The log of a predictor and polynomial regression

Question

I’m working with primate brain data as a predictor in regression models. In the primate brain literature it is custom to log brain data, but it is unclear to me why. It has been argued that since one gram brain tissue means different things to primates of different size, it is the relative brain volume (among species) that is interesting, hence taking the log (does that make sense?).

By taking the log you enhance the impact of differences in small values in a regression, and the opposite for larger values?

In another post it has been argued that logging a predictor can be approximated with a polynomial regression such that

$𝛽_{0}+𝛽_{1}log(𝑃)≈𝛽_{0}+𝛽_{1}(𝑃−1)−𝛽_{1}(1/2)(𝑃−1)^2+𝛽_{1}(1/3)(𝑃−1)^3−𝛽_{1}(1/4)(𝑃−1)^4\\=𝛽^∗_0+𝛽^∗_1𝑃+𝛽^∗_2𝑃^2+𝛽^∗_3𝑃^3+𝛽^∗_4𝑃^4$

If this is correct it means that logging a predictor correspond to a certain hypothesis about the relationship between X and Y. And that’s all it means, or am I missing something?

I have been trying to illustrate this and this is the code I came up with. However, I was expecting the regression lines to be more similar. Am I doing it right?

x <- rlnorm(100, 1)
log.x = log(x)
y <- rnorm(100, x, 5)

a = x-1
b = (1/2*(x-1))^2
c = (1/3*(x-1))^3
d = (1/4*(x-1))^4

f1 <- lm(y ~ a + b + c + d)
f2 <- lm(y ~ log.x)

a <- seq(min(a), max(a), length.out=1000)
b <- seq(min(b), max(b), length.out=1000)
c <- seq(min(c), max(c), length.out=1000)
d <- seq(min(d), max(d), length.out=1000)
t <- predict(f1, newdata = data.frame(a = a, b = b, c = c, d = d))

plot(y ~ log.x)
lines(log(a), t, col = "red")

s <- seq(min(log.x), max(log.x), length.out=1000)
t <- predict( f2, newdata = data.frame(log.x = s))
lines(s, t, col = "blue")

Answer 1

Much of your question is addressed extensively on this page. Basically, log transforms bring out fractional rather than arithmetic differences in original variable values, whether the variable is a predictor or and outcome.

So yes, it does make sense to take logs if it is "relative brain volume (among species) that is interesting." And it is true that "logging a predictor corresponds to a certain hypothesis about the relationship between X and Y." In particular, the hypothesis (if X is the predictor and Y is the outcome) is that equal fractional changes in X are associated with equal additive changes in Y.

I haven't checked your code in detail. Note that the original sampling of $x$ was from a lognormal distribution itself rather than from a normal distribution, with a meanlog value of 1 (corresponding to $x=e$ on a linear scale). The Taylor approximation you use for $x$ holds in the vicinity of $x=1$, which might be part of your problem.

Answer 2

See below for a modification to your code. Some comments:

• Your f1 model estimates separate coefficients for each of a, b, c and d, so this is not a faithful demonstration of the Taylor series expansion.
• You had a small error in your implementation of the Taylor series. You shouldn't apply the polynomial term to the fraction but rather only to the $P-1$ term.
• The true data generating model has $E[Y|X=x] = x$, but you are fitting models that regress $Y$ against $\log(x)$ or an approximation thereof, so it's not reasonable to expect that the lines that you plot, which demonstrate your fitted models, will recreate the points that you plot, which illustrate the true model.
• As EdM points out, your log-normal generating distribution can have values that are quite far from $X=1$, where the Taylor series expansion is centered at, and so the approximation quickly deteriorates. I had to decrease the variation in the predictor in order to create a sensible approximation.

x <- rnorm(5e2, mean = -0.5, sd = 0.5) # this is log(P)
y <- rnorm(length(x), x, 0.3)

taylor.log = function(x) 
{ 
  (x - 1) - 
    (x - 1)^2 / 2 + 
    (x - 1)^3 / 3 - 
    (x - 1)^4 / 4
}

f1 <- lm(y ~ x)
f2 <- lm(y ~ taylor.log(exp(x)))

new_x = seq(quantile(x, p = 0.025), 
            quantile(x, p = 0.975), 
            length = 1001);

yhat1 <- predict(f1, newdata = data.frame(x = new_x))
yhat2 <- predict(f2, newdata = data.frame(x = new_x))

plot(x, y, cex = 0.5, pch = 16)
lines(new_x, yhat1, col = "red", lwd = 2)
lines(new_x, yhat2, col = "blue", lwd = 2)