Transformations and the error standard deviation

In a scatterplot of Y against X, transforming X moves the crosses horizontally, but does not affect the spread of response values at each value of X.

If the error standard deviation is the same for each x in a plot of Y against X, it will also be constant in a plot of Y against any transformation of X.

Transformation of X therefore does not affect whether or not the linear model's assumption of constant error standard deviation holds.

However,

Transformation of the response, Y, not only affects linearity of the relationship, but also affects whether or not the error standard deviation is constant.

This is more easily explained in an example than with words.

Toxicity of syphilis drug

The scatterplot below results from an experiment in which the toxicity of neoarsphenamine, a drug that was once used for the cure of syphilis, was tested in mice. Each mouse was given an intravenous injection of a solutions containing one of seven different concentrations of the drug. The grey line is the least squares line fitted to the data.

The residual plot on the right highlights two problems. Most importantly, the standard deviation of the survival time is lower at high drug concentrations. There is also a suggestion of curvature — the survival time would be expected to level off at high doses since it is impossible to get negative survival times.

Drag the red line on the vertical axis upwards to apply a power transformation to the survival time. Observe that a power of -0.5 both linearises the relationship reasonably well and also gives residuals with fairly constant spread. A log transformation (power between 0.01 and -0.01) also works reasonably well. (Use the arrow keys on the keyboard to make fine adjustments to the power.)


Fortunately, the same transformation of the response that linearises the relationship often also results in fairly constant error standard deviation.