Prediction using transformed variables

To predict Y at any value x, we use the least squares line that was fitted to the transformed data.

  1. Transform x
  2. Put this transformed x-value into the least squares equation to obtain a prediction of the transformed Y.
  3. Apply the inverse transformation to the one that was used on Y to obtain the prediction of Y itself

Example

In the example below, a linear model can be used to describe the relationship between log10(y) and log10(x). The diagram shows how it can be used to predict y from x.

Prediction intervals can be obtained in a similar way. A prediction interval is found using the transformed variables, then its end-points are back-transformed into values of Y on its original scale.

For example, if the square root of Y is linearly related to X, we find a prediction interval for sqrt(Y), then square both ends of this interval to get a prediction interval for Y itself.