Statistical modeling with two predictor variables that are dependent, and a response variable that is dependent on both of the predictor variables?

Question

So I am doing a mathematical project about tennis serve speed. The question I want to dig into is that whether player's height and weight has a correlation with average first serve speed. So the predictor variables are height, and weight, and the response variable is average first serve speed. I tried using multiple linear regression, but it did not turn out well because I do not think height and weight are completely independent. In fact, I found a r value of 0.64 with my data. So is there a way for modeling the three variables together then?

Thanks

Answer 1

It does not matter for the linear regression that your variables are correlated, as long as they are not perfectly dependent (which they aren't).

I think that linear regression in this case is a good choice, and if weight and height indeed contribute to the first serve speed, you will get a nice, easy to communicate quantitative dependency (for example, given that the weight is constant, each 1 cm adds 1+-0.1 m/s).

Also, please add self-study tag if this is for homework.

Answer 2

Linear regression will pick apart the two variables. However, even if they are not identical, if they are correlated - as height and weight are - you may get the problem that the statistical significance (measure by the t- or p-value on each coefficient) are not very good, since the effect gets split across two similar variables.

Try running with one or the other, separately, to see if that helps if that is your problem.

More to the point, if the $R^2$ (you said $R$ but I suspect you mean $R^2$) is .64 that isn't too bad on its own. I am actually surprised that the explanatory power is that good.

There is a big difference between having a statistically significant set of coefficients (i.e., they are close the the 'actual' number) and having a high $R^2$ (explaining 64% of what determines the speed).

This is one of the things people get wrong very often in statistics. Suppose you have lots of data, and make a statistically significant finding that mothers who eat cheese during pregnancy are more likely to have difficult deliveries. But, maybe 25% of mothers have difficult deliveries overall. If cheese eating makes one more mother out of every thousand have a difficult delivery, the effect is statistically significant but pretty useless since out of 1,000 births it means not eating cheese drops the number of difficult births, on average, from 250 to 249.