24 Scaling predictor variables

When predictors have a natural scale, interpreting them can be relatively straightforward. However when predictors are on an arbitrary scale, or when multiple predictors are on different scales, then interpreting the model (or comparing between models) can be hard. In these cases scaling or standardising predictors in the model can make it easier to interpret the coefficients that are estimated.

Standardising

‘Standardising’ predictors, by subtracting the mean and dividng by the standard deviation, is a common way to make interpreting regression models easier, and particularly to make comparisons between predictors — e.g. regarding their relative importance in predicting the outcome.

@gelmanscaling_2008 covers in detail the advantages and disadvantages of standardising regression coefficients. Based on the observation that we often wish to compare continuous with binary predictors, they recommend standardisation by subtracting the mean and dividing by _two standard deviations (rather thant the normal one SD). The arm package implements this procedure, and makes it easy to automatically scale the predictors in a linear model.

First, we run the linear model:

m1 <- lm(mpg ~ wt + am, data=mtcars)
m1

Call:
lm(formula = mpg ~ wt + am, data = mtcars)

Coefficients:
(Intercept)           wt           am
37.32155     -5.35281     -0.02362  

And then use arm::standardize to standardize the coefficients:

arm::standardize(m1)

Call:
lm(formula = mpg ~ z.wt + c.am, data = mtcars)

Coefficients:
(Intercept)         z.wt         c.am
20.09062    -10.47500     -0.02362  

This automatically scales the data for m1 and re-fits the model.

An alternative is to use the MuMIn::stdizeFit although this applies scaling rules slightly differently to arm, in this case standardising by a single SD:

MuMIn::stdizeFit(m1, mtcars)
Registered S3 method overwritten by 'MuMIn':
method         from
predict.merMod lme4

Call:
lm(formula = mpg ~ wt + am, data = mtcars)

Coefficients:
(Intercept)           wt           am
37.32155     -5.35281     -0.02362  

Check the help file for MuMIn::stdize for a detailed discussion of the differences with arm::standardize.

Dichotomising continuous predictors (or outcomes)

Dichotomising a continuous scale is almost always a bad idea. Although it is sometimes done to aid interpretation or presentation, there are better alternatives (for example estimating means from a model using Stata’s margins command and plotting them, something we will do in the next session). As the Cochrane collaboration puts it:

The down side of converting other forms of data to a dichotomous form is that information about the size of the effect may be lost. For example a participant’s blood pressure may have lowered when measured on a continuous scale (mmHg), but if it has not lowered below the cut point they will still be in the ‘high blood pressure group’ and you will not see this improvement. In addition the process of dichotomising continuous data requires the setting of an appropriate clinical point about which to ‘split’ the data, and this may not be easy to determine.

See http://www.cochrane-net.org/openlearning/html/mod11-2.htm and also @peacock_dichotomising_2012. Also note that trichotomising (splitting into 3) is likely to be a better better/more efficient approach, see @gelman2009splitting.