Type:	Package
Title:	ROC Curve Inference with and without Covariates
Version:	1.0-9
Date:	2024-05-09
Imports:	stats, grDevices, graphics, splines, np, Matrix, moments, nor1mix, spatstat.univar, lattice, MASS, pbivnorm, parallel
Description:	Estimates the pooled (unadjusted) Receiver Operating Characteristic (ROC) curve, the covariate-adjusted ROC (AROC) curve, and the covariate-specific/conditional ROC (cROC) curve by different methods, both Bayesian and frequentist. Also, it provides functions to obtain ROC-based optimal cutpoints utilizing several criteria. Based on Erkanli, A. et al. (2006) <doi:10.1002/sim.2496>; Faraggi, D. (2003) <doi:10.1111/1467-9884.00350>; Gu, J. et al. (2008) <doi:10.1002/sim.3366>; Inacio de Carvalho, V. et al. (2013) <doi:10.1214/13-BA825>; Inacio de Carvalho, V., and Rodriguez-Alvarez, M.X. (2022) <doi:10.1214/21-STS839>; Janes, H., and Pepe, M.S. (2009) <doi:10.1093/biomet/asp002>; Pepe, M.S. (1998) http://www.jstor.org/stable/2534001?seq=1; Rodriguez-Alvarez, M.X. et al. (2011a) <doi:10.1016/j.csda.2010.07.018>; Rodriguez-Alvarez, M.X. et al. (2011a) <doi:10.1007/s11222-010-9184-1>. Please see Rodriguez-Alvarez, M.X. and Inacio, V. (2021) <doi:10.32614/RJ-2021-066> for more details.
License:	GPL-2 | GPL-3 [expanded from: GPL]
NeedsCompilation:	no
Packaged:	2024-05-31 10:31:41 UTC; mrodriguez
Author:	Maria Xose Rodriguez-Alvarez [aut, cre], Vanda Inacio [aut]
Maintainer:	Maria Xose Rodriguez-Alvarez <mxrodriguez@uvigo.es>
Repository:	CRAN
Date/Publication:	2024-05-31 13:30:30 UTC

ROC Curve Inference with and without Covariates

Description

Estimates the pooled (unadjusted) Receiver Operating Characteristic (ROC) curve, the covariate-adjusted ROC (AROC) curve, and the covariate-specific/conditional ROC (cROC) curve by different methods, both Bayesian and frequentist. Also, it provides functions to obtain ROC-based optimal cutpoints utilizing several criteria. Based on Erkanli, A. et al. (2006) <doi:10.1002/sim.2496>; Faraggi, D. (2003) <doi:10.1111/1467-9884.00350>; Gu, J. et al. (2008) <doi:10.1002/sim.3366>; Inacio de Carvalho, V. et al. (2013) <doi:10.1214/13-BA825>; Inacio de Carvalho, V., and Rodriguez-Alvarez, M.X. (2022) <doi:10.1214/21-STS839>; Janes, H., and Pepe, M.S. (2009) <doi:10.1093/biomet/asp002>; Pepe, M.S. (1998) <http://www.jstor.org/stable/2534001?seq=1>; Rodriguez-Alvarez, M.X. et al. (2011a) <doi:10.1016/j.csda.2010.07.018>; Rodriguez-Alvarez, M.X. et al. (2011a) <doi:10.1007/s11222-010-9184-1>. Please see Rodriguez-Alvarez, M.X. and Inacio, V. (2021) <doi:10.32614/RJ-2021-066> for more details.

Details

Author(s)

Maria Xose Rodriguez-Alvarez and Vanda Inacio Maintainer: Maria Xose Rodriguez-Alvarez <mxrodriguez@uvigo.es>

References

Erkanli, A., Sung M., Jane Costello, E., and Angold, A. (2006). Bayesian semi-parametric ROC analysis. Statistics in Medicine, 25, 3905–3928.

Faraggi, D. (2003). Adjusting receiver operating characteristic curves and related indices for covariates. The Statistician 52, 179–192.

Gu, J., Ghosal, S., and Roy, A. (2008). Bayesian bootstrap estimation of ROC curve. Statistics in Medicine, 27, 5407–5420.

Inacio de Carvalho, V., Jara, A., Hanson, T. E., and de Carvalho, M. (2013). Bayesian nonparametric ROC regression modeling. Bayesian Analysis, 8, 623–646.

Inacio de Carvalho, V., and Rodriguez-Alvarez, M. X. (2022). The Covariate-Adjusted ROC Curve: The Concept and Its Importance, Review of Inferential Methods, and a New Bayesian Estimator. Statistical Science, 37, 541 -561

Janes, H., and Pepe, M.S. (2009). Adjusting for covariate effects on classification accuracy using the covariate-adjusted receiver operating characteristic curve. Biometrika, 96, 371–382.

Pepe, M.S. (1998). Three approaches to regression analysis of receiver operating characteristic curves for continuous test results. Biometrics 54, 124–135.

Rodriguez-Alvarez, M. X. and Inacio, V., and (2021). ROCnReg: An R Package for Receiver Operating Characteristic Curve Inference with and without Covariate Information. The R Journal

Rodriguez-Alvarez, M.X., Tahoces, P. G., Cadarso-Suarez, C., and Lado, M.J. (2011). Comparative study of ROC regression techniques–Applications for the computer-aided diagnostic system in breast cancer detection. Computational Statistics and Data Analysis, 55, 888–902.

Rodriguez-Alvarez, M.X., Roca-Pardinas, J., and Cadarso-Suarez, C. (2011). ROC curve and covariates: extending induced methodology to the non-parametric framework. Statistics and Computing, 21, 483–499.

Nonparametric Bayesian inference of the covariate-adjusted ROC curve (AROC).

Description

This function estimates the covariate-adjusted ROC curve (AROC) using the nonparametric Bayesian approach proposed by Inacio de Carvalho and Rodriguez-Alvarez (2018).

Usage

AROC.bnp(formula.h, group, tag.h, data, standardise = TRUE, 
  p = seq(0, 1, l = 101), ci.level = 0.95, compute.lpml = FALSE, compute.WAIC = FALSE, 
  compute.DIC = FALSE, pauc = pauccontrol(), density = densitycontrol.aroc(),
  prior.h = priorcontrol.bnp(), mcmc = mcmccontrol(),
  parallel = c("no", "multicore", "snow"), ncpus = 1, cl = NULL)

Arguments

Details

Estimates the covariate-adjusted ROC curve (AROC) defined as

AROC\left(p\right) = Pr\{1 - F_{\bar{D}}(Y_D | \mathbf{X}_{D}) \leq p\},

where F_{\bar{D}}(\cdot|\mathbf{X}_{\bar{D}}) denotes the distribution function of Y_{\bar{D}} conditional on the vector of covariates \mathbf{X}_{\bar{D}}.

The method implemented in this function combines a single-weights linear dependent Dirichlet process mixture model (De Iorio et al., 2009) to estimate F_{\bar{D}}(\cdot|\mathbf{X}_{\bar{D}}) and the Bayesian bootstrap (Rubin, 1981) to estimate the outside probability. More precisely, and letting \{(\mathbf{x}_{\bar{D}i},y_{\bar{D}i})\}_{i=1}^{n_{\bar{D}}} be a random sample from the nondiseased population, our postulated model for the conditional distribution function takes the following form

