I've been noticing that statisticians might be using the term "linear" in a bit sloppy way.

A linear model is a model of the form:

$$Y = \beta_1X_1+\beta_2 X_2 + \cdot \cdot \cdot+ \beta_nX_n$$

Which in mathematical terms, is a linear combination (betas are scalars, $X$s are vectors).

However, statisticians allow a linear model to have other than $\le 1$ order polynomial terms. Such as quadratic terms $X_i^2$. In mathematics, polynomials of order $> 1$ are not linear functions. To me this sounds like the statistics concept of a linear model is a bit sloppy.

What do you think of the terms and how do you interpret "linearity" in linear models? Is it the same as in mathematics?


Actually it only means that the model is linear in the parameters (beta). The important part is that there exists a linear relationship between the data points in X. Any transformation of X is ok as long as X is linear in the parameters (and more strictly the other regression assumptions hold).


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