# Normal.VUS: Volume under ROC surface (VUS) and partial VUS estimate and... In DiagTest3Grp: Diagnostic test summary measures for three ordinal groups

## Description

Estimate volume under ROC surface (VUS), partial VUS and associated variance assuming a diagnostic test follows normal distribution in all three ordinal diagnosis groups D_-, D_0, D_+.

## Usage

 1 2 3  Normal.VUS(x, y, z, p = 0, q = 0, alpha = 0.05, subdivisions = 50000, lam.minus = 1/3, lam0 = 1/3, lam.plus = 1/3, typeIerror = 0.05, margin=0.05,FisherZ=FALSE,...) 

## Arguments

 x A numeric vector, a diagnostic test's measurements in the D- (usually healthy subjects). y A numeric vector, a diagnostic test's measurements in the D0 (usually mildly diseased subjects). z A numeric vector, a diagnostic test's measurements in the D+ (usually severely diseased subjects). p A numeric value, the minimum required specificity, 0<=p<1, for calculation partial volume under ROC surface. Default, p=0. q A numeric value, the minimum desired sensitivity,0<=q<1, for calculation partial volume under ROC surface. Default, q=0. p=q=0 will give the complete VUS estimate, otherwise give the partial VUS estimate satisfying specificity no less than p and sensitivity no less than q. alpha A numeric value, the (1-alpha)*100% Confidence interval of the VUS estimate under normal assumption. Default, alpha=0.05. subdivisions A numeric value, the number of subintervals for integration using adaptive quadrature in the R function integrate. Default, subdivisions=50000. lam.minus A numeric value, the expected population proportion of the D^- group, used for sample size calculation. Default, lam.minus=1/3. The proportions of the three ordinal groups (lam.minus,lam0,lam.plus) should sum to 1. lam0 A numeric value, the expected population proportion of the D^0 group, used for sample size calculation. Default, lam0=1/3. The proportions of the three ordinal groups (lam.minus,lam0,lam.plus) should sum to 1. lam.plus A numeric value, the expected population proportion of the D_+ group, used for sample size calculation. Default, lam.plus=1/3.The proportions of the three ordinal groups (lam.minus,lam0,lam.plus) should sum to 1. typeIerror A numeric value, (1-typeIerror)*100% confidence interval (CI) in sample size calculation. Default typeIerror=0.05, i.e., calculate 95% CI. margin A numeric value, the margin of error on the VUS estimates in sample size calculation. Default, margin=0.05. The (1-typeIerror)% CI on VUS estimate under normality is (VUS-Z_a*SE(VUS),VUS+Z_a*SE(VUS)), then margin= Z_a*SE(VUS) or half of the CI's length, where Z_a is the normal quantile, Z_a=1.95 given default typeIerror=0.05. FisherZ A logic value. Default=FALSE. If set to TRUE, the VUS estimate, associated variance, CI and sample size will be calculated on the logit scale, i.e., after the Fisher Z transformation. In sample size calculation, we place the margin of error "delta" on the CI of the VUS estimate (if FisherZ=FALSE). If VUS is close to 1, the resulting CI on the original scale will have insufficient nominal coverage and under-estimated sample size. Therefore, the VUS estimate, variance, CI and sample size calculation should be implemented on the logit scale via the Fisher Z transformation. ... Other arguments that can be passed to the R function integrate, e.g., abs.tol,rel.tol,stop.on.error etc.

