# Comparing sample fluorescence at different temperatures

First of all, sorry if this format of question has already been answered - but I've had a look at previous answers and still can't figure this out for the life of me.

So, I have these 15 samples of bacterial colonies. Leftmost column are 3 replicates of a cell line that I think should be more shiny when they get hot, all 4 columns to the right are triplicate controls for comparing against. (2nd column from the left is the direct comparison to the leftmost column as this cell line expresses the same fluorescence protein, but shouldn't get shinier when hot.)

I have taken this single plate of these colonies and put it into 0 degrees before measuring the fluorescence, and then I put the plate into 45 degrees before measuring fluorescence. What I'm struggling now with is how to process and present the data. My first instinct was to make a 3x5 table where each entry is equal to the fluorescence of its corresponding well at 45 degrees divided by its fluorescence at 0 degrees. From there I would then take the average and standard deviations for each column, before plotting those on a graph. I turned them into relative percentage changes by calculating =100*(1-(proportion)) and =100*(standard deviation) for the standard deviations. (In terms of what bar represents what columns on the plate: dTomato = 2nd column from left. dTomatoELF3 = 1st column on the left. GFP = 4th column from the left. GFPELF3 = 3rd column from the left. 5th column is not represented in this graph.)

This resulted in some really nice graphs that actually show the trend I was hoping for with no standard deviation overlaps for dTomato and dTomatoELF3 and so some possible significance. To test this hypothesis I did a simple Students T-Test in excel, 2 tails, type 2 test, where the two arrays corresponded to the proportions for each column. The p-value was 0.034722 so I wanted to accept the alternative hypothesis that the dTomatoELF3 samples are shinier in the heat; however this p-value is a little suspicious and nice seeing as I have only 3 replicates.

A colleague looked at these graphs, wasn't convinced, and asked for the results but processed in a different way. Rather than calculating proportions first and then averaging the proportions from 0 to 45 degrees, he wanted me to just calculate the average fluorescence across the triplicates, and make a graph of the averages for 0 and 45 degrees side-by-side. He wasn't interested in the value of the proportions. (Sorry this isn't as refined a graph; GFP = 4th column from left, GELF = 3rd column from left, DTOM = 2nd column from left, DTELF = 1st column on left.) Sadly my promising results were lost when the data was presented in terms of absolute data instead of proportions, but I guess that's not surprising with only 3 replicates.

So finally I've got to the problem.

1. I feel that absolute values is a more accurate way to represent the data than proportions, but I honestly don't know why and so I don't know how to explain it in my discussion section (and I would definitely like to try as I think it's important for the write-up). The person who asked me to present it this way didn't have much to say other than that the first graph was unconvincing. My maths graduate friend tried to figure out an explanation, but he wasn't entirely sure either and pointed me to this website.

2. I don't know what stats test I should be using. The same friend mentioned "I think the discrepancy comes from the weirdness that happens when you divide one stat by the other", (i.e. the fluorescence at 45 degrees divided by fluorescence at 0 degrees.) "I think you want to treat the 0 and 45 as separate trials. Combining them into a combined statistic is [probably] invalid for this sort of hypothesis test". But this has only confused me more, as this means a T-test isn't correct. But I have no idea what other test I'd use.

For context, I've dabbled in stats for previous write-ups and I can generally follow a good online guide for how to do stats tests like One-way ANOVA on Excel. But unfortunately I have no R experience. I would very much like to get some, but my write-up is due soon and I don't have time to learn.

Thank you very much for reading! If you have any suggestions or any commentary at all please let me know what you think I should do :) (Oh and unfortunately due to time constraints, repeating this experiment with lots more replicates isn't feasible)

I'm not confident I followed all the details, so I summarize my understanding first:

The experiment involves one 3x5 plate with 15 wells: 5 cell lines with 3 replicates. Fluorescence is measured at 0° separately in each well, then the plate is heated to 45° and fluorescence is measured again. The hypothesis is that one cell line gets shinier when heated.

And there are two questions:

• What is the proper order of operations: compute statistic (comparing fluorescence at 0° and 45°) and then average (across replicates). Or average first and then compute statistic?
• What is the appropriate statistic: ratio or differences of fluorescence values?

If this is the correct setup, then the measurements before and after heating are paired since the data comes from the same wells. Then averaging first and making a comparison second is not the right approach; instead make comparison first (between the before and after fluorescence values within a well) and then average. We observe the same wells under two different conditions, so we make a pre-post comparison of paired measurements.

Now about what you should compare: ratios (value after heating divided by value before heating) or differences (value after heating minus value before heating). This is best determined by domain knowledge: do you expect the effect of heating on shininess to be multiplicative or additive? If the multiplicative, then you can look at the logs rather than the raw values, so let's assume that you want to compare differences of fluorescence values.

We've abstracted the problem to the following analysis: we want to determine if there are differences in post-heating fluorescence between groups (columns) given the pre-heating fluoresence. I think it's reasonable to contrast only two groups: the first and the second column on the left because they are comparable before heating: the cell lines express the same fluorescence protein.

The most straightforward analysis is to compare change scores (post-fluoresence - pre-fluoresence) by cell line (column in plate). This you can do it with a two-sample Wilcoxon rank sum test. Better but requires more statistics, is to use ordinal regression to analyze post-fluoresence as a function of pre-fluorescence and cell lines.