# What is "empirical regression"?

I was asked how much I know of "empirical regression." I have never heard the expression. A web search yielded nothing useful. I suspect it is a term coined by someone to refer to some ad hoc procedure which might be known by other name. Has anyone have a reference?

• On the contrary, I suspect it's just regular old regression, with an emphasis on empirical use rather than its theoretical properties. Aug 1 '14 at 4:16
• Could you provide us with a little more context? Aug 1 '14 at 6:56
• Context would be helpful: I suspect it's the model that's empirical rather than some special regression technique's being used. An empirical model is one flexibly parametrized to cope well enough with the unknown form of the true relationship between predictors & response (e.g. using polynomials or splines to allow for non-linearity); a theoretical model is one whose form & parameters are determined by theory (e.g. estimating $V_\mathrm{max}$ & $K_\mathrm{M}$ for the Michaelis-Menten kinetic model from substrate concentration & initial rates). Aug 1 '14 at 9:10
• They might mean nonparametric regression, e.g. kernel regression or as others have said, using things like splines. Probably best to respond back to that person, no I haven't heard of it - can you explain what it is in more detail? Aug 1 '14 at 19:51
• The context does not help. I was being probed on my modeling background. I was given a laundry list of of topics to self-asses my experience with them. One of the items was "Regression (empirical and Bayesian)." I answered I hadn't heard of "empirical regression." When I search for the term I could not find anything. Thanks for the answers. Aug 14 '14 at 18:09

## 2 Answers

Empirical regression is a technique in non-parametric regression. The key idea is to estimate the joint density of X, Y, denotes p(X,Y). From this joint density one can derive conditional p(Y|X), and conditional mean E(Y|X). This E(Y|X) is empirical regression. This empirical regression technique was proposed by Schmerling and Peil (1985).

It is described in Hardle (1990): Applied Nonparametric Regression. Here is a link to the page

Here is the original reference

• Is there an R package which (reliably) implements empirical regression? Dec 11 '19 at 13:28

If this was in the context of cognitive psychology or especially psycholinguistics and/or eye-tracking, they may have been talking about linear regression with an emirical logit transformation, which is a way of avoiding having to do logistic regression.

Empirical logit regression involves creating bins of observations and applying a modified logit transformation that adds $\frac1{2n}$ successes to each bin to avoid (negative and positive) infinitity for bins with $0$ successes or $0$ failures, then doing linear regression using a weighted least squares.

This is potentially attractive in some situations where it's hard to fit logistic regression, and there have been some arguments for using it with data where the sample rate exceeds the rate of the phenomenon you're sampling (as with modern eye-trackers, which far exceed human saccade frequency). There are some drawbacks to not actually doing logistic regression though.