Transformations

You can also perform transformations inside the model formula. For example, log(y) ~ sqrt(x1) + x2 is transformed to log(y) = a_1 + a_2 * sqrt(x1) + a_3 * x2. If your transformation involves +, *, ^, or -, you'll need to wrap it in I() so R doesn't treat it like part of the model specification. For example, y ~ x + I(x ^ 2) is translated to y = a_1 + a_2 * x + a_3 * x^2. If you forget the I() and specify y ~ x ^ 2 + x, R will compute y ~ x * x + x. x * x means the interaction of x with itself, which is the same as x. R automatically drops redundant variables so x + x become x, meaning that y ~ x ^ 2 + x specifies the function y = a_1 + a_2 * x. That's probably not what you intended!

Again, if you get confused about what your model is doing, you can always use model_matrix() to see exactly what equation lm() is fitting:

df <- tribble(
~y, ~x,
 1,  1,
 2,  2, 
 3,  3
)
model_matrix(df, y ~ x^2 + x)
#> # A tibble: 3 x 2
#>   `(Intercept)`     x
#>           <dbl> <dbl>
#> 1             1     1
#> 2             1     2
#> 3             1     3
model_matrix(df, y ~ I(x^2) + x)
#> # A tibble: 3 x 3
#>   `(Intercept)` `I(x^2)`     x
#>           <dbl>    <dbl> <dbl>
#> 1             1        1     1
#> 2             1        4     2
#> 3             1        9     3

Transformations are useful because you can use them to approximate non-linear functions. If you've taken a calculus class, you may have heard of Taylor's theorem which says you can approximate any smooth function with an infinite sum of polynomials. That means you can use a polynomial function to get arbitrarily close to a smooth function by fitting an equation like y = a_1 + a_2 * x + a_3 * x^2 + a_4 * x ^ 3. Typing that sequence by hand is tedious, so R provides a helper function: poly():

model_matrix(df, y ~ poly(x, 2))
#> # A tibble: 3 x 3
#>   `(Intercept)` `poly(x, 2)1` `poly(x, 2)2`
#>           <dbl>         <dbl>         <dbl>
#> 1             1     -7.07e- 1         0.408
#> 2             1     -7.85e-17        -0.816
#> 3             1      7.07e- 1         0.408

However there's one major problem with using poly(): outside the range of the data, polynomials rapidly shoot off to positive or negative infinity. One safer alternative is to use the natural spline, splines::ns().

library(splines)
model_matrix(df, y ~ ns(x, 2))
#> # A tibble: 3 x 3
#>   `(Intercept)` `ns(x, 2)1` `ns(x, 2)2`
#>           <dbl>       <dbl>       <dbl>
#> 1             1       0           0    
#> 2             1       0.566      -0.211
#> 3             1       0.344       0.771

Let's see what that looks like when we try and approximate a non-linear function:

sim5 <- tibble(
x = seq(0, 3.5 * pi, length = 50),
y = 4 * sin(x) + rnorm(length(x))
)

ggplot(sim5, aes(x, y)) +
geom_point()

I'm going to fit five models to this data.

mod1 <- lm(y ~ ns(x, 1), data = sim5)
mod2 <- lm(y ~ ns(x, 2), data = sim5)
mod3 <- lm(y ~ ns(x, 3), data = sim5)
mod4 <- lm(y ~ ns(x, 4), data = sim5)
mod5 <- lm(y ~ ns(x, 5), data = sim5)

grid <- sim5 %>% 
data_grid(x = seq_range(x, n = 50, expand = 0.1)) %>% 
gather_predictions(mod1, mod2, mod3, mod4, mod5, .pred = "y")

ggplot(sim5, aes(x, y)) + 
geom_point() +
geom_line(data = grid, colour = "red") +
facet_wrap(~ model)

Notice that the extrapolation outside the range of the data is clearly bad. This is the downside to approximating a function with a polynomial. But this is a very real problem with every model: the model can never tell you if the behaviour is true when you start extrapolating outside the range of the data that you have seen. You must rely on theory and science.