Are we using the right statistics?

Naturally speaking. Hi, I'm Graham. And I'm Rachel. And today's episode of Naturally Speaking is on Statistics.

Or we're using the right statistics. It is a contentious question in science. Statistics may seem like a hard objective discipline, but there are in fact many different opinions and methods when it comes to analyzing our data. Today on Naturally Speaking, we are discussing the two main schools of statistics, frequentist and Bayesian.

We also discuss if we are teaching statistics correctly and what are the obstacles for biology students faced with learning statistics. In the frequentist corner, we are joined by Sophie Smitharis. Sophie teaches undergraduate statistics, introducing students, often for the first time, to the world of statistics. She lays the groundwork of statistical theory, building on skills and understanding of the frequentist approach.

And in the Bayesian corner is Jason Matthew-Oppler. Jason teaches Bayesian stats as a master's module, building on the framework provided by Sophie to explain Bayesian theories and the practicalities of this field. We hope you enjoy this very interesting episode. Thank you very much for joining us both here.

And I'm actually speaking to discuss a frequentist and Bayesian statistics. I'm going to start off with an attempt at a statistics joke. Bayesian and frequentist are to be executed. The judge asked them what were the last wishes.

The Bayesian replied that he would like to give the frequentist one more lecture. The judge granted the Bayesian's wish and then turned to the frequentist for his last wish. The frequentist quickly responded that he wished to hear the lecture again and again and again. So first question, I'd like to ask you both to try and define what are frequentist and Bayesian statistics and maybe what are the key differences?

So I've got a very harsh answer for this coming from years of teaching the Bayesian course. So the way we introduce it is by saying that the first frequentist name comes to the notion that a probability is defined as a relative frequency of an event occurring given an infinite number of identical experiments. So if you think about a relative frequency, that's the pollute of the name, as the number of times that the event has occurred out of the total number of times that you've tried the experiment, that's a proportion. And then as you repeat the number of experiments, an infinite number of times, that proportion tends to the true underlying probability.

So the frequentist is a bit like a platonic philosopher who says that there is an ideal notion of a probability somewhere out there and either. That's pretty determined and existing, depending on the way of us. And all we're doing is we're repeating the same experiment as faithfully as possible, trying to determine what that probability is. And the Bayesian comes along and says that, well, I'm not really going to think of probability as an objective kind of quantity, but exists outside of me.

I'm going to think of probability as a subjective opinion. That reflects my experience of the world. And I'm going to update and adjust this opinion as I experience the world. OK, so that's kind of the philosophical distinction as people like to present it, which I think pits the two viewpoints in kind of opposite apparently, opposite philosophical camps, which is where some of the conversations lead more antagonistic conversations between the two schools of thought come from.

But I like your way of thinking a lot more in the sense that the two are actually completely complementary. And the Bayesian viewpoint is a natural extension of the frequentist point and we can kind of talk about that a little bit more. There's a much more grander spirit of reconciliation, I think, is much easier to adopt in thinking about these two and teaching them. That is the thing that when we actually go and say which one of the two you actually need for the things that you need to do.

Do you need to do both? Obviously not. They're not complementary in that sense. They are absolutely, I think, in the sense that.

So we're both in Greece. And in Greece, when I go into Greece and you hear about those two sort of soon schools of thought being discussed, it's almost as if there's like an ongoing battle between the camps. When you speak to the British statisticians here, they're very pragmatic about it. They just go, you know, I'll use the frequentist approach when I use the Bayesian approach when it suits me.

And when does it suit you? Well, invariably when you have independent information in addition to your data, that's the crux. So if you, if in addition to the data you're just collected, you have colleagues who have collected data somewhere else and some of them. Some of them are telling you have strong biological opinions or you have experts who have worked as empiricists in the system for decades.

It seems to shame not to bring that additional information into the fold to kind of help yourself. So you basically supplement with extra existing information or notions. Exactly. Exactly.

And that's it in an answer. Really the difference. So so given your teaching, do you have to define positions probability? We are teaching always, well, define probability.

We really don't go into the philosophy. I mean, we do talk about proper at some point and positivism and the different schools, but I got through very quickly and it's very applied, very applied the way we teach it. So do you think at undergrad level we should have a bit more of a background to these applied things? Should I be included in a portion or is it better just to forget all this philosophy when you're trying to learn sets and focus on the numbers?