F_{\bar{D}}(y_{\bar{D}i}|\mathbf{X}_{\bar{D}}=\mathbf{x}_{\bar{D}i}) = \sum_{l=1}^{L}\omega_l\Phi(y_{\bar{D}i}\mid\mu_{l}(\mathbf{x}_{\bar{D}i}),\sigma_l^2),

where \Phi(y|\mu, \sigma^2) denotes the cumulative distribution function of the normal distribution, evaluated at y, with mean \mu and variance \sigma^2. The regression function \mu_{l}(\mathbf{x}_{\bar{D}i}) can incorportate both linear and nonlinear (through B-splines) effects of continuous covariates, categorical covariates (factors) as well as interactions. Interactions between categorical and (nonlinear) continuous covariates are also allowed (factor-by curve interactions). For the sake of simplicity we write \mu_{l}(\mathbf{x}_{\bar{D}i}) = \mathbf{z}_{\bar{D}i}^{T}\mathbf{\beta}_l (l=1,...,L), where \mathbf{z}_{\bar{D}i} is the ith column of the design matrix (possibly containing a basis representation of some/all continuous covariates). Here L is a pre-specified upper bound on the number of mixture components. The \omega_l's result from a truncated version of the stick-breaking construction (\omega_1=v_1; \omega_l=v_l\prod_{r<l}(1-v_r), l=2,\ldots,L; v_1,\ldots,v_{L-1}\sim Beta (1,\alpha); v_L=1, \alpha \sim \Gamma(a_{\alpha},b_{\alpha})), \mathbf{\beta}_l\sim N_{Q}(\mathbf{m},\mathbf{S}), and \sigma_l^{-2}\sim\Gamma(a,b). It is further assumed that \mathbf{m} \sim N_{Q}(\mathbf{m}_0,\mathbf{S}_0) and \mathbf{S}^{-1}\sim W(\nu,(\nu\Psi)^{-1}). Here \Gamma(a,b) denotes a Gamma distribution with shape parameter a and rate parameter b, W(\nu,(\nu\Psi)^{-1}) denotes a Wishart distribution with \nu degrees of freedom and expectation \Psi^{-1}, and Q denotes the dimension of the vector \mathbf{z}_{\bar{D}i}. It is worth mentioning that when L=1, the model for the conditional distribution of the test outcomes (in the healthy population) reduces to a normal regression model (where continuous covariates effects are modelled either parametrically or nonparametrically). For a detailed description, we refer to Inacio de Carvalho and Rodriguez-Alvarez (2018).

Regarding the area under the curve, we note that

AAUC = \int_{0}^{1}AROC(p)dp = 1 - E\{U_D\},

where U_D = 1 - F_{\bar{D}}(Y_D |\mathbf{X}_D). In our implementation, the expectation is computed using the Bayesian bootstrap (using the same weights as those used to estimate the AROC, see Inacio de Carvalho and Rodriguez-Alvarez (2018) for details). As far as the partial area under the curve is concerned, when focus = "FPF" and assuming an upper bound u_1 for the FPF, what it is computed is

pAAUC_{FPF}(u_1)=\int_0^{u_1} AROC(p)dp = u_1 - E\{U_{D,u_1}\},

where U_{D,u_1} = min\{u_1, 1 - F_{\bar{D}}(Y_D |\mathbf{X}_D)\}. Again, the expectation is computed using the Bayesian bootstrap. The returned value is the normalised pAAUC, pAAUC_{FPF}(u_1)/u_1 so that it ranges from u_1/2 (useless test) to 1 (perfect marker). Conversely, when focus = "TPF", and assuming a lower bound for the TPF of u_2, the partial area corresponding to TPFs lying in the interval (u_2,1) is computed as

pAAUC_{TPF}(u_2)=\int_{AROC^{-1}(u_2)}^{1}AROC(p)dp-\{1-AROC^{-1}(u_2)\}\times u_2.

Here, the computation of the integral is done numerically. The returned value is the normalised pAAUC, pAAUC_{TPF}(u_2)/(1-u_2), so that it ranges from (1-u_2)/2 (useless test) to 1 (perfect test).

Finally, it is important referring that with respect to the computation of the DIC, when L=1, it is computed as in Spiegelhalter et al. (2002), and when L>1, DIC3 as described in Celeux et al. (2006) is computed. Also, for the computation of the conditional predictive ordinates (CPO) we follow the stable version proposed by Gelman et al. (2014).

Value

As a result, the function provides a list with the following components:

Note

The input argument formula.h is similar to that used for the glm function, except that flexible specifications can be added by means of the function f(). For instance, specification y \sim x1 + f(x2, K = 3) would assume a linear effect of x1 (if x1 continuous) and the effect of x2 would be modeled using B-splines basis functions. The argument K = 3 indicates that 3 internal knots will be used, with the quantiles of x2 used for their location. Categorical variables (factors) can be also incorporated, as well as interaction terms. For example, to include the factor-by-curve interaction between age and gender we need to specify, e.g., y \sim gender + f(age, by = gender, K = c(3, 5)). Note that, in this case, the number of knots can be different for each level of the factor. The order of the vector K of knots should match the levels of the factor.

References

Celeux, G., Forbes, F., Robert C. P., and Titerrington, D. M. (2006). Deviance information criteria for missing data models. Bayesian Analysis, 1, 651–674.

De Iorio, M., Johnson, W. O., Muller, P., and Rosner, G. L. (2009). Bayesian nonparametric nonproportional hazards survival modeling. Biometrics, 65, 762–775.

Geisser, S. and Eddy, W.F. (1979) A Predictive Approach to Model Selection. Journal of the American Statistical Association, 74, 153–160.

Gelman, A., Carlin, J.B., Stern, H.S., Dunson, D.B., Vehtari, A., and Rubin, D.B. (2014). Bayesian Data Analysis, 3rd ed. CRC Press: Boca Raton, FL.

Gelman, A., Hwang, J., and Vehtari, A. (2014). Understanding predictive information criteria for Bayesian models. Statistics and Computing, 24, 997–1010.

Inacio de Carvalho, V., and Rodriguez-Alvarez, M. X. (2022). The Covariate-Adjusted ROC Curve: The Concept and Its Importance, Review of Inferential Methods, and a New Bayesian Estimator. Statistical Science, 37, 541 -561.

Rubin, D. B. (1981). The Bayesian bootstrap. The Annals of Statistics, 9, 130–134.

Speigelhalter, D. J., Best, N. G., Carlin, B. P., and van der Linde, A. (2002). Bayesian measures of model comparison and fit. Journal of the Royal Statistical Society, Ser. B, 64, 583–639.

Watanabe, S. (2010). Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory. Journal of Machine Learning Research, 11, 3571–3594.

See Also

AROC.bnp, AROC.sp, AROC.kernel, pooledROC.BB, pooledROC.emp, pooledROC.kernel, pooledROC.dpm, cROC.bnp, cROC.sp or AROC.kernel.

Examples

library(ROCnReg)
data(psa)
# Select the last measurement
newpsa <- psa[!duplicated(psa$id, fromLast = TRUE),]

# Log-transform the biomarker
newpsa$l_marker1 <- log(newpsa$marker1)

AROC_bnp <- AROC.bnp(formula.h = l_marker1 ~ f(age, K = 0),
              group = "status", 
              tag.h = 0, 
              data = newpsa,
              standardise = TRUE,
              p = seq(0,1,l=101),
              compute.lpml = TRUE,
              compute.WAIC = TRUE,
              compute.DIC = TRUE,
              pauc = pauccontrol(compute = TRUE, focus = "FPF", value = 0.5),
              density = densitycontrol.aroc(compute = TRUE, grid.h = NA, newdata = NA),
              prior.h = priorcontrol.bnp(m0 = rep(0, 4), S0 = 10*diag(4), nu = 6, Psi = diag(4),
              a = 2, b = 0.5, alpha = 1, L =10),
              mcmc = mcmccontrol(nsave = 500, nburn = 100, nskip = 1))

summary(AROC_bnp)

plot(AROC_bnp)

Nonparametric kernel-based estimation of the covariate-adjusted ROC curve (AROC).

Description

This function estimates the covariate-adjusted ROC curve (AROC) using the nonparametric kernel-based method proposed by Rodriguez-Alvarez et al. (2011). The method, as it stands now, can only deal with one continuous covariate.

