How important is that variable?

When modeling any phenomena by including explanatory variables that highly relates the variable of interest, one question arises: which of the auxiliary variables have a higher influence on the response? I am not writing about significance testing or something like this. I am just thinking like a researcher who wants to know the ranking of variables that influence the response and their related weight.

There are a variety of methods that try to answer that question. The one inducing this thread is very simple: isolate units from variables. Assume a linear model with the following structure (for the sake of simplicity, assume only two explanatory variables):

$$y = \beta_1 x_1 + \beta_2 x_2 + \varepsilon$$

If you assume this model as true and $\beta_i > 0$, then the influence of variable $x_i$, over response $y$ could be found when isolating measure units from variables. Then, one could fit a model over the standardized variables (explanatory and response) and then directly comparing the regression coefficients. Another way to do this is by means of the following expression:

$$I(i) = \frac{\beta_i}{sd(\beta_i)} = \beta_i\frac{ sd(x_i)}{sd(y)}$$

For example, let’s consider the following model $y = -500 x_1 + 50 x_2 + \varepsilon$, then the relative importance of the first and second variable is around 500/(500+50) = 0.9, and 50/(500+50) = 0.1, respectively. The following code shows how to perform this simple analysis in R.

n <- 10000

x1 <- runif(n)
x2 <- runif(n)
y <- -500 * x1 + 50 * x2 + rnorm(n)

model <- lm(y ~ 0 + x1 + x2)

# 1a. Standardized betas
summary(model)$coe[,2]
sd.betas <- summary(model)$coe[,2]
betas <- model$coefficients
imp <- abs(betas)/sd.betas
imp <- imp/sum(imp)
imp

# 1b. Standardized betas
imp1 <- abs(model$coefficients[1] * sd(x1)/sd(y))
imp2 <- abs(model$coefficients[2] * sd(x2)/sd(y))

imp1 / (imp1 + imp2)
imp2 / (imp1 + imp2)

# 2. Standardized variables
model2 <- lm(I(scale(y)) ~ 0 + I(scale(x1)) + I(scale(x2)))
summary(model2)
abs(model2$coefficients)/sum(abs(model2$coefficients))