Well, we are both from Greece and we do like philosophy. And yes, in many ways I would love that. So I understand and get to do one more on it. I'm not sure how it would go down with students, but usually students do enjoy more the background aspects.

I don't know if. They have to quite know what really gets students go and students at different levels get excited by different things. So if you try to be conceptual with a class of undergraduate, you get 40, 50 poker faces looking back again. And in amongst them, there's three or four that you just switched on and you don't know it for another five or six years.

Five or six years later, the come along is here. That actually again was really interesting. So I don't really know. What exactly should go on?

What exactly should go on? What exactly should go on? What exactly should go on? What's better for them?

So for me, for instance, the continuous battle with the undergraduate teaching is whether we should be actually teaching them the least squares approach or go directly into the likelihood and the generalize leader model because that's what they are going to be using in their pre-quentist stats. Can you briefly explain what you're doing. Yes. So we have the usual Gaussian distributed response variables that students usually, we usually approach with the, well, in the past, we were analyzing with least squares approach, which is all the stats that are being reported in papers with the P values, the F ratios and all that, which we still find useful to teach because the students might still encounter them in publications.

So it might be useful for them to know what it is when they see this F ratio. But more and more, we realized that of course this is a bit of a set because there's the broader framework of the likelihood that the generalizing a model study can use when you have many different types of distributions in your response variable. And that you can test, you can do models selection with different ways and more and more publications come out using these approaches. So you know, the question is, do you really need to spend a lot of time teaching them about the least squares approach and just go directly into the meat and potatoes.

So I think, I mean, I think this is the point I was quite keen to make today, which is that a lot of the debate around the frequentism Bayesian dichotomy, which is really an dichotomy is dressed with associated topics, but very often have nothing to do with frequentism versus Bayesian. But just a historical artifact of the time, the peak time periods of the above which these kinds of approaches will be used for doing science. So you know, a favorite point to make by many Bayesians, the Bayesian statistics probably exists, I mean, it's sort of the basics of it, I've been around for 300 years or more. So despite that, business statistics are only being discussed actively in the last four to 50 years.

Frequentist statistics have been around engaged in research much longer than that. At a time when computers didn't really exist accessibly for scientists like us anyway. So the whole epistemology that went with the frequentist golden era was very different. To the way that we think about science now.

But a lot of these things have nothing to do with Bayesian versus frequentist. So to give an example, you mentioned, you mentioned least squares and you mentioned testing. So significance hypothesis testing was the way to do statistics in the last century. A more modern view of statistics is rather than ask whether something is there or not, is to say how big is it, what color does it have, is moving, quantify it rather than just test the hypothesis that it isn't there.

So I've got into arguments with people over this quite long in the past where I say, well, significance hypothesis testing is really not a very good way to do science. Because if you think about what you're doing, you're setting out a hypothesis that you really don't care about, which you call it the null hypothesis. This is the uninteresting explanation for the world. And you set out to disprove an explanation for the world that you're really not interested in with only one or maybe two alternatives.

That's a really weak, powerless way to do science. Essentially, what you want to say is, what do I think about the world today? What do the new data tell me? How do I update my opinions to deduce the relative support of both my previous and new evidence for a whole range of different explanations of the world?

And of course, I'm biased, they're not selling this, but hopefully it's a no brainer, right? The way of doing science is much more colorful, more balanced way. But you can, if you really want to, do hypothesis testing with basic success. And you can very easily do estimation and model-fitting and so on with frequent success.

It just so happens that we happen to do the article, the stuff in a frequentist kind of setting and more modern stuff in a Bayesian setting. I don't think we should be confusing to a school of thought with the various methods that happened to be active at the time when they were being thought about. And the final point, sorry, is that you mentioned least squares versus likely. Well, it's nice to be able to see things as generalizations of simpler versions of our two blocks.

And that's the way that really good science always progresses. So if you're able to say to a student that least squares is essentially a special form of a likelihood. And there's other likelihoods that generalize this idea. And what is a likelihood is basically your model for how the world works.

So when you say that this is a normal distribution and in my data, our normal distribution, essentially, you're making some quantitative statements and qualitative statements about the thing that you mentioned, that it's got its bulk in the middle, that there's think-tales at the end, that it's symmetric, and so on and so forth. So this is your likelihood and you're saying, that's how I think, though that particular aspect of the world looks like. Then the Bayesian builds on that, like that and says, I will now bring in the experts who have been working this for a while, and I'll just combine that with the data, with these opinions, and I'll get slightly hopefully more refined result at the other end. So very quickly, you can kind of unify all these things that seem disparate and unconnected into a unified framework.