Usage

AROC.kernel(marker, covariate, group, tag.h, 
    bw = c("LS", "AIC"), 
    regtype = c("LC", "LL"),
    pauc = pauccontrol(), 
    data, p = seq(0, 1, l = 101), B = 1000, ci.level = 0.95,
    parallel = c("no", "multicore", "snow"), ncpus = 1, cl = NULL)

Arguments

Details

Estimates the covariate-adjusted ROC curve (AROC) defined as

AROC\left(p\right) = Pr\{1 - F_{\bar{D}}(Y_D | X_{D}) \leq p\},

where F_{\bar{D}}(y|x) = Pr\{Y_{\bar{D}} \leq y | X_{\bar{D}} = x\}. In particular, the method implemented in this function estimates the outer probability empirically (see Janes and Pepe, 2009) and F_{\bar{D}}(y|x) is estimated assuming a nonparametric location-scale regression model for Y_{\bar{D}}, i.e.,

Y_{\bar{D}} = \mu_{\bar{D}}(X_{\bar{D}}) + \sigma_{\bar{D}}(X_{\bar{D}})\varepsilon_{\bar{D}},

where \mu_{\bar{D}}(x) = E(Y_{\bar{D}} | X_{\bar{D}} = x) is the regression funcion, \sigma^2_{\bar{D}}(x) = Var(Y_{\bar{D}} | X_{\bar{D}} = x) is the variance function, and \varepsilon_{\bar{D}} has zero mean, variance one, and distribution function G_{\bar{D}}. As a consequence,

F_{\bar{D}}(y | x) = G_{\bar{D}}\left(\frac{y - \mu_{\bar{D}}(x)}{\sigma_{\bar{D}}(x)}\right).

By default, both the regression and variance functions are estimated using the Nadaraya-Watson estimator (LC), and the bandwidths are selected using least-squares cross-validation (LS). Implementation relies on the R-package np. No assumption is made about G_{\bar{D}}, which is empirically estimated on the basis of the standardised residuals.

The area under the AROC curve is

AAUC=\int_0^1 AROC(p)dp,

and there exists a closed-form estimator. With regard to the partial area under the curve, when focus = "FPF" and assuming an upper bound u_1 for the FPF, what it is computed is

pAAUC_{FPF}(u_1)=\int_0^{u_1} AROC(p)dp,

where again there exists a closed-form estimator. The returned value is the normalised pAAUC, pAAUC_{FPF}(u_1)/u_1 so that it ranges from u_1/2 (useless test) to 1 (perfect marker). Conversely, when focus = "TPF", and assuming a lower bound for the TPF of u_2, the partial area corresponding to TPFs lying in the interval (u_2,1) is computed as

pAAUC_{TPF}(u_2)=\int_{AROC^{-1}(u_2)}^{1}AROC(p)dp-\{1-AROC^{-1}(u_2)\}\times u_2.

Here, the computation of the integral is done numerically. The returned value is the normalised pAAUC, pAAUC_{TPF}(u_2)/(1-u_2), so that it ranges from (1-u_2)/2 (useless test) to 1 (perfect test).

Value

As a result, the function provides a list with the following components:

References

Hayfield, T., and Racine, J. S. (2008). Nonparametric Econometrics: The np Package. Journal of Statistical Software, 27(5). URL http://www.jstatsoft.org/v27/i05/.

Inacio de Carvalho, V., and Rodriguez-Alvarez, M. X. (2022). The Covariate-Adjusted ROC Curve: The Concept and Its Importance, Review of Inferential Methods, and a New Bayesian Estimator. Statistical Science, 37, 541 -561.

Janes, H., and Pepe, M.S. (2009). Adjusting for covariate effects on classification accuracy using the covariate-adjusted receiver operating characteristic curve. Biometrika, 96, 371–382.

Rodriguez-Alvarez, M. X., Roca-Pardinas, J., and Cadarso-Suarez, C. (2011). ROC curve and covariates: extending induced methodology to the non-parametric framework. Statistics and Computing, 21, 483–499.

See Also

AROC.bnp, AROC.sp, AROC.kernel, pooledROC.BB, pooledROC.emp, pooledROC.kernel, pooledROC.dpm, cROC.bnp, cROC.sp or AROC.kernel.

Examples

library(ROCnReg)
data(psa)
# Select the last measurement
newpsa <- psa[!duplicated(psa$id, fromLast = TRUE),]

# Log-transform the biomarker
newpsa$l_marker1 <- log(newpsa$marker1)

m2 <- AROC.kernel(marker = "l_marker1", 
covariate = "age",
group = "status", 
tag.h = 0,
data = newpsa, 
bw = "LS",
regtype = "LC",
pauc = pauccontrol(compute = TRUE, focus = "FPF", value = 0.5),
B = 500)

summary(m2)

plot(m2)

Semiparametric frequentist inference for the covariate-adjusted ROC curve (AROC).

Description

This function estimates the covariate-adjusted ROC curve (AROC) using the semiparametric approach proposed by Janes and Pepe (2009).

Usage

AROC.sp(formula.h, group, tag.h, data, 
    est.cdf.h = c("normal", "empirical"), pauc = pauccontrol(),
    p = seq(0, 1, l = 101), B = 1000, ci.level = 0.95, 
  	parallel = c("no", "multicore", "snow"), ncpus = 1, cl = NULL)

Arguments

Details

Estimates the covariate-adjusted ROC curve (AROC) defined as

AROC\left(p\right) = Pr\{1 - F_{\bar{D}}(Y_D | \mathbf{X}_{D}) \leq p\},

F_{\bar{D}}(y|\mathbf{x}) = Pr\{Y_{\bar{D}} \leq y | \mathbf{X}_{\bar{D}} = \mathbf{x}\}.

The method implemented in this function estimates the outer probability empirically (see Janes and Pepe, 2009) and F_{\bar{D}}(\cdot|\mathbf{x}) is estimated assuming a semiparametric location regression model for Y_{\bar{D}}, i.e.,

Y_{\bar{D}} = \mathbf{X}_{\bar{D}}^{T}\mathbf{\beta}_{\bar{D}} + \sigma_{\bar{D}}\varepsilon_{\bar{D}},

where \varepsilon_{\bar{D}} has zero mean, variance one, and distribution function G_{\bar{D}}. As a consequence, we have

F_{\bar{D}}(y | \mathbf{x}) = G_{\bar{D}}\left(\frac{y-\mathbf{x}^{T}\mathbf{\beta}_{\bar{D}}}{\sigma_{\bar{D}}}\right).

In line with the assumptions made about the distribution of \varepsilon_{\bar{D}}, estimators will be referred to as: (a) "normal", where a standard Gaussian error is assumed, i.e., G_{\bar{D}}(y) = \Phi(y); and, (b) "empirical", where no assumption is made about the distribution (in this case, G_{\bar{D}} is empirically estimated on the basis of standardised residuals).

The area under the AROC curve is

AAUC=\int_0^1 AROC(p)dp,

and there exists a closed-form estimator. With regard to the partial area under the AROC curve, when focus = "FPF" and assuming an upper bound u_1 for the FPF, what it is computed is

pAAUC_{FPF}(u_1)=\int_0^{u_1} AROC(p)dp,

where again there exists a closed-form estimator. The returned value is the normalised pAAUC, pAAUC_{FPF}(u_1)/u_1 so that it ranges from u_1/2 (useless test) to 1 (perfect marker). Conversely, when focus = "TPF", and assuming a lower bound for the TPF of u_2, the partial area corresponding to TPFs lying in the interval (u_2,1) is computed as

pAAUC_{TPF}(u_2)=\int_{AROC^{-1}(u_2)}^{1}AROC(p)dp-\{1-AROC^{-1}(u_2)\}\times u_2.

Here, the computation of the integral is done numerically. The returned value is the normalised pAAUC, pAAUC_{TPF}(u_2)/(1-u_2), so that it ranges from (1-u_2)/2 (useless test) to 1 (perfect test).

Value

As a result, the function provides a list with the following components:

References

Janes, H., and Pepe, M.S. (2009). Adjusting for covariate effects on classification accuracy using the covariate-adjusted receiver operating characteristic curve. Biometrika, 96(2), 371 - 382.

See Also

AROC.bnp, AROC.sp, AROC.kernel, pooledROC.BB, pooledROC.emp, pooledROC.kernel, pooledROC.dpm, cROC.bnp, cROC.sp or AROC.kernel.

