The trinormal ROC model is a parametric model in three-class ROC
analysis. It is based on normality in each of the trhee classes D_-
(healthy), D_0 (intermediate) and D_+ (diseased) with denoted distributions
\(N(\mu_-,\sigma_-^2)\), \(N(\mu_0,\sigma_0^2)\) and
\(N(\mu_+,\sigma_+^2)\). A classifier of a trinormal ROC model classifies
individuals into one of the three ordered classes based on two cut-off points
\(c_- < c_+\). We define \(t_-=F_-(c_-)\) and \(t_+
=1-F_+(c_+)=G_+(c_+)\). Now, the ROC surface can be written as
$$ROCs(t_-,t_+) = \Phi \left(\frac{\Phi^{-1} (1-t_+) +d}{c} \right) -
\Phi \left(\frac{\Phi^{-1} (t_-)+b}{a} \right)$$
whith parameters a, b, c and c given by \(a =
\frac{\hat{\sigma}_0}{\hat{\sigma}_-}, b = \frac{ \hat{\mu}_- -
\hat{\mu}_0}{\hat{\sigma}_-}, c = \frac{\hat{\sigma}_0}{\hat{\sigma}_+}, d =
\frac{ \hat{\mu}_+ - \hat{\mu}_0}{\hat{\sigma}_+} \). It is a surface in the
unit cube that plots the probability of a measurement to get assigned to the
intermediate class as the two thresholds \(c_-,c_+\) are varying.
Based on the reference standard, the trinormal based ROC test can be used
to assess the discriminatory power of such classifiers. It distinguishes
between single classifier assessment, where a classifier is compared to some
hypothetical distributions in the classes, and comparison between two
classifiers. The latter case tests for equality between the parameters a, b,
c and d of the ROC curves. The data can arise in a unpaired or paired
setting. If paired is TRUE, a correlation is introduced which
has to be taken into account. Therefore the sets of the two classifiers have
to have classwise equal size. The data can be input as the data frame
dat or as single vectors x1, y1, z1, ....
As the Chi-squared test is by definition a one-sided test, the variable
alternative cannot be specified in this test. For this 'goodness of
fit' test, we assume the parameters \(a_1, \dots , d_1\) and \(a_2, \dots , d_2\) to have a
pairwise equivalent normal distribution (in large sample sets).