# Statistical difference between curves

I have 12 curves (three replicates for each treatment), see attached picture. X=days; Y=percentage. The experiment has 2 variables: A: 4 cases; B: 3 cases.

I would like to know, if there is a method to ascertain, if there is statistical significance between these curves.

Is interpolation always necessary in such cases?

NB: sorry, I am not using R.

• Oct 25, 2022 at 19:17

If I understand correctly, you measure some kind of percentage on different individuals for up to 18 days (but occasionally, you don't observed all 18 days), and each individual gets randomly assigned (non random assignment would make this a lot harder - or perhaps almost impossible given the extremely small number of individuals) to a different treatment. So, something like 12 individuals, 3 of which are assigned to each of 4 treatments?

This would seem reasonably well suited to some form of repeated measures binomial regression (if you have a denominator for the percentages e.g. 20 out of 400 cells examined under the microscope had some property) or repeated measures beta-regression (if there is not exactly an integer denominator such as percentage of skin on the back that is irritated). You'd want a repeated measures (aka random effects) type of approach to correctly reflect that measurements from the same individuals are correlated and usually more closely correlated the closer in time they are taken. Given the low number of separate individuals using parametric modeling and making some kind of assumption about smooth curves would seem like an obvious thing to do, but it's hard to say what would be appropriate here (ideally you'd inform that by previous studies, other data or theoretical considerations in order to not overfit the present small set of data too much).

Such models would then allow you to answer some kind of questions about how the curves are different. E.g. do you believe that they should plateau off and may plateau at different levels for different treatments? Or do you have some other kind of question on how the curves might differ in mind (e.g. reaching some specific percentage faster)?

• Sorry, I forgot to mention that it is a germination test. I measured germination for each treatment using 3 plates with 50 seeds each. The "missing" measurements are not really missing. I just omitted the "missing" points when the curves reached the plateau. I have edited the post. Oct 25, 2022 at 16:32
• Ah, now that changes things a bit, because I guess a seed cannot un-germinate, so it's a time-to-event thing where once a seed has germinated it's germinated, right (and the curve can never go down)? I guess you may also speculate that the plateauing may indicate that some may never germinate? In that case some kind of time-to-event (aka survival model) with a non-germinating fraction (aka "cure fraction") could be a sensible thing to use here. Again, most software packages have methods for parametric cure (fraction) model estimation, ... Oct 25, 2022 at 18:31
• ... you'd just want a version that can account for plate effects. That could be either via a fixed plate effect or a random plate effect. The latter might be more robust for your particular problem since there seem to be some plates with 0 germination that might otherwise be hard to deal with. Oct 25, 2022 at 18:33

My approach would be to first try to model the percent germination as a function of time. A one-phase exponential curve would sort of kind of fit the data. But maybe it needs a fancier model with a lag parameter?

Then fit all the data with nonlinear regression. Then take the parameter of interest (rate constant, or may its reciprocal time constant) and compare between the four treatments with one-way ANOVA and followup tests.

You can probably figure out a way to fit the entire model, including the treatment, so the comparison of treatments is fit by the nonlinear regression. But this would require a fancier model.

Are you only interested in the difference in rate? Or also in the difference in plateaus?

• reading various works I started to think that modelling the curves could have been a possible solution. I would have many questions about modelling since I never used this kind of statistics. For example, for each treatment, is modelling the mean equal to model the replicates? Which could be a software (possibly other than R...) to perform the analysis you suggested? For the plateau I had compared the means of the final germination via anova. Oct 25, 2022 at 17:31
• I'm totally biased (since I founded the company and wrote most of the explanations) but take a look at GraphPad Prism for fitting curves with nonlinear regression. Better to fit all the data, and not preprocess. But nonlinear regression is also in R and most other stats packages. Oct 25, 2022 at 18:43