… most companies would be taking a huge step forward if they got somebody who knows how to do linear regression.

— Hacker News user ‘mindcrime’ on the necessary skills for data science

When you have predictors of a scalar-valued outcome with observations indexed by and residuals denoted , a model of the form

or (equivalently) in matrix notation

is best^{[1]} estimated using ordinary least squares, the workhorse of linear regression. The underlying math is a fair bit of matrix algebra which, when all is said and done, returns

**This was the one equation my graduate school program director urged every student to know by heart.**

*X prime X inverse X prime y*, “prime” meaning transpose, yields OLS estimates of linear regression coefficients. On the left-hand side, the hat symbol denotes an estimate from a sample as opposed to a true value in the population.

We can walk through a real-data example using `mtcars`

, an automobile-themed dataset built into R. Regressing fuel economy (mpg) on weight (wt) and number of cylinders (cyl),

by

summary(lm(mpg ~ wt + cyl, data=mtcars))

will give you

Estimate Std. Error t value Pr(>|t|) (Intercept) 39.6863 1.7150 23.141 < 2e-16 *** wt -3.1910 0.7569 -4.216 0.000222 *** cyl -1.5078 0.4147 -3.636 0.001064 **

wherein the estimates (first column of numbers) from top to bottom correspond to

Per the foregoing all-important equation, only two objects are necessary to compute the estimates manually: (1) the matrix and (2) the vector . Both are easy to extract from `mtcars`

.

## getting `X` the easy but obfuscated way X = model.matrix(mpg ~ wt + cyl, data=mtcars) ## getting `X` a more transparent way: ## grab only the `wt` and `cyl` columns from mtcars ## and prepend a column of ones to represent the intercept X = cbind(1, mtcars[, c("wt", "cyl")]) X = as.matrix(X) # transform data.frame to matrix type ## getting `y` y = mtcars[, "mpg"]

Given and , all that’s left are matrix operations. Mathematically they are described in this tutorial from Harvey Mudd College. Computationally, `t()`

transposes, `solve()`

inverts, and `%*%`

multiplies matrices.

Now for the moment of truth—*X prime X inverse X prime y*.

solve(t(X) %*% X) %*% t(X) %*% y

[,1] 1 39.686261 wt -3.190972 cyl -1.507795

This formula doesn’t get us *p*-values but who needs those anyway. š

^{[1]}OLS is BLUE—the best linear unbiased estimator—under certain assumptions that are very important but beyond the scope of this post.