And I think that's the way we should be kind of approaching our teaching and our thinking. I agree. And this is what you taught us in the first summer school when you came to Lesus and you were talking about this unified framework. And we are all looking at you like that because then I only knew about an overs and regressions.

And then Jason comes and says, ah, but here, there's this whole unified umbrella. And we started thinking very differently. And also with using R more and more, I think that that helped a lot to help us think about this more unified framework. And if we were to introduce a more philosophical approach to the way we do science, I think that this is the kind of, you know, thinking we should be having like a continuum of cases of how we approach problems.

And the methods are just binding across this continuum. As opposed to just thinking of the differences between frequencies and Bayes language is just bleeding our world instead of putting it together. Yeah, and we aren't moving in that direction. I mean, the education systems moving in that direction, Sophie wouldn't think of no, you know, I've seen your courses and you wouldn't really think of presenting a linear model as a completely separate thing to the generalizing anymore.

Even the name is nested within the other name. But if I think back to the way stats, sorry to go back to the bible days, but the way stats was taught to biologists, where I was doing that, only as a way of just demonstrating how much progress was made in our respect. What we were given at the time were think books with apparently dissociative recipes for doing different things. So we had an endless catalog of parametric and non-parametric significant hypothesis tests with mystical magical test statistics, incomprehensible probability distributions and look at tables.

And the whole task of doing stats was to determine which obscure hypothesis tests you should pull out of the book book to deploy it as a one-off thing. The fact that we've now got a unified framework from doing stats, thanks to this concept of the leg note, is marvelous and rather complicated than the life of our students, we're simplifying it because once they've got this one concept right, they can adapt it and deploy it to do all the stuff that we used to need a book for it. Dan Heidner has a lot of his slides and approach teaching and he has my favorite slide of his is a sponge that is absorbing water and he says, this water is your variation. Your model is the sponge and your goal is to absorb without sponge this variation.

And that's the goal here. Absolutely. Within these unified frameworks, we're breaking down this dichotomy between the two, which I really like because what I was understanding was that there was this dichotomy. If we are unifying it together, you touch another start when we were defining these different approaches.

Within the unified theory, is there a time to use one approach or leaning over to one approach rather than the other? Because from my understanding, people tend to think that if you're in a laboratory setting where you had a high amount of control and you can repeat experiments and be directly into the number of times, that would mean that you'd maybe use more of a good approach to approaches or if you're out in the field in the real world, things are messy, things don't happen twice, so you're observing it and having once, people tend to say that you should use Bayesian statistics. Is that right? Or is that wrong?

I mean, I do not use Bayesian for my lab experiments, but given what Jason said, that if you have some prior idea of what is happening in your system, then I would find it interesting to constrain my priors. That's a key word actually constrained here because in many cases you don't know, you don't have quantitative view of what's happening, but at least you know they can't go below or exceed certain, so for example, you're talking about population sizes. It really doesn't make sense looking at negative values for those. That doesn't mean that you can't do constrained optimization in a frequenting setting.

So there are good ones there. So going back to your question, I think it's probably true what you said. So if you have enough data that you can dominate whatever prior opinions you have with those data and you have unlimited resources to collect more, then you really don't need to think very much about a Bayesian approach. But if you were to use a Bayesian approach automatically, the data would swamp your preconceptions anyway, they would dominate those.

So if you happen to use a Bayesian approach, you wouldn't lose anything. Now, how many times are we really operating in a world of super abundant data? Certainly not in ecology. And it still seems a shame to not be able to use the findings of a non-clinical study simply because it was conducted in a different country at a different time.

So even if you have abundant data, being able to bring in some of those prior findings still feels like a useful thing to you. Why would you use frequentist as well? Can I just say, I just want you to ask for you to define priors or to explain what priors are. So there's essentially one component in frequentist stats.

So I should clarify, we're talking about estimation here, which essentially ignores much of the stuff to do with hypothesis testing. Simply because I think it's a much more useful thing. You can do hypothesis testing when estimating models, but you can't do modelling just with hypothesis testing. So let's take a more general approach.

