X on Y Regression Line
Equation of the Regression Line
The x on y regression line, also known as the inverse regression line, is a statistical tool used to predict the value of an independent variable (x) based on a given dependent variable (y). This is in contrast to the more commonly used y on x regression line, which predicts y based on x.
The equation for the x on y regression line is:
$x = a + by$
Where:
- $x$ is the independent variable
- $y$ is the dependent variable
- $a$ is the x-intercept
- $b$ is the slope of the line
It's important to remember that the x on y regression line is different from the y on x regression line. They will have different equations and slopes, except in the case of perfect correlation (r = 1 or r = -1).
There are two ways to calculate the gradient of a regression line
- You can either plot out the points on an axes and then draw out a line of best fit
- Put the values into the Statistics section of the GDC and then copy the data very carefully onto your exam paper
Using the Equation for Prediction
Once we have the equation of the x on y regression line, we can use it to predict values of x for given values of y. This is done by simply plugging the y value into the equation and solving for x.
Using the equation from our previous example: $x = 1 + y$
If we want to predict x when y = 6: $x = 1 + 6 = 7$
So when y = 6, we predict x to be 7.
When making predictions, it's generally best to stay within or close to the range of your original data set. Extrapolating too far beyond your data can lead to unreliable predictions.
Comparison with Y on X Regression
The x on y regression line differs from the y on x regression line in several key ways:
- Equation Form:
- Y on X: $y = mx + c$
- X on Y: $x = a + by$
- Prediction Direction:
- Y on X predicts y from x
- X on Y predicts x from y