## Details

For three ordinal group diagnostic test, there are two underlying cut-point t_- and t_+ with t_-<t_+ based on which patients are divided into the three ordinal groups. Patients with a diagnostic test below t_- will be assigned to D^-; those with the test above t_+ will be assigned to D^+ and the remaining fall into D^0. Following the specificifity and sensitity definition as in diagnostic test for two groups, we call the probability of the first two events as specificity x=Pr_-\{T ≤ t_-\}=F_-(t_-) and sensitivity.y=Pr_+\{T > t_+\}=1-F_+(t_+)=G_+(t_+) where the Pr_i and F_+ denotes the probability density function (PDF) and cumulative density function (CDF) of a diagnostic test in D^i, i=-,0,+ separately. Then, the probability that a patient randomly selected from the D^0 group has the test result between the two cut-points can be expressed as, z=P_0\{t_- ≤ T ≤ t_+\}=F_0(t_+)-F_0(t_-)=F_0(G_+^{-1}(y))-F_0(F_-^{-1}(x))

where the notation H^{-1}(.) denotes the inverse function of H. z is a function of the specificity and sensitivity, i.e., z=z(x,y), which constitutes a ROC surface in the three-dimensinal space (x,y,z). The volume under the ROC surface (VUS) defined by z can be written as,

V_{00}=\int\int_{D_{00}} {F_0(G_+^{-1}(y))-F_0(F_-^{-1}(x))}d_x d_y=\int_{-∞}^{+∞} F_-(s)G_+(s)f_0(s)d_s

The integration domain is D_{00}=\{0≤ x ≤ 1,0≤ y ≤ G_+(F_-^{-1}(x))\}. Partial VUS requires a marker has specificity at least p and sensitivity at least q. The equation of partial VUS will be similar to the above but the integration domain is D_{pq}=\{p≤ x ≤ 1,q ≤ y ≤ G_+(F_-^{-1}(x))\}.

Details on the sample size calculation in the original scale and in the logit scale (after Fisher's Z transformation) can be found in the reference papers.

## Value

A list of components, including

 dat A list of components “x”,“y”, “z”, recording the original marker measurement inputted into the function. dat.summary A numeric data frame with three rows (D^-,D^0, D^+) and three columns (“n”, “mu”,“sd”),summariz number of observations, mean and SD estimated for each diagnosis group from samples. estimate A numeric value. A numeric value. If p=q=0, VUS gives the complete VUS estimate, otherwise give the partial VUS estimate satisfying specificity no less than p and sensitivity no less than q. variance A numeric value. The variance of VUS estimate under normal assumption. CI A named numeric vector of length 2. The (1-alpha)*100% CI on VUS or partial VUS. sampleSize The sample size on investigating a diagnostic test based on the pilot measurments x, y, z. The sample size calculated such that based on given population proportion of the three diagnosis groups, the margin of error on the (1-typeIerror)*100% confidence interval of the VUS estimate under normality is within given margin of error (by default 0.05). partialDeriv A numeric data frame with one row and multiple columns, containing relevant parameters (a,b,c,d) and the partial derivatives of VUS estimate w.r.t the relevant parameters which are outputted for performance of statistical tests on markers under normal method or NA under nonparametric method.

## Note

Bug reports, malfunctioning, or suggestions for further improvements or contributions can be sent to Jingqin Luo <[email protected]>.

Jingqin Luo

## References

Xiong, C. and van Belle, G. and Miller, J.P. and Morris, J.C. (2006) Measuring and Estimating Diagnostic Accuracy When There Are Three Ordinal Diagnostic Groups. Statistics In Medicine 25 7 1251–1273. Luo, Jingqin and Xiong, Chengjie (2012) DiagTest3Grp: An R Package for Analyzing Diagnostic Tests with Three Ordinal Groups. Journal of Statistical Software (submitted)

VUS NonParametric.VUS NonParametric.VUS.var
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26  data(AL) group <- AL$group table(group) ##take the negated FACTOR1 marker measurements factor1 <- -AL$FACTOR1 x <- factor1[group=="D-"] y <- factor1[group=="D0"] z <- factor1[group=="D+"] ##Complete VUS vus <- Normal.VUS(x=x, y=y, z=z) ##complete VUS estimate vus$VUS ##variance of the complete VUS estimate vus$variance ##sample size vus\$sampleSize ##partial VUS with p>=0.4 and q>=0.5 partial.vus <- Normal.VUS(x=x, y=y, z=z,p=0.4, q=0.5)