So in the general approach, the estimation, essentially what you say is I've got a perception or a notion of how the world works and we're going to formalize that into some sort of mathematical statement and that statement is going to have some parameters and these parameters determine the quantitative behaviour of my perception of the world. Let's call that model. In other words, it's like that. So the likelihood that you develop essentially in English says the following.

What is the likelihood of our data being observed given the parameter values that exist in this model? Now if we're going to setting, that is interpreted as exactly the same as the inverse question, which says what is the probability of our parameters being true given the data? Now if you set a parameter, you replace that with a conceptually easier to handle hypothesis. Let's rephrase that.

What's the probability that my hypothesis is true given my hypothesis? If my hypothesis true was probably not my data, that's the likelihood. The flip side of that, which is really what we're interested in, is what is the probability that my hypothesis is true given the data that I've observed? And that's usually the scientific question of him.

You collect data to see if the way you think the world works agrees is consistent with the data that you collected. But in a frequency setting, those two are two examples of the same thing. You can kind of see that if I want, I struggled to put these things in the right order just there. This is because even in my mind, quite similar.

But in a Bayesian setting, you always begin from the forward bit, which says what's the probability that a particular set of observations comes given the parameters. And mixed to that, you stick another component, which is called a prior, that says that before I collected the data, I felt that my hypothesis, sorry, my hypothesis was true, even that this probability applied to my hypothesis, I felt that this was the case with 90% support of belief. And that's the prior. Now, if you generalize that to various values of parameters in an estimation setting, what you're saying is I'm going to associate a degree of belief for every single value of that parameter, because I think some are more likely than that.

And that's a prior. So what you're probably going to say, you update the prior with the likelihood, which says given any given one parameter, what is the probability of the data coming through. And it so turns out, and this is the sort of Bayesian principle that that product multiplication, when you bolt these two things together, it gives you the thing that you're really interested in what is the probability that a parameter was that hypothesis to give into data, but it just doesn't know. Right?

So if you drop the prior, you get the frequentist approach. If you bring in the prior, you get the Bayesian approach. So you can see that it's just bolting on another component, which brings in your punches or previous data and so on. And not just it just expands on the frequentist frame, the problem, predicted.

But the prior is not quite arbitrary, if you based on feeling the kind of being. And there's two answers to that very briefly. Sometimes the prior is not arbitrary. So if I tell you that, I just got to truncate my prior to exclude negative values.

So if the parameter that I'm looking at is a probability, for example, it makes complete sense to truncate that at zero and one and say that I'm only going to look at the flies in between zero and one. So if your parameter is a variance, it doesn't make sense to have negative values for it. It's kind of negative. So even that little bit of intervention based on first principles helps a great deal.

But then you might have good data. So if somebody conducted an experiment somewhere else about that particular parameter and they recorded a mean and a variance around that, well, there's your two characteristics for a prior distribution. You can characterize how believable, how supported the different values are. But you're right, there are cases where you really have very little information, very little evidence.

What do you do then? You need to keep precautionary. So you need to kind of inflate your uncertainty and your degree of belief and you can check how sensitive your results are on these prior, which makes the whole thing a little bit more in labor intensive. It's a reason for people not wanting to use Bayesian stress because you have to think about what you're doing.

I think based on how you were talking about before that these are unified approaches and I think you've explained that well, but also saying that Bayesian is more of a bolt-on to frequentist. Do you think that it's appropriate to be teaching it with frequentists just in undergrad and Bayesian in masters where it kind of sets up that hierarchy between the two and maybe already sets of prior in a student's mind that that's something that's more difficult, that's maybe less attainable for them? We were actually discussing just now with students getting their feedback about stats. The thing is that the more we teach them and give them this global framework of understanding, the more they want to learn about it.

They keep asking us why don't you introduce more of these earlier points. So there is this continuum that we try to, we do think that we need to introduce this gradually so can maybe they need to learn about a t-test in the first years. And then we build and build and then we make them understand that t-test is essentially any other model with a factor by a little bit of an explanatory end. But then we feel that they are more and more demanding about understanding the linear model much earlier.

So to me it's really a question whether we should be compartmentalizing. I mean, yes, this gradient or rather introducing this whole thing from the very start, to be honest. I don't really have an opinion that I mean, there is a little bit, well, it's considerably more investment in both teaching and learning this stuff. So you kind of split your hours according to how much stats education you've allocated in a degree.

