I have stimulated macrophages to become pro-inflammatory (M1) and anti-inflammatory (M2) and I have unstimulated control group that was treated with with vehicle, PBS (M0). I would like to compare if M1 and M2 are different from eachother, but also if M1 and M0 are different and if M2 and M0 are different. As M0 is only a control and my actual question is if making cells M1 and M2 was successfull, I do not need to perform ANOVA.
My readout is cytokine secretion, for instance IL-6. I have repeated the experiment for 3 times making my n=3. Normally I would use Mann-Whitney U test as my n number is so low, I cannot assume Gaussian distribution.
I use ELISA to detect the concentration of IL-6 in the cell supernatants. It turns out that in most cases the M0 cells do not secrete detectable amounts of IL-6, same goes with M2. As they are below detection, I have set the value as the lowest standard. I run into problems because I have identical values and thus I do not think Mann-Whitney is working correctly because it is comparing ranks. Is it better to do a T-test or Mann-Whitney or something completely different?
I am using GraphPad Prism 8 to perform my statistical calculations.
Here is my cytokine concentration values from the three different experiment:
PBS: 15.625, 15.625, 15.625
M1: 4399.92, 5160.707, 4143.947
M2: 15.625, 15.625, 15.625
Thank you, in advance!
"Normally I would use Mann-Whitney U test as my n number is so low"
That's a strange argment. The U-test has virtually no power for small sample sizes. It is also difficult to interpret. Without the assumption of homogeneous variances (an assumption that is very obviousely invalid for protein expression data), the tested hypothesis is whether the distributions are equal, or that Pr(a>b)=0.5, where a and B are two randomly selected values from the two groups A and B. Rejecting this does not help you to say that the expected gene expression is higher (or lower) in A compared to B. It also does not help you to conclude that the median expression if higher (or lower) in A vs. B. And, to complicate things further, it can be the case that the median shift is in the opposite direction than the mean shift. How would you interpret this?
To your problem:
It's very obvious that M1 > PBS and that M1 > M2. But you don't have any information to compare M2 and PBS.
Since you don't have quantitative information about PBS and M2, you cannot make any quantitative comparison. But you could go back to a qualitative comparison, comparing the event whether or not a measurement was above the detection limit (or another lower limit). For the comparisons of M1 vs.PBS or M2 you have 3 values "above the limit" in M1 and 3 values "below the limit" in the other group(s). Under the null hypothesis, you would expect these events being evenly distributed across the samples of both groups, so Pr("above the limit") = 0.5. The probability, that all 3 measurements of M1 are above and all 3 measurements of the other group are blow the limit is 0.5^6 = 0.016. The same extreme case is the opposite distribution, that also happens with a probability of 0.016. Assuming the null hypothesis, the probability to get a distribution at least as extreme as the one observed is thus 2 x 0.016 = 0.032, what is your p-value (this is called a binomial test).
There is nothing you can do to (statistically) compare PBS and M2. You don't have values for any of these samples, so there is no way (no information) to guess which one is higher or lower. You can state that both have barely detectable expression.
Thank you for this answer, Jochen Wilhelm. This was very helpful. I am aware that my understanding of statistics is very limited. I have always based using the Mann-Whitney U test to an assumption that if I have low n number, I cannot assume Gaussian distribution and thus I should use non-parametric test to avoid an error, but clearly I have not understood what I am doing. I will follow your suggestion.
There is a lot of confusion about assumptions and tests out there, so this is likely not your fault!
Assumptions are never "correct" or "wrong". Instead, they are either "reasonable"/"justified" or "unreasonable"/"unjustified".
The amount of data you have has nothing to do with whether or not any assumption is reasonable. It's only difficult or impossible to see if an assumption is very unjustified (because the data looks entirely different than expected by the assumption; e.g. the data shows a strongly skewed distribution but you assumed that the distribution is symmetric).
Assumptions should be made mainly based on our understanding of the variable (what was measured). For instance, if I measure protein concentration in cells, it would be very unreasonable to assume a symmetric distribution, no matter how my observed data looks like. If I made hopefully reasonable assumptions, and if I have quite a lot of data, I may peek into the data and cross-check that my reasonable thoughts about the data were not too absurd. If I find that the data behaves really different than I think it should, I must go back and get a better understanding. After that, I may either come up with different assumptions or new insights, possibly telling me that my original idea of analyzing the data makes not much sense the way I thought, or that the principle of measurement is suboptimal or faulty. Painful, but extremely beneficial long-term.
Assumptions are not generally bad, they can be very helpful and good. Disregarding absolutely reasonable assumptions is usually a very stupid thing to do. You pay that with a loss of power.
People may argue that bad assumptions are in fact dangerous, as they can lead you to very wrong conclusions. However, not understanding assumptions and therefore using tests that are not understood either (what precisely is the hypothesis tested in the U-test and what will a rejection allow us to conclude?) lead to the very same problems, not less severe. So if you (the general "you", not you personally!) think you should use a t-test but you don't know if the required assumtions are justified, and because of this you would "just use the U-test", you should stop and gain a better understanding of your data and of (possibly) of methods of statistical analysis (or co-operate with a statistician). Otherwise you "avoind an error", but you just do another.
I hope this post was sufficiently provokative ;)
- Badges
- Science topic