Broadly, I want to compare ratiometric intensity distributions between different groups.
Each age group (let's say D2 and D6) has 20+ animals across one or more biological replicates. For each animal I have measured intensity ratios (intensity in red channel divided by that in cyan) for 1000+ individual puncta, which I have graphed as violin plots. I have also pooled all the values within each replicate and within each age group. Abbreviated versions of these plots are in the examples, e.g. 5 individual D2 animals each from replicates 1 and 2, the pooled data for each replicate, and the pooled data for the whole age group.
The distributions reflect what's in the images: on D2, almost all objects are cyan (corresponding to low red/cy ratio) but there are also a minor population of small deep red puncta (high red/cy ratio); while on D6, there is a larger population of both red puncta (though not as high a ratio as D2) and pink (medium red/cy ratio) puncta.
Looking at both pictures and graphs, it's clear there is a big difference between D2 and D6 puncta that goes beyond just the mean and median. The distributions are obviously not Gaussian for either age, but the standard deviations are also significantly different (according to Welch's t), so does that mean both parametric tests and Kruskal-Wallis/Mann-Whitney would be inappropriate? Which statistical tests would be best when we want to consider the "shape" of distributions? Should different ages be compared age-pooled, replicate-pooled, or each animal treated as an individual replicate within one age group? How should substantial within-group variance be handled? I'm only really familiar with GraphPad Prism so tests possible in that suite would be preferred, although I recognize very large complex datasets may be better handled in R.
The D6 distributions are interesting – that shape suggests to me that there are two underlying distributions involved. Either that or some sampling or measurement factor is preventing observation of data in the middle.
Ratios are also horrible things to work with. Even for normal distributions, the resulting distribution (Cauchy) has bizarre properties – the probability of extreme values is so high that increasing sample size does not actually improve estimation of the mean!
So I'd start by asking why the heck the D6 looks like that.
The animals are your sampling units (the entity that has been randomized to the treatment), not the "punctua". It is not meaningful or sensible to run a hypothesis test to compare the distributions of these punctua as if each "punctum" (what's the signular?) was a statistically independent observation. You need to summarize one aspect of the distribution of puntua per animal (randomization unit) to test a statistical hypothesis about this aspect. You can do this manually, for some aspects (like the mean) this could also be part of a statistical model accounting for this kind of replication (hierarchical aka multilevel aka mixed-effects models).
Hi, thanks for the advice! The shape of the D6 distribution is actually what we expected it to be, given the nature of the reporter (pH, dual-channel ratiometric) and the physiology of the animals. The cyan objects are unacidified, the pink are partially acidified, and the reddest are fully acidified. There aren't as many "intermediate" puncta simply because acidification is generally quick, but as the animal ages the acidification efficiency starts to decline.
The reason I've included all the puncta in the graphs is to demonstrate the issue I'm having summarizing the data on a per-animal or per-age group basis. If I had normally-distributed data and consistent SDs between animals I would just do a parametric ANOVA analyzing the means. But since the distributions are strongly bimodal in at least some the animals the mean wouldn't be meaningful and I'd need a non-parametric test, and since the SDs are very heterogeneous I think it wouldn't be appropriate to use Kruskal-Wallis.
Previously I had been binning each puncta in each animal into one of the three levels of acidification and calculating the proportion of the total puncta area in a given category. Then I'd run a t-test or ANOVA on those values to compare, e.g., the mean partially-acidified area proportion of D2 vs D6 animals. Should I go back to this method? Jochen Wilhelm Ronán Michael Conroy
I'd summarize the data as proportion of punctua > 0.5 par animal. As far as I understood, the resulting variable then encodes the relevant information, and its distribution can be assumed beta, so you can use a beta-model, a quasi-binomial model, or a normal-model using logit-transformed values.
I would not use binning (categorization).
Perhaps it's just me but I don't think that your question is clear. There are a lot of undefined terms such as ratiometric, etc. i find it is impossible to say because I don't have a clue what you are doing or what you want to know.
David Booth
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