Arguments

• data
  A data frame containing the biomarker and status variables.

• biomarker
  Character string; name of the numeric column containing biomarker values.

• status
  Character string; name of the column containing binary disease status.

• diseased
  The value in status indicating the diseased class (e.g. "carrier").

• kernel
  Character string; kernel function for smoothing. One of "gaussian", "biweight", or "epanechnikov".

• bw_method
  Character string; bandwidth selection method. One of:

  • "pdf" – density-based rule-of-thumb (Silverman),
  • "BHP" – CDF-based normal-reference (Bowman–Hall–Prvan),
  • "AR" – adjusted CDF reference bandwidth,
  • "AL" – Altman–Leger CDF plug-in,
  • "PB" – Polansky–Baker multistage plug-in.

• alpha
  Numeric; significance level for \((1 - \alpha)\) confidence intervals (default: 0.05).

• logtrans
  Logical; if TRUE, applies a log-transformation to biomarker values prior to ROC estimation (useful for right-skewed biomarkers). Default: FALSE.

• grid_n
  Integer; number of grid points for evaluating the ROC curve (default: 1000).

Returned value

smoothROC() returns an object of class "smoothROC" with components including:

• curve
  Data frame with columns FPR, TPR, threshold, and Youden index \(J\), one row per unique false-positive rate (deduplicated, keeping the maximum TPR at each FPR), with the corner points \((0,0)\) and \((1,1)\) always included.

• AUC, AUC_ci, AUC_ci_lo, AUC_ci_hi
  Kernel-smoothed AUC estimate and its confidence interval.

• J, J_ci, J_ci_lo, J_ci_hi
  Kernel-smoothed Youden index estimate and its confidence interval.

• t0
  Estimated optimal cutoff associated with the Youden index.

• sensitivity, specificity
  Sensitivity and specificity at the Youden cutoff.

• kernel, bandwidth_method
  The chosen kernel and bandwidth selection method.

• hX, hY
  Selected CDF bandwidths for the non-diseased and diseased groups.

• plot
  A ggplot2 ROC plot object including the Youden point and a textual annotation.

Print, summary, and plot methods are available:

print.smoothROC()
summary.smoothROC()
plot.smoothROC()

and are invoked automatically via print(), summary(), and plot().

Kernel CDF estimators and the smooth ROC curve

To obtain a smooth ROC curve while remaining nonparametric, smoothROC() applies kernel-based CDF estimators of the form \[ \hat F(x) = \frac{1}{m} \sum_{i=1}^m K\!\left(\frac{x - X_i}{h_m}\right), \quad \hat G(x) = \frac{1}{n} \sum_{j=1}^n K\!\left(\frac{x - Y_j}{h_n}\right), \] where \(K(u) = \int_{-\infty}^u k(v)\,dv\) is the integrated kernel, and \(h_m\), \(h_n\) are bandwidths for the non-diseased and diseased groups, respectively. The smooth ROC curve is then obtained by plugging \(\hat F\) and \(\hat G\) into the ROC functional.

Three univariate kernels are implemented:

• Gaussian kernel
  \(k(u) = (2\pi)^{-1/2}\exp(-u^2/2)\) with CDF \(K(u)\) equal to the standard normal distribution function. This kernel has infinite support and is a default choice in many smoothing problems.

• Epanechnikov kernel
  \(k(u) = \tfrac{3}{4}(1-u^2)\mathbf{1}_{\{|u|\le 1\}}\), with compact support on \([-1,1]\) and optimal second-order efficiency under many mean squared error criteria.

• Biweight kernel
  \(k(u) = \tfrac{15}{16}(1-u^2)^2\mathbf{1}_{\{|u|\le 1\}}\), a higher-order, compactly supported kernel that produces more rounded estimates near the boundaries.

Bandwidth selection strategies

Bandwidth selection is critical for balancing bias and variance in the smoothed CDFs and the resulting ROC curve. smoothROC() focuses on bandwidths that are optimal for CDF estimation (rather than densities), which aligns more directly with ROC functionals.

The bw_method argument implements several strategies:

• "pdf"
  A density-based rule-of-thumb (Silverman) using a kernel density bandwidth. Convenient and widely used, but it does not satisfy the usual asymptotic conditions for CDF estimation.

• "BHP"
  A CDF-based normal-reference bandwidth that approximately minimizes the integrated mean squared error of \(\hat F\). It uses a robust scale estimate based on \(\min(\mathrm{SD}, \mathrm{IQR}/1.34)\).

