Abstract

This supplementary material contains the two appendix for the blog post Retracing Prenatal Testing Algorithms, which contains an implementation of the Merz et al. (2016) method for the risk calculation for Down syndrome based on combined first trimester testing. Appendix 1 contains a detailed description of how the likelihood ratio is computed based on the concept of “degree of extremeness” of an observation. We exemplify this by showing the computations for the NT measurement. Appendix 2 contains an account of our difficulties to reproduce the NT curve given in Fig. 3 of the paper.

This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License. The markdown+Rknitr source code of this blog is available under a GNU General Public License (GPL v3) license from github. The specific source file of this appendix is _drafts/fts-appendix.Rmd.

Appendix 1 - Computing the Likelihood Ratios

For each biomarker the likelihood ratio between the euploid and uneuploid population is in Merz et al. (2016), after a possible suitable transformation, computed as a ratio between two Gaussian densities. To this end Merz et al. (2008) introduced the concept of “degree of extremeness” (DoE) of an observation.

The degree of extremeness of an observation

Let $$x$$ denote the crown-rump-length (CRL) of a fetus and let $$y$$ denote the measured biomarker value, say, NT. The DoE is then defined as $\operatorname{DoE}(y) = \left\{ \begin{array}{ll} \displaystyle{\frac{y-\hat{y}(x)}{y^*(x) - \hat{y}(x)}} & \text{if } y \geq \hat{y}(x) \\ \displaystyle{-\frac{\hat{y}(x)-y}{\hat{y}(x) - y_*(x)}} & \text{if } y < \hat{y}(x) \end{array} \right. ,$ where $$\hat{y}(x)$$ is the predicted value (i.e. the conditional expectation) in a polynomial regression model regressing NT on CRL and $$y^*(x)$$ and $$y_*(x)$$ are the so called reference bands from the regression model. These are obtained by subtracting a value $$\sigma_*$$ from $$\hat{y}(x)$$ and adding a value $$\sigma^*$$ to $$\hat{y}(x)$$ in order to get the 5% and 95% quantiles of the distribution, respectively. Note that this means that the distribution of NT does not have to be symmetric around the mean, but since $$\sigma_*$$ and $$\sigma^*$$ are fixed values, the quantiles are assumed to be parallel over the whole range of CRL. There are thus similarities with quantile regression based on a generalized model for location, scale and shape (GAMLSS), which would have been an alternative approach towards the problem and without the discontinuity the $$y\leq \hat{y}(x)$$ condition gives (Fenske et al. 2008). Since we were unable to obtain any kind of reasonable curve with the 10th order polynomial specified in the upper right column of p. 22 of Merz et al. (2016) (see Appendix 2), we instead inverse graphed the curve from the Fig. 3 of the article using the open-source Engauge Digitizer program.