Summary: A well-known and useful method for generalised regression analysis when a linear covariate \(x\) is available only through some approximation \(z\) is to carry out more or less the usual analysis with \(E(x\,|\,z)\) substituted for \(x\). Sometimes, but not always, the quantity var\((x\,|\,z)\) should be used to allow for overdispersion introduced by this substitution. These quantities involve the distribution of true covariates \(x\), and with some exceptions this requires assessment of that distribution through the distribution of observed values \(z\). It is often desirable to take a nonparametric approach to this, which inherently involves a deconvolution that is difficult to carry our directly.
However, if covariate errors are assumed to be multiplicative and log-normal, simple but accurate approximations are available for the quantities \(E(x^k\,|\,z)\) \((k=1,2, \dots)\). In particular, the approximations depend only on the first two derivatives of the logarithm of the density of \(z\) at the point under consideration and the coefficient of variation of \(z\,|\,x\). The methods will thus be most useful in large-scale observational studies where the distribution of \(z\) can be assessed well enough in an essentially nonparametric manner to approximate adequately those derivatives. We consider both the classical and Berkson error models. This approach is applied to radiation dose estimates for atomic-bomb survivors.