• "AR"
  An adjusted CDF reference bandwidth for the Gaussian kernel, obtained by shrinking the normal-reference constant to reduce oversmoothing for non-Gaussian data while preserving the \(m^{-1/3}\) CDF rate.

• "AL"
  A fully data-driven CDF-based bandwidth in the spirit of Altman and Leger, where the unknown roughness functional is estimated using an auxiliary kernel estimator.

• "PB"
  A multistage plug-in CDF-based bandwidth (a two-stage version of Polansky and Baker) that uses an initial normal-reference pilot followed by a data-driven refinement.

In simulation studies (not shown here), the "PB" method often provides stable performance across a range of underlying distributions, especially when sample sizes are small to moderate.

AUC estimation and kernel DeLong-type variance

Given the kernel CDFs, smoothROC() computes the AUC by trapezoidal integration of the smoothed ROC curve, \[ \hat \delta = \int_{-\infty}^{\infty} \hat F(x)\, d\hat G(x) \approx \sum_i \tfrac{1}{2}\big(\mathrm{TPR}_i + \mathrm{TPR}_{i-1}\big)\big(\mathrm{FPR}_i - \mathrm{FPR}_{i-1}\big), \] which is asymptotically equivalent to the empirical AUC based on the Mann–Whitney statistic. This link ensures that classical large-sample results for the empirical AUC remain valid in the smoothed setting.

To quantify uncertainty, smoothROC() implements a kernel-smoothed analogue of DeLong’s variance estimator. Instead of using empirical placement values, the variance expression replaces them by their kernel-smoothed counterparts. This typically yields a more stable variance estimate in small samples while retaining the large-sample properties of the original DeLong method. The confidence interval is then centered at the trapezoidal AUC estimate above, using this kernel-smoothed DeLong-type standard error.

The resulting AUC estimate and confidence interval are available via:

roc$AUC
roc$AUC_ci

Youden index and optimal cutoff

The Youden index \[ J = \max_t \{\mathrm{sensitivity}(t) + \mathrm{specificity}(t) - 1\} = \max_t \{F(t) - G(t)\} \] provides a summary of the optimal trade-off between sensitivity and specificity, with \(J \in [0,1]\). The corresponding optimal cutoff is \[ t_0 = \operatorname*{arg\,max}_t \{F(t) - G(t)\}. \]

Using the kernel CDFs, smoothROC() computes a smoothed Youden index \(\hat J\) and the maximizing cutoff \(\hat t_0\) on a grid of thresholds. When multiple cutoffs achieve the same maximum, secondary rules (favoring higher sensitivity or specificity) or median-based summaries can be used. A Delta-method approximation provides the variance of \(\hat J\), from which a Wald-type confidence interval is constructed.

These quantities are returned as:

roc$J
roc$J_ci
roc$t0
roc$sensitivity
roc$specificity

Bootstrap percentile confidence intervals

The confidence intervals returned by smoothROC() for the AUC and the Youden index are kernel-smoothed DeLong-type variance and Delta-method, respectively. These asymptotic approximations can be less reliable in small samples. As an alternative, smoothROCboot() computes bootstrap percentile confidence intervals for the AUC, the Youden index, and its optimal cutoff, by independently resampling the non-diseased and diseased groups with replacement and recomputing the full smoothed analysis on each replicate.

boot <- smoothROCboot(
  data      = dystrophy,
  biomarker = "CK",
  status    = "Class",
  diseased  = "carrier",
  kernel    = "biweight",
  bw_method = "PB",
  alpha     = 0.05,
  logtrans  = TRUE,
  B         = 1000,
  grid_n    = 1000,
  seed      = 1691
)

print(boot)

smoothROCboot() shares the same data/biomarker/status/diseased/ kernel/bw_method/alpha/logtrans/grid_n arguments as smoothROC(), plus:

• B
  Number of bootstrap replicates (default: 1000).

• seed
  Optional integer seed, for reproducible bootstrap resampling.

It returns an object of class "smoothROCboot" with:

boot$J             # Youden index estimate
boot$J_ci          # Bootstrap percentile CI for J
boot$t0            # Optimal cutoff estimate
boot$t0_ci         # Bootstrap percentile CI for the cutoff
boot$AUC           # AUC estimate (trapezoidal)
boot$AUC_ci        # Bootstrap percentile CI for AUC
boot$sensitivity   # Sensitivity at the Youden point (observed data)
boot$specificity   # Specificity at the Youden point (observed data)

along with the raw bootstrap replicate vectors boot$boot_J, boot$boot_t0, and boot$boot_AUC, for users who want to inspect the bootstrap distribution directly (for example, plotting a histogram of boot$boot_AUC).

