We need to choose the correct regression line because each line is designed to minimise a different

Question

We need to choose the correct regression line because each line is designed to minimise a different type of prediction error. When we use $x$ to estimate $y$, the regression line of $y$ on $x$ is chosen because it makes the errors in the predicted $y$-values as small as possible.

On the other hand, when we use $y$ to estimate $x$, the regression line of $x$ on $y$ is used because it minimises the errors in the predicted $x$-values. Therefore, the choice of regression line depends on which variable we are trying to predict. Using the wrong line may not give the smallest possible errors for the variable of interest.

Scenario	Given $x$ to estimate $y$	Given $y$ to estimate $x$
$y$ depends on $x$	Given $x$, use $y$ on $x$ to estimate $y$. Given $y$, use $x$ on $y$ to estimate $x$.
$x$ depends on $y$	Given $x$, use $y$ on $x$ to estimate $y$. Given $y$, use $x$ on $y$ to estimate $x$.
No observable dependence	Use the $y$ on $x$ line equation.	Use the $x$ on $y$ line equation.

Accepted Answer

Choose the regression of $y$ on $x$ to predict $y$ from a given $x$, and the regression of $x$ on $y$ to predict $x$ from a given $y$. This choice follows the prediction direction, regardless of a suggested causal direction.

Scenario	Given \(x\) to estimate \(y\)	Given \(y\) to estimate \(x\)
\(y\) depends on \(x\)	Given \(x\), use \(y\) on \(x\) to estimate \(y\). Given \(y\), use \(x\) on \(y\) to estimate \(x\).
\(x\) depends on \(y\)	Given \(x\), use \(y\) on \(x\) to estimate \(y\). Given \(y\), use \(x\) on \(y\) to estimate \(x\).
No observable dependence	Use the \(y\) on \(x\) line equation.	Use the \(x\) on \(y\) line equation.

Choice of Regression Lines

Choice of Regression Lines

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