using either units. This situation leads to unstable estimates. I tried a log10 transformation, but it does not help. You have an abrupt change in the response between 2 and 0.2 nM, and so the sigmoidal shape is not observed. You would presumabley have to expand the concentration between 2 and 0.2 nM to define the transition better on order to use this model.

[#3]

Thanks for your reply !

From my understanding, the Rodbard fitting is specially designed for the non log-transformed data fitting. This is why we usually choose this model. 

If for any reason we would have to use log-transformed data, we usually go for the 4P-Hill model.

Actually, if you change the visualisation of the scale, (using a log visualisation of the axis without log transformation for the fitting), we indeed have a good fit, as you can see below:

(But my thinking can be wrong, I am not a mathematician :).)

However, your second point make sense to me, it is true that for this sample, there is a big gap between 2 and 0.2pM, whcih could be an issue for the approximation of the inflection point.

This is also something we notice internally, this is usually why we can have inconsistency between pM and nM approximation but also between different software (using the same model at first sight).

Thank you.

[#4]

Your data shows a step function, not a sigmoidal curve. You have the lower and upper asymptotes, but no information (data) about the transition between them. This data might have been selected as a survey to locate the transition, but it is incapable of modeling the transition. I suggest a second study the zooms in on a narrower range that still includes both asymptotes but also exhibits the transition for several concentrations.

[#5]

Thanks for your insight !

I understand that from those data the dilution would need to be changed to find some value in the middle of the curve in order to have a better fit and approximation of an inflection point.

It was just strange to me that JMP was fitting this model differently using 2 different units when other software give similar answer with both units (but different than JMP). I guess the fact that the data are not precise is the main reason.

[#6]

No, it is not a matter of imprecise data. That fact is handled by the random effect called error implied in the non-linear model. It is due to the poor choice of the model, the lack of data to support the selected model, and the finite precision of the numerical methods implemented by the software and hardware that you are using.