Examples

library(ROCnReg)
data(psa)
# Select the last measurement
newpsa <- psa[!duplicated(psa$id, fromLast = TRUE),]

# Log-transform the biomarker
newpsa$l_marker1 <- log(newpsa$marker1)

m3 <- AROC.sp(formula.h = l_marker1 ~ age,
group = "status", 
tag.h = 0,
data = newpsa,
est.cdf.h = "normal",
pauc = pauccontrol(compute = TRUE, focus = "FPF", value = 0.5),
p = seq(0,1,l=101), 
B = 500)

summary(m3)

plot(m3)

Nonparametric Bayesian inference for the covariate-specific ROC curve (cROC).

Description

This function estimates the covariate-specific ROC curve (cROC) using the nonparametric Bayesian approach proposed by Inacio de Carvalho et al. (2013).

Usage

cROC.bnp(formula.h, formula.d, group, tag.h, data, 
    newdata, standardise = TRUE, p = seq(0, 1, l = 101), ci.level = 0.95,
    compute.lpml = FALSE, compute.WAIC = FALSE, compute.DIC = FALSE, 
    pauc = pauccontrol(), density = densitycontrol(),
    prior.h = priorcontrol.bnp(), prior.d = priorcontrol.bnp(), 
    mcmc = mcmccontrol(),
    parallel = c("no", "multicore", "snow"), ncpus = 1, cl = NULL) 

Arguments

Details

Estimates the covariate-specific ROC curve (cROC) defined as

ROC(p|\mathbf{x}) = 1 - F_{D}\{F_{\bar{D}}^{-1}(1-p|\mathbf{x})|\mathbf{x}\},

where

F_{D}(y|\mathbf{x}) = Pr(Y_{D} \leq y | \mathbf{X}_{D} = \mathbf{x}),

F_{\bar{D}}(y|\mathbf{x}) = Pr(Y_{\bar{D}} \leq y | \mathbf{X}_{\bar{D}} = \mathbf{x}).

Note that, for the sake of clarity, we assume that the covariates of interest are the same in both the healthy and diseased populations. The method implemented in this function estimates F_{D}(\cdot|\mathbf{x}) and F_{\bar{D}}(\cdot|\mathbf{x}) by means of a single-weights linear dependent Dirichlet process mixture of normals model (De Iorio et al., 2009). More precisely, and letting \{(\mathbf{x}_{\bar{D}i},y_{\bar{D}i})\}_{i=1}^{n_{\bar{D}}} and \{(\mathbf{x}_{Dj},y_{Dj})\}_{j=1}^{n_{D}} be two independent random samples from the nondiseased and diseased populations, respectively, our postulated model for the conditional distribution in each group function takes the following form

F_{\bar{D}}(y_{\bar{D}i}|\mathbf{X}_{\bar{D}}=\mathbf{x}_{\bar{D}i}) = \sum_{l=1}^{L_{\bar{D}}}\omega_{l\bar{D}}\Phi(y_{\bar{D}i}\mid\mu_{l\bar{D}}(\mathbf{x}_{\bar{D}i}),\sigma_{l\bar{D}}^2),

F_{D}(y_{Dj}|\mathbf{X}_{D} = \mathbf{x}_{Dj}) = \sum_{l=1}^{L_{D}}\omega_{lD}\Phi(y_{Dj}\mid\mu_{lD}(\mathbf{x}_{\bar{D}i}),\sigma_{lD}^2),

where \Phi(y|\mu, \sigma^2) denotes the cumulative distribution function of the normal distribution, evaluated at y, with mean mu and variance \sigma^2. The regression function \mu_{ld}(\mathbf{x}_{di}) can incorportate both linear and nonlinear (through B-splines) effects of continuous covariates, categorical covariates (factors) as well as interactions. Interactions between categorical and (nonlinear) continuous covariates are also allowed (factor-by curve interactions). For the sake of simplicity we write \mu_{ld}(\mathbf{x}_{di}) = \mathbf{z}_{di}^{T}\mathbf{\beta}_{ld}, where \mathbf{z}_{di} is the ith column of the design matrix (possibly containing a basis representation of some/all continuous covariates), d \in \{D, \bar{D}\}. Here L_d is a pre-specified upper bound on the number of mixture components. The \omega_{ld}'s result from a truncated version of the stick-breaking construction (\omega_{1d} = v_{1d}; \omega_{ld} = v_{ld}\prod_{r<l}(1-v_{dr}), l=2,\ldots,L_{d}; v_{d1},\ldots,v_{L_{d}-1}\sim Beta (1,\alpha_{d}); v_{Ld} = 1, \alpha_d \sim \Gamma(a_{\alpha_d},b_{\alpha_d})), \mathbf{\beta}_{ld}\sim N_{Q_d}(\mathbf{m}_{d},\mathbf{S}_{d}), and \sigma_{ld}^{-2}\sim\Gamma(a_{d},b_{d}). It is further assumed that \mathbf{m}_{d} \sim N_{Q_k}(\mathbf{m}_{0d},\mathbf{S}_{0d}) and \mathbf{S}_{d}^{-1}\sim W(\nu,(\nu_k\Psi_d)^{-1}). Here W(\nu,(\nu\Psi)^{-1}) denotes a Wishart distribution with \nu degrees of freedom and expectation \Psi^{-1}, Here \Gamma(a,b) denotes a Gamma distribution with shape parameter a and rate parameter b, and Q_d denotes the dimension of the vector \mathbf{z}_{di}. It is worth mentioning that when L_d=1, the model for the conditional distribution of the test outcomes reduces to a normal regression model (where continuous covariates effects are modelled either parametrically or nonparametrically). For a detailed description, we refer to Inacio de Carvalho et al. (2013).

The covariate-specific area under the curve is

AUC(\mathbf{x})=\int_{0}^{1}ROC(p|\mathbf{x})dp.

When the upper bound on the number of mixture components is 1, i.e., L_d = 1 (d \in \{D, \bar{D}\}), there is a closed-form expression for the covariate-specific AUC (binormal model), which is used in the package. In contrast, when L_D > 1 or L_{\bar{D}} > 1, the integral is computed numerically using Simpson's rule. With regard to the partial area under the curve, when focus = "FPF" and assuming an upper bound u_1 for the FPF, what it is computed is

pAUC_{FPF}(u_1|\mathbf{x})=\int_0^{u_1} ROC(p|\mathbf{x})dp.

As for the AUC, when L_d = 1 (d \in \{D, \bar{D}\}), there is a closed-form expression for the pAUC_{FPF} (Hillis and Metz, 2012), and when L_D > 1 or L_{\bar{D}} > 1 the integral is approximated numerically using Simpson's rule. The returned value is the normalised pAUC, pAUC_{FPF}(u_1|\mathbf{x})/u_1 so that it ranges from u_1/2 (useless test) to 1 (perfect marker). Conversely, when focus = "TPF", and assuming a lower bound for the TPF of u_2, the partial area corresponding to TPFs lying in the interval (u_2,1) is computed as

pAUC_{TPF}(u_2|\mathbf{x})=\int_{u_2}^{1}ROC_{TNF}(p|\mathbf{x})dp,

where ROC_{TNF}(p|\mathbf{x}) is a 270^\circ rotation of the ROC curve, and it can be expressed as ROC_{TNF}(p|\mathbf{x}) = F_{\bar{D}}\{F_{D}^{-1}(1-p|\mathbf{x})|\mathbf{x}\}. Again, when L_d = 1 (d \in \{D, \bar{D}\}), there is a closed-form expression for the pAUC_{TNF} (Hillis and Metz, 2012), and when L_D > 1 or L_{\bar{D}} > 1 the integral is approximated numerically using Simpson's rule. The returned value is the normalised pAUC, pAUC_{TPF}(u_2|\mathbf{x})/(1-u_2), so that it ranges from (1-u_2)/2 (useless test) to 1 (perfect test).

It is worth referring that with respect to the computation of the DIC, when L=1, it is computed as in Spiegelhalter et al. (2002), and when L>1, DIC3 as described in Celeux et al. (2006) is computed. Also, for the computation of the conditional predictive ordinates (CPO) we follow the stable version proposed by Gelman et al. (2014).

