

Regression after Linearisation

In cases where the direct application of linear regression is impossible (due to non-linearity of the data) one can try to linearize the data before calculating the regression:

First, transform the curvilinear model to a linear model by applying a proper transformation to both the independent and the dependent variable. For the univariate case, you may visually check the linearity after the transformation by plotting the transformed variables against each other. Next, the regression parameters for the linearized model have to calculated; these parameters are then transformed back to the original (curvilinear) function.

Below is a table of the transformations for linearizing some common relationships.

Non-Linear Model Linearized Model Back Transformation
y = abx lg y = a* + b*x a = 10a* b = 10b*
y = axb lg y = a* + b* lg x a = 10a* b = 10b*
y = aebx ln y = a* + b*x a = ea* b = b*
y = ae(b / x) ln y = a* + b* (1/x) a = ea* b = b*
y = a + b/x y = a* + b* (1/x) -- y is plotted against (1/x) a = a* b = b*
y = a / (b + x) (1/y) = a* + b*x a = b/a* a = 1/b*
y = a + bxn y = a* + b*xn a = a* b = b*

Please note that the regression parameters obtained after the linearisation do not exactly match those obtained from a direct application of the method of least squares because the linearisation changes the metrics of the data space. Furthermore, the assumption of homoscedasticity may not be fulfilled in both approaches.

Last Update: 2012-10-08