ROCsurvcomp

MM Rahman and LE Bantis

2026-05-05

About the ROCsurvcomp Package

The ROCsurvcomp package provides statistical tools for comparing survival distributions under non-proportional hazards (non-PH). Traditional methods such as the log-rank test rely on the proportional hazards assumption and may lose power when this assumption is violated. Alternative approaches, including the Fleming–Harrington family of weighted log-rank tests, often require prior specification of weight functions that depend on the unknown pattern of hazard differences. Incorrect specification of this pattern may lead to a substantial loss of statistical power.

To address these issues, the package implements a novel nonparametric and semiparametric framework for comparing two survival curves based on the length of the ROC curve and the overlap coefficient, along with a joint inference approach that combines both measures. These methods:

Overview of the Methods

ROC Length

The first method is based on the length of the Receiver Operating Characteristic (ROC) curve. The ROC curve is one of the most widely used statistical methods for evaluating continuous classifiers that attempt to distinguish between two groups, typically a control group (group 1) and a diseased group (group 2) (Pepe 2003). The length of the ROC curve under the biomarker framework has been discussed by Maswadeh and Snyder (2015), Martı́nez-Camblor et al. (2017), Franco-Pereira, Nakas, and Pardo (2020), and Bantis et al. (2021).

Let \(X_1\) and \(X_2\) denote the continuous scores of the classifier for groups 1 and 2, respectively, with survival functions \(S_1(\cdot) = 1 - F_1(\cdot)\) and \(S_2(\cdot) = 1 - F_2(\cdot)\), where \(F_1\) and \(F_2\) are the cumulative distribution functions (CDF) of the two groups with a common support. The corresponding probability density functions (PDF) are denoted with \(f_1(\cdot)\) and \(f_2(\cdot)\). The implied ROC curve is then given by,

\[\begin{eqnarray} \text{ROC}(p) = S_2(S_1^{-1}(p)), \quad p \in (0,1). \end{eqnarray}\]

The length of ROC curve, denoted by \(l_{\text{ROC}}\), is defined as:

\[\begin{eqnarray} l_{\text{ROC}} = \int_{-\infty}^{\infty} \sqrt{1+\left( \frac{d \text{ROC}(p)}{dp}\right)^2} \, dp = \int_{-\infty}^{\infty} \sqrt{f_1^2(x) + f_2^2(x)} \, dx. \end{eqnarray}\]

The range of ROC length is: \(\sqrt{2} \leq l_{\text{ROC}} \leq 2\). The lower bound \(\sqrt{2}\) is achieved if and only if the distributions of two groups are exactly identical. On the other hand, \(l_{\text{ROC}}=2\) if and only if the two distributions are perfectly separated.

Overlap Coefficient (OVL)

The second method utilizes the overlap coefficient (OVL), which measures the similarity between two distributions using the common area between two probability density functions (Franco-Pereira et al. 2021). It was first introduced by Weitzman (1970), and further studied by Inman and Bradley (1989), Reiser and Faraggi (1999), and Clemmons and Bradley (2000). The overlap area between the density curves of \(f_1\) and \(f_2\), defined as OVL, is given by,

\[\begin{eqnarray} \text{OVL} = \int_{-\infty}^{\infty} \min\left[ f_1(x), f_2(x) \right] \, dx. \end{eqnarray}\]

The range of OVL is: \(0 \leq OVL \leq 1\). OVL \(=0\) if and only if the distributions of the two groups are completely disjoint, and OVL \(=1\) if and only if the two distributions are exactly identical.

Joint ROC length-OVL-based Method

A joint inference framework is used to combine both ROC length and OVL measures. This method is based on the convex hull of the estimates obtained from permuted samples, constructed using their Euclidean distances from the origin. Before applying this procedure, both measures are standardized to a common scale using their corresponding percentile ranks. Details of this method are provided in Rahman and Bantis (2026, manuscript under review).

