 recording and welcome back everyone on Moodle as well. So today we will be talking about statistical testing and I will do an example on microarrays. The thing that I wanted to discuss is not so much like very advanced linear models and things. Now I want to start very, very basic. So I first want to talk a little bit about experimental design and about some of the questions that you can answer when you use microarrays. Then I'm going to talk a little bit about the normalization of microarrays and then we are going to talk about probabilities and calculating more or less p-values using things like a student's t-test and other nonparametric tests. Of course, when we're talking about different statistical tests, we should also talk about the assumptions of each of these tests. So every statistical test that exists has a certain assumption underlying it. It can be that it assumes that your data is a normal distribution or that your data is drawn from a Poisson distribution. I want to talk a little bit about correlation as well and, of course, I want to talk about multiple testing, which is very important when you are analyzing microarrays because you're not just doing a single statistical test, but microarrays, they measure all genes in the genome. So you generally want to correct for the fact that you do 20,000 or 30,000 tests which are all more or less independent tests. And then I also want to tell you for today where you can get some free microarray data. So in case you want to work on some micro data at home and want to get a high impact publication using data that other people collected, then head to free microarray data or sources where you can get free microarray data are pretty important. And you can get some data here so that you can kind of work on it yourself. All right, so the first thing that I want to say about project planning is not something that I said, but it's something that Ronald Fisher said. So, and he said that to consult the statistician after an experiment is finished is often merely to ask him to conduct a post-mortem examination. He can perhaps say what the experiment died off. So the thing that he wanted to get across is that when you are planning to do a scientific experiment, you should consult the statistician directly before you start doing anything because statistics is fundamental in the analysis. And you need to kind of have an idea on what you are going to analyze, which effect sizes you're going to expect and how many individuals or how many samples you need to have before you can say that something is significantly changed or significantly different. And I see this a lot in my line of work that people with no proper or with no real understanding of statistics, they design an experiment and then they do the experiment. Sometimes the experiment takes like a couple of months to perform and then they do their data analysis. They don't really find anything and then they come to me and then they ask, can you help me? And at this point it is too late. I once had a partner who was working on apple trees and these trees take around 10 years to grow. And then in the end, I had to tell them that they should have planted five more trees. So that they were short five trees and that's why they couldn't prove anything. And of course this is very, very sad because a lot of money is invested, a lot of time is invested. And then to tell people after 10 years of working on a project that they should have planted five more trees is not a very happy message. So as a word of caution to you guys, if you start doing a project or planning a project and you are controlling how this project is designed, talk to as many people as you can, get input, not just from your professor or from people that you work with, also try to find someone who is knowledgeable in statistics. And because it is very important beforehand to make sure that you put enough samples in the ground or that you measure enough fish or that you order enough mice to do your experiment, because in the end it's really a shame if you finish your experiment and you end up realizing that you're five plants or five animals short. So some of the questions have for experimental design. For example, if we do microarray, so the questions that we can answer using microarrays are questions like which genes are differentially expressed between healthy tissue and disease tissue. And this is of course very important in cancer research where you want to compare things like lung cancer towards like the primary tissue, towards lung tissue and want to find the differences so that you can have some idea of which genes might be worth targeting. Had questions like which genes are expressed in which tissue? For example, we as a group, we work a lot on obesity. So of course the genes that we're interested in are the genes that are active in metabolically active tissue, right? So genes which are active in muscles or genes which are active in fat tissue. And we're not so much interested in genes which are active in your toenail or genes which are active in your spinal cord. Although they might play a role, a priori you would not know if they have and you want to know for your tissue of interest which genes are expressed and the genes that are not expressed probably don't play a role. So microarrays can help you figure out in which tissue which gene is expressed. One of the other things that microarrays can also help you is to kind of figure out what a drug does. So imagine that we're testing a new drug for lung cancer and then of course we want to see if giving the drug changes the expression of genes after we gave a drug