 Okay, cool. So, everybody, I'm happy to be talking about my work today called Journal of Non-Pyrametric Tests of Differential Genomex, sorry, Genexpression for a single cell and bulk genomics. So, I want to start by motivating this piece of work. So, as we all know, differential analysis is a key component of single cell and bulk genomics, and there is more of a need for general methods that can encode flexible alternatives when running the analysis. So, for example, in a recent work by Shyam et al., it's shown that changes in gene regulation that are owing to aging or senescence, they're actually more likely to be picked up by variability indices rather than mean or median indices. And, moreover, in the bulk RNA-seq setting, it's also recently shown that a lot of current approaches actually have met the false discovery rate. And so, there are, of course, some approaches that can detect alternatives that are characterized by multimodality, but these methods rely on parametric models, typically. So, given these issues, now I would like to present this general two-sample test to you. So, the way I'm going to do it is to first explain how we construct our test statistic. So, essentially, this general test is really a test on a simplex. So, I'm going to walk you through step-by-step how this works. So, suppose we're given two samples, and these samples are labeled by, say, some condition of interest, then what we do to the two samples is we take the smaller sample, so for instance, x, and we're going to rank the values of x and set the points of them as bars, and then we're going to slot the second sample's points in between the consecutive bars, so kind of looking like this. And so, from this procedure, we actually get a count vector where we have s1 to sk plus 1 counting number of yj's that are contained between two bars. And then, so this count vector actually sums to n, and so we then have to actually divide throughout by n so that we can get a vector that actually lies in the simplex. So, after division, we get some vector lies in the simplex, and then finally we construct a test statistic by taking some linear combination of the p-power of the entries of this vector. So, now we've seen a test statistic, what is this null distribution? So, the null distribution is exactly when the two samples come from the same underlying distribution. And when this happens, the distribution of the normalized count vector that I described in the previous slide is actually uniform over the simplex. And so this allows us to compete the p-value under this uniform null hypothesis. So, the key insight, however, is that the choice of the test statistic will actually help to boost the power of the test against alternatives of interest to the user. So, remember that the test statistic actually depends on two user defined parameters, the weights and the exponents p. And so the generality of these choices actually will help increase the power against alternatives that the user has in mind. So, as two examples, we know the standard Wilcoxon test, where the alternative that we have in minus-locastic dominance, we actually can use the following parameterization. However, if let's say we're interested in an alternative that is characterized by more of a skill shift rather than a stochastic dominance. So, the mean or the median could be the same, but then the variance could be shifted, for example, right? Then we would use another parameterization as shown in this slide. So, as to reiterate this point, in general, we would have some kind of alternative distribution shift of interest. This could be, for example, changes in the zero-inflated negative binomial dispersion parameter or some other alternative of interest. From there, we can actually work out the optimal weights and exponent of interest. And then we can then use the test using these weights. So, before I talk about some of the applications to single cell genomics, I would just want to point out that because this test is non-parametric, so, first of all, it does control the type 1 error. And more over, we did the kind of semi-synthetic, we ran an experiment on some semi-synthetic data. We showed that it actually also controls the false discovery rate. And, of course, finally, for the choice of weights and exponent that was shown in the previous slide, we do show that the test actually is strictly more powerful than men with me. So, now I'm going to talk about some applications to genomics, and I'm going to keep it short so that we have some time for questions. So, we actually ran this test on both bulk RNA-seq as well as single cell RNA-seq. So, I'll be talking about the single cell RNA-seq application today, and we ran it on the tabular mirror center data where the condition of interest is the age group. So, first specifically, we performed the analysis with respect to three different age groups that were available in the spleen, the kidney, and the lung tissues. And we, first of all, optimized the parameters for detecting a scale shift. And this is really because, as was mentioned in the first, in the very beginning, there are these papers that report signatures of aging or senescence to be governed by shifts in spread rather than centrality measures like the mean or the median. And on top of that, we also consider multiple data normalizations. So, we consider the raw counts, we consider the log counts, and then we consider also what is called SE-transform, the variance stabilizing transform that uses the Pearson residuals. So, just to show you one, you know, example of finding that we have, if you look at this plot over here, you can see that this is a plot of some differentially expressed genes that were called by either running the Wilcoxon test or our test, which we call Mochis. And on the X and the Y axis, I'm actually plotting the full changes as well as the spread changes. So, if you look at the red points, which are the points that are called DE genes by running the Mochis test. Mochis is the name of our software. We see that a lot of these genes actually are lying in parts of this graph where spread changes are more extreme than the full changes. Now, if you look at the blue points, however, which are the DE genes that are called, where we run the Wilcoxon test, you'll see that, you know, they're sort of, it's less clear. The pattern is, you know, less clear. You see that there are some genes that are characterized by a more extreme change in spread than fold. You also see some other genes that are characterized by more extreme changes in the fold than the spread. So, I actually do have some more plots, but feel free to ask me questions during Q&A, and I'm happy to share more plots. But I would like to end the presentation here by acknowledging the following individuals for helpful discussions leading up to this work. Thank you. Okay, thank you, Alan. Just remind everybody online if you have any questions to please put them in the chat, and we will get them up there, and I can read them. And at the end, we'll also walk around with microphones for people who have questions in the room. Please make sure that you also say who the question is for. I see Ryan has a question about the availability of the code. So, that's a great question. So, actually, it's over here in this slide. Here, can everyone see this? So, we basically have the package on GitHub. So, right now, it's still work in progress, but we have the function that actually can run the p-value computation, and it's already packaged up in the tar ball. So, you can just download it and install it as an R package as you would a typical R package in a tar ball. And just to show you an example of how we can run the main function. So, over here, we actually have it documented already. So, it looks kind of like this. So, we call it the Mochis test, and there are a bunch of correctors, and you can just specify the parameters of interest, and here's an example of how you would run it on some simulated two samples. Okay. All right. Well, we're going to move on to the next speaker, and then we'll take all more questions at the end. Okay. Thank you, Alan.