Okay, in this video, I'm going to demonstrate and analysis in which we transform our data. I've already created another video where I go where I discuss in depth how she perform an analysis for one factor general, general linear model. And in this video we're going to museum one factor general linear model as well. I want to say that in this video and transformation, I'm going to focus specifically on the aspects that have to do with transformation. Which means that I'm going to skip a lot of the discussion that I made in previous videos when I was talking about how to perform analyses generally. So when I skip those bits in this video, it's not because I think they run important. It's just because those bits don't pertain to this specific topic that we're interested in, which is how we transform data and how does that influence our interpretation? Okay? So I just want to make that clear at the beginning to know so you know what I'm really trying to explain. So if I skip something that you've seen me talk about in another video, it's not because that's an important it just because it doesn't have to do directly with data transformation. Okay, we're going to use a data set that's already built into R. It's cold, the InsectSprays data set. Okay. I'll just show you what it looks like here. Okay? So there's 22 data points and you can see that we have two columns, count and spray. Okay, within spray, there are a, B, C, D, E, F, six groups. Okay. Or another way getting that is just like this, okay? So you can see we have six groups. We have 12 data points in each. And when this experiment was being performed, I assume since I wasn't the one who performed it, I assume that this experiment was performed as a way to determine whether a different types of InsectSprays would influence the number of insects that we find on plants. So again, this experiment, what they presumably did is they would have taken 12 plants and randomly allocated, or sorry, they would've taken 72 plants and randomly allocated 12 of them to each of our six different sprays. And then for each plant, they counted the number of insects that they found on the plant afterward, after it had been sprayed. Okay. So those are the kinds of data we're dealing with. We have count in this column and spray in this column. Normally our first stage of our analysis would be to think carefully about our experimental design to determine whether or not these data were appropriate for a one-factor general linear model. Skipping that step, and we're just going to assume that they are. Again, that's because our focus is on transformation. Okay? The second step after determining whether or not our approach to analyzing data is generally suitable is to plot our data. And so we're going to plot our data, just did a box plot. We're not going to bother putting on the individual data points. Again, we're streamlining. And so to create a box plot, we want to first list our dependent variable. Our hypothesis is that the sprays will differ in their ability to affect the number of insects that are found on plants. And so that implies that the number of insects we have on plants will depend on the sprayed that's used. So that implies, and the count is our dependent variable. Since I'm actually plotting this, you might guess, are there things we learned from the plot that have to do with data transformation? And there we go. Alright, so there's just a couple of things I want to highlight here. Again, we're focusing on the aspects that have to do with data transformation. When I look at this, the first thing that jumps out at me is that the breadth of these boxes is highly variable. So four sprays, C, D, and E are box plots are relatively narrow. Whereas for Sprays a, B, an F. The box plots tend to be relatively wide open. You might notice that it's the sprays that have a high average count number. Those also have high variance or so it appears. Okay, whereas the sprays that gave low average count numbers also tend to have lower variance or so it appears. So based on this, we would expect, or we would predict that our data will not meet the assumption of equal variance. Whether or not the data meet the assumption of normality. That's harder to tell because the box plots generally look relatively symmetrical. This one certainly looks nice and symmetrical. This one's okay. The meeting is a little bit high. This one, the meetings a little bit low. Here. This also looks generally symmetrical except for that unusual point. The meetings a little bit low. Here we find the opposite scenario, the medians a bit high. So it's a, it's a bit hard to predict what we'd find in terms of normality, but certainly we would predict there's a problem with variance. Okay? So said we're not going to go through the stage of trying to predict what the differences will be among the R groups. Because again, that's, that's not the focus of our analysis. We're just going to jump at this point into analyzing the data. So we're going to perform a general linear model. Will see the output in this object, insect LM. And so we'll say linear model. We give our dependent variable followed by independent variable, sorry. And then we tell our dataset. And we will run this. And now we want to look at our residuals to see whether or not our data meet the assumptions. So let's just plot that. That is worrisome. Remember there's two plots that allow us to assess the assumption of equal variance. This is the first one. And what I want you to see is that, well, first of all, we have six columns of residuals. Those corresponds to our six different treatments. So remember we had plots a, B, an F that had the high means, and it also tended to have higher variance. Well, that's what we see here. We have three plots. It presumably referred to AB at F. I'm not sure which is which here, but AB and f will be included here. And you can see they have more variation there residuals. Then we see down for the remaining