The R Book

Misspecified Link Function

Although each error structure has a canonical link function associated with it (see p. 514), it is quite possible that a different link function would give a better fit for a particular model specification. For example, in a GLM with normal errors we might try a log link or a reciprocal link using quasi to improve the fit (for examples, see p. 513). Similarly, with binomial errors we might try a complementary log-log link instead of the default logit link function (see p. 572).

An alternative to changing the link function is to transform the values of the response variable. The important point to remember here is that changing the scale of y will alter the error structure (see p. 327). Thus, if you take logs of y and carry out regression with normal errors, then you will be assuming that the errors in y were log normally distributed. This may well be a sound assumption, but a bias will have been introduced if the errors really were additive on the original scale of measurement. If, for example, theory suggests that there is an exponential relationship between y and x,

then it would be reasonable to suppose that the log of y would be linearly related to x:

Now suppose that the errors in y are multiplicative with a mean of 0 and constant variance, like this: ...