Polynomial Regression interpretation

[lancearmstrong1313]

I have 2 questions that I need help interpreting:

1.
Y = 47.31051 + 1.06914(X) - .01383(X^2)

Because X is used twice (and one of them is a squared term) how do I interpret this? Something along the lines of as X increases by so many units, Y increases by....?

2.
Y = -834.7872 + .44028(X) + 28.32715(Z) + .00054574(Z^2) - .01476(X*Z)

Again, how do I not only interpret the squared term but the interaction term as well? As Z increases by 1 unit, what is the effect on Y?


Thanks

[Dragan]

Qualitatively, as X increases so does Y ---but at a diminishing rate. The key is the fact that the coefficient associated with X^2 is negative, so your prediction curve is an "upside-down U" shape (or parabola).

[jrai]

I have 2 questions that I need help interpreting:

1.
Y = 47.31051 + 1.06914(X) - .01383(X^2)

Because X is used twice (and one of them is a squared term) how do I interpret this? Something along the lines of as X increases by so many units, Y increases by....?

2.
Y = -834.7872 + .44028(X) + 28.32715(Z) + .00054574(Z^2) - .01476(X*Z)

Again, how do I not only interpret the squared term but the interaction term as well? As Z increases by 1 unit, what is the effect on Y?


Thanks

1) This is a non-linear equation therefore, the marginal effects are not constant & vary with each point on the curve. You can have marginal effect by differentiating y wrt x.

dy/dx= 1.06914-2*0.01383 X. Usually the marginal effects are evaluated at mean of X. Therefore, you can read this as increase in x by 1 unit will lead to increase in y by 1.06914-2*0.01383 X_bar units, where x_bar is mean of X.

2) For interpreting the marginal effect wrt Z in the second equation you have to hold X constant at a value. X can be a continuous or a categorical variable. Usually, if X is continuous then binning it into categories helps consolidating the problem. Say X is continuous variable (say income) with range 100-100000. Then you can calculate marginal effects wrt Z for any of the possible values between 100-100000 because the interaction term is essentially saying that response of y because of z depends on x. But if you bin them into High, medium & low categories then you'll have only 3 parabolas on the graph.

The marginal effect wrt z: dy/dz= 28.32715 + 2*.00054574 Z_bar - .01476(X*Z_bar).