# Continuous independent variable divided into two levels without discretion

As far as I know, for differentiating between different levels of an independent continuous variable in regression model, you can convert it into a dummy variable with a reference level and other levels taking values one or zero. However, in Parkin and Rotherham (2010) they estimate a linear regression model that evaluates speed against gradient, and they split their gradient variable into negative and positive levels, without turning it into a binary dummy variable. Their model is:

$$\text{Speed} = \text{constant} + a \times \text{Uphill Gradient} + b \times \text{Downhill Gradient}.$$

For instance, if an observation has $\text{Gradient} = 0.15$, they input this as $\text{Uphill Gradient} = 0$ and $\text{Downhill Gradient} = 0.15$. What I do not understand is how they use gradient as two variables (uphill and downhill gradients) but they do not use dummy variables, and instead substitute the real value of gradient.

My question: Can we divide continuous variable into levels like this without making them a dummy variable?

• Could you provide a bit more context as the link appears to be an abstract and paywall? You'll need to explain more precisely what they are doing. – ReneBt Jun 12 '18 at 3:56
• Perhaps searching for broken stick regression or segmented linear regression will help you to clarify things. – mdewey Jun 12 '18 at 11:29
• This formulation can be seen as an interaction between a dummy for up/down hill (coded as $(1,0)$ and $(0,1)$) and the gradient. It can also be seen as a regression against two nonlinear transformations of the gradient $X$: namely, its positive part $\max(X,0)$ and its negative part $\max(-X,0).$ It's readily related to yet another regression with independent variables $X$ and $|X|.$ Evidently, then, one can "divide a continuous variable into levels like this." If you seek some deeper meaning or underlying theory, it's probably not there--but is this the sort of analysis you're after? – whuber Jun 14 '18 at 2:23