Simple Linear Regression - Transformations

$Figure\; 5\; shows\; a\; graph\; of\; data\; for\; which\; a\; simple\; linear\; least\; squares\; fit\; would\; be\; inappropriate.$

A line fit to these data would produce overestimates of Y for small Xs and underestimates of Y for large Xs. The trend would be obvious in a plot of studentized residuals versus X. In such circumstances it is often possible to still apply linear least squares methodology, only to transformations of Y and/or X rather than Y and X themselves. For the data in figure 5 transforming X to lnX does the trick (fig. 6).

The appropriate equation to fit would thus be

Y = b₀ + b₁ (lnX)

and the formulas presented earlier would be applied to Y and lnX to estimate b₀ and b₁.

In some instances theoretical knowledge of the process or phenomenon behind the relationship between Y and X will lead to the appropriate linearizing transformation(s). For example, for many biological organisms the size of various organism parts have been shown to be related by

Y = a X^b

which is known as the allometric equation. Data conforming to this equation may be analyzed using linear least squares methodology applied to

lnY = b₀ + b₁ lnX

where b₀ = ln a and b₁ = b. Formulas presented earlier would be applied to lnY and lnX to estimate b₀ and b₁.

Unfortunately, in most circumstances theoretical knowledge is lacking and different combinations of transformations must be tried until an appropriate one is discovered. Charts like those in appendix table 10 of your textbook can be helpful in choosing a linearizing transformation.

Transformations are sometimes employed to make assumption 6. more believable as well.