# What is the meaning of 'regression' in 'linear regression'?

I can see why linear regression is linear, i.e., because it is represented by a line, but what does regression have to do with the term as a whole?

What is the meaning this word contributes to the term?

• wiki said : The term "regression" was coined by Francis Galton in the nineteenth century to describe a biological phenomenon. The phenomenon was that the heights of descendants of tall ancestors tend to regress down towards a normal average. – user354674 Jul 28 '16 at 12:39
• What a coincidence. I was about to do some reading on regression. I think we have Galton to thank for the term. – Em. Jul 28 '16 at 12:39
• Galton later observed that the heights of ancestors of tall descendants also tended to be closer to the average, but by that stage his Eugenicist regression had been adopted by other statisticians – Henry Jul 29 '16 at 8:30

## 2 Answers

Semi-intuitive explanation:

Assume that we are interested in one's IQ. Not in a score that he might get in some IQ test, rather in his true IQ value. So, we have to assume that there exists such a value. Lets denote it by $\mu$. However, it is impossible to measure it directly. As such, we can use IQ tests to estimate it. Denote by $X_i$ his score in the $i$th test. We can model this score by $X_i = \mu +\epsilon_i$, where $\epsilon_i$ is the random error of the $i$th test with $\mathbb{E}\epsilon_i = 0$ and $var(\epsilon_i) = \sigma^2$ . That is, the score of the $i$th test is composed of his real value (signal) and some random error (noise). Because $\mathbb{E}X_i=\mu$, his score will (in some sense) tend to his real IQ value. Therefore, after $n$ such tests we will take his average score as the estimator of this value. This average will indeed tend to $\mu$ in a sense that as larger the number of tests that he takes - the more accurately will the sample average estimate the real IQ.

What exactly is the random error is more philosophical rather than statistical question. That is, it may stem from some imperfections of the tool (test) that may vary in its difficulty or it may stem from some subject related factors (tiredness, mood etc.) or even may be some inherent property of the IQ itself (i.e., it is not a scalar but rather a random variable. Then you may be interested in its mean value.). Or some combination of the above.

Formally: The linearity of a regression model does not mean that it is straight line. Namely, any model that can be written in matrix notations as $$Y=X\beta +\epsilon,$$
is called linear. Special cases like $y=\beta x +\epsilon$ and $y=\beta_0 + \beta_1x +\epsilon$ are indeed estimated by straight lines. Or can be viewed as straight lines (signal) that is interrupted by some noise $\epsilon$. Linearity is defined by independence of the first derivative of $Y$. I.e., if $\partial y / \partial \beta_j = x_j$, $j=1,...,p$ with $x_1 = 1$, then it is linear. If at least one of the derivatives depends on the parameters, then it is non-linear. While, the observations may be interpret as random fluctuations around the mean $\mathbb{E}Y = X\beta$, where $\mathbb{E}\epsilon = 0$. The estimation methods are try to estimate this means by minimizing (mostly) the squared error, i.e., the (squared empirical) deviation from this unknown mean.

• Where in your answer did you actually answer my question? – Michael Smith Jul 29 '16 at 12:51
• Regression -> regression to the mean. – V. Vancak Jul 29 '16 at 12:57

The idea of linear regression is to infer from dataset the dependence of some random variable in other variable, what the term regression mean in this context is that from the dataset we regress to some kind of dependency that brought us to that data.

• yes ! it's what we must understand. But Galton introduced it with another meaning. – user354674 Jul 28 '16 at 12:41
• Could you please elaborate a little bit? I can't seem to grasp what you're talking about. – Michael Smith Jul 28 '16 at 15:23
• not false and the answer recalls what we need to remember – user354674 Jul 28 '16 at 22:39