Centering and Scaling Continuous Rating Predictors

Why Center and Scale Rating Predictors?

A continuous rating variable may be measured in units that are inconvenient for interpretation. A mileage coefficient expressed per mile can be very small, while a driver-age coefficient expressed per year may not match the business question, which may concern a ten-year age difference. Centering and scaling provide a simple way to make those quantities easier to read. They are changes of coordinates, not nonlinear transformations of the predictor.

The key distinction is between changing a model's parameterization and changing its fitted predictions. With an intercept and an unconstrained coefficient, subtracting a constant from a predictor changes the intercept but leaves the linear predictor, and therefore the fitted mean, unchanged. Dividing by a scale changes the numerical coefficient to reflect the new units. These statements require an affine, one-to-one re-expression of the predictor and the same model structure.

For the examples, suppose the response is a hypothetical policy's positive annual pure premium \(Y\), and a Gamma GLM is fitted with a log link. Use the parameterization \(Y\mid x\sim\operatorname{Gamma}(k,\theta)\), where \(k=5\) is the shape, \(\theta=\mu/5\) is the scale, \(E[Y\mid x]=\mu\), and \(\operatorname{Var}(Y\mid x)=\mu^2/5\). The assumed dispersion is therefore \(0.20\). The response distribution and dispersion do not determine the algebra of centering or scaling; they simply make the example's GLM assumptions explicit. There is no interest or mortality assumption in these non-life pure-premium examples.

Centering: The Intercept Moves, the Fitted Mean Does Not

Start with a one-predictor GLM whose linear predictor is \(\eta=\beta_0+\beta_1x\). Under a log link, the conditional mean is \(\mu=\exp(\eta)\). Define \(x_c=x-c\), so \(x=x_c+c\). Substitution gives the centered parameterization:

Thus the new intercept is \(\alpha_0=\beta_0+\beta_1c\), and the slope is \(\alpha_1=\beta_1\). For each observation, the new and old linear predictors are equal. Applying the same inverse link to equal linear predictors gives equal fitted means. This argument also holds for another GLM link, because it relies on preserving \(\eta\), not on the log link specifically.

Scaling: The Coefficient Changes Units

Suppose the predictor is scaled as \(z=x/s\), without centering. Then \(x=sz\), and the linear predictor becomes \(\eta=\beta_0+(s\beta_1)z\). The new slope is \(s\beta_1\), not \(\beta_1/s\). If \(z=m/1000\), a one-unit increase in \(z\) is an increase of 1,000 miles, so its coefficient is 1,000 times the coefficient per mile. The fitted mean remains unchanged because the product of coefficient and predictor is unchanged.

More generally, if \(z=(x-c)/s\), then \(x=c+sz\), and \(\eta=(\beta_0+\beta_1c)+(\beta_1s)z\). Thus the transformed intercept is \(\beta_0+\beta_1c\), and the transformed slope is \(s\beta_1\). Under a log link, a change of \(\Delta z\) has relativity \(\exp((s\beta_1)\Delta z)\), equal to the original relativity \(\exp(\beta_1(s\Delta z))\).

Centering and Scaling Together: Driver Age

A useful center is often a meaningful or typical rating value. After centering, scaling can put a coefficient on a useful comparison unit. For age, \(z=(a-40)/10\) makes \(z=0\) correspond to age 40 and makes a one-unit increase correspond to ten years. Under a log link, the coefficient on \(z\) is the log relativity for a ten-year age increase, with other predictors held fixed.

What Is Preserved, and What Can Go Wrong?

For an ordinary GLM with an intercept and no coefficient constraints, an affine re-expression of a predictor preserves the model's set of possible linear predictors. Each original coefficient pair maps to a transformed pair by the equations above, and the inverse mapping exists when \(s>0\). Consequently, for the same observations and response distribution, the likelihood as a function of fitted means is unchanged; fitted means and residual-based summaries are unchanged as well. The individual coefficient values and their units do change, so coefficient interpretation must match the predictor's current coding.

The intercept's meaning is especially easy to misread. With several predictors, the intercept is the linear predictor when all predictors equal zero. After centering, zero corresponds to the chosen center for each centered variable. Under a log link, exponentiating the intercept gives the fitted mean at those reference predictor values, provided the other terms in the model are also evaluated at their zero values.

Scaling also changes the coefficient's standard error in the same units. If \(z=x/s\) and the transformed slope is \(\alpha_1=s\beta_1\), then its standard error is \(s\) times the standard error of \(\beta_1\). The ratio of estimate to standard error, and thus the Wald test for a zero slope, is unchanged for a positive scale. A confidence interval must be converted using the same scale factor if it is reported in the original units.

The preservation result has limits. If a model has no intercept, centering generally adds a constant contribution that the model cannot absorb, so fitted values may change. Constraints on coefficients or some penalized fitting procedures can also make the transformed fitting problem differ unless the constraints or penalty are adjusted consistently. Finally, centering and scaling only re-express a linear effect; they do not make a linear predictor flexible enough to capture a nonlinear relationship.

For practical rating work, centered predictors can give a useful reference profile and more readable intercepts. Scaled predictors can make coefficient tables easier to compare and communicate. Neither operation improves fit by itself or changes a fitted indication under the stated conditions. Record the center and scale wherever coefficients are reported; without them, a coefficient's unit and reference point are ambiguous.