Quadratic model

General
1

Dear all,
Can anyone help explain what a quadratic model actually tells us? Let’s say I have annual discharge data and I want to fit both a linear and a quadratic trend. What exactly does the second-order term (x²) tell us? I haven’t found much detail beyond statements saying that x² measures acceleration. I’d like to understand what the quadratic coefficient actually means and how we should interpret it in terms of acceleration.


quad_model <- lm(y ~ x + I(x^2))
2

Suppose we have discharge = \alpha + \beta x + \gamma x^2. If \gamma=0 then each additional unit of x increases discharge by \beta. But if \gamma \neq 0, then the effect of x rises (\gamma>0) or falls as x rises.

Or more formally, \frac{\partial discharge}{\partial x} = \beta+2\gamma x.

3

This might help a bit, Interaction Effects in Regression in R

4

The quadratic isn't an interaction, but that's a very nice link.

5

Yes, thanks.
About a 5 minutes after I sent it I realized I'd misread the OP but I was already out of the house. I'll have to see if I can find a a decent interaction post.

6

Dear @startz thank you! I just tested this in R using a sample data:

    dat$t <- dat$Year - min(dat$Year)

    model <- lm(Discharge ~ t + I(t^2), data = dat)

    summary(model)

Coefficients:
         Estimate  Std. Error  t value  Pr(>|t|)
   (Intercept)  103.2       1.1        93.8     <2e-16
           t              1.10      0.12        9.2     <2e-16
        I(t^2)         0.015     0.003       5.0     1e-05

From this is the acceleration 0.015 or 2*0.015 units year ^2