spray types where there seems to be less variation. So this is worrisome. This does suggest that we have problems with variance. Normality is actually not a complete disaster. We have a lot of points off the dotted line here. But this is, this is certainly not the end of the world. Okay? Nothing's the end of the world. It comes to data analysis. World's not going to end. Just being dramatic for fun. This is our second plot that gives us a stronger sense of whether or not the data meet the assumptions of equal variance. And you can see here we have the strong general trend for the red line increase as we go from left to right. Okay? And that is worrisome. Okay? So this generally tells us that the data likely violate the assumption of equal variance. The points are generally evenly distributed around the red line. And that's one thing to look for. And so that's, that's nice. But the fact that we have this strong general trend in the red line, that's, that's worrisome. Okay, So this figure and, or this plot and this plot here, both indicate to us that we should worry about equal variance as we predicted. We're not going to pay attention this mascot for outliers. Okay, so what do we do? Well, I said I'm going to show you how to transform data. So that's what we're going to do. We're going to try transforming our data in order to see whether or not that transformation helps us or helps our data meet the assumptions of our one-factor general linear model. So I'm just going to copy that, pop it down there. Let's first try log transformation. So I'm just going to change the name of my objects that saves the output. And what I'm gonna do is I'm going to take this variable here. Remember this, this points to a column in the dataset, InsectSprays. So this variable here, so count, this refers to the column in the DataFrame. Insectsprays a contains all of our accounts. What we'd like to do is you would like to log transform all of those data that are in this column. We could do that by going back to the original spreadsheet and created a new column and then log transforming those data and saving them in that column and that new column. That's cumbersome. And it has some other disadvantages as well. So what we're going to do instead is something much more elegant. We're going to log transform our data directly in the lm function like this. So what this does, what we're doing here is we're saying OK, Oh, OK. We want you to take all of the data in the column count and take a log of all of those data. And then those log transformed data, those are the data that will be used in our general linear model. Okay? So that's the logic here. That's how that works. When we include or when we log transform or when we just transform our data. Generally, in the lm function has an additional benefit, which is that the fact the data were log transformed, it ends up being saved and the information in our output object. And that information could get passed on to functions that we're going to use later on, particularly when we use the EM Means function. And at that stage we can make use of the fact that the data were log transformed in order to increase our options for how we want to analyze the data. The main point here is, I'm just saying that if you wants to transform your data, this is the approach that I would suggest. Just transform your data directly in your general linear model. Let's run this. Okay? So down here that says it, we're getting an error message. And specifically it says that when we try to log transform the data, we're either getting missing values or infinite values for our y variable. Why is that? Let's just look up here. Look at this data point. We have a count of 0. Can you take the log of 0? No, the log of 0 is negative infinity. So when we tried to take the log of this data point That caused us to get this error message. So how can we fix that? Well, what we're gonna do is we're just going to add one to all of our data points before we take the log. And we can do that right there. Okay, So now what we're doing is we're taking the data in our column count. We're adding one to everything. And then we're taking the log of all of our data which have all had one added to them. And what this will do is, I mean, we're adding one to all of our data. This is particularly helpful for that instance where we had account to 0 because that 0 will change into one. And everything else waltzes, increments up by one as well. But when we change that 0 to a one and we can now take the log of it, because the log of one is equal to 0. So by doing this, we're creating data where we can get around situation that causes error message. So let's try this again. Okay? Now, no error message this time, so that's good. Let's plot our residuals. That's looking much better, much better than we saw before. There is an ever so slight trend, I think, where you can see that we kind of have reversed the issue we saw earlier, early, we had tight data, so very little variation and residuals at this end, and much more spread out data on this end. With the log transformation. We've kind of reverse that a little bit. We're now it looks like we have a little bit more variation at this end on the left that may do at this end on the right. But I would say that overall, my hunches, this looks better than we saw before. Normality. Looks beautiful, That's really nice. The points really lie nicely along that line. What about our other plot for checking equal variance? This is okay. I think you can see how this red line is not perfectly flat as we would like. And it goes up a little bit here, and I think it goes up here on this left-hand side because of what we spotted just to two pictures back, where you end up having what looks like a little bit more variation or variance here on the left, here on the right. So overall, I think that this is much improved over the first situation where we did not log transform our data. It may not be absolutely