Value

As a result, the function provides a list with the following components:

Note

The input arguments formula.h and formula.d are similar to that used for the glm function, except that flexible specifications can be added by means of the function f(). For instance, specification y \sim x1 + f(x2, K = 3) would assume a linear effect of x1 (if x1 continuous) and the effect of x2 would be modeled using B-splines basis functions. The argument K = 3 indicates that 3 internal knots will be used, with the quantiles of x2 used for their location. Categorical variables (factors) can be also incorporated, as well as interaction terms. For example, to include the factor-by-curve interaction between age and gender we need to specify, e.g., y \sim gender + f(age, by = gender, K = c(3, 5)). Note that, in this case, the number of knots can be different for each level of the factor. The order of the vector K of knots should match the levels of the factor.

References

Celeux, G., Forbes, F., Robert C. P., and Titerrington, D. M. (2006). Deviance information criteria for missing data models. Bayesian Analysis, 1, 651–674.

De Iorio, M., Johnson, W. O., Muller, P., and Rosner, G. L. (2009). Bayesian nonparametric nonproportional hazards survival modeling. Biometrics, 65, 762–775.

Geisser, S. and Eddy, W.F. (1979) A Predictive Approach to Model Selection, Journal of the American Statistical Association, 74, 153–160.

Gelman, A., Carlin, J.B., Stern, H.S., Dunson, D.B., Vehtari, A., and Rubin, D.B. (2014). Bayesian Data Analysis, 3rd ed. CRC Press: Boca Raton, FL.

Gelman, A., Hwang, J., and Vehtari, A. (2014). Understanding predictive information criteria for Bayesian models. Statistics and Computing, 24, 997–1010.

Hillis, S. L. and Metz, C.E. (2012). An Analytic Expression for the Binormal Partial Area under the ROC Curve. Academic Radiology, 19, 1491–1498.

Inacio de Carvalho, V., Jara, A., Hanson, T. E., and de Carvalho, M. (2013). Bayesian nonparametric ROC regression modeling. Bayesian Analysis, 8, 623–646.

Speigelhalter, D. J., Best, N. G., Carlin, B. P., and van der Linde, A. (2002). Bayesian measures of model comparison and fit. Journal of the Royal Statistical Society, Ser. B, 64, 583–639.

Watanabe, S. (2010). Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory. Journal of Machine Learning Research, 11, 3571–3594.

See Also

AROC.bnp, AROC.sp, AROC.kernel, pooledROC.BB, pooledROC.emp, pooledROC.kernel, pooledROC.dpm, cROC.bnp, cROC.sp or AROC.kernel.

Examples

library(ROCnReg)
data(psa)
# Select the last measurement
newpsa <- psa[!duplicated(psa$id, fromLast = TRUE),]

# Log-transform the biomarker
newpsa$l_marker1 <- log(newpsa$marker1)

cROC_bnp <- cROC.bnp(formula.h = l_marker1 ~ f(age, K = 0),
               formula.d = l_marker1 ~ f(age, K = 0),
               group = "status", 
               tag.h = 0,
               data = newpsa,
               standardise = TRUE, 
               p = seq(0, 1, len = 101),
               compute.lpml = TRUE, 
               compute.WAIC = TRUE,
               compute.DIC = TRUE, 
               pauc = pauccontrol(compute = TRUE, value = 0.5, focus = "FPF"),
               density = densitycontrol(compute = TRUE, grid.h = NA, grid.d = NA),
               mcmc = mcmccontrol(nsave = 500, nburn = 100, nskip = 1))
summary(cROC_bnp)

plot(cROC_bnp)

Nonparametric kernel-based estimation of the covariate-specific ROC curve (cROC).

Description

This function estimates the covariate-specific ROC curve (cROC) using the nonparametric kernel-based method proposed by Rodriguez-Alvarez et al. (2011). The method, as it stands now, can only deal with one continuous covariate.

Usage

cROC.kernel(marker, covariate, group, tag.h, 
  bw = c("LS", "AIC"), regtype = c("LC", "LL"), 
  data, newdata, pauc = pauccontrol(),  
  p = seq(0, 1, l = 101), B = 1000, ci.level = 0.95,
    parallel = c("no", "multicore", "snow"), ncpus = 1, cl = NULL)

Arguments

Details

Estimates the covariate-specific ROC curve (cROC) defined as

ROC(p|x) = 1 - F_{D}\{F_{\bar{D}}^{-1}(1-p|x)|x\},

where

F_{D}(y|x) = Pr(Y_{D} \leq y | X_{D} = x ),

F_{\bar{D}}(y|x) = Pr(Y_{\bar{D}} \leq y | X_{\bar{D}} = x).

Note that, for the sake of clarity, we assume that the covariate of interest is the same in both healthy and diseased populations. In particular, the method implemented in this function estimates F_{D}(\cdot|x) and F_{\bar{D}}(\cdot|x) assuming a nonparametric location-scale regression model for Y in each population separately, i.e.,

Y_{D} = \mu_{D}(X_{D}) + \sigma_{D}(X_{D})\varepsilon_{D},

Y_{\bar{D}} = \mu_{\bar{D}}(X_{\bar{D}}) + \sigma_{\bar{D}}(X_{\bar{D}})\varepsilon_{\bar{D}},

where \mu_{D}(x) = E(Y_D | X_D = x), \mu_{\bar{D}}(x) = E(Y_{\bar{D}} | X_{\bar{D}} = x) (regression function), \sigma^2_{D}(x) = Var(Y_D | X_D = x), \sigma^2_{\bar{D}}(x) = Var(Y_{\bar{D}} | X_{\bar{D}} = x) (variance functions), and \varepsilon_{D} and \varepsilon_{\bar{D}} have zero mean, variance one, and distribution functions G_{D} and G_{\bar{D}}, respectively. In this case, the covariate-specific ROC curve can be expressed as

ROC(p|x) = 1 - G_{D}\{a(\mathbf{x}) + b(\mathbf{x})G_{\bar{D}}^{-1}(1-p)\},

where a(x) = \frac{\mu_{\bar{D}}(x) - \mu_{D}(x)}{\sigma_{D}(x)}, b(x) = \frac{\sigma_{\bar{D}}(x)}{\sigma_{D}(x)}, and G_{D} and G_{\bar{D}} are the distribution functions of \varepsilon_{D} and \varepsilon_{\bar{D}}, respectively. By default, for both the healthy and diseased population, both the regression and variance functions are estimated using the Nadaraya-Watson estimator (LC), and the bandwidth are selected using least-squares cross-validation (LS). Implementation relies on the R-package np. No assumptions are made about G_{D} and G_{\bar{D}}, which are empirically estimated on the basis of standardised residuals.

The covariate-specific area under the curve is

AUC(\mathbf{x})=\int_{0}^{1}ROC(p|\mathbf{x})dp,

and is computed numerically (using Simpson's rule). With regard to the partial area under the curve, when focus = "FPF" and assuming an upper bound u_1 for the FPF, what it is computed is

pAUC_{FPF}(u_1|\mathbf{x})=\int_0^{u_1} ROC(p|\mathbf{x})dp,

where again the integral is approximated numerically (Simpson's rule). The returned value is the normalised pAUC, pAUC_{FPF}(u_1|\mathbf{x})/u_1 so that it ranges from u_1/2 (useless test) to 1 (perfect marker). Conversely, when focus = "TPF", and assuming a lower bound for the TPF of u_2, the partial area corresponding to TPFs lying in the interval (u_2,1) is computed as

pAUC_{TPF}(u_2|\mathbf{x})=\int_{u_2}^{1}ROC_{TNF}(p|\mathbf{x})dp,

where ROC_{TNF}(p|\mathbf{x}) is a 270^\circ rotation of the ROC curve, and it can be expressed as ROC_{TNF}(p|\mathbf{x}) = F_{\bar{D}}\{F_{D}^{-1}(1-p|\mathbf{x})|\mathbf{x}\}=G_{\bar{D}}\{\frac{\mu_{D}(x)-\mu_{\bar{D}}(x)}{\sigma_{\bar{D}}(x)}+G_{D}^{-1}(1-p)\frac{\sigma_{D}(x)}{\sigma_{\bar{D}}(x)}\}. Again, the computation of the integral is done via Simpson's rule. The returned value is the normalised pAUC, pAUC_{TPF}(u_2|\mathbf{x})/(1-u_2), so that it ranges from (1-u_2)/2 (useless test) to 1 (perfect test).