I don't think that undergraduate students are incapable of grasping these topics, not at all. Because we don't teach business stats to undergrad, I've often, including my current undergraduate student, I'd have to give them a crash question on how to understand and fix Bayesian models which lasts about three hours. So they get a three hour state of this and they're actually able to run models and understand the diagnostics straight afterwards. You can do a quick and dirty job like that.

It's not ideal. I mean, I don't think that there's much to discuss there. There's nothing magical about that, that break point between undergrad and postgrad. I think that's the only clear thing in my mind.

If we had some extra time, I would rather we'd talk students about probability and thinking logically about outcomes and experiments and how to understand conditional and joint probability which are really useful topics in formulating ideas about science. So you touched earlier on how when students are reading papers, it's important that they can understand those components and that's why you teach them a lot of the stuff which are still being published. Do you think that there needs to go wider? I'm sorry, you also talked about how there was this 20th century hypothesis testing and removing on to this more unified advanced understanding of not just the yes or no, but what's actually going on in terms of relationships.

It needs to be a bigger change in science and must be published for us to be able to make a change in what we're actually teaching or I think that we need to change what we're teaching for it to be a bigger change in science. I think that the change in science is happening. It's a lot. It's very fast the way I have experience.

Like what I was taught in university starts in the way of ANOVAs and regressions and pushing different buttons for every single thing. I'm all. So yes, my experience with this approach was introduced to me by Jason actually for Stanford 2015. It completely changed my way of actually doing the science, like collecting the data to begin with, but also analyzing them and interpreting what I read in the future.

But there's still no papers up there that's very valid ones that the students need to understand what is being presented and because it is a sub case of what we are doing now. So yeah, I think that presenting is like with a continuum of all of this evolution of how the one binds into the other. I think that it will help students understand from day one though we should not be delaying into the fourth year of study or later on, but we should be producing this thing from the very start because yes, it does affect the way that you said. I agree.

I think we can make good use of vantage points in teaching in general, by which I mean, just because it took us 300 years to develop a sort of sort of scientific statistical toolbox that we have and we use now doesn't mean that we should take 300 years to teach it. You know, the sort of ontology of the development of a discipline doesn't mean you have to follow that particular route. That's why we can teach, you know, physics and chemistry in two years of high school because we take the correct vantage point where we're seen from that vantage point. The scientific facts coherently make a lot more sense.

So you don't have to take the sort of meandering route that the researchers took to get there. So simply like that. So I agree entirely with yourself that the methodological advances are galloping ahead and the teachings always struggling to keep up. We're not going to precipitate a change in research through teaching.

People in science are self-stocked to such an extent that undergraduate and postgraduate degrees are always struggling to catch up. But I think that if we pick the right vantage point and I think the one that you're using for your lectures is the right vantage point which gives people that unified feeling, you can ensure this back compatibility with past papers by saying, you know, guys, this thing that you see in this paper is actually very special form of this kind of model that I talked yesterday, who rather than starting from the antiquated non-parametric test or whatever and then building up to the generalized linear model. I think the vantage point that you have, the point of entry for your students is really empowering for them and it gives them some back compatibility and as a simple observation, I would say that notions are talking to these days don't read papers beyond five years in the past anyway. That's true.

Yes. I don't think we should encourage it, but it's a fact. Yeah. That's definitely not a reason not to be introducing this much earlier.

Hopefully it's cool as well. That's where I don't know what is happening in schools because students do come to the Uni knowing about anavasic regressions. Some students do that sometimes. Yes.

And then it's kind of difficult to make them understand that what you have learned is actually this, but you can actually put building blocks together as explanatory variables and here is the linear model. So you don't have to think about all these different things are separate, but it's really hard to make them understand that later on. The more you will delay introducing it, the more difficult it becomes for them to also then understand that they're not using a framework that is just another building block in a way. It's another, sorry, extra that you're adding.

So yeah, do find that students tend to be quite receptive and willing to learn this because I think even when I was minus five years ago now, people didn't want to be reluctant to learn stats and I think even now there's people that are not a stats, that's not a don't do stats, but from what you've been saying, it seems like students are quite interested and that's what I understand that we don't put enough faith in them to understand and grasp these things, but to be honest, because it is so much more logical way to understand things this more unified brain but our linear model. We should be trusting that they will understand it when we teach to them earlier on, but we currently don't have that faith in us yet. I think it's a pity. So I mean, we need a more way to break down barriers between preventers and basic statistics.