ideal, but it's certainly better. Okay? And again, ignore this last situation or this last plot. Often when we try transforming our data, we want to try more than one transformation just to see whether or not one transformation might work better than another. So we're going to try another approach. Try square root transforming our data as well. So I've done something dangerous here where I've copied and pasted. That's always dangerous because you might forget to change something. Okay? So if you copy and paste, be very careful to make sure you've made all the necessary changes. So let's just walk through this, whoops. A change log square root. I've taken away the needs to add one because unlike log transformation or unlike taking a log of something, we can take the square root is 0. So there's no need to add one to our accounts here. And we said that we're going to see this output in the out put object insect LM square root. And we're going to plot, we've changed the name of the object. We're using a plot of residuals. Said it looks okay. Let's run this. How does this look? That I think looks better. So here I think that the, the variation that we have in the left hand side is more similar to the variation that we have on the right. So the spread amongst our residuals on the left is more comparable to the spread that we saw, that we see on the right-hand side of this plot. I think this looks better than this one which we had with the log transformation. It's ever so slightly better. What about normality? Normality is beautiful. So those points really fall nicely along the line. So normality is really nice. What about our second plot for assessing equal variance? This I also think is pretty good. I'd say it's overall, it's probably a little bit more flat than we saw with the log transformation. Let's just compare. So this one tends to slope up ever so slightly towards the left. So as a general weak tendency to go that way, I'm sorry, There's a week tendency for the red line to increases. We go to the left. Here. Overall, it's, I'd say more constant. I wouldn't say there's a big difference though between these two approaches. Here. With the square root transformation overall, I'd say the red line is, is probably the best who've seen so far. And you can see that the data are nicely or the residuals are relatively nicely centered around the red line. Okay? So overall, I would say that log transformation or square root transformation would probably be adequate. If there is, if I was forced to make a choice that I might say the square root transformation is ever so slightly better. Having said that, to continue our analysis, I'm going to use a log transformation. And that's because there's a feature of EM means that's particularly useful with a log transformation. And so we're going to use the output from the log transformed data as we go forward, just because it gives us the opportunity to highlight this nice feature that I want you to see in an IEP means, okay. All right, so we've checked our residuals, were generally happy that I the retransmission should be. Okay. So let's now check our results. So we're going to say summary. In, say, let's look at it for log. Okay? All I really want you to see here is that the overall format of the data or format of the output is exactly as we've seen in our previous video. So for example, in our analysis, the chick weight data, the one thing that I want you to notice is that all of the numbers here for the standard, sorry, for our estimates of the coefficients and for our standard errors. These are all based upon log transformed data. Okay, we can see that if a compare back to our box plot, because Let's focus on this top line. Remember that the intercept in a one-factor general linear model is going to give us information about the mean value of one of the groups which is chosen as a reference. Here. It's spray letter or spray type a that's been chosen as our reference point or as our intercept. So this estimate here refers the mean value of the data in spray type a. Can see here that this suggests that the estimator, the mean values around 2.7. If you look at the boxplot of the original data that were not log transformed, we would expect the mean value to be somewhere around 15. Okay? So really all I wanted to highlight here is that when you log transform the data, your estimates and their standard errors that you get in your summary output will also be calculated based upon those log transformed data. We can get our p-value from this output though, we can see our p-value is nice and small. So this means that we have strong evidence that there are differences among the average values for, among our various groups. Okay, So with that in mind, we can move on and do our post hoc tests. And we're going to use the EM means library to do this. And we're going to start out. So in our previous videos, we said that when we use EM means to conduct our post hoc tests, they're kind of three stages. The first is we use the function EN means in order to calculate the mean values and standard errors for each of our groups. The second step is then to use the pairs function to compare among those various groups. And then the third stage, or in a third stage we use week, sorry, we calculate confidence intervals for our effect sizes. And we're just going to work through that process just here. Okay? So I'm going to create a new object. I'm going to say the output from our EM Means function in insect EM Means. And my mind is just gone blank, right? Reason the EM Means function. What this function first needs, it needs, it needs the output from our general linear model, which we called insect linear model log. So this is the output based on the log transformed data. The next thing we need to tell EM means it. Oops, the next thing we need to tell EM means is the name of a column that contains the groups for which we want to calculate the averages. And that column was cold spray. So