Correlation and Regression Statistics 2126

Introduction Means etc are of course useful We might also wonder, “how do variables go together?” IQ is a great example It goes together with so much stuff

A scatterplot You tend to put the predictor on the x axis and the predicted on the y, though this is not a hard and fast rule A scatterplot is a pretty good EDA tool too eh Pick an appropriate scale for you axes Plot the (x,y) pairs

So what does it mean If, as one variable increases, the other variable increases we have a positive association If, as one goes up, the other goes down, we have a negative association There could be no association at all

Linear relationships BTW, I am only talking about straight line relationships Not curvilinear Say like the Yerkes Dotson Law, as far as a the stuff we will talk about, there is no relationship, yet we know there is

The strength is important too The more the points cluster around a line, the stronger the relationship is Height and weight vs height in cm vs height in inches We need something that ignores the units though, so if I did IQ and your income in real money or IQ and your income in that worthless stuff they use across the river, the numbers would be the same

The Pearson Product Moment Correlation Coefficient

Properties of r <= r <= The sign indicates ONLY the direction (think of it as going uphill or downhill) |r| indicates the strength So, r = -.77 is a stronger correlation than r =.40

Some examples

EDA is KEY

Check these out.. All of these have have the same correlation R =.7 in each case Note the problem of outliers Note the problem of two subpopulations

Remember this Correlation is not causation I said, correlation is not causation Let me say it again, correlation is not causation Birth control and the toaster method

Wouldn’t it be nice If we could predict y from x You know, like an equation Remember that in school, you would get an equation, plug in the x and get the y Well surprise surprise, there is a method like this in statistics

If we are going to predict with a line Well, we will make mistakes We will want to minimize those mistakes

There is a problem, a common problem Those prediction errors or residuals (e) sum to 0 Damn Though guess what we could do… Why square them of course So we get a line that minimizes squared residuals

The line will look like this

In general the equation of the line is….. Y hat (predicted y) Y interceptslope

This might help

So…. With a regression line you can predict y from x Just because it says that some value = a linear combination of numbers it does not mean that there is necessarily a causal link Don’t go outside the range Linear only