For example, if I have the dependent variable Height and independent variables Age, Weight, and Shoe Size, how do I answer the following questions:
- Controlling for Age and Weight, is Shoe Size related to Height?
- Allowing for Age and Weight, is Shoe Size a useful predictor for Height?
Is question #2 just another way of asking question #1?
If so then, how do I find out if they're related (or that Shoe Size is a useful predictor)?
Let $y=\beta_0+\beta_1x_1 + \beta_2x_2 + \beta_3x_3$
$x_1$ = Age, $x_2$ = Weight, and $x_3$ = Shoe Size
Then I should test the null hypothesis $\beta_1=\beta_2=0$ with $\alpha = 0.05$. If the p-value is less than $\alpha$, I can reject the null hypothesis, and thus, Shoe Size size is NOT related to Height. Is this correct?