Tolerance interval from exponential decay

[#1]

I have several samples taken from a population that are stressed over time.

At certain intervals measurements are taken.

All samples show an asymptotic exponential decay as y = a + b*exp(-c*time).

What I have to predict from this experiment is where my population will be (tolerance interval?) after long time under this stress condition.

I have attached an examples that should reflect my actual data. In this example, I have 5 samples that start at a value of 100 and exponentially decay reaching a value of around 70 at time=100. The samples have slightly different values every time, adding randomness to the experiment.

How can I estimate the tolerance interval at time 1000?

[#2]

maybe try log transforming your time variable then use the stability platform to get the lower 95% CI (not tolerance interval) for the crossing time.

 

[#3]

Hi @bbenny7 : I'm afraid there is no easy answer for such tolerance intervals. And I know of no software package where that is an option (it's a research topic). That doesn't mean it's not doable...may be doable using JSL, but I don't know enough about JSL to confirm that. That said, there are folks working on such problems because it is needed in applications!.  See link below (in particular, section 5.2).

 file:///C:/Users/m290593/Downloads/v36i05.pdf

More here.

https://www.informahealthcare.com/doi/abs/10.1080/03610918.2012.689064

Good luck!

[#4]

Hi @bbenny7 : On refection, I'm curious to know exactly what you mean by "where my population will be after long time under this stress condition".

So, using the nonlinear platform and expanding a, b, and c across Samples (or Fit Curve by Sample), each sample has somewhat different values for a, b, and c. The asymptote is "a". Can your question, perhaps, then be rephrased as follows? Given the variability in "a" between these samples, what can we say about the spread of a in the future population of many many such samples?

Edit:

The problem as I’ve stated it above can give rise to a mixed nonlinear model (e.g., a is random, rather than fixed, and varies with sample) and modeling that directly complicates things even further. But, we may be able to simplify things and find a “fit for purpose” solution depending on how you answer the above questions.

Or, are you interested in these 5 samples only, and want a tolerance interval for the population of future observations from each of those 5 samples, respectively?

Or, are you interested in all future samples, and want a tolerance interval for the population of future observations from all future batches?

Or…?

[#5]

Hi @MRB3855 ,

Thanks for your post.

I am interested in is all future samples, so I would like to estimate the tolerance interval for the population of future observations.

[#6 · Accepted solution]

Hi @bbenny7 : I'm afraid there is no easy answer for your question.  For linear mixed models it is challenging enough.

See here for linear mixed models.

https://onlinelibrary.wiley.com/doi/full/10.1002/sim.8386

But, for nonlinear mixed models? That's a research topic and there is no option for that in any stat software package that I'm aware of.  A google search highlights the issues, with confidence intervals, prediction intervals, and tolerance intervals for nonlinear mixed models. One such article is below.

https://www.researchgate.net/publication/220504625_New_Prediction_Interval_and_Band_in_the_Nonlinear...

Great question by the way!