Okay, in this video, I'm just going to make a few closing comments to summarize our discussion of analyses that have one covariate or one continuous independent variable. In this context, I'm going to issue one warning before summarizing the things you talked about so far. And that warning has to do with extrapolation. One of the really nice features of regression is that it allows us to make predictions about our data. So what I mean by that? Here's an example that we're going to talk about more generally to moments. But for, for the time being, I want you to just focus on the area of this figure where we actually have data. So we have data ranging from about this point. So it looks like from the year 19, well, 1900 up to 2012, roughly. Okay. You can see that we do not have, even though those those data span a 112 years, we do not actually have a 112 data points. Okay? So there are gaps in our data. There's no data, for example, for this blue line represents males. There's no data between this point and that 0.1 of the nice features about regression is that because we're fitting a line to our data, we can use that line to predict what the results would be for these intermediate values where we don't actually, we can, while I was going to say we can predict for these intermediate values, but we can actually make predictions for what we expect. What we expect our y value to be anywhere here within the range of our data. Okay, and that's a really nice feature of regression to be able to make those kinds of predictions about what a y variable would be based for a particular x value that we're interested in. The point of this slide, however, it is that we really don't want to take that process too far. And this example, which is taken from literature, really exemplifies this. But he's, you can see from this, this title is saying women sprinters are closing the gap on mend and May 1 day overtake them. And so these are the data from females in red and use the data from males in blue. And you can say the winning time. And so in general, the times to run a race are, have so far generally been longer for females. That's why there they lie at this higher level of the y-axis than there are for males. But you can also see that the amount of time that it takes to win a race or the winning time. It has been decreasing over the years as people have been getting faster and faster, presumably because of better training. And, but what this notes is that the slope seems to be different for females and males. And what they've done is they've said, well, let's extrapolate. And they can follow these dotted lines and you can see that at some point in the future, they expect the females to be finishing races faster than males because the predicted line lies, the predicted line for females lies beneath predicted line for males. I just want to point out we can't do that. When we fit a model. When we are fitting a line through our data, we only have information about the shape of that line. Within the range of data that we actually have. We do not know what form the data would take if we we're to move outside of the range of data that we actually have. Ok? So if I were to predict what would actually happen in the future for to predict these lines, I would expect that these lines might continue to decrease. But at some point they're going to asymptote because there will be some physical limit to the ability for humans to run at a particular speed. Okay? At some point, physics is going to takeover. And so although currently our data can be modeled nicely by a straight line, at some point, we expect that to no longer be true. So extrapolating, taking this information for a range of data where we have a straight line. And then imagining that straight line continues for infinity along our X axis is not a wise thing to do. I mean, at an extreme, if we were, let, let's just consider what would happen if we went far enough. Since these slopes are negative. Eventually, if we were to look far enough into the future, this extrapolation procedure would predict that eventually we would be, we would have negative winning times. So essentially be winning races. I don't know by using a time machine, by winning them even before, by winning the race, even before the race started. That's clearly ridiculous. Ok, so the main point that I'm making here is be careful. If you're making predictions, you should really only just be making predictions within the range of your data. I would argue that also holds true when making conclusions about your Y intercept. If your data do not actually reach 0 on the x axis. Then we can't say for certain exactly how the data would be, what, exactly how the data would behave near that value of 0. And so if we're fitting a straight line to our data, we do not know whether or not a straight line will still be appropriate near that value of x being equal to 0. So when you get a Y intercept as part of your output, if your data, if your x values did not include 0, then be skeptical about your interpretation of that y-intercept. On that note, let's just summarize what we've talked about so far with general linear models with one continuous independent variable. We've seen that the analysis is very similar to analyses with one, with a one factor, one factor general linear model. When we have a continuous independent variable, we're saying that the X variable is independent, but we're also same implying that there may be some sort of cause related to that x variable. And the y variable is the response. The y variable is always is our dependent variable, where, but we're imagining that the y variable is responding to the x. And what we can do with these kinds of analyses is we can quantify a relationship between the x and the y variables. And so we can essentially estimate the equation for a line in order to describe this relationship. And we can quantify when we characterize that line. We can also characterize the uncertainty in that line. So for example, we can get standard errors for the intercept and the slope. I haven't talked about this earlier, but I'll mentioned it very briefly. We can use these standard errors to test whether or not our intercept and slope differ from particular value of interest. So let's consider the slope. The default null hypothesis for a regression is that we're testing whether the slope is significantly different from 0. So our no hypothesis is that the slope is going to equal 0. Sometimes a slope of 0 is not actually what interests us. Maybe for some biological reason, we want to know whether or not our slope is significantly different from one. In that case, we can use the measurement of our standard error. And we can use the machinery of a t-test. For example, in order to test whether or not our slope is significant, is significantly different from one. Okay? Alternatively, we can just get, look at the effect size for our slope and look at 95% confidence intervals for our slope. And then use that to be able to determine whether or not our, we have evidence that our estimated slope is likely to be different from some other value that we're interested in. This next point is just reiterate where we talked about earlier, where you can, with regression, we can predict y values from our X using the line that we have estimated. But be very careful about extrapolating beyond the data. The data that we have is the only thing we know about. And any situations that lie beyond our data, we simply don't know about yet. So we can't know what lies beyond your data unless we actually collect the New appropriate data. And finally, we have introduced this idea of r-squared, where we can interpret the R squared as the percentage or proportion of the variation in and our Y variable that is explained by our X variable. I'll close by mentioning that there is also a term, an adjusted R squared. And we're not going to talk about that in depth here. And that's just because I want to point out that the adjusted R squared applies. In particular when we have multiple independent variables. So for example, if we were doing something like a multiple regression where we had more than one continuous independent variable. So that's our summary and our warnings about, about extrapolation. Hope that's been helpful. And I'll say thank you very much.