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When we say, to regress $Y$ against $X$, do we mean that $X$ is the independent variable and Y the dependent variable? i.e. $Y =aX + b$.

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    $\begingroup$ It depends on the person talking, unfortunately. I think "I regressed Y on X" more commonly means Y is the left hand side variable, but some people mean the opposite. $\endgroup$
    – Bill
    Commented Dec 4, 2014 at 21:48
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    $\begingroup$ Nearly always, yes ... but you probably mean E(Y)=aX+b, otherwise you don't need regression at all (since if you really meant the equality you gave, every point would be on the line). $\endgroup$
    – Glen_b
    Commented Dec 5, 2014 at 0:27
  • $\begingroup$ > Personally, I don't find the independent/dependent variable language to be that helpful. Those words connote causality, but regression can work the other way round too (use Y to predict X). The independent/dependent variable language merely specifies how one thing depends on the other. Generally speaking it makes more sense to use correlation rather than regression if there is no causal relationship. If one thing is not causing the other, there is not much point in using it to predict the other thing (at least not from a scientific standpoint), and simply inverting the relation whenever you $\endgroup$
    – user96247
    Commented Nov 25, 2015 at 15:00
  • $\begingroup$ The isn't much substantive difference between correlation and regression; certainly nothing to do with causation. $\endgroup$ Commented Nov 25, 2015 at 15:07
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    $\begingroup$ This is only part of the story. Causation and prediction don't go hand in hand even in science. For example, a large chunk of the environmental sciences is devoted to using effects to predict or infer causes, e.g. past temperatures from proxies which are affected by temperature. Sometimes the mutual predictability of two variables is of interest regardless of causation, e.g. with different measures of the "same" property. Even if two variables are on the same footing, there can be linear fits that don't depend on distinguishing different roles for $y$ and $x$ (reduced major axis, etc.) $\endgroup$
    – Nick Cox
    Commented Nov 25, 2015 at 15:25

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It typically means finding a surface parametrised by known X such that Y typically lies close to that surface. This gives you a recipe for finding unknown Y when you know X.

As an example, the data is X = 1,...,100. The value of Y is plotted on the Y axis. The red line is the linear regression surface.

enter image description here

Personally, I don't find the independent/dependent variable language to be that helpful. Those words connote causality, but regression can work the other way round too (use Y to predict X).

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Probably, Yes. Many times we need to regress a variable (say Y) on another variable (say X). In Regression, it can therefore be written as $Y = a+bX$; regress Y on X: regress true breeding value on genomic breeding value, etc.

bias=lm(TBV~GBV)
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As long as you keep in mind that when x is regressed against y, y is a proposed dependent variable and x is a proposed independent variable, there's no problem.

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