or a certain treatment. And so these are like three main questions that we can answer when we use microarrays. So what is a microarray? So a microarray is actually a very simple thing. Have we have things like spotted microarrays so they can measure the expression level of more than 20,000 genes in a single experiment. And the way that they are is you have kind of a solid surface which in many cases is just a little piece of glass. And on this piece of glass there are little pieces of DNA. And these little pieces of DNA are designed to be complementary to the DNA of the target genes that you want to measure. So it's kind of a fishing expedition. So these little spots on the glass surface or on the solid surface are called probes or oligos. And what you do is you extract RNA from your target tissue. You do reverse transcriptase to go from RNA to DNA. And then you label your DNA using fluorescent dyes. So you have, for example, a green color on healthy tissue and a red color on disease tissue. And then what you do is you take your healthy tissue and your disease tissue and you mix them on the microarray. And then what happens is that the gene sequence will flow across the glass plate and it will bind a complementary strand when it finds one on the glass plate. So, hey, this is done by hydrogen bonding. So it's a reversible binding. But once they bind, they are kind of stuck. And of course, if there is more of one color binding here, you can see a difference in color, right? If a spot turns green, you know it was active in healthy tissue. If a spot turns red, you know it was active in disease tissue. So the color of the spot tells you something about how much of a certain gene or certain fragment of a gene was inside of your sample. So the way that it kind of works is here we see the microarray. So we have CDNA probes, we do amplification and we have a target, we have a sample target and we have a control target like this disease tissue and the healthy tissue. We extract RNA, we do reverse transcription and then we label with fluorescent dyes. We then mix both samples in equal quantities. We put them on the microarray. We use a laser to excite. Actually, we use two lasers, one from the red channel and one for the green channel. In the end, we end up with two pictures and here we see a microarray more or less scanned and if we zoom into a single microarray, what we see is we see all of these little dots. Yellow dots, of course, are dots where a gene was expressed equally in healthy tissue compared to disease tissue. Red dots like this one here are sequences which are highly expressed in the disease tissue but lowly expressed in the healthy tissue and of course the green dots are the opposite. So when we get microarray data or when you have done all of this scanning and hybridization and these kinds of things, generally what you get from either a company doing this for you or when you do it yourself, you then flatten this image information into a matrix and the matrix generally has samples, so the different samples that you used in the columns and all of the genes that you've measured are in the rows and then each of these little cells just contains the expression level of a gene. So that might mean 15,000 intensity for red divided by 2,000 intensity for the green channel and so it's just a big matrix and we have two factors. So we have gene names and we have sample names and of course we have sample annotation as well because we need to know if a certain tissue was healthy tissue or disease tissue and for the genes the same thing, we need to know for example where a certain gene is located and if there's any annotation or these kinds of things. But in the end, we're only dealing with the big matrix of data and this data is just the expression levels of the individual genes in the experiment. So there's a lot of things which can go wrong when you're doing microarray. So here we see a picture of a good quality microarray and we see very clear dots but sometimes this happens. So you look at your photo and you see that there's this kind of a scratch or a fiber which is on top of the microarray. So these things happen. So there's a lot of little things that can go wrong and of course we now just have to process the data and say well all of the dots which are underneath this thing or close to this thing, we're just going to ignore them and they are going to be turned into missing values. Besides that things which happen because they are little glass plates and it's very minuscule is things like this where we see that we have no data on a certain part and this is generally an air bubble on the microarray between the slide. So we have like no information on this microarray for these probes here. So all of these things we can kind of see when we look at the pictures and then we can transform this information and say well these data points are missing. Although the original measurement of course is zero intensity because this is a bubble, we can say no it's not zero intensity, it's just that there was a bubble. So it's a missing value. So something went wrong. There's also something called the edge effect which you can very clearly see here. So here you see a microarray, you see that there's kind of a green haze across the microarray which is not too bad but here at the top you see that there's something going wrong and this is the edge effect. So it's where the samples or the fluid that you put on of the microarray it hits the edge and it kind of doesn't really hybridize very well. You can still see the dots but of course because of this kind of overlaying like reddish haze on this