Now that we have a formal statistical framework, we can interpret our regression coefficients with respect to that framework. Take for example, the intercept. It's the expected value Y given that the regressor is 0. And here I'm simply plugging into the formula. Note that the regressor being equal to zero is often not of interest in the study. For example, if the regression variable is blood pressure, probably you're not interested in the response for among people with blood pressure of zero. However, there is an easy fix for this. Consider just shifting our regression variable by a constant a. And here in this equation, I've just subtracted off an a here. Which means I have to add an a beta 1 over here to have done nothing, to have simply added 0. Well this is exactly rewriting the aggression equation as beta not tilde plus beta 1 times x minus a. A new regression line with a new intercept, the same slope, and my regressor now shifted by a constant a. So this goes to show you that if you shift your regressor by a constant,, it doesn't change the slope, however it does change the intercept. Now the intercept is inter, interpreted not when x is zero, but when x is this value that you shifted it by a. A very common constant to due is the average. Set a equal to the average of the X variable. Then your intercept will be interpreted as the expected value of the response when the regressor is at the average value of the Xs. Let's see if we can figure out a model-based interpretation of our regression coefficient. Now, it's a slope so it must have an interpretation that is of the form, the change in Y per unit change in X. So let's see if we can work with that. Take the expected value of Y, given that X takes the value x plus 1, a one unit increase in X. And subtract off from the expected value of Y, given that X takes the value x. Well, that works out to be beta naught plus beta 1 times x plus1, minus beta naught plus beta 1 times x. Do the arithmetic and that works out to be beta 1. So, beta 1 is the expected change in the response per unit change in the regression variable. And that, so it has a nice interpretation, it's not that different from the regular interpretation of a line, just now with the expected value, the conditional expected value of the response. Now, remember, if we shift the X variable, it doesn't change the slope, we discussed that on the previous slide. What if we scaled the regression variable? And it usually only we're usually scaling by number, say a, that's positive. So, what we can do is simply multiply and divide by a, and we can rewrite our progression equation as beta naught plus beta 1 tilde times X times a plus epsilon. So, we've multiplied X by a and our regression slope has now been divided by a. So in other words, multiple, multiplication of a regression variable by a factor a results in dividing the coefficient by a factor of a. Let me give you a simple example, so to hopefully solidify these points. Imagine if X's height measured in meters and Y's weight measured in kilograms, a pretty simple regression problem. Then beta 1 is interpreted in kilograms per meter. Converting X to centimeters implies multiplying X by 100 centimeters per meter. To get beta 1 in the right units, now we simply divide it by 100 centimeters per meter to get it, to, to, to get it back on the right scale. And here I just wrote a, a little equation to remind us.