Value

As a result, the function provides a list with the following components:

References

Hayfield, T., and Racine, J. S.(2008). Nonparametric Econometrics: The np Package. Journal of Statistical Software 27(5). URL http://www.jstatsoft.org/v27/i05/.

Rodriguez-Alvarez, M. X., Roca-Pardinas, J., and Cadarso-Suarez, C. (2011). ROC curve and covariates: extending induced methodology to the non-parametric framework. Statistics and Computing, 21, 483–499.

See Also

AROC.bnp, AROC.sp, AROC.kernel, pooledROC.BB, pooledROC.emp, pooledROC.kernel, pooledROC.dpm, cROC.kernel or cROC.sp.

Examples

library(ROCnReg)
data(psa)
# Select the last measurement
newpsa <- psa[!duplicated(psa$id, fromLast = TRUE),]

# Log-transform the biomarker
newpsa$l_marker1 <- log(newpsa$marker1)

cROC_kernel <- cROC.kernel(marker = "l_marker1",
               covariate = "age",
               group = "status", 
               tag.h = 0,
               data = newpsa, 
               bw = "LS",
               regtype = "LC",
               p = seq(0, 1, len = 101),
               pauc = pauccontrol(compute = TRUE, value = 0.5, focus = "FPF"),
               B = 500)

plot(cROC_kernel)

summary(cROC_kernel )

Parametric and semiparametric frequentist inference of the covariate-specific ROC curve (cROC).

Description

This function estimates the covariate-specific ROC curve (cROC) using the parametric approach proposed by Faraggi (2003) and the semiparametric approach proposed by Pepe (1998).

Usage

cROC.sp(formula.h, formula.d, group, tag.h, data, 
  newdata, est.cdf = c("normal", "empirical"),
  pauc = pauccontrol(), p = seq(0, 1, l = 101), B = 1000, ci.level = 0.95,
  parallel = c("no", "multicore", "snow"), ncpus = 1, cl = NULL)

Arguments

Details

Estimates the covariate-specific ROC curve (cROC) defined as

ROC(p|\mathbf{x}) = 1 - F_{D}\{F_{\bar{D}}^{-1}(1-p|\mathbf{x})|\mathbf{x}\},

where

F_{D}(y|\mathbf{x}) = Pr(Y_{D} \leq y | \mathbf{X}_{D} = \mathbf{x}),

F_{\bar{D}}(y|\mathbf{x}) = Pr(Y_{\bar{D}} \leq y | \mathbf{X}_{\bar{D}} = \mathbf{x}).

Note that, for the sake of clarity, we assume that the covariates of interest are the same in both healthy and diseased populations. In particular, the method implemented in this function estimates F_{D}(\cdot|\mathbf{x}) and F_{\bar{D}}(\cdot|\mathbf{x}) assuming a (semiparametric) location regression model for Y in each population separately, i.e.,

Y_{D} = \mathbf{X}_{D}^{T}\mathbf{\beta}_{D} + \sigma_{D}\varepsilon_{D},

Y_{\bar{D}} = \mathbf{X}_{\bar{D}}^{T}\mathbf{\beta}_{\bar{D}} + \sigma_{\bar{D}}\varepsilon_{\bar{D}},

such that the covariate-specific ROC curve can be expressed as

ROC(p|\mathbf{x}) = 1 - G_{D}\{a(\mathbf{x}) + b G_{\bar{D}}^{-1}(1-p)\},

where a(\mathbf{x}) = \mathbf{x}^{T}\frac{\mathbf{\beta}_{\bar{D}} - \mathbf{\beta}_{D}}{\sigma_{D}}, b = \frac{\sigma_{\bar{D}}}{\sigma_{D}}, and G_{D} and G_{\bar{D}} are the distribution functions of \varepsilon_{D} and \varepsilon_{\bar{D}}, respectively. In line with the assumptions made about the distributions of \varepsilon_{D} and \varepsilon_{\bar{D}}, estimators will be referred to as: (a) "normal", where Gaussian errors are assumed, i.e., G_{D}(y) = G_{\bar{D}}(y) = \Phi(y) (Faraggi, 2003); and, (b) "empirical", where no assumptios are made about the distribution (in this case, G_{D} and G_{\bar{D}} are empirically estimated on the basis of standardised residuals (Pepe, 1998)).

The covariate-specific area under the curve is

AUC(\mathbf{x})=\int_{0}^{1}ROC(p|\mathbf{x})dp.

When Gaussian errors are assumed, there is a closed-form expression for the covariate-specific AUC, which is used in the package. In contrast, when no assumptios are made about the distributionis of the errors, the integral is computed numerically using Simpson's rule. With regard to the partial area under the curve, when focus = "FPF" and assuming an upper bound u_1 for the FPF, what it is computed is

pAUC_{FPF}(u_1|\mathbf{x})=\int_0^{u_1} ROC(p|\mathbf{x})dp.

Again, when Gaussian errors are assumed, there is a closed-form expression (Hillis and Metz, 2012). Otherwise, the integral is approximated numerically (Simpson's rule). The returned value is the normalised pAUC, pAUC_{FPF}(u_1|\mathbf{x})/u_1 so that it ranges from u_1/2 (useless test) to 1 (perfect marker). Conversely, when focus = "TPF", and assuming a lower bound for the TPF of u_2, the partial area corresponding to TPFs lying in the interval (u_2,1) is computed as

pAUC_{TPF}(u_2|\mathbf{x})=\int_{u_2}^{1}ROC_{TNF}(p|\mathbf{x})dp,

where ROC_{TNF}(p|\mathbf{x}) is a 270^\circ rotation of the ROC curve, and it can be expressed asROC_{TNF}(p|\mathbf{x}) = F_{\bar{D}}\{F_{D}^{-1}(1-p|\mathbf{x})|\mathbf{x}\}=G_{\bar{D}}\{\frac{\mu_{D}(\mathbf{x})-\mu_{\bar{D}}(\mathbf{x})}{\sigma_{\bar{D}}(\mathbf{x})}+G_{D}^{-1}(1-p)\frac{\sigma_{D}(\mathbf{x})}{\sigma_{\bar{D}}(\mathbf{x})}\}. Again, when Gaussian errors are assumed, there is a closed-form expression (Hillis and Metz, 2012). Otherwise, the integral is approximated numerically (Simpson's rule). The returned value is the normalised pAUC, pAUC_{TPF}(u_2|\mathbf{x})/(1-u_2), so that it ranges from (1-u_2)/2 (useless test) to 1 (perfect test).

Value

As a result, the function provides a list with the following components:

References

Faraggi, D. (2003). Adjusting receiver operating characteristic curves and related indices for covariates. The Statistician 52, 179–192.

Hillis, S. L. and Metz, C.E. (2012). An Analytic Expression for the Binormal Partial Area under the ROC Curve. Academic Radiology, 19, 1491–1498.

Pepe, M.S. (1998). Three approaches to regression analysis of receiver operating characteristic curves for continuous test results. Biometrics 54, 124–135.

See Also

AROC.bnp, AROC.sp, AROC.kernel, pooledROC.BB, pooledROC.emp, pooledROC.kernel, pooledROC.dpm, cROC.bnp, cROC.sp or AROC.kernel.

Examples

library(ROCnReg)
data(psa)
# Select the last measurement
newpsa <- psa[!duplicated(psa$id, fromLast = TRUE),]

# Log-transform the biomarker
newpsa$l_marker1 <- log(newpsa$marker1)

# Covariate for prediction
agep <- seq(min(newpsa$age), max(newpsa$age), length = 50)
df.pred <- data.frame(age = agep)


cROC_sp_normal <- cROC.sp(formula.h = l_marker1 ~ age,
                          formula.d = l_marker1 ~ age,
                          group = "status", 
                          tag.h = 0,
                          data = newpsa,
                          newdata = df.pred,
                          est.cdf = "normal",
                          pauc = list(compute = TRUE, value = 0.5, focus = "FPF"),
                          p = seq(0, 1, l = 101), 
                          B = 500)
summary(cROC_sp_normal)

plot(cROC_sp_normal)

Selects an adequate set of points from a data set for obtaining predictions.

