Variable Transformation

Study guides: Exam SRM, Exam PCPA

A Variable Transformation replaces a predictor $x$ with a function of it, $h(x)$ , so that its relationship with the response is captured by a model that is linear on the scale of the link. In a Generalized Linear Model the response is not transformed — the link and the error distribution handle its scale — so transformation is about the predictors.

$g(\mu) = \beta_0 + \beta_1\,h(x) + \cdots$

$h(x) = \ln x \quad \Rightarrow \quad \mu \propto x^{\beta_1} \text{ (log link)}$

$\mu = E[Y]$ , $g$ is the link, and $h$ is the transformation. Under a log link a logged predictor gives a power curve: doubling $x$ multiplies the mean by $2^{\beta_1}$ , whatever the starting value of $x$
Log: for positive, right-skewed predictors whose effect is proportional (amount of insurance, building value). With a log link this is usually the natural first choice
Binning (banding): cut a continuous variable into groups and fit a level for each, treating it as a categorical predictor. Captures any shape, but spends a parameter per band and ignores the ordering within one
Polynomial ( $x$ , $x^2$ , …) and piecewise linear (hinge) terms: fit a curve with a few parameters. Polynomials can behave badly at the edges of the data
Capping extreme predictor values before fitting stops a handful of points from driving the fit (see Outlier); standardizing, $(x - \bar{x})/s_x$ , puts predictors on a common scale and helps when interactions or penalties are used
The need for a transformation shows up in Exploratory Data Analysis and in a Residual Plot against the predictor — a curve where there should be none

Interpreting a Logged Predictor A severity GLM with a log link includes $\ln(\text{Amount of Insurance})$ with coefficient $\hat{\beta} = 0.6$ . By how much does expected severity change when the amount of insurance doubles, and when it rises by $50\%$ ?
Answer
 $\begin{align*} \mu &\propto \text{AOI}^{0.6} \\ \text{Doubling:}\quad 2^{0.6} &= 1.516 \\ \text{50\% increase:}\quad 1.5^{0.6} &= 1.275 \end{align*}$ 
Doubling the amount of insurance raises expected severity by about $52\%$ , and a $50\%$ increase raises it by about $27\%$ . Severity grows less than proportionally ( $\hat{\beta} < 1$ ): larger properties rarely burn to the ground, so their average claim is a smaller share of the amount insured.

Binning a Non-Linear Effect A frequency model's residuals show that the effect of driver age is high for young drivers, falls through middle age, and rises again after $75$ . A single linear term for age is fitted. What is wrong, and what are the options?

Answer A linear term forces the effect to move in one direction, so it cannot capture a U-shape; it will understate the risk at both ends. The options:

Bin age (for example $16$ – $20$ , $21$ – $25$ , …, $76+$ ) and fit a level per band. This is flexible and easy to explain, but the bands need enough exposure to be credible.

Add polynomial or hinge terms (such as $\max(0, \text{age} - 75)$ ) to bend the line with fewer parameters.

Either way, choose between them on hold-out performance rather than training fit, which always favours more parameters.