with this information, we can calculate our EM Means. And let's see, we get, so this kind of output will seem familiar from our previous videos. What you can see here is we've got an indication of which spray we're looking at. And then for each spray we have an estimate of the mean value. For each spray. We also have an estimated standard error for each of those means. And then we also have confidence intervals for each of these means are estimated for our various spray types. Okay? So all that's pretty straightforward. The thing that I really want to draw your attention to though, is that EM means is really kind. And it gives us this beautiful message reminding us that these results are given to us on the transformed scale. So these means, you can see they're much smaller than we would expect based on the raw original data. So for the original insect counts, we would expect to have means of around 1550, maybe something higher than that and we don't see anything. We don't see any mean guys that come anywhere close to that in this output from EM means. So all of these, all of this output was based on calculations. Performed upon log transformed data. That's really what I wanted you to see there. That was the main point. Now, we're going to use these mean values and their standard errors to conduct pairwise comparisons. Okay? So to do that, insect pairs and we'll use the pairs function. And here we go. So now I want you to see just type here. Now we get a whole lot of output because we had a whole lot of treatments to compare. And what I want you to see here is that this output is very similar to what we've seen before. So we have a contrast which tells us which means are being compared. And here we're comparing a and B. So the mean of a against the mean of B. And you can see that this contrast comes in the form of subtracting the mean value from B, from the mean value of a. And this estimate here refers to the difference between the mean of B and the mean of eight. And you can see it's negative and that's going to be because the mean of B, let me slightly larger the mean of a. And we can see that here, the mean of B is 2.7, whereas the mean of a is 2.69. Okay? Here we have a stand, whoops. Here we have the standard error for this difference. And here we have some statistics. So t-ratio p-value that test whether or not this estimate is likely different from 0. In other words, this PVA and this t-ratio basically amounts to a test of whether or not the mean of b is different from the mean of a. All of that's familiar. Again, the thing that I really want to highlight here is that all of these results that are given here are given on the log scale, not the original scale. Notice that EM means refers to the original scale as the response scale. We're going to make use of that term and a little bit. Okay, so now the last thing to do up here is we're going to create our confidence intervals or calculate confidence intervals. Whoops. Oh my goodness, lots of typos. Alright, so let's just calculate this. And this is pretty much exactly the same thing. We're getting confidence intervals. But again, all of this output is based on calculations using the log transformed data. Hey, these are not on the response scale, not on the original scale of the data. Okay? Having said that title, Why Just hit that? So there's two things, two main points I want you to get from what I've just shown you. The first is to note that we can obtain all these different output on the log transform scale. And that's it. We've got. The other thing to note is that we should report this output somewhere in a publication. So if we were to try and publish these results, so they'd want to include these log transformed results somewhere in our manuscript. It could be in supplemental materials, it doesn't matter. We need to make these data available to our readers. And that's because enlarge part these types output maybe particularly useful for people who want to conduct meta-analyses. Case we want to give them the original data on the scale at which the analysis was performed. Okay. So those are the two points that I wanted you to get from this first look at EM means. First of all, we can get all the output on the log scale, but also highlighting that he wants to report all of these results. When we did that, we want to be crystal clear about what scale these are reported on and we want to be crystal clear that these data are all reported on the log scale. Now, I don't know about you. I find it kind of hard to think on the log scale. If I see that the estimate for the difference between a and B, or sorry, a and C on the log scale is 1.7. I find it difficult to think about exactly what this means because my brain doesn't work very well in the log scale. So it would be really nice if we could have some way to represent these results on a scale that's closer to the original scale, the data. So that's what I'm going to show you next. Okay. I'm going to show you now how to get results from EM means that are on a scale that is more consistent with the original scale. Notice I'm not saying that it's exactly the same as the original scale because what I'm gonna show you is not exactly the same, but we'll get to that in a moment. Okay? But I'm gonna show you a method to get results like these that are on a scale much closer to the original scale. And to do that, all we have to do is you want to say type equals response. And we do that in our EM Means function. Okay? So when we do that, when we add type equals response, that is going to cause EN means to begin to back transform our data to make the numbers that we see much more similar. The numbers that we expect based on the original data. I'm just going to change this to just add dot RESP so that we know that we're dealing with output or the response scale. Okay? So what does this output look like? Okay, here is the same general form of the output, but now look at this comment. Now it says that