Transformation and Weighting

By the early 1980s, regression with homoscedastic errors was well understood, but methodology for handling heteroscedastic noise was just being developed. There were two general approaches. In the first, ​​​ studied by Carroll and Ruppert (1981 [TW-1], 1984 [TW-3]), the response is transformed to homoscedasticity. In the second, studied by Carroll and Ruppert (1982 [TW-2]) and Davidian and Carroll (1987 [TW-4]), one uses a variance function that ​​​ specifies the conditional variance of the response given the covariates. Transformation has the added feature that it can also reduce skewness of the errors, but transformation is useful only when the conditional variance is of a special form and, in particular, is a function of the conditional mean; this is a common occurrence, but there are many applications where it does not occur. Transformation and variance functions can be combined into a very general methodology as described ​​​ briefly below.

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