Formula for polynomial function / Finding Area under the Curve

[#1]

So, I would like to find the Area under some curves. Actually quite many of them, so using the Trapezoid method manually won't work, and can't imagine it would be valid enough. There is some kind of function for this for ROC-curves, but is it usable for finding the AUC in curves, measureing blood substances over time (five times)?

However, if I get a formula for a fitted Curve, I can use it to find AUC (Integral) in other software. For example, a Sixth degree polynomial. Here, I get the the function but I can't make sense of it (pic 1). 

Polynomial functions are:
But I can't make sense of the Formula JMP provides me. Using only Time=3? 

By using a flexible smoothing spline fit, I can get some coeffecients, but those won't help me (pic 2)? 



Would be grateful for tips getting the equation integral-ready and/or finding the AUC easily, for many subjects/curves.

[#2]

When fitting a polynomial JMP will center the independent variable to reduce collinearity. That is why you see (time - 3) in the equation (it is not an equal sign).
You should still be able to work with the equation in this form. But if you must have an uncentered form you can save the prediction formula, open the formula in the formula editor, and select the simplify option under the red triangle menu.

[#3]

Thank you. Hmm so "26 - 1,3333333*Time - 1,25*(Time-3)^2 + 0,3333333*(Time-3)^3 + 0,25*(Time-3)^4 + 0*(Time-3)^5 + 0*(Time-3)^6" would be the predicted formula? Will try. 

[#4]

Ok, seems like I had to change the commas to dots, then I got something out of it.

(Forth degree, not sixth, would be enough btw)

First:Then efter simplify:



Hm is that a valid polynomial function? Having trouble integrating it atleast.

[#5 · Accepted solution]

the Fit Special dialog has a check box for centering.

Uncheck to get the simpler looking form of the equation

It looks like this dialog stops at degree 5.

[#6]

Ah! Yeah, thank you. 

Now I got the formula looking like: 26 - 11,833333*Time + 9,25*Time^2 - 2,6666667*Time^3 + 0,25*Time^4
Will try to exchange Time for X and integrate..

[#7]

Wow, seems like it might be solved. Via an integral calculator.

However a little time consuming for multiple curves but could do. Thank you!

[#8]

You only have to symbolically integrate the polynomial once, then evaluate it with different sets of fitted parameter estimates.

[#9]

Hmm. I'm about to compare the mean-AUC for some parameters, for 20 subjects divided into two groups. 

[#10]

Anyone wondering about the centering: I think it helps numerical accuracy, potentially a lot. JMP uses double precision floating point numbers that can represent 15-16 digits. In this example you can see the two formulas agree to about 9 digits. 

Centered and uncentered formulas

For Katie (row 1) the calculations look like this:

term	centered	uncentered
^0	-100.391805727981	-3850.46871657937
^1	190.861948723194	11664.6866311208
^2	2.9439683570111	-11635.8007229098
^3	-0.852640295610253	3914.14427943114
Sum	92.5614710566141	92.5614710627333

For the uncentered column, the most significant digits in all of the terms are much bigger than the sum, and those most significant digits mean there is no room to represent more least significant digits.