Because each replicate refits bandwidths and re-evaluates the smoothed ROC curve from scratch, smoothROCboot() is more computationally intensive than smoothROC(); B = 1000 with grid_n = 1000 is typically a reasonable default, but both can be reduced for exploratory work.

Displaying the result

The print and summary methods provide a concise summary:

roc
#> Kernel-smoothed ROC
#> Kernel: biweight
#> Bandwidth: PB
#> AUC = 0.8456 (95.0% CI: 0.7915, 0.8997)
#> Youden Index, J = 0.5724 (95.0% CI: 0.4543, 0.6905)
#> Youden point: (FPR = 0.1305, TPR = 0.7029)
#> Threshold (cutoff) = 4.0707
#> At Youden point Sensitivity = 0.7029, Specificity = 0.8695
summary(roc)
#> Kernel-smoothed ROC
#> Kernel: biweight
#> Bandwidth: PB
#> AUC = 0.8456 (95.0% CI: 0.7915, 0.8997)
#> Youden Index, J = 0.5724 (95.0% CI: 0.4543, 0.6905)
#> Youden point: (FPR = 0.1305, TPR = 0.7029)
#> Threshold (cutoff) = 4.0707
#> At Youden point Sensitivity = 0.7029, Specificity = 0.8695

They report the kernel and bandwidth method, AUC with confidence interval, the Youden index and its confidence interval, and the Youden point (FPR, TPR, cutoff, sensitivity, specificity).

We can visualize the ROC curve. By default, plot() shows the full ROC curve, including:

• the 45-degree reference line (no-discrimination),
• the smooth ROC curve,
• the Youden point marked in red, and
• a label showing AUC, its confidence interval, the Youden index, and the Youden cutoff.

plot(roc)

ROC curve with Youden point and annotation

The label and youden arguments toggle the annotation and the Youden point (with its guide segments) independently:

plot(roc, label = FALSE)

ROC curve with Youden point, no annotation

plot(roc, youden = FALSE)

ROC curve with annotation, no Youden point

plot(roc, label = FALSE, youden = FALSE)

ROC curve only

The underlying ROC data and key summaries can be accessed directly:

head(roc$curve)     # FPR, TPR, threshold, J (one row per unique FPR)
#>            FPR       TPR threshold         J
#> 1 0.000000e+00 0.3107176  5.187889 0.3107176
#> 2 3.008076e-08 0.3134597  5.177544 0.3134597
#> 3 9.766935e-07 0.3162222  5.167200 0.3162212
#> 4 4.589869e-06 0.3190053  5.156855 0.3190007
#> 5 1.248845e-05 0.3218091  5.146511 0.3217966
#> 6 2.611099e-05 0.3246338  5.136166 0.3246077
roc$AUC             # AUC estimate
#> [1] 0.8455996
roc$AUC_ci          # AUC confidence interval
#> [1] 0.7914592 0.8997401
roc$J               # Youden index estimate
#> [1] 0.5723763
roc$J_ci            # Youden index CI
#> [1] 0.4542678 0.6904848
roc$t0              # Youden cutoff
#> [1] 4.07068
roc$sensitivity     # Sensitivity at Youden point
#> [1] 0.7028534
roc$specificity     # Specificity at Youden point
#> [1] 0.8695229
roc$hX              # Bandwidth for non-diseased CDF
#> [1] 0.3146762
roc$hY              # Bandwidth for diseased CDF
#> [1] 0.9802306

Log transformation

Setting logtrans = TRUE applies a natural log transformation to the biomarker prior to ROC estimation. This is often appropriate for biomarkers with strong right skew or multiplicative variability (e.g. enzyme concentrations, cytokines). When logtrans = TRUE, biomarker values must be strictly positive.

Controlling the ROC grid

The argument grid_n controls the resolution of the ROC curve. Larger values produce a smoother ROC curve and more precise localization of the Youden cutoff, at the cost of increased computation. Reasonable values include:

• grid_n = 500 – fast and adequate for exploratory work;
• grid_n = 1000 – default, smoother curve and better stability;
• grid_n = 3000 – more refined.

Choosing a bandwidth method

For most applications, a good starting point is:

• kernel = "biweight",
• bw_method = "PB".

The "PB" method tends to perform well across a range of scenarios and sample sizes. When computation time is a concern, the "AR" method offers a simple, robust alternative that is easy to compute.

In large samples, the differences between bandwidth methods may be minor. However, in small or moderate samples, bandwidth selection can substantially affect ROC shape, AUC estimates, and the stability of the Youden index.