Dummy variable for level measurement I'm looking at the PH level in a lake (x) and the number of fishes (y). If the PH level goes below a certain level, say 4.5, the severity for the fishes theoretically increases.
Model: $y=\beta+\alpha*x+Dummy_{x<4.5}$
Question: Is it okay to include a dummy variable for a variable (x) and the actual variable itself as in the regression above? Note: the example is indeed just an example.
 A: Assuming you also fitted a coefficient for the dummy variable, the model you described would be:
$$y = \beta_0 + \beta_1x + \beta_2\mathbb{1}_{x < 4.5} + \epsilon$$
Assuming the fitted model has $\beta_2\neq 0$, this would model the relationship between $x$ and $y$ as being discontinuous. For instance, if you had $\beta_0 = \beta_1 = 1$ and $\beta_2 = -1$, then the fitted value for $x = 4.5$ would be 5.5 while the fitted value for $x = 4.5-\delta$ for arbitrarily small $\delta > 0$ would be $4.5-\delta$.
In most applications such discontinuities don't make sense, as an infinitesimal change in $x$ would be expected to yield an infinitesimal change in the fitted value of $y$.
You might consider instead using segmented regression, which models $y$ as a piecewise linear function of $x$. Then you could have a different slope for the relationship between $y$ and $x$ in the region where $x < 4.5$ and the region where $x \geq 4.5$ but you could have a continuous relationship between the two variables. Segmented regression is implemented in R in the segmented package.
