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