将二次拟合应用于未知数

Question

I'm trying to build a spreadsheet to find a quadratic fit for a set of control data, then apply that fit to a set of unknowns to get a calculated concentration. For my quadratic curve calculation, I have this:

=LINEST(F28:F33,A28:A33^{1,2},TRUE,TRUE)

An example of relevant control data (where 0-40 would be found in the A column, and the 0.001-0.575 in the F column) is:

0   0.001

2   0.030

5   0.076

10   0.156

20   0.310

40   0.575

This is giving me a curve solution that matches the software currently being used to analyze the data (SoftMax 4.7):

A: -5.1E-05 

B: 0.016 

C: -0.002

Using this formula to apply the curve to data (where E16 represents any individual datapoint I'm solving for and Blank1 is a set of negative controls):

=(-CurveB+SQRT((CurveB^2)-(4*CurveA*(CurveC-(E16-AVERAGE(Blank1))))))/(2*CurveA)

However, when I apply the curve using the formula to a set of data, eg:

0.275      0.269     0.266

0.217      0.193     0.194

0.011      0.013     0.011

0.004      0.006     0.003

I get output:

17.835     17.426    17.221

13.922     12.333    12.399

0.796      0.919     0.796

0.369      0.491     0.308

Compared to SoftMax's output:

17.827     17.405    17.215

13.918     12.333    12.393

0.785      0.950     0.797

0.353      0.487     0.298

My problem is, I can't find enough documentation on how SoftMax applies the quadratic fit to the data so I don't know which set of results is more accurate. I've checked to see if it's a rounding error (ie Softmax is rounding the displayed results but calculating using unrounded figures or possibly the other way around), I've tried throwing the whole mess through Solver, letting Excel change the curve variables and the blank factor (I also tried removing the blank factor and solving, and adding independent blank factors for each column and solving) and solving for a minimum total variance from the Softmax results, but I cannot find a solution that produces the same results as the Softmax software (or even closer than 0.58% or so average variance from the Softmax results).

Can anybody tell me if this is an error in my calculations (I'm specifically skeptical of my formula to apply the curve to data-is there a more graceful way to apply a quadratic fit to a set of unknowns in Excel?) or is it an error with the calculations produced by the other program, eg solving using approximations or rounded values somewhere?

Answer 1

Summary : I think you're seeing rounding errors.

Details . I used your Excel equations and the data provided and reproduced your curve parameters, so that seems OK. I then plugged the SoftMax Pro output (17.827, 17.405, 17.215, 13.918, ...) and your output (17.835, 17.426, 17.221, 13.922, ...) into y=AX^2+BX+C and calculated y-values. The pair-wise differences were in the 4th decimal place or smaller --- biggest (abs) difference was ~ -0.0005, so that's consistent with a rounding/truncation of the X-data that's hidden from you.

Final Comment : I suspect you should not subtract blanks. The standard curve appears to have been created using not-blank-subtracted data (at zero input the output is non-zero) so it seems like you need to treat samples the same way as standards. It may not make much difference ...

Hope that helps.