How can I test if two fitted nonlinear curves are statistically different?

[#1]

Hi,

Can someone help me with this?

I did experiments to determine a certain feature (Open Porosity) in 3 different directions, in 21 little volumes.

So, I had 21 volumes and each volume yielded 3 values, i.e. 3 times an open porosity value.

I've fitted non-linear curves through the data, and now I want to know if they are significantly different from each other.

Is this possible to do this in JMP?

Here I fitted the model y=a*x^b, JMP gave me the best possible estimation of the model parameters (a and b). Measurements are not shown here, these are just the fitted nonlinear curves in the 3 directions.

Here, I fitted again 3 curves but with another model, there was just one model parameter that was needed to be fit.

I know the curves are quite similar and that they probably will not be significantly different, but I need to know it 100% sure, as well as the strategy to test if they are statistically different or not

Thanking you in advance!

[#2]

Hi,
If you use the Fit Curve platform (standard nonlinear models), there are Test Parallelism, Compare Parameter Estimates, and Equivalence Test options for any model that you have fit.
If you use the Nonlinear platform (customer nonlinear models), there is the option to calculate confidence intervals for parameter estimates. Non-overlapping confidence intervals can be used as an indicator of statistical significance.
Take a look at the help documentation for these platforms for more details.
Regards,
Phil

[#3]

Hi Phil,

Thanks for your answer!

I used the Nonlinear platform (Analyze > Specialized modeling> Non-lineair), but are there any tests I can apply to check whether they differ significantly or not?

Or are confidence intervals for parameter estimates the only thing that can be used to test this?

Regards,

VinceL

[#4]

I am not aware of any other feature for testing statistical significance. Apart from those that I have mentioned in the Fit Curve platform.

You could see if it is possible to fit your models using Fit Curve. Or you could take a look at the statistical details for the tests in Fit Curve and try to apply them to your results from Nonlinear.

Is there a problem with using the confidence intervals to test for statistical significance differences?

[#5]

Curve similarity is a complicated subject, and it depends on what your goal is:

• Do want to conclude that the fitted curves themselves are statistically similar?
• Do the parameter estimates in the model have a physical meaning, and can you demonstrate similarity by focusing on the parameter estimates?  If so, then the confidence interval approach that @Phil_Kay mentions could be used.  However, in non-linear models, the parameter estimates tend to be correlated, which creates some complications, statistically.
• Is the curve itself important, or is more about the similarity of the profile of the data for each condition?  If the data are aligned in the X variable, you can just look at the distribution differences between the Y values for each condition.

For your particular problem, an easier approach is to linearize the model by taking the log of both sides of the equation, that is, fit this model

Log(Y) = A + B Log(X)

A=log(a) and B=log(b) in your non-linear model.

Then you can use a very standard analysis of covariance to compare the slopes and intercepts of the log-log models (see attached example)

[#6]

In addition to all that has been contributed by @Phil_Kay and @MathStatChem, you state you need "...100% sure." My question is 100% sure of 'what'? All significance tests force the decision maker to pick a test statistic and critical value for said statistic to declare, "statistical significance". Picking that critical value is entirely the decision maker's choice...but also opens up the discussion of Type I and Type II errors. Which make it impossible to be 100% sure of anything when making statements about parameter estimates 'significance'.