And the columns correspond to individual predictors.

Is that. Putting these two, predictors together.

This, that's the beauty of multiple regression and building a model with multiple predictors, is we can account for more variance with this set of predictors.

With both predictors in it. Is doing better.

And we're just gonna look at two predictors.

We could try and come up with more reliable predictors or just add in more predictors to the model. So we did that.

And this, this, this, this particular linear combination of predictors.

Generally speaking, past fixes and a bit less so past changes are good predictors for future fixes.

And then, we can have multiple predictors and multiple regression coefficients.

To get a sense for how much each one of these predictors by themselves would explain in faculty salary.

And we'll predict the outcome variable on the basis of multiple predictors.

So this is the sum to squares for X1, this is sum to squares for X2, and that's a sum of cross products for the two predictors,

And, as you can see, I could extend this out to K predictors.

Now, we can get a better correlation because we're adding in multiple predictors.

Now, that doesn't mean that these individual predictors are correlated that strongly, alright?

Today, we'll see how multiple regression, multiple regression works where we have multiple predictors.

Now, let's look at what happens when I put both predictors in the model.

That is, the more complex model with both predictors in the equation is accounting for significantly more variance than the simple model.

So there's only two cross products, because we just have two predictors and one outcome.

And we're going to predict faculty salary not from one predictor but from multiple predictors.

And to see if there's moderation, we just take the product of those two predictors.

So if you put in a lot of predictors, the , eventually they should all suffer.

We said one predictor in the equation. Today, we're just going to enter in more predictors.

What we're doing now is we're going to add in as many predictors as we like.

The general linear model is this equation, it's assuming that all the predictors are additive, right?

Just two simple regressions and model three is the multiple regression with both predictors in there.

That's the way to interpret the regression coefficients when there are multiple predictors in the model.

We'll have multiple predictors, So the, the models will be more than just this one equation.

If you can start to spot patterns in your residuals, then you can start to think about how you can improve your predictors, or you can think of other predictors to add to the regression equation.

If your predictors aren't correlated, if they're orthogonal to one another, then, these different approaches won't matter.

What we're doing is we're finding the best linear combination of these predictors to explain the outcome.

Now we have two predictors, its X1 X2. Remember this is a linear model and its additive.

And then, I'll talk about the difference between what I'll call a standard multiple regression, where we enter all predictors in together, and what I call a sequential multiple regression, where we enter predictors in a particular order.

So, I like this, this title, regression, it's, it, it, it gets better because the goal is to produce better models , to come up with better predictors or more predictors, that w ill explain more of the variance in our outcome.

But, when there are multiple predictors in the equation, then it's the predicted change in y for the average level of every other variable in the equation.

And then, we'll do multiple regression. And we're not gonna get crazy, we're just gonna put two predictors in.

What that tells us is, is each one, each one of these predictors are they significantly predicting salary or not?

So if we just take the sum of the squares and cross products for the X variables, that's just a two by two matrix, because we just have two predictors.

That's how we get predicted scores on Y, from a set of values on, set of predictors, X1, X2, and X3.

Then we walked through the matrix algebra that allowed us to estimate the regression coefficients with multiple predictors in one equation.

So, it's important to know that if the predictors themselves are not correlated, then it won't matter how you run the regression.

The model R squared will take into account the entire proportion of var, of variants that's shared with the entire set of predictors.

And then, this idea of doing things sort of sequentially or just putting all of the predictors into an equation all at once

look below the diagonal, if you're a pessimist laugh two and 0.6 those are the cross products relating the predictors x1, x2 to y.

It's more common when you run a multiple regression analysis and you add in more predictors, that each individual predictor gets weaker in strength.