topic Re: Formula for polynomial function / Finding Area under the Curve in Discussions

Formula for polynomial function / Finding Area under the Curve

JohanJakobsson — Mon, 09 Jul 2018 10:52:46 GMT

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.

Re: Formula for polynomial function / Finding Area under the Curve

Dan_Obermiller — Mon, 09 Jul 2018 11:17:19 GMT

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.

Re: Formula for polynomial function / Finding Area under the Curve

JohanJakobsson — Mon, 09 Jul 2018 11:48:37 GMT

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.

Re: Formula for polynomial function / Finding Area under the Curve

JohanJakobsson — Mon, 09 Jul 2018 12:08:44 GMT

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.

Re: Formula for polynomial function / Finding Area under the Curve

Craige_Hales — Mon, 09 Jul 2018 12:11:19 GMT

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.

Re: Formula for polynomial function / Finding Area under the Curve

JohanJakobsson — Mon, 09 Jul 2018 12:19:35 GMT

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..

Re: Formula for polynomial function / Finding Area under the Curve

JohanJakobsson — Mon, 09 Jul 2018 12:25:12 GMT

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

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

Re: Formula for polynomial function / Finding Area under the Curve

Mark_Bailey — Mon, 09 Jul 2018 13:35:34 GMT

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

Re: Formula for polynomial function / Finding Area under the Curve

JohanJakobsson — Mon, 09 Jul 2018 13:44:33 GMT

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

Re: Formula for polynomial function / Finding Area under the Curve

Craige_Hales — Mon, 09 Jul 2018 14:14:47 GMT

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.

Re: Formula for polynomial function / Finding Area under the Curve

Mark_Bailey — Mon, 09 Jul 2018 14:20:16 GMT

Good point, Craige.

The centering also minimizes the correlations between the estimates. Otherwise, the correlations increase the standard errors of the estimates. The larger standard errors produce wider confidence interval estimates and smaller t-ratios for the same estimate. The smaller t-ratio will have a higher p-value. So correlations compromise the power of the decision about the estimate being significantly different from zero.