Bias in risk estimation: application to Down's syndrome screening

In this paper we consider the bias associated with parametric estimation of a univariate or bivariate Gaussian density, and also the induced bias when these Gaussian densities are used to determine a likelihood ratio. Algebraic approximations are derived that accurately predict the relative biases obtained, verification being achieved by a simulation exercise. The expressions confirm that when estimating a univariate Gaussian density there are four Z-scores for which there is zero bias and that relative bias increases rapidly beyond two standard deviations from the mean. The results are then extended to determine approximate confidence intervals for both the true density and the likelihood ratio. A simulation exercise confirms that the derived 95 per cent confidence intervals have coverage that ranges from 94 to 97 per cent. The results are applied to a Down's syndrome screening programme where 95 per cent confidence intervals are established for a woman's posterior odds of carrying a Down's foetus. It is shown that patients with similar posterior odds can give rise to confidence intervals for their true posterior odds that have very different widths, thus emphasizing that not all risks are of equal quality.