the intervals, so these values are back transformed from this original scale. Okay? That's useful because now you can see these numbers are much closer to what we expect based on the box plots. And so it becomes much easier to think about results on this scale. Okay? And you can see here this value here for a, this might represent the mean value for our counts for spray type a. Okay. Now I'm being a little bit disingenuous when I say that this is a mean value for the counts in spray type a, but it's not the kind of mean value that we're normally used to. Thereof, a whole variety of ways of calculating a mean. What we're used to when we calculate the mean by adding numbers, adding up some numbers and dividing by the number of numbers we add it up. We call that type of mean an arithmetic mean. That's the type of mean that we're used to thinking about. Okay? This mean here is called a geometric mean, okay, when we back transform data using this approach that we've used here by a this type equals response. The means that we get back are not arithmetic means. In other words, they are not the exact values that we would get if we had just gone into a calculator and calculated the mean of our original data by hand by adding up all the values within a treatment and then dividing by the number of data points. Okay? So in general, when we use this approach to back transform our data in EM means, the results that we get here, or what's called generalized means. And there are a number of different types of generalized means depending on the form of back transformation. When we're back transforming log transformed data than the type of generalized mean that we have here as a specific name. And it's called a geometric mean. Okay? So this value here is the geometric mean of the number of insects that were found on plants in spray type a. And this is the geometric mean for spray type B, et cetera. Okay. So what do we have here? We have data or we have output that is much closer to the original scale. And that helps us because it helps us to wrap our heads around the results a little bit more easily. But we need to keep in mind that these means are not arithmetic means. And so when we report them, or if we were to report them in a manuscript, we need to be clear about what type of means these are. You would say that these are generalized means and these confidence intervals are based on generalized means. As is the standard error. And if you're dealing with log transformed data to give you more specific and say that these are geometric means, okay? Not arithmetic means. It's a huge and important point, which is why I've been hammering on it so hard. Okay, so that's the first bit that we get here. Let's see what happens when we run pairs, because through our pairwise comparisons, say insects pairs response. So pairs and insects EM Means response. That's what we want. So let's just run this. And now let's look at the output of this insect pairs response. Okay? Let's start at the bottom. Okay? First thing to note is we no longer have our nice little message saying that these results are on the transform scale, because they're no longer on the transform scale. These results are much closer to the original scale. Okay? The other thing to note, however, is that the tests that were performed to generate these p-values, they were performed on the log scale. And that was done because that's the right thing to do. Remember, we log transform the data in the first place to make sure that our tests give us reliable output. Okay? And so because it's necessary to log transform our data in order to get reliable conclusions. It only makes sense. These p-values that we calculate here will be based upon the log transformed data. Because those are the data that are appropriate for calculating p-values. Okay? So these the data or the information in this column, p-values. This is based on log transform data because that's what's appropriate for conducting our comparisons. The other bits though, where we have contrast ratio and standard error, these are based on back transformed data. Okay? So the output here is a mixture of output based on log transformed data and data that has been backed transformed. So let's talk about what this output means. We should notice immediately that something's different here. So previously, Let's go back up to our previous bit here. First time we ran, pairs are contrast was expressed in terms of a difference. So take you to one mean and subtracting that from the other. Now he has something totally different. Now a contrast comes in the form of a ratio, or we have one mean divided by the other. Why is that? I'm going to explain that, okay. Back up here. When we're dealing with log transformed data, we can generally think of what we're doing is we say we have the mean of a minus the mean of B8 minus me to be. So you have a minus b. But remember we're dealing with log transformed data. So I'm going to say log of b, oops, excuse me. And log of a. Ok. So this can be generally how we might think about this kind of contrast. I want you to think way back to high school. In high school I'm going to assume that you learned log rules. That's where I learned about log rules. And when I was in high school, I learned that this expression is mathematically equivalent to log of a over a, a over B. Because of rules for logarithms, what we have on the left here is equivalent to this on the right. You might be starting to see the connection between these results now. And these results. Because these are equivalent. Em Means can take advantage of this form of the contrast. Because with this form of the contrast, I can basically just take this ratio or take this value and then back transform or do the inverse log of this, which makes that log to go away. And what we're left with now is the ratio a over b. Okay? That's why when we back transform log transformed data. Getting contrasts in the form of a ratio. And that is why