Estimation of ROC-length and OVL

Kernel density estimation is used to estimate the densities of the two underlying groups. A Box-Cox transformation is applied to the data prior to density estimation, as Silverman’s bandwidth (and other similar closed-form bandwidths) is optimized for approximately normal densities. Thus, transforming the data may improve density estimation, especially when Gaussian kernels are involved. In the case of the right-censored data, the kernel density estimator is adjusted by replacing the empirical CDF with the Kaplan-Meier estimator. For left- and doubly censored data, a similar approach is applied using the Turnbull estimator. Moreover, if the last observation is right-censored and/or the first observation is left-censored, censored observations beyond the last and/or before the first ordered uncensored event time are imputed by utilizing the Box-Cox transformed estimated parameters, following the approach described by Bantis et al. (2021). Note that imputation is used only to complete the tails and not for the main body of the data.

Inference Procedure

Let \(S_1(t)\) and \(S_2(t)\) denote survival probabilities at time \(t \geq 0\) for the two groups. The hypothesis to be tested is:

\[\begin{eqnarray*} H_0 &:& S_1(t) = S_2(t) \quad \forall t \nonumber \\ H_1 &:& S_1(t) \neq S_2(t). \nonumber \end{eqnarray*}\]

Note that the above hypothesis is equivalent to testing \(H_0: l_\text{ROC} = \sqrt{2}\) vs. \(H_1: l_\text{ROC} \neq \sqrt{2}\) or \(H_0: \text{OVL} = 1\) vs. \(H_1: \text{OVL} \neq 1\).

Statistical significance is assessed using a permutation-based test. The estimation procedure described above is applied to obtain \(\hat{l}_{ROC}\) and \(\hat{OVL}\) from the original sample. To conduct permutation-based testing, all observations from both groups are first pooled into a single dataset. Then, \(n_1\) observations are randomly selected without replacement to form group 1 under permutation, and the remaining \(n_2\) observations are assigned to group 2. For each permuted sample \(p = 1, \ldots,P\), where \(P\) denotes the total number of permutations, \(\hat{l}^{(p)}_{\text{ROC}}\) and \(\hat{OVL}^{(p)}\) are computed using the same estimation procedure.

The null hypothesis is rejected at a nominal level of \(5\%\) if the \(\hat{l}_\text{ROC}\) is greater than the 95th percentile value of the kernel-based \(\hat{l}^{(p)}_{\text{ROC}}\) obtained from \(P\) number of permutations. A similar procedure is applied for the OVL-based test. For joint inference using the combination of ROC length and OVL-based methods, the permutation-based pairs \(\left( 1 - \hat{OVL}^{(p)}, \hat{l}_{ROC}^{(p)} - \sqrt{2} \right)\) are standardized to a common scale using their corresponding percentile ranks. Then the convex hull is constructed that contains only those pairs of values that fall within the 95th percentile of the Euclidean distance from the origin \((0, 0)\). The null hypothesis is rejected at a nominal level of \(5\%\) if the percentile-ranked observed point corresponding to \(\left( 1 - \hat{OVL}, \hat{l}_{ROC} - \sqrt{2} \right)\) lies outside the convex hull.

surv.comp() function

library(ROCsurvcomp)
library(survival)
library(ggplot2)
library(PWEXP)
## Loading required package: fastmatch
# Generating right-censored data with crossing survivals
set.seed(126)
n_trt <- 50
break_trt <- c(2, 4)
rate_trt <- c(log(2)/3, log(2)/7, log(2)/20)
rate.censor_trt <- c(log(2)/55, log(2)/62, log(2)/68)
event_trt <- PWEXP::rpwexp(n_trt, rate = rate_trt, breakpoint = break_trt)
censor_trt <- PWEXP::rpwexp(n_trt, rate = rate.censor_trt, breakpoint = break_trt)

n_ctrl <- 50
rate_ctrl <- log(2)/10
rate.censor_ctrl <- log(2)/58
event_ctrl <- rexp(n_ctrl, rate = rate_ctrl)
censor_ctrl <- rexp(n_ctrl, rate = rate.censor_ctrl)