Description

Selects an adequate set of points from a data set for obtaining predictions

Usage

cROCData(data, names.cov, group)

Arguments

Value

A data frame containing selected values of all needed covariates. For those that are continuous, 30 different values are selected.

See Also

AROC.bnp, cROC.bnp, cROC.sp, cROC.kernel, compute.threshold.cROC or compute.threshold.AROC.

Examples

library(ROCnReg)
data(psa)
# Select the last measurement
newpsa <- psa[!duplicated(psa$id, fromLast = TRUE),]

# Log-transform the biomarker
newpsa$l_marker1 <- log(newpsa$marker1)

newdf <- cROCData(newpsa, "age", "status")

summary(newdf)

AROC based threshold values.

Description

This function implements methods for estimating AROC-based threshold values.

Usage

compute.threshold.AROC(object, criterion = c("FPF", "YI"), FPF, newdata,
  ci.level = 0.95, parallel = c("no", "multicore", "snow"), ncpus = 1, cl = NULL)

Arguments

Details

Estimates AROC-based threshold values based on two different criteria, namely, the Youden index (YI) and the one that gives rise to a pre-specified FPF. Before proceeding, we would like to mention that when the accuracy of a test is not affected by covariates, this does not necessarily imply that the covariate-specific ROC curve (which in this case is the same for all covariate values) coincides with the pooled ROC curve. It does coincide, however, with the AROC curve. Consequently, in all cases where covariates affect the test, even though they might not affect its discriminatory capacity, inferences based on the pooled ROC curve might be misleading. In such cases the AROC curve should be used instead. This also applies to the selection of (optimal) threshold values, which, as will be seen, might be covariate-specific (i.e., possibly different for different covariate values).

For the AROC curve, the Youden Index is defined as

YI = \max_{p}\{AROC(p) - p\},

The value p^{*} (FPF) that achieves the maximum is then used to calculate the optimal (covariate-specific) YI threshold as follows

c^{*}_{\mathbf{x}} = F_{\bar{D}}^{-1}(1-p^{*}|\mathbf{x}),

where

F_{\bar{D}}(y|\mathbf{x}) = Pr(Y_{\bar{D}} \leq y | \mathbf{X}_{\bar{D}} = \mathbf{x}).

In a similar way, when using the criterion for a fixed FPF, the covariate-specific threshold values are obtained as follows

c^{*}_{\mathbf{x}} = F_{\bar{D}}^{-1}(1-FPF|\mathbf{x}).

In both cases, we use the notation c^{*}_{\mathbf{x}} to emphasise that this value depends on covariate \mathbf{x}.

Value

As a result, the function provides a list with the following components:

References

Inacio de Carvalho, V., and Rodriguez-Alvarez, M. X. (2022). The Covariate-Adjusted ROC Curve: The Concept and Its Importance, Review of Inferential Methods, and a New Bayesian Estimator. Statistical Science, 37, 541 -561.

Rodriguez-Alvarez, M. X., Roca-Pardinas, J., and Cadarso-Suarez, C. (2011). ROC curve and covariates: extending induced methodology to the non-parametric framework. Statistics and Computing, 21, 483–499.

Rutter, C.M. and Miglioretti, D. L. (2003). Estimating the Accuracy of Psychological Scales Using Longitudinal Data. Biostatistics, 4, 97–107.

Youden, W. J. (1950). Index for rating diagnostic tests. Cancer, 3, 32–35.

See Also

AROC.bnp, AROC.kernel or AROC.sp

Examples

library(ROCnReg)
data(psa)
# Select the last measurement
newpsa <- psa[!duplicated(psa$id, fromLast = TRUE),]

# Log-transform the biomarker
newpsa$l_marker1 <- log(newpsa$marker1)

AROC_bnp <- AROC.bnp(formula.h = l_marker1 ~ f(age, K = 0),
              group = "status", 
              tag.h = 0, 
              data = newpsa,
              standardise = TRUE,
              p = seq(0,1,l=101),
              prior = priorcontrol.bnp(m0 = rep(0, 4), 
              S0 = 10*diag(4), nu = 6, Psi = diag(4),
              a = 2, b = 0.5, alpha = 1, L =10),
              mcmc = mcmccontrol(nsave = 500, nburn = 100, nskip = 1))

### Threshold values based on the YI
th_AROC_bnp_yi <- compute.threshold.AROC(AROC_bnp, criterion = "YI")

# Plot results
plot(th_AROC_bnp_yi$newdata$age, th_AROC_bnp_yi$thresholds[,"est"], 
	type = "l", xlab = "Age", 
	ylab = "log(PSA)", ylim = c(0,3), 
	main = "Threshold values based on the Youden Index")
lines(th_AROC_bnp_yi$newdata$age, th_AROC_bnp_yi$thresholds[,"qh"], lty = 2)
lines(th_AROC_bnp_yi$newdata$age, th_AROC_bnp_yi$thresholds[,"ql"], lty = 2)

### Threshold values for a fixed FPF
th_AROC_bnp_fpf <- compute.threshold.AROC(AROC_bnp, criterion = "FPF", FPF = 0.1)

# Plot results
plot(th_AROC_bnp_yi$newdata$age, th_AROC_bnp_fpf$thresholds[["0.1"]][,"est"], 
	type = "l", xlab = "Age", 
	ylab = "log(PSA)", ylim = c(0,3), 
	main = "Threshold values for a FPF = 0.1")
lines(th_AROC_bnp_yi$newdata$age, th_AROC_bnp_fpf$thresholds[["0.1"]][,"qh"], lty = 2)
lines(th_AROC_bnp_yi$newdata$age, th_AROC_bnp_fpf$thresholds[["0.1"]][,"ql"], lty = 2)

Covariate-specific ROC based threshold values.

Description

This function implements methods for estimating covariate-specific ROC-based threshold values.

Usage

compute.threshold.cROC(object, criterion = c("FPF", "TPF", "YI"), FPF, TPF, newdata,
  ci.level = 0.95, parallel = c("no", "multicore", "snow"), ncpus = 1, cl = NULL)

Arguments

Details

Estimates covariate-specific ROC-based threshold values based on three different criteria, namely, the Youden index (YI), one that gives rise to a pre-specified FPF, and one that gives rise to a pre-specified TPF.

In the conditional case, the Youden index is defined as

YI(\mathbf{x}) = \max_{c}|TPF(c|\mathbf{x}) - FPF(c|\mathbf{x})| = \max_{c}|F_{\bar{D}}(c|\mathbf{x}) - F_{D}(c|\mathbf{x})|,

where

F_{D}(y|\mathbf{x}) = Pr(Y_{D} \leq y | \mathbf{X}_{D} = \mathbf{x}),

F_{\bar{D}}(y|\mathbf{x}) = Pr(Y_{\bar{D}} \leq y | \mathbf{X}_{\bar{D}} = \mathbf{x}).

The value c^{*}_{\mathbf{x}} that achieves the maximum is called the optimal covariate-specific YI threshold. Regarding the criterion for a fixed FPF, the covariate-specific threshold values are obtained as follows

c^{*}_{\mathbf{x}} = F_{\bar{D}}^{-1}(1-FPF|\mathbf{x}),

and for a fixed TPF we have

c^{*}_{\mathbf{x}} = F_{D}^{-1}(1-TPF|\mathbf{x}),

In all cases, we use the notation c^{*}_{\mathbf{x}} to emphasise that this value depends on covariate \mathbf{x}.

Value

As a result, the function provides a list with the following components:

References

Inacio de Carvalho, V., de Carvalho, M. and Branscum, A. J. (2017). Nonparametric Bayesian Covariate-Adjusted Estimation of the Youden Index. Biometrics, 73, 1279-1288.

