Linearising the relationship between Y and X
Even if X and Y are nonlinearly related, some transformation of X may be linearly related to some transformation of Y. For example, the data may satisfy the model
y2 = β0 + β1 log x + ε
ε ~ normal (0 , σ)
The parameters of this model could again be estimated by least squares, based on the transformed values of the two variables, and confidence intervals and hypothesis tests would be valid.
Example: Transformation of X
For this data set, a linear model seems reasonable after a log transformation of X.