A new way for ranking functional data with applications in diagnostic test

Abstract

This is a two faces paper. Firstly, it investigates diagnostic tests in situations when the observed variables are functional, that is, diagnostic tests that use functional variables as biomarkers. A procedure based on functional version of ROC analysis is proposed, the main question being linked with a suitable way for ranking the sample of functional data. The second facet of this paper is to present a general new way for ordering functional data in a self-contained way allowing for a wide scope of applications overpassing the former diagnostic test problem. Finite sample analysis highlight how this ranking procedure behaves for diagnostic test.

Annex I: Supplementary material

1.1 Simulation study with only replication

We carried out an initial analysis based only on one replication to understand how the functional diagnostic test works on a particular dataset. Specifically, for the family of mean curves (F1), one trial with $n_i=30$ independent trajectories for each group was produced. In this case, the parameters $\sigma$ and $\sigma _{1}$ take values 0.04 and 0.06, respectively although other values could be consider. The smoothed curves, obtained according to it was indicated in Sect. 4, are shown in the left part of Fig. 7. In this situation, the curves of both groups present a similar shape although the data of Affected subjects take in general higher values than data of Non-Affected ones, such as we have supposed in Sect. 3.1.

Pairing the different location measures r.t with the different proximities measures d.t (see Table 2), we have obtained the corresponding empirical ROC curves and their AUC values, which are shown in Table 6.

Table 6 Area under each empirical ROC curve (AUC) and minimal distance from the empirical ROC curves to the point (0, 1) for several values of r.t and d.t

It can be seen that values of AUC, for $L_2$, Semi.PCA, Semi.hshift and Semi.mplsr $(d.t \in \{1,5,6,8\})$, are around 0.87, 0.87, 0.88 and 0.96 independently of representative curve type. For the other options of semi-metrics there are more variability depending of r.t. We observe that $r.t=4$ (CC) leads to the worst result, in terms of discriminating power, for all semi-metric. In addition, the best combination in terms of AUC is: FMean-Semi.Basis ($AUC=0.996$), whereas other reasonable results have been obtained combining FMean-Semi.Deriv ($AUC=0.988$), for example.

Also for each combination of (r.t, d.t), we compute $c_0$ as the value in $\mathbb {R}$ that minimizes the distance from corresponding ROC curve to the point (0, 1) (North-West corner criterion). Such minimum distances, shown in Table 6, lead the analogous results as those observed for AUC measures. Following, to get the cutoff curve $\chi _{c_0}$ and hence the diagnostic rule, we choose the combination (r.t, d.t) that maximizes the correspondent AUC, taht is $r.t=1$ (FMean curve) and $d.t=2$ (Semi.Basis with first deriv).

Once the parameters $d.t=2$ and $r.t=1$ have been chosen, the optimal cutoff curve $\chi _{c_0}$ is obtained and the diagnostic test is established. In this case, we achieve a $3.33\%$ of false positive and a $96.67\%$ of true positive ($(\alpha _{c_0}, 1-\beta _{c_0})=(0.033, 0.967)$). Finally, in the left plot of Fig. 7 the optimal cutoff curve $\chi _{c_0}$ (thin line in black) was added to sample data and the representative curves for each group (mean curves for A and NA). The right side of Fig. 7, is presenting an analogous graphic with the first derivatives of each curve (functional data, mean curves and optimal cutoff curve) because the selected semi-metric has been $d.t=2$ (Semi.Basis) with first derivative.

Fig. 7

Left graphic show the smoothed simulated dataset (thin line) for scenario F1 and mean curves for each group (thick line). Optimal cutoff curve (black line) together with the sample data (dashed lines) and the representative curves for each group (thick lines)

1.2 Other graphs of simulation study

Fig. 8

Means of area under ROC curve (AUC) for several values of r.t and d.t when F1 scenario is considered. Sample size $n_1=n_2=30$ and $\sigma =0.08$ and $\sigma _1=0.1$ (A bottom chart); $\sigma =0.06$ and $\sigma _1=0.06$ (B central chart); $\sigma =0.04$ and $\sigma _1=0.04$ (C top chart)

Fig. 9

Means of area under ROC curve (AUC) for several values of r.t and d.t when F2 scenario is considered. Sample size $n_1=n_2=50$, $\sigma =0.01$ and $\sigma _1=0.03$

Fig. 10

Means of area under ROC curve (AUC) for several values of r.t and d.t when F2 scenario is considered. Sample size $n_1=n_2=50$, $\sigma =\sigma _1=0.06$

Fig. 11

Means of AUC, 1-specificity and sensitivity for several values r.t and d.t when F3 scenario is considered. Sample size $n_1=n_2=30$ and $\sigma =0.2$ and $\sigma _1=0.8$

Fig. 12

Means of AUC, 1-specificity and sensitivity for several values r.t and d.t when F4 scenario is considered. Sample size $n_1=n_2=20$ and $\sigma =1.4$ and $\sigma _1=1.8$