J K Morris1, N J Wald. 1. Wolfson Institute of Preventive Medicine, Barts and the London Queen Mary's School of Medicine and Dentistry, London EC1M 6BQ, UK. j.k.morris@qmul.ac.uk
Abstract
OBJECTIVE: The screening performance of tests involving multiple markers is usually presented visually as two Gaussian relative frequency distributions of risk, one curve relating to affected and the other to unaffected individuals. If the distribution of the underlying screening markers is approximately Gaussian, risk estimates based on the same markers will usually also be approximately Gaussian. However, this approximation sometimes fails. Here we examine the circumstances when this occurs. SETTING: A theoretical statistical analysis. METHODS: Hypothetical log Gaussian relative distributions of affected and unaffected individuals were generated for three antenatal screening markers for Down's syndrome. Log likelihood ratios were calculated for each marker value and plots of the relative frequency distributions were compared with plots of Gaussian distributions based on the means and standard deviations of these log likelihood ratios. RESULTS: When the standard deviations of the distributions of a perfectly Gaussian screening marker are similar in affected and unaffected individuals, the distributions of risk estimates are also approximately Gaussian. If the standard deviations differ materially, incorrectly assuming that the distributions of the risk estimates are Gaussian creates a graphical anomaly in which the distributions of risk in affected and unaffected individuals plotted on a continuous risk scale intersect in two places. This is theoretically impossible. Plotting the risk distributions empirically reveals that all individuals have an estimated risk above a specified value. For individuals with more extreme marker values, the risk estimates reverse and increase instead of continuing to decrease. CONCLUSION: It is useful to check whether a Gaussian approximation for the distribution of risk estimates based on a screening marker is valid. If the value of the marker level at which risk reversal occurs lies within the set truncation limits, these may need to be reset, and a Gaussian model may be inappropriate to illustrate the risk distributions.
OBJECTIVE: The screening performance of tests involving multiple markers is usually presented visually as two Gaussian relative frequency distributions of risk, one curve relating to affected and the other to unaffected individuals. If the distribution of the underlying screening markers is approximately Gaussian, risk estimates based on the same markers will usually also be approximately Gaussian. However, this approximation sometimes fails. Here we examine the circumstances when this occurs. SETTING: A theoretical statistical analysis. METHODS: Hypothetical log Gaussian relative distributions of affected and unaffected individuals were generated for three antenatal screening markers for Down's syndrome. Log likelihood ratios were calculated for each marker value and plots of the relative frequency distributions were compared with plots of Gaussian distributions based on the means and standard deviations of these log likelihood ratios. RESULTS: When the standard deviations of the distributions of a perfectly Gaussian screening marker are similar in affected and unaffected individuals, the distributions of risk estimates are also approximately Gaussian. If the standard deviations differ materially, incorrectly assuming that the distributions of the risk estimates are Gaussian creates a graphical anomaly in which the distributions of risk in affected and unaffected individuals plotted on a continuous risk scale intersect in two places. This is theoretically impossible. Plotting the risk distributions empirically reveals that all individuals have an estimated risk above a specified value. For individuals with more extreme marker values, the risk estimates reverse and increase instead of continuing to decrease. CONCLUSION: It is useful to check whether a Gaussian approximation for the distribution of risk estimates based on a screening marker is valid. If the value of the marker level at which risk reversal occurs lies within the set truncation limits, these may need to be reset, and a Gaussian model may be inappropriate to illustrate the risk distributions.