# Observed time and censoring status (0 = event, 1 = right-censored)
time_trt <- pmin(event_trt, censor_trt)
status_trt <- ifelse(event_trt <= censor_trt, 0, 1)
time_ctrl <- pmin(event_ctrl, censor_ctrl)
status_ctrl <- ifelse(event_ctrl <= censor_ctrl, 0, 1)
time <- c(time_trt, time_ctrl)
status <- c(status_trt, status_ctrl)
group <- c(rep(1, n_trt), rep(2, n_ctrl))

# Plot Kaplan-Meier survival curves
plot(survfit(Surv(time, status == 0) ~ group), col = c("red", "blue"), lwd = 2,
     xlab = "Time", ylab = "Survival Probability",
     main = "Kaplan-Meier Curves")

# Run `surv.comp` function
# Note: n_perm = 10 is used here only for illustration purposes.
# Highly recommended n_perm to be set at 50,000 or more.
surv.comp(
  time = time,
  status = status,
  group = group,
  censor_type = "right",
  method = "roc_length",
  n_perm = 10,
  progress = TRUE,
  plot = FALSE
)
## Hypothesis testing method: ROC length
## Significance level: 5% (two-sided)
## Number of permutations used for inference: 10 
## 
##            estimate p_value
## ROC Length   1.4907       0

References

Bantis, L. E., J. V. Tsimikas, G. Chambers, M. Capello, S. Hanash, and Z. Feng. 2021. “The Length of the ROC Curve and the Two Cutoff Youden Index Within a Robust Framework for Discovery, Evaluation, and Cutoff Estimation in Biomarker Studies Involving Improper ROC Curves.” Statistics in Medicine 40 (7): 1767–89.
Clemmons, T. E., and E. L. Jr. Bradley. 2000. “A Nonparametric Measure of the Overlapping Coefficient.” Computational Statistics & Data Analysis 34 (1): 51–61.
Franco-Pereira, A. M., C. T. Nakas, and M. C. Pardo. 2020. “Biomarker Assessment in ROC Curve Analysis Using the Length of the Curve as an Index of Diagnostic Accuracy: The Binormal Model Framework.” AStA Advances in Statistical Analysis 104 (4): 625–47.
Franco-Pereira, A. M., C. T. Nakas, B. Reiser, and M. C. Pardo. 2021. “Inference on the Overlap Coefficient: The Binormal Approach and Alternatives.” Statistical Methods in Medical Research 30 (12): 2672–84.
Inman, H. F., and E. L. Jr. Bradley. 1989. “The Overlapping Coefficient as a Measure of Agreement Between Probability Distributions and Point Estimation of the Overlap of Two Normal Densities.” Communications in Statistics - Theory and Methods 18 (10): 3851–74.
Martı́nez-Camblor, P., N. Corral, C. Rey, J. Pascual, and E. Cernuda-Morollón. 2017. “Receiver Operating Characteristic Curve Generalization for Non-Monotone Relationships.” Statistical Methods in Medical Research 26 (1): 113–23.
Maswadeh, W. M., and A. P. Snyder. 2015. “Variable Ranking Based on the Estimated Degree of Separation for Two Distributions of Data by the Length of the Receiver Operating Characteristic Curve.” Analytica Chimica Acta 876: 39–48.
Pepe, Margaret S. 2003. The Statistical Evaluation of Medical Tests for Classification and Prediction. Oxford University Press.
Reiser, B., and D. Faraggi. 1999. “Confidence Intervals for the Overlapping Coefficient: The Normal Equal Variance Case.” Journal of the Royal Statistical Society: Series D (The Statistician) 48 (3): 413–18.
Weitzman, M. S. 1970. “Measures of Overlap of Income Distributions of White and Negro Families in the United States.” 22. U.S. Bureau of the Census; U.S. Government Printing Office.