We need to have more faith in our students. We also need to take a broader view or teach them a broader view of the landscape of statistics. Is there any other crucial components that you think are important to really develop our teaching here at the University of the University of the University and is there any key barriers to change to bringing about this? I guess for me, the main barrier is that I would like to be communicating more with other people teaching statistics, like really because I think that the developments are quite rapid and not only with respect to methodologies, but the actual philosophy.

And I would like me to be in the same room as Dan, Paul, Jason and discuss these things, because I feel it helps me develop as a teacher. That's my, you know, what I feel it would help me. Yeah, I'm kind of struggling to make very concise recommendations because obviously, you know, every concise, every easy improvement that could be made, we think we've made, you know, in our teaching here. So real obstacles, I don't think so.

There's definitely a fear of anything quantitative in biology students, almost, in some cases, by definition, people who went into biology, were escaping from something quantitative, or that used to be the case historically. It's horrible way to frame a biology students, but it seems to be true. But even as adult scientists, I think we have an aversion towards quantitative frameworks, because they're actually designed to reveal the mistakes in our intuition, which is another way for, you know, another word for good science. But it's a painful experience to discover that your pet hypothesis is disproven by theory or by the data.

I wonder if that's an element why, in addition to the technical overheat of doing that, we're kind of afraid of putting our ideas to a quantitative test, maybe a fun, simple idea. I don't want to shift blame onto the Scottish Action Education System, but do you think students should be coming to biological degrees with a background of this stuff, you think that some people do, but imagine that that's what you think of the physics of the high school, which you think you should be going to university? It depends on the requirements we set for our entrance. I think you wouldn't dream of accepting a student to do theoretical physics, for example, without a background, you know, so a strong background in maths.

And every scientific degree requires, for better or for worse, requires a strong quantitative background. You can't really do science with our waving, experiments and quantification, or tools. And trade. So perhaps we should be a little bit more demanding on the level of quantitative education we expect to our students, or at least we should be honest with our students that if they accept the things showing and they join the degree that they will be treated to a good diet of training as for their own good, if they want to prosper in this area.

Yeah, I agree. I think that more than having some background in quantification is necessary, of course. But I think that that is always the case when they just leave high school and go to the beginning. But more important is the expectations, because I'm not sure that all students have the same understanding of why we do, why we teach quantitative approaches.

And they, some rare, some are surprised of the amount that they are getting, like why, why. But that is an expectation that needs to be set right from the start. I don't know how to be honest for me to give in, but apparently it's not for everybody. I wish students would come to see into our own degrees, especially in the geology and the science of variation, like there's no way that you will not use this.

You need to treat variation somehow. So we don't have model species all the time with zero variation. We have to be with variation. So they somehow need to understand that.

And do you think that would help with fairing? That's kind of a fair round of stats. But if the expectations change and the idea of what they're going into changes, then maybe that less fit for what they've been doing. More prepared, I'd say.

More prepared, the basis of something about is needed. There's some we all other way around it. Put a lot at some point to be able to tap into the motivations of why our students come into the science. It'd be nice to be able to ask them, what is it you're looking to get out of this?

What do you think science is? How do you imagine it? And if they say something like, I'd like to discover something that nobody else has found, if it's the sole explorers instinct that attracts people to do insane things, then you can definitely dive around and tell them, well, if you want to be empowered to do this, that's the only way you're going to do it. These are the, your climbing tools.

And once they get that, they really engage. Once they treat it as a means to your name, they really engage. And this is all right. And then cliche that, you know, if you present the quantitative sciences independent of their application, scientists stop engage.

But we've always tried to do that. You know, we always lead by presenting examples and so on. So if people, our students are disengaging, it isn't really because we presented in a dry, unapplied way. It is something else.

So there are obviously a lot of students with multivestures other than the explorers' mistakes. So it'd be nice to know what these other multivestures are. Okay. Well, I think that was great.

Thank you so much for coming to speak to us. I think that's given us both and hopefully I was listening a clear idea of maybe not the differences between pre-medicine and ethnicities and the way that they evolve from each other and overlap. And some of the ways that people should be approaching them and students can better use them. So thank you.

That's really helpful. Thank you very much for coming and speaking to us today. Thank you. Thank you for listening to the Naturally Speaking Talk channel.

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