edge you can't use the values at this point. There's something called background haze as well. You see a little bit of background haze here but here you very clearly see. So these are four different microarrays and you can clearly see that there's something really, really weird going on with this part of the microarray that there's like an intense green background color which can have various reasons and not only that but we see kind of a scratchy thingy here as well on the bottom where it seems that there's something which is on top of the microarray. And so all of these things we need to take into account and we need to more or less do quality control on the data that we get. So we look at the pictures and then based on that we make a decision is this a value that we can still use or is it a value that we just have to ignore and just say well it was missing. So unfortunately we don't have any information like that. So the common microarray workflow kind of looks like this. So we create our oligo arrays which is part of a bioinformaticians job. So a bioinformatician is responsible for selecting which genes we want to measure and which part of which gene. So we have to create these little oligo probes that we put on the array. Then generally a biologist will acquire samples so they will go fishing and take the fish and extract RNA from the fish. There's this DNA or this RNA to DNA reverse transcription step where we use a protein which transforms RNA into DNA. Then there's an optional PCR step. We do labeling using Psi3 which is the green color and Psi5 which is the red color. Then we do hybridization and scanning and hybridization and scanning gives us something called a TIFF file which is these image files that I showed you here. So these are just TIFF images and you get that from the scanning machine. Besides that there's something, there's the next step so going from these TIFF images we can go to a cell format and this is the standard format for storing microarray data. So if you go to these three microarray data sources which I will tell you about at the end of the lecture these generally store data in a cell format. So the cell format contains the intensity and the scanning but it also contains information of the probes that were there so that you can use things like correct for GC percentage of the probe or other effects that might occur. So what do we do then? Well we take these cell files, one for each sample or one for each microarray that we ran and then we extract the expression levels and this normally goes into a TXT file so a comma separated file or a tab separated file which is just a text based file where we see the intensity value. So this has more or less the structure that I showed you before where we have samples in the columns and we have the different genes or the different probe expression levels in the rows. And then the next steps that I wanna talk about is about the data normalization step which again results in a text file. Then we have a gene expression analysis step which generally results in a text file as well and then the data interpretation step so the gene expression analysis is looking or doing statistical testing to figure out which gene is different from disease tissue to healthy tissue and then the data interpretation step is clustering for example these things together to see if a certain pathway is involved or if there's something else going on like it doesn't have to be pathway but it could also be that for example a lot of microRNAs are up-regulated or a lot of proteins that have to do with cell cycle are down-regulated and that is the data interpretation step. But the thing that we will be focusing mostly today is data normalization. So when I get my data in this matrix format how can I now transform it so that we can do statistics on it and then after we've done statistics I think there are one or two slides about how you can interpret the data or the statistical results. So why do we normalize microarray data? Well the idea behind normalization is that you want to remove technical variation from noisy data because there are many different steps when you do one of these microarray analysis and every step introduces random errors. Right, the making of the microarray might include some air bubbles. When you do your DNA to RNA step it might be that for one sample you used a little bit more transcriptase. So it just has a better efficiency so there's more DNA in this sample compared to the other samples. So all of these things are things which increase the variability in our experiment but the assumption here is that these global changes across samples are due to kind of unwanted technical variability. So the source of it is not real biological variation but it is variation which comes in because some of the microarrays were scanned on Thursday when it was really warm outside and the rest of the microarrays were done on Friday and on Friday it was really cold and there is an influence of temperature. And so the fact that all of the microarrays on Tuesday are a little bit higher expressed than the ones on Friday has nothing to do with biological variation but has to do with the temperature in the scanning room at this point. And of course at each of these steps variation is introduced and we assume that all of this variation is kind of random that there's no, that it's not coupled to a single sample or to a single gene. No, this is just variation that comes in from the outside and we want to get rid of it. And so we remove these differences. However, we always have to keep in mind that by removing these global differences by kind