I said earlier I wants to go on with our analyses using the log transformed data instead of the square root transformation. I wanted to have an opportunity to point this out to you. Okay? So now because our contrast is given in form, in the form of a ratio, we interpret all of this slightly differently. Okay? So this value here, you can see it says ratio instead of estimate, which is what we had up here with the non back transformed data. So this value here represents the ratio of a over b, and you can see it's relatively close to one. So if this were exactly equal to one, that would mean that the mean of a was equal to the mean of B. Because if you take one thing and divide it by itself, that equals one. Case, the fact that this ratio is close to one implies that the average of a is pretty close to the average of B. This standard error, this refers to the standard error of this ratio estimate, okay? And these test statistics, they essentially can be interpreted as tests of whether or not this ratio is different from one. Okay? And you can see that our p value is very large here, which would suggest we have no reason to believe that this ratio is different from one. What about this situation here, this next contrast? So here we have a divided by C, we have a ratio of about 5.7. What that tells us is that the average insect count and a, is about 5.7 times greater than the average insect count in C. So that means, okay, and we have a standard error for that ratio. And this p-value here, which this p-value here tells us, we have strong evidence to believe that this ratio is different from one, which essentially tells us that the mean of a is different from the mean of c. Okay? So that's how we interpret the output for our pairwise comparisons when we're dealing with back transforms data like this. Now the last thing to see is it, of course we can calculate current whoops, confidence intervals. And so we'll just do that here. And once again, it means it's been very nice to us here in telling us that the intervals are back transformed on the log scale. Okay? So it'd be a good idea to report these kinds of results in a manuscript. Which is the hint, there's anything else that I want to say about that. I don't think there is. Remember that when you i'll I'll I'll admit one thing that I'm unsure about, and that is I'm unsure about whether or not this ratio and the standard error would also be considered a generalized values as well. My hunches they are, but I can't say for sure. So if you were to report these values, add a manuscript, just be absolutely clear how will they were obtained so that readers can determine exactly what the mean. That's what I was pausing to think about for a moment. Okay. There's one other points that I want to make about this before we finish this part of the video. And that is that although I've used this batch transformation approach for the data in this experiment, I want to point out that I actually wouldn't recommend this approach for this particular dataset. Why is that? That's because when we conducted our model, we hads to add one to all of our data. And that influence of that, of that value of one that was added. That influence is present in this ratio. And it needs values that we get here and will also be present in the EM means, our geometric means. Okay? How do I know that? I know that because if I did a little experiment where I just repeated the analysis here, but instead of adding one, I added 15. And when I did that, I got very different estimates for the ratios that are listed here. Okay. And you'd also get different estimates for the responses and things as well. I believe I didn't actually look there. The first place I thought, look, we look at the ratios. So that's important. The fact that these values that had been back transformed include this plus one that we've included is another reason to be concerned. Or sorry, I should say that that way. Because all of these output based on the back transform to include the plus 1. That again makes it a little bit more difficult to interpret these results in terms of the original data. The data before we did any form of transformation. And so for that reason, I would suggest that you, that if you have to add one to some counts data before you, or if you have to add one to some data before we take a log, then I would recommend not back transforming your data in a way that you did. However, if you could log transform your data just like this, don't have to add anything on. Then. Everything that I've shown you here should be absolutely just fine. I hope that's clear. There's one last thing that I want to point out to you. And that is, I think it might be or my guess is it'll be a bit unsatisfying for back transformation to give you generalized means when ideally what we'd like to get is additive means. I just want to show you where to go to learn how to do that. So this website are Seek.org saves my life and a regular basis. This basically is Google specifically for our. So in this little search box, we can do searches, Google searches. But where the returns will specifically relates to our SI, want to learn about back transforming an EM means. So I'm going to say EM Means and back transforming. Okay? We'll just go to our first link that we're given here. And this webpage is one of many webpages that are associated with EM means. Em Means has exceptional documentation to help you learn how to use it. You can click on here to get a sense of all the vignette topics are all the information about EM means, but we're not going to do that. What I'd like you to point you to is this last option on this page here where it talks about bias adjustment. And it's here that you can learn about how to obtain back transformed data on the additive scale. Okay? I'm not going to go into that, but this is where you can go to learn more. I'm going to stop the video there. So I hope it's been helpful. And I'll say, thank you very much.