Step Functions

1. Introduction

• Polynomial regression supposes a global relationship between $Y$ and $X$, while step functions avoid such assumption.
• Step functions break the range of $X$ into bins, and fit a different constant in each bin.

2. Construction

Assume that we create cutpoints $c_1,\dots ,c_K$ over $X$, and thus obtain $K+1$ new variables:

$$\begin{array}{rl}{C}_0(X)& =I(X<c_1),\\ C_1(X)& =I(c_1\le X<c_2),\\ & \cdots \\ C_{K-1}(X)& =I(c_{K-1}\le X<c_K),\\ C_K(X)& =I(c_K\le X),\end{array}$$

where $I(\cdot )$ is the indicator function.

Then we can fit the model:

$$y_i={\beta }_0+{\beta }_1{C}_1(X_i)+\cdots +{\beta }_K{C}_K(X_i)+{ϵ}_i,$$

where $C_0(X)$ is excluded given that ${\sum }_{i=0}^K{C}_i(X)=1$, for the same reason as dummy variable traps.

Note that, for a given $X$, at most one of the $C_i(X)$ can be nonzero. Moreover, if $X<c_1$, then $\forall C_i=0$, thus ${\beta }_0$ is the mean value of $Y$ for $X<c_1$.

3. Notes

• The selection of breakpoints will influence the outcome, and thus might miss some important trends, as the curve between the breakpoints is flat.
• It is popular in biostatistics and epidemiology.

Related Notes

• Regression Splines
• Smoothing Splines