of normalizing our data we might remove interesting biological-driven variation. And I think that during the rest of the presentation there will be an example where we are actually removing biologically interesting data from our data set. But in general, the assumption is is that normalization is good because it gets rid of all of these things which might have gone weird or are slightly different for each sample in our analysis. So there are three major ways of normalization or standardization. So the first one is relatively easy. It is called mean centering. So because every sample, the mean, so the average expression of each microarray is a little bit different. So I do microarray one and the average expression is 2,000 intensity units. The next microarray that I do is 2,100 intensity units. The next one that I do is like 1,915 intensity units. And so each of these microarrays is a slightly different average. So mean centering just means what do we do? Well, we take all of the old measurement values and then we subtract the overall mean from the microarray and then we just say this is the new intensity value. So we just take the old value, subtract the mean which is calculated across the whole microarray and then we have our new measurement value. So now of course the mean, after doing this, the mean of every microarray will be zero and they will all have the same mean because we just subtracted the average out of the whole data set. This is a relatively good technique because you get rid of like big variances in the mean. So things like having used a little bit more DNA for one sample compared to the other sample, this will be taken care of using mean centering. There's a different way of doing it and that is also correcting for the variance, right? Because it can be that some microarray or that one of the microarrays has a variance so the lowest expressed gene has an intensity of 100, the highest expressed gene has an intensity of 16,000. And then when we look at the next microarray, the mean is very similar or slightly different but now the range is completely different. So instead of going from 100 to 16,000, this one goes from 200 to 20,000. And so the standard deviation is something that we can also more or less harmonize across the arrays. So how do we do that? Well, we can use the standard score or it's called the student's T statistic. So what do we say? We take the old measurement, we subtract the mean, so we mean centering first and then after mean centering, we divide by the standard deviation of the whole group. So this now means that when we look at the new data, right? So the new data, when we calculate the mean of the new data, the mean is zero and the standard deviation is one because we divided out the standard deviation. So now we have data which is relatively clean. Why? Because every microarray has the same mean and every microarray has the same standard deviation. Another way of doing this is called quantile normalization and this is a kind of nonparametric method where it uses the rank of the individual kind of probes to harmonize the data into different quantiles. So instead of saying, well, no, we just divide out the standard deviation, this quantile normalization is a slightly more advanced method where it looks at the different quantiles and how they are organized and then organizes so that every quantile for every array becomes the same. So instead of just dividing by the standard deviation, it uses a rank-based method to get rid of variants between different arrays. So just as an example, here we see some microarray data that we got in our group. So I'm going to explain to you what HT and GF means. So here we see HT 2001, 2010. So HT means hypothalamus. So that's just a part of the brain. So we took the brain out of the mouse and then what we did is we measured for mouse 2001. We measured the expression of all of the genes. So what we see here is we see here that the data varies a little bit. So the mean of every microarray is slightly different and that is something that we want to get rid of. What we also see is that some microarrays have a very different range because this one ranges from around two all the way up to 16. But you can see that there are some outliers in this array. So when we do quantile normalization, we go from a picture which looks like this. So we have the hypothalamus samples, the gonadophat samples. So what do we do? Well, we normalize it so that every array gets the same mean value and every microarray now has the same more or less variants or the same structure of the data, right? So this is just to get rid of some unwanted variants. However, if you look very closely, you can see that there's something going on here because you can see that the hypothalamus samples, so the HT samples, and we have also extracted GF, which is gonadophat. So when we look at the gonadophat samples, we see that when we compare these four gonadophat samples towards these four hypothalamus samples, we can see that they are all lower compared to the hypothalamus. And this makes sense because there's a lot more genes active in your brain than there are in your fat tissue. Fat tissue is relative simple tissue. There's just one thing while brains are relatively complex tissue, which does a lot of different things. So by normalizing, we are actually normalizing away some of the real biological variants, right? Because if we are interested in genes which are different to the express between gonadophat and hypothalamus, we are removing some because what we are doing is some of the genes which are actually expressed in hypothalamus get pulled