Okay, in this video, we're going to introduce the topic of generally near models that have an independent variable that is continuous. And an independent variable that is continuous is often called a covariate. So I may use those terms interchangeably. That the purpose of this video is to provide a general introduction to analyses that have continuous independent variables, that have covariates. And we're also going to make a very general comparison between these types of models. And what we've dealt with in previous videos, which has been one factor models. So, so far. In this series of videos, we've focused on situations where our dependent variable, so the variable that we'll be plotting on the y-axis has been continuous. We've assumed that the data are related to the normal distribution. But where our explanatory variables or the independent variables are factors. And factors. I mean, we've been couldn't take considering cases where we might want to know how. Let's, let's imagine some physiological process like the rate of metabolism. We might want to know how a particular drug influences metabolism. So you might have a factor with several different treatments. So for example, where seven different levels. You might have one drug. You might have a placebo control, and then you might have another treatment where there is no placebo but there's just nothing done. So you'd have three different groups being compared in an experiment like that. So that's an example of a case where we have a factor as our independent variable and we'd want to know how, whether or not it influences our dependent variable being our rate of metabolism. But very often our independent variables are not or cannot be categorized into discrete groups like those three different treatments I was just describing. Very often our independent variables are continuous. So what do I mean by that? Let's, let's consider a couple of examples. So let's imagine we want to know what affects the heart rate of a mouse. Ok. And we're not imagined in a case where we might be giving a mouse a particular kind of drug or apply in a particular kind of experimental manipulation. Instead, we might be interested in how factors that I have listed here influence heart rate. For example, we might want to know how does the heart rates depend on body size, where body size is continuously distributed. We can measure body size in grams or milligrams. We might want to know how, how heart rate depends on food availability, which you might quantify as the number of calories that are ingested per day, for example, it's another continuous variable. We might want to know how temperature influences heart rate or the mouse's age. Or we might measure hormone levels. These are all things that are continuously distributed. They might follow a normal distribution, they might follow some other kind of continuous distribution. But the point is that they do not fall into discrete categories. They're all things that are continuous. And we might want to know how these continuous variables influence the heart rate of a mouse. Similarly, we might want to know what affects plant height or the height at which a plant grows too. We could quantify the amount of light that's available. Using a particular instrument. We might want to quantify the nutrient availability and see how the amount of nutrients that plants are exposed to influences their height. Or we could quantify the amount of herbivore damage. So we can estimate the percentage of leaf, of leaf matter that's produced, that's been eaten by herbivores. Or maybe we can quantify the number of deleterious mutations that plants have and look at how the expression of these deleterious mutations influences height. Or we might measure 40, the rate of photosynthesis. Again, these are all continuous variables. We might argue that number of deleterious mutations is not continuous because you can't have half a mutation. But functionally, for these kinds of analyses we could treat. If we had enough mutations, enough categories of mutations, then we could treat this as a continuous variable. Ok, so our goal in this next series of videos is to understand how we can understand the effect of some continuous variable on a contiguous response. So now let's just do a very general overview to compare one factor analyses, which we've spent a lot of time discussing in previous videos, with an analysis that just has one covariate. Okay? So where we have just one explanatory or independent variable that is continuous. First of all, I'll point out that the way in which we do the analyses is identical. Also, the assumptions are identical. So withheld these things in mind. Basically, moving from a one factor analysis to one co-varied analysis is really straightforward. What's new is how we interpret the output. And that's because when we model the effective a covariate on some response, we're able to address a new question. Because what we're doing in this case is we're quantifying a relationship. Between two different continuous variables. And so we can ask, for instance, with the covariate, we can do things like we can estimate the equation of a line that we can fit to our data and whole exemplify this in a moment. Okay? So here's another way of making this comparison between a one factor general linear model and a general linear model that has one covariate. So what I'm showing you here, it's really the situation that we've been used to so far. We have a factor which I'm calling drug, and it has three different levels is either drug one drug to or drug three. And we're interested in how growth differs or whether or not it differs. Among subjects that have experience. He's three different drugs. So in this case our y value is continuous, but x is a discrete factor. There are just three different categories for this drug treatment. And our model would look like this. We'd say our growth is equal to the effect of the drug plus random error, which is our residual error. If this doesn't look familiar, I'll just say that all the models that we've been doing using the LM function so far, when we've been dealing with factors, they've all been doing this whether you've appreciated it or not. When we've run these models, what we've done is the model has fitted values. So these lines here, it has fitted a mean value for each of the different groups. So for drug one, drug-to-drug 3x3. And then we've basically, the point of this kind of analysis is to try to assess any evidence for any potential differences between these fitted values, between these means of our various other various drug types. And also we want to know, we want to estimate the size of those differences and which are our effect sizes. And he wants to characterize them. So we want to, for instance, get 95% confidence intervals for these effect sizes. Okay? So this is the perspective that we've been used to so far. Here's our new perspective that which, which we're going to be exploring in the next few videos. Now we're considering a case where still our y variable is continuous. But now our x variable or our independent variable is also continuous. So I've just made up these data where we're imagining a case where we're looking at the thickness of an exon and we want to know how exon thickness, right? The, the, the fiber diameter influences conduction velocity for that axon. Ok? So in this case, our model is speed or conduction velocity as a function of the axon thickness plus error, which is very similar to the kind of thinking we had earlier when we had discrete factors. In this case, however, when we're fitting, our values were no longer fitting a mean for each different group. Instead, what we're doing is we're fitting a line and we're going to be fitting the line that fits best through our data. Okay? And so we have this standard equation for a line where we have Y, which is our conduction velocity, is equal to x, which is our thickness, multiplied by the slope plus the y-intercept. That's what C represents. And in this case, we're not asking whether or not means differ from one another, which is what we did with the this factor perspective. Instead, we're asking whether or not there is a relationship between x and y, okay? And we want to quantify that relationship. And this type of analysis is often called regression. So you can think of regression and a one factor, or sorry, a, a, generally you can think of regression and a general linear model with one covariate as being synonymous. Okay? So that is our general introduction to modeling data that have a covariate. In other words, that's our introduction to modelling data, where we're trying to explain some continuous dependent variable. We're trying to understand how that dependent variable depends or whether or not it depends on some other continuous X variable or independent variable. And we'll stop a video there and say, I hope this has been helpful. Thank you.