Scaling variables in R

A walkthrough of a tip I recently learned

Herein I present a walkthrough of a recent improvement I made to my analyses when scaling (i.e., normalizing) variables.

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Scaling or normalizing variables in R is super convenient by using the scale(x, center = TRUE, scale = TRUE) function and default arguments. Here, the variable x is mean centered and scaled to unit variance, also called a Z-score.

Let’s do this on the mtcars dataset variable hp:

data(mtcars)
mtcars$hp_z <- scale(mtcars$hp)

Here’s proof that the newly scaled hp_z variable was indeed scaled:

library(tidyverse)
mtcars %>% 
  summarise(across(contains("hp"), list(mean = mean, sd = sd))) %>%
  round(2)

  hp_mean hp_sd hp_z_mean hp_z_sd
1  146.69 68.56         0       1

As we can see, the M=0 and Standard Deviation(SD)=1 for hp_z.

The scale() function, however, does something really useful that I used to ignore for the longest time; it saves the scaling factors as attributes in the column/variable. This is seen in a few ways:

class(mtcars$hp_z)

[1] "matrix" "array" 

mtcars$hp_z

             [,1]
 [1,] -0.53509284
 [2,] -0.53509284
 [3,] -0.78304046
 [4,] -0.53509284
 [5,]  0.41294217
 [6,] -0.60801861
 [7,]  1.43390296
 [8,] -1.23518023
 [9,] -0.75387015
[10,] -0.34548584
[11,] -0.34548584
[12,]  0.48586794
[13,]  0.48586794
[14,]  0.48586794
[15,]  0.85049680
[16,]  0.99634834
[17,]  1.21512565
[18,] -1.17683962
[19,] -1.38103178
[20,] -1.19142477
[21,] -0.72469984
[22,]  0.04831332
[23,]  0.04831332
[24,]  1.43390296
[25,]  0.41294217
[26,] -1.17683962
[27,] -0.81221077
[28,] -0.49133738
[29,]  1.71102089
[30,]  0.41294217
[31,]  2.74656682
[32,] -0.54967799
attr(,"scaled:center")
[1] 146.6875
attr(,"scaled:scale")
[1] 68.56287

The attribute for the mean-centering (i.e., the mean) can be accessed like this:

attr(mtcars$hp_z, "scaled:center")

[1] 146.6875

And the attribute for the scaling factor (i.e., the SD) can be accessed like this:

attr(mtcars$hp_z, "scaled:scale")

[1] 68.56287

In sum, the output of scale() is a "matrix" "array" with the centering and scaling factors, if requested, attached as attributes. Notice how they match the M and SD of the original mtcars$hp variable above.

The problem

However, there are a few issues with using scale(). The "matrix" "array" output is little annoying because it prints differently:

as_tibble(select(mtcars, mpg, contains("hp")), rownames = "car")

# A tibble: 32 × 4
   car                 mpg    hp hp_z[,1]
   <chr>             <dbl> <dbl>    <dbl>
 1 Mazda RX4          21     110   -0.535
 2 Mazda RX4 Wag      21     110   -0.535
 3 Datsun 710         22.8    93   -0.783
 4 Hornet 4 Drive     21.4   110   -0.535
 5 Hornet Sportabout  18.7   175    0.413
 6 Valiant            18.1   105   -0.608
 7 Duster 360         14.3   245    1.43 
 8 Merc 240D          24.4    62   -1.24 
 9 Merc 230           22.8    95   -0.754
10 Merc 280           19.2   123   -0.345
# ℹ 22 more rows

and it does not play nicely with predict():

mod <- lm(mpg ~ 1 + hp_z, data = mtcars)
new_data <- data.frame(hp_z = seq(-2, 2, length.out = 100))
predict(mod, newdata = new_data, type = "response", se.fit = TRUE)

Error: variable 'hp_z' was fitted with type "nmatrix.1" but type "numeric" was supplied

In the past, I would normally strip these features by wrapping the call with as.numeric():

mtcars$hp_zn <- as.numeric(scale(mtcars$hp))

Although this will work, you lose the awesome feature of having access to the centering/scaling attributes.

The solution

I found a nice elegant solution to this issue via this Stack Overflow post.

The drop() function will drop the formatting of scale() without the loss of attributes:

mtcars$hp_z <- drop(scale(mtcars$hp)) # using drop
class(mtcars$hp_z)

[1] "numeric"

as_tibble(select(mtcars, mpg, contains("hp")), rownames = "car")

# A tibble: 32 × 5
   car                 mpg    hp   hp_z  hp_zn
   <chr>             <dbl> <dbl>  <dbl>  <dbl>
 1 Mazda RX4          21     110 -0.535 -0.535
 2 Mazda RX4 Wag      21     110 -0.535 -0.535
 3 Datsun 710         22.8    93 -0.783 -0.783
 4 Hornet 4 Drive     21.4   110 -0.535 -0.535
 5 Hornet Sportabout  18.7   175  0.413  0.413
 6 Valiant            18.1   105 -0.608 -0.608
 7 Duster 360         14.3   245  1.43   1.43 
 8 Merc 240D          24.4    62 -1.24  -1.24 
 9 Merc 230           22.8    95 -0.754 -0.754
10 Merc 280           19.2   123 -0.345 -0.345
# ℹ 22 more rows

Using drop() no longer causes an issue with predict():

mod <- lm(mpg ~ 1 + hp_z, data = mtcars)
new_data <- data.frame(hp_z = seq(-2, 2, length.out = 100))
pred <- predict(mod, newdata = new_data, type = "response", se.fit = TRUE)
new_data <- cbind(new_data, pred)

# plots
ggplot(new_data, aes(hp_z, fit)) + 
  geom_line () +
  geom_ribbon(aes(ymin = fit-se.fit, ymax = fit+se.fit), alpha = 1/3) +
  labs(x = "HP (in SD units)", y = "Predicted MPG", caption = "SE shading.") +
  scale_x_continuous(breaks = seq(-2, 2, .5)) +
  theme_bw()

But what if we want to represent the x-axis in its original units? No problem! Just use the attributes from the scaled column to convert back to the original unstandardized units:

m <- attr(mtcars$hp_z, "scaled:center")
sd <- attr(mtcars$hp_z, "scaled:scale")
new_data$hp_o <- new_data$hp_z * sd + m

# plots
ggplot(new_data, aes(hp_o, fit)) + 
  geom_line () +
  geom_ribbon(aes(ymin = fit-se.fit, ymax = fit+se.fit), alpha = 1/3) +
  labs(x = "HP (in original units)", y = "Predicted MPG", caption = "SE shading.") +
  theme_bw()

Thank you drop()!

Summary

When scaling variables in R via scale(), be sure to use the drop() function to strip away the dimensions of scale()’s output. This new variable/column will play nicely with predict() and retain the center and scale attributes for later use.