down, so they will not be expressed anymore. Well, some genes in gonadophat, which were not really expressed, now get pushed up a little bit. So it seems like they are expressed. So normalization between samples, it is good because we get rid of unwanted variants which we don't want, but we run a great risk of also removing variants in which we are really interested in. And of course, R is perfect for this because R allows you to do plots and allows you to look at your data. You can look at your raw data more or less and you can look at your normalized data. And based on your raw data, you see that this might not be the best way of normalizing your data. Hey, you might want to keep these two things separate and normalize gonadophat tissue differently than hypothalamus tissue. So to still have the difference in mean after normalization. Hey, but this is something that like no one can tell you how to really handle this. Hey, but the general idea is that you want to get rid of as many technical variation as possible, but you want to keep real biological differences in your data. All right, so the thing that we saw here is real microarray data. And of course, microarray data doesn't come in a scale from zero to 15, right? We're using lasers to scan. So that means that the range is dependent on the amount of lumen or the amount of lux that we get back. And this is very different for different probes, right? Because the different dyes have a different dynamic range. The Psi5, the lowest Psi5 is around like 5000, around 6000 and the highest intensity value that you get on your array is around 16,000. So the intensity of the Psi5, so of the red channel is not as big as the one from Psi3. The green channel has a much larger dynamic range because the minimum values are in the order of like 2000, while the maximum values, the maximum intensity values are in the order of 40,000. Green is just a much more intense color than red is. So the way to deal with this is just to say, well, we're going to divide it, right? So we're going to take the intensity of the red channel and then divide it by the intensity of the green channel. And then we get a ratio. So ratios are nothing else than dividing one number by another number and in microarrays, it's very common to take the red channel and divide it by the green channel. So the ratio in this case is the mean of tissue one divided by the mean of tissue two or the mean of the first, so the colors, of course, are the mean of the red channel, which is the disease tissue divided by the mean of the green channel, which is the non-disease tissue. Of course, we now run into a different issue because when we look at ratios, if I look here where both have more or less the same intensity, and then here we can see, oh, this is one, right? So red and green are equally expressed, but then when we look at the slightly greenish dot, we see that we get a ratio of one divided by two, so half. But if we look at the slightly yellowish, so the same intensity change at the other side, now we get a value of two divided by one. And now, of course, we break linearity because going from half to one is plus 0.5, but going from one to two is actually plus one. So we now all of a sudden have a range which is much bigger in the red side of the data than it is in the green side of the data. Going from one to one eighth is minus 7 eighths, but going from one to eight is plus seven. So that's like a massive difference in range. So head to kind of circumvent that. We say, well, we don't directly look at the ratios, but we look at the log two ratios, right? Because what happens now, if we take the log two of the ratios, now we have linearity again. And that means that taking a step from zero to one is the same as taking a step from zero to minus one. So one divided by two is minus one. Two divided by one is one. And so that now means that every step is linear again. So going from equal to completely green is now minus three values. Going from equal to completely red is now positive three values. And that is the reason why when we talk about microarrays, we always have to talk about ratios. Ratios is, of course, to get rid of the difference in ability of a certain dye to generate color, right? Because the green color is just always more intense than the red color. So that's why we do a ratio. And then to make it linear again, so to have a linear relationship, we take a log two ratio. I hope that's clear. I always found this a little bit difficult to explain and to understand. But in the end, it's nothing more than just taking your data and doing two transformations on it. So to have data, which now is unaffected by how well your dye functions. And also, just using the ratio is not good enough because we also want our data to be linear. So if we say plus one, then the minus one, so that the step from here to there should be equally big as the step from here to here, right? OK, I hope that's clear. All right, so now we have microarray data. We know that we have the different intensity colors. We can normalize it. We can take the values, the intensity values, transform them to log twos. And now, what do we want to do? Well, after we've done all of this, we want to use statistics, right? Because we want to show that there's a real difference in the data. And that the data is not just by chance. Tester-Sardis has a question. Do you also have that shirt? No, no, no, no, no. I think it's the wrong way around. If I would have a shirt like this, I would probably want to have it say my F statistics is bigger than your F statistic. The smaller than is a little bit weird, I think. But has