2 Multiple Linear Regression - The linear model

Well, hello, my friends. We will move on in multiple linear regression to look at the regression model that we will be developing. Again, multiple linear regression is founded in correlational analysis. You just can't get away from those Pearson r correlation coefficients, can you? You should have a lot of fun with those. Since multiple linear regression is founded in correlational analysis, the correlation coefficients, or the various variables will be of great interest to us. And with an adequate correlation, we will develop an n-th level linear model when we have n independent variables. In other words, we can have a two-dimensional line with an x and a y. We can have a three-dimensional line, a four-dimensional line, a fifth-dimensional line and so forth. So we will be developing those models. Now, I know that appears about as clear as mud right now, but hang on. I think it's like athlete's foot. It'll grow on you. Now the multiple linear regression model will identify beta values, like a beta 1, a beta 2, all the way up to beta n. If we have n variable independent variables, then we will have n betas. And these betas will match down to the independent variables so that we can develop a linear model. These beta values will be used to predict the dependent variable y as follows. Now, here's the formula. The predicted value of y is equal to the first beta times the first independent variable's value plus the second beta times the second independent variable and so on and so forth, plus some arbitrary constant, C. This is a pretty powerful linear model. And we will develop some of these. And I think it will help you as we move forward to see what that actually looks like. The predicted value y, of course, is given by the following formula. Here's the formula again. This is the linear model. It is an n-th dimensional linear model. It's a line in [INAUDIBLE] space. And of course, once we find all the beta values and multiply them by the independent variable values at that point and at a constant, we have a predicted value of y. The formula provides the multiple linear regression model of best fit for the indicated data sets. Do you remember, best fit means that it minimizes the distance from the points. This is a very powerful model. And notice that multiple linear regression becomes simple linear regression when you have only one independent variable. Because what happens with one independent variable is all of this stuff right here, from beta 2 all the way up to beta n goes away. And you have a value times the variable plus C when you only have one variable. So simple linear regression is multiple linear regression with only one independent variable. Well again, I thank you very much for your support. May the odds be ever in your favor. We wish you the best. We will be moving forward now to look at the assumptions of multiple linear regression and actually run a multiple linear regression here in just a moment. Have a good one.