Sabermetrics 101: Correlation

are you gauging the “worth” based on the Beta of the resulting linear model, or something else?

(Stolen from Wikipedia who took it from somewhere else)

Something can be correlated and give a bad correlation coefficient. Looking at the data vs the model is the only good way to figure out if things are being accounted for well.

As well as showing my ignorance, but it would be really beneficial to have a few examples for these posts of what actually means what.

If I understand right…
Swings & misses probably have a correlation to strikeouts of close to 1.
Ground balls probably have a correlation to GIDP of 0 < X < 1
How you prefer your eggs cooked probably has a correlation of roughly 0
Contact pitchers probably have a correlation to strikeouts of -1 < X < 0
Being drunk probably has a correlation to hand eye coordination of -1 (drinking games “talent” excluded from sample set)

Forgive my ignorance… I’m a bit of a numbers guy and a bit of a baseball guy, but an expert in neither… It would just help me considerably to see the two connected in black and white (to the admittedly limited extent you can be black and white with these) examples of how basic baseball behaviors relate to all these funky numbers, letters and graphs.

I want to point out a few important issues here as a “real” statistician.

In a correlation calculation, it doesn’t matter which variable is the independent variable and which is the dependent variable, you are trying to measure how well the data conforms to a line

The value of r^2 is hugely important here. Take the correlation coefficient and square it. This result tell you how much of the variance in one variable is explained by the variance in the other. This in particular is important when you see correlation coefficients around 0.5. Many people think that a correlation of 0.5 is really high – not exactly, the line that the data clusters around only explains a quarter of the variance.

What we are measuring with the correlation coefficient is how well knowledge of one variable helps you predict the value of the other. One example is the fact that beer sales at the ball park are highly correlated with temperature. The usual way to use this information is to look at the thermometer and predict how much beer will be sold at Safeco. On the other hand the other way is useful, too. Suppose you lost your temperature data, but knew how much beer was sold at Safeco that day – you could use the regression equation (the equation that describes the line) to estimate what the temperature was that day. Would it be perfect, no, but it would be as good as using that line to predict beer sales given the templ

“Correlation” is often used to refer to two (slightly) different concepts:

1) The numerical correlation (R value) between two variables. This measures the degree to which a straight line describes the relationship between them (with the sign of R indicating the direction). In fact, R=1 if and only if one variable is a linear function of the other.

2) A more general dependence between two variables. For example, in the third row of the set of plots above, we would informally say that the two variables are “correlated”, since the values on the Y axis clearly depend on how far along the X axis you travel. However, since this dependence (“correlation”) is not a linear relationship, the R value is zero. This is why one of the important things taught in an introductory statistics course is that R=0 does not automatically mean that two variables are independent.

If you get the joke then you get the correlation vs causation distinction.

Statement: Regression can be used to model anything as long as you know all the relevant variables.

Reasoning: Any function can be expressed as an infinite sum polynomials. For example, you can model the function y=exp(x) using a string of polynomials where the accuracy depends on the number of terms you use. This method is often referred to as a Taylor series.

Therefore any data that depends on a variable should be able to be modeled using a large number of polynomials (Taylor series). We can not use an infinite sum of terms because we have a finite number of data points however we should be able to describe the system well without too many terms (as long as we have a big data set). If we know that strikeouts are only a function of swinging strikes (not true, only for example purpose) but we don’t know how they are related, we can use a string of polynomials to model the system and adjust the constants to make it fit the data. This is only when you are trying to find a relationship for something like y = function(x).

Question: If we have an equation where n= f(x,y,z), we should still be ok I think. In this case we need to do a taylor expansion in every variable and then every combination of variables. For example n = f(x,y) = a + bx + cx^2 + dy + ey^2 + fyx where I have expanded it out to the quadratic form. If I extended the series to infinity it should be able to model any function (where a normal talyor series would work) where n = f(x,y).

If we did this sort of expansion for runs (using every single possible relevant parameter) we should be able to fit the data as well as anything out there now if we were able to get enough data free of too much bias. Then we could simply take partial differentials of the equation to figure out the value of a single or double for example. The value this method provides should be close to the best metrics out there now if done correctly.

This method probably isn’t the best way to do it but it seems like it could work just as well as long as you had a big enough data set and knew every single variable that was relevant to the final answer. I’m not suggesting anybody do this because this technique is not very elegant or intuitive. Also, since it is difficult to imagine every single relevant variable that could be used to provide unique information, this approach is fraught with possible errors.