statistics are there to determine which things are significant? Oh, god. Always be smart, right? Don't do P values. Do F statistics. They go in the opposite direction. But what is significant? Well, significant just means nothing more than that it is unlikely to have occurred by chance. And that's it. That's what a P value represents. It represents how likely it is that this is due to chance or not. And for the rest, the P value doesn't really mean anything. So a lot of people think that a P equals 1 times 10 to the minus 16 is a better P value than 1 times 10 to the minus 14. But in this range that we're talking about, like 10 to the minus 14 versus 10 to the minus 16, we're talking about chances which are so small that they're actually more or less the same thing, right? Because the chances of it occurring by chance are just so much smaller than that you want. And so a result is called statistically significant if it has been predicted as unlikely to have occurred by chance alone. So how does this work? So here you see the probability density function. And here we have the set of all possible results. So what does a P value represent? Well, we have the most likely observations in the middle. So we now have observed the data point which is on this side of the distribution. And that means that it is near the very unlikely observations. And then the distance of it towards the really unlikely observation, this is what is called the P value. So it's kind of the area under the curve of this kind of probability density function. So you have to kind of know which probability density function. And then you can use your data point to transform your observed measurement based on the kind of observations which you could have observed. And then had the extremity of this point towards the rest of the data that you have gives you your P value. So probability is a measurement of deviation from random. So that's what it's called a probability value. And in biology, we generally say that we reject the null hypothesis. So the null hypothesis, it occurs just because of chance. Because if I measure something, then it's not always exactly the same thing. So the null hypothesis is rejected if the probability is below a predefined significance level. And we call this the alpha level. So in biology, we agree that the alpha level of 0.05 is good enough. The smaller the P value, the larger the significance. But it doesn't change anything from the results. Because being significant with a P value of 1 times 10 minus 3 versus being significant 1 times 10 minus 4 doesn't mean anything. It just means that both times you have observed something which is unlikely to have occurred by chance. But the smaller the P value, the larger the significance, I think that everyone knows this. Just a little word about the alpha level, right? The alpha level here is a little bit weird in biology. Because if you come from a physics side, right, and you would want to show that gravity exists, right? So you would do an experiment, and you would say, well, I have something which weighs something, and now I drop it. And then it falls down. And then I write down, OK, so I dropped it, and it fell down. And I'm going to then repeat my experiment, right? Because I need to collect observations. So if I drop this thing 20 times and 19 out of 20 times, it falls down to the ground. And one time it just flies off into space, then a biologist would still conclude gravity exists. And that is, of course, very weird. Because if one out of 20 observations that you have would mean that something flies just up into space, that that's not, like for physics, this alpha value is not good enough, right? So for physics, when we want to conclude that gravity is real, we want to have not one out of 19 observations flying into space, we want to have almost zero observations flying into space. Even if we would drop something 1,000 times, the thing flying into space is not going to be good enough, right? Even if we observe once that it flies into space, we would have already concluded, like, that's very strange. That shouldn't happen, right? So the alpha value in biology is relatively high. Many other fields have much more restrictive p-values, like p-values in the order of 1 times 10 minus 6, which is a very basic p-value in physics. And every field has their own kind of when is it significantly different from random? And biology, in that sense, is on the lower side. So we, as a biologist, quite quickly say that this is unlikely to have occurred by chance. For physicists, they are much, much stricter. So statistics are used to determine if significant effects, right? So we are talking about the likelihood that the differences observed are real and unlikely to have occurred by chance alone. So a likelihood is kind of related to probability, but it is not the same. Probability is the measurement of deviation from random. And likelihood is that the differences that you observed are real and unlikely to have occurred by chance. So there's a difference between probability and likelihood. And probability is coming always or likelihood always comes with a confidence interval. Probability itself doesn't have a confidence interval. So when someone says that they have a p-value plus minus something, Matteo Adon, thank you for following. Can people hear the sound for following or not? I don't think there was a sound for you guys. What's going on here? No. All right, so what's going on with my desktop audio then? Oh, it's on my speakers, and I want to have it on my headset. All right, so the next person following should generate a sound. So if anyone could