Kitt Kroeger

You may try to do the exceedance probability (EP) analysis.

Preprint Exceedance probability analysis: a practical and effective a...

The EP analysis described in the paper assumes normality. However, it can be used for any distributions.

The subject of the study was very interesting.

However, this is simply done using a GraphPad Prism.

Kitt - 

I see you don't have a lot of data, so prediction intervals will be large.  I suggest that you use software which will plot prediction intervals around (in the y-direction) each of your nonlinear regression lines (curves).  You might also want confidence intervals around the exponential parameters. 

Also, are you sure this is exponential, or maybe you have quadratic linear regression?  Perhaps exponential makes more sense from a subject matter perspective? 

If you have only one parameter to compare, perhaps you could get a confidence interval around the difference between the two cases?  How much 'confidence' could you then have before the interval included zero?

Maybe something in the following might be useful, even though it is for comparing alternative models for one set of data, not comparing two models with two sets of data: 

https://www.researchgate.net/publication/313925533_Comparing_Model_Performances_Graphically.

Remember also that the circumstances under which you collected the data may not be fully 'representative' of what you want for an underlying population, so that can make an unknown difference.  However, as in some cross-validations, if you could collect some more data, that could be particularly helpful.  A larger sample size will tend to reduce the length of prediction intervals and confidence intervals, but if you have not covered the breadth of conditions allowable, then doing so will get you more realistic intervals, but covering allowable situations better also will tend to lengthen intervals ... likely less so than they are shortened by more data. The more reasonable inclusion of various conditions will shift the prediction or estimate (where y-values, being a random variable, are predicted, and parameters are estimated).

Cheers - Jim

Kitt -

By the way, I think Hening has an interesting idea, if it answers the question you are really asking yourself, but I think it is probably very sensitive to the situations of your data collection being highly representative of the circumstances that will be encountered. When you put all of your analysis into one consideration, it has to be meaningful, but you might want to check out his paper. I am more inclined to consider graphs as a way to get a better overall picture, however.

Abolfazl considers the interesting question "Does one curve adequately fit all the data sets?" But I think how that question is answered needs explanation.

Thank you for the interesting question/discussion.

Cheers - Jim

Hi Kitt,

It is hard to me why do you think of exponential distribution? However, I suggest transform the data by log function and compare their new versions using ordinary regression analysis.

Good luck

Babak Jamshidi It's exponential because duckweed has exponential growth since it clones. I'll try using an ordinary analysis on a log graph though!

James R Knaub Thanks for the reference! I was kinda stuck on what I would be looking for, so thank you for giving me a nod in the right direction!

I know I don't have a ton of data; this was kind of done behind the scenes so I didn't dedicate a ton of time to it, and it was done in very small jars; at about 2 weeks the data usually stops being reliable since the duckweed has at that point filled up the whole surface area. I'm glad I was able to contribute to an interesting conversation!

Any way I can try and convince my superior to let me just use a dishwasher to clean 100s of small jars and literally save hours of my time is worth trying.

HI Kitt,

Exponential growth is not relevant to exponential distribution.

Jochen Wilhelm

Jochen Wilhelm

You're right, this graph is only showing averages, but I only had 2 jars per category. That was about all I was able to allocate time and space for.

To explain the difference, it's referencing 2 different species of duckweed; Lemna minor and Spirodela polyrhiza. Spiro has much larger fronds and (based on another experiment I did to determine doubling time) reproduces slower than lemna, with a doubling time (in the media we're growing it in) at about 11 days and lemna at about 9 days. They can slow their reproduction once they start to realize they're getting crowded, and since they're in small glass jars, spiro gets crowded much faster. They're absolutely fascinating plants!