just quickly unfollow and refollow, then they should be able to test it. Now don't all of you guys go following and unfollowing and just making a sound. So there's a difference between probability and likelihood. Probability is, is it different from random? And likelihood is, are the differences that we observed big enough so to say that there is a real difference? And likelihood always comes with a confidence interval, while probability is just a single number. It's just a fixed number. So there is a difference. And remember that there is a difference. All right, so the first statistical test that I want to talk to you about is the student's t-test. It's the most commonly used statistical test in the world, and it compares the difference between two groups. So it is a likelihood test, not a probability test. So the thing that is here, the main thing of the student's t-test is that it requires data, input data, to be normally distributed. In R, it is just the t.test function. So the t.test function in R tells you, if Skorita, thank you for following. This time you hear it, right? I think so. I hope people hear it. I'm not seeing any feedback. Give me some feedback, people. I'm deaf now. Oh, is it that loud? All right. I'm sorry. I will put it slightly down. I heard. OK. No, just kidding. OK, good. Do I still have the? I could have just tested it like this. So. All right, so t-test come in two different ways, or actually in multiple different ways. But the t-test is there's two types of t-test. So you can do a single-sided t-test where your hypothesis is that one group is smaller than a certain group, or that one group is larger than a certain group. But if you don't have any a priori assumptions on what is happening with the groups or with the effect, then you have to do a two-sided t-test. So as an example here, we are studying the Berlin Fettmaus. And it's called Berlin Fettmaus, because the mice generally tend to be bigger than the reference mouse strain. So when we test, if our mice are really bigger than the reference mouse strain, the first time that we tested this, we had to use a two-sided test. Why? Because we observe that they are slightly fatter than the other ones, but we have no proof yet. We have no prior assumption that they are fatter than the normal mouse. So the first time that we did these measurements, we used a two-sided t-test, because we just wanted to show that the group of fat mice was different from the group that was not fat. So after this test succeeded and we proved that indeed these two groups of mice are not equal, then we use a single-sided t-test to prove that the Berlin Fettmaus are really heavier than the standard reference mouse strain. So this is a two-step procedure. And you can only use a single-sided test when you have a priori data. So you have already previous results which show that one group is bigger than the other one or that one group is smaller than the other one. If you don't have prior information, then you always have to do a two-sided t-test. All right, I'm going to five minutes. Been recording for 43 live for OK. I'm going to do another slide and then we're going to take a quick break. So the most common test that you would want to do is the one-sample t-test. So it tests the hypothesis that the measurements that you did is equal to a specific value. So imagine that I have measured 5,000 humans. And now my assumption is that the 5,000 humans that I've measured are, on average, 1 meter, 65 centimeters. That's the thing that I want to know. Then we do a one-sample t-test. So I don't have two groups. I just have one group of measurements. And I have a specified value which I think is the real mean. So what can I then do? And so then the t-test is more or less this. So I have the overall mean of my sample minus the mean that I expected to be divided by S, which is the standard deviation of the sample, divided by the square root of n, which is the square root of the number of samples that I have. And then I get a t-value. And again, I can make this probability density function of the t-distribution. So when you were growing up in high school like me, then you were not allowed to use a graphical calculator, but you had these little books. And in these books, you could look up the critical value for the t-distribution. I don't think that they use that anymore. I think everyone nowadays is allowed to use. But at these books, they just had this probability density function. And what it would tell you is you could look up, and it would tell you that, well, the lowest 5% is at minus 1. The lowest 10% is at minus 0.5. The middle is at 0. And the top 10% is at 0.5 and so on. And then you would do this test. You would get your t-value. Then you would look this t-value up in your table. And then in the table, you would see, OK, so my t-value was 0.6. That means that my p-value, so the distribution, the point in the distribution where this value occurs, is at a p-value of 0.1 something or a significant p-value. But this is the most easy test that you can do. You don't have to measure two groups of individuals or two groups of experimental data. You just measure one group of experimental data. And then you test the hypothesis that the population mean is equal to a specific value. So for example, human height, you assume that all humans on average are like 1 meter 65. And then you can just use this t-statistic to calculate if this is really true. All right. Then just look at the slide quickly. I'm going to stop the recording for like 10 minutes. So I will be back at like 3.10. And I hope you guys enjoy the next round of slides. So I will.