Rodriguez-Alvarez, M. X., Roca-Pardinas, J., and Cadarso-Suarez, C. (2011). ROC curve and covariates: extending induced methodology to the non-parametric framework. Statistics and Computing, 21, 483–499.

Rutter, C.M. and Miglioretti, D. L. (2003). Estimating the Accuracy of Psychological Scales Using Longitudinal Data. Biostatistics, 4, 97–107.

Youden, W. J. (1950). Index for rating diagnostic tests. Cancer, 3, 32–35.

See Also

cROC.bnp, cROC.kernel or cROC.sp

Examples

library(ROCnReg)
data(psa)
# Select the last measurement
newpsa <- psa[!duplicated(psa$id, fromLast = TRUE),]

# Log-transform the biomarker
newpsa$l_marker1 <- log(newpsa$marker1)

cROC_bnp <- cROC.bnp(formula.h = l_marker1 ~ f(age, K = 0),
			  formula.d = l_marker1 ~ f(age, K = 0),
              group = "status", 
              tag.h = 0, 
              data = newpsa,
              standardise = TRUE,
              p = seq(0,1,l=101),
              mcmc = mcmccontrol(nsave = 500, nburn = 100, nskip = 1))

### Threshold values based on the YI
th_cROC_bnp_yi <- compute.threshold.cROC(cROC_bnp, criterion = "YI")

# Plot results
	# Threshold values
	plot(th_cROC_bnp_yi$newdata$age, th_cROC_bnp_yi$thresholds[,"est"], 
		type = "l", xlab = "Age", 
		ylab = "log(PSA)", ylim = c(0,3), 
		main = "Threshold values based on the Youden Index")
	lines(th_cROC_bnp_yi$newdata$age, th_cROC_bnp_yi$thresholds[,"qh"], lty = 2)
	lines(th_cROC_bnp_yi$newdata$age, th_cROC_bnp_yi$thresholds[,"ql"], lty = 2)

	# Youden Index
	plot(th_cROC_bnp_yi$newdata$age, th_cROC_bnp_yi$YI[,"est"], 
		type = "l", xlab = "Age", 
		ylab = "log(PSA)", ylim = c(0,1), 
		main = "Threshold values based on the Youden Index")
	lines(th_cROC_bnp_yi$newdata$age, th_cROC_bnp_yi$YI[,"qh"], lty = 2)
	lines(th_cROC_bnp_yi$newdata$age, th_cROC_bnp_yi$YI[,"ql"], lty = 2)

### Threshold values for a fixed FPF
th_cROC_bnp_fpf <- compute.threshold.cROC(cROC_bnp, criterion = "FPF", FPF = 0.1)

# Plot results
	# Threshold values
	plot(th_cROC_bnp_yi$newdata$age, th_cROC_bnp_fpf$thresholds[["0.1"]][,"est"], 
		type = "l", xlab = "Age", 
		ylab = "log(PSA)", ylim = c(0,3), main = "Threshold values for a FPF = 0.1")
	lines(th_cROC_bnp_yi$newdata$age, th_cROC_bnp_fpf$thresholds[["0.1"]][,"qh"], lty = 2)
	lines(th_cROC_bnp_yi$newdata$age, th_cROC_bnp_fpf$thresholds[["0.1"]][,"ql"], lty = 2)

Pooled ROC based threshold values.

Description

This function implements methods for estimating pooled ROC-based threshold values.

Usage

compute.threshold.pooledROC(object, criterion = c("FPF", "TPF", "YI"), FPF, TPF, 
  ci.level = 0.95, parallel = c("no", "multicore", "snow"), ncpus = 1, cl = NULL)

Arguments

Details

Estimates pooled ROC-based threshold values based on three different criteria, namely, the Youden index (YI), one that gives rise to a pre-specified FPF, and one that gives rise to a pre-specified TPF.

The Youden Index is defined as

YI = \max_{c}\{TPF(c) - FPF(c)\} = \max_{c}\{F_{\bar{D}}(c) - F_{D}(c)\},

where

F_{D}(y) = Pr(Y_{D} \leq y),

F_{\bar{D}}(y) = Pr(Y_{\bar{D}} \leq y).

The value c^{*} that achieves the maximum is called the optimal YI threshold. Regarding the criterion for a fixed FPF, the threshold value is obtained as follows

c = F_{\bar{D}}^{-1}(1-FPF).

and for a fixed TPF we have

c = F_{D}^{-1}(1-TPF).

Value

As a result, the function provides a list with the following components:

References

Rutter, C.M. and Miglioretti, D. L. (2003). Estimating the Accuracy of Psychological Scales Using Longitudinal Data. Biostatistics, 4, 97–107.

Youden, W. J. (1ci.level*1000). Index for rating diagnostic tests. Cancer, 3, 32–35.

See Also

pooledROC.BB, pooledROC.emp, pooledROC.kernel or pooledROC.dpm.

Examples

library(ROCnReg)
data(psa)
# Select the last measurement
newpsa <- psa[!duplicated(psa$id, fromLast = TRUE),]

# Log-transform the biomarker
newpsa$l_marker1 <- log(newpsa$marker1)

m0_dpm <- pooledROC.dpm(marker = "l_marker1", group = "status",
            tag.h = 0, data = newpsa, standardise = TRUE, 
            p = seq(0,1,l=101), compute.WAIC = TRUE, compute.lpml = TRUE, 
            compute.DIC = TRUE, 
            prior.h = priorcontrol.dpm(m0 = 0, S0 = 10, a = 2, b = 0.5, alpha = 1, 
            L =10),
            prior.d = priorcontrol.dpm(m0 = 0, S0 = 10, a = 2, b = 0.5, alpha = 1, 
            L =10),
            mcmc = mcmccontrol(nsave = 400, nburn = 100, nskip = 1))


## Threshold values based on the YI
th_m0_dpm_yi <- compute.threshold.pooledROC(m0_dpm, criterion = "YI")

th_m0_dpm_yi$threshold
th_m0_dpm_yi$YI

### Threshold values for a fixed FPF
th_m0_dpm_fpf <- compute.threshold.pooledROC(m0_dpm, criterion = "FPF", FPF = 0.1)

th_m0_dpm_fpf$threshold

  

(Conditional) density estimates of test outcomes

Description

This function is used to set various parameters controlling the estimation of the (conditional) density (densities) of test outcomes in both the healthy and diseased groups.

Usage

densitycontrol(compute = FALSE, grid.h = NA, grid.d = NA)

Arguments

Details

The value returned by this function is used as a control argument of the cROC.bnp and pooledROC.dpm functions.

Value

A list with components for each of the possible arguments.

See Also

cROC.bnp and pooledROC.dpm

Examples

library(ROCnReg)
data(psa)
# Select the last measurement
newpsa <- psa[!duplicated(psa$id, fromLast = TRUE),]

# Log-transform the biomarker
newpsa$l_marker1 <- log(newpsa$marker1)

cROC_bnp <- cROC.bnp(formula.h = l_marker1 ~ f(age, K = 0),
               formula.d = l_marker1 ~ f(age, K = 0),
               group = "status", 
               tag.h = 0,
               data = newpsa,
               standardise = TRUE, 
               p = seq(0, 1, len = 101),
               compute.lpml = TRUE, 
               compute.WAIC = TRUE,
               compute.DIC = TRUE, 
               pauc = pauccontrol(compute = TRUE, value = 0.5, focus = "FPF"),
               density = densitycontrol(compute = TRUE, grid.h = NA, grid.d = NA),
               mcmc = mcmccontrol(nsave = 500, nburn = 100, nskip = 1))

Conditional density estimates of test outcomes in the healthy population

Description

This function is used to set various parameters controlling the estimation of the conditional densities of test outcomes in the healthy group.

Usage

densitycontrol.aroc(compute = FALSE, grid.h = NA, newdata = NA)

Arguments

Details

The value returned by this function is used as a control argument of the AROC.bnp function.

Value

A list with components for each of the possible arguments.

See Also

AROC.bnp

Examples

library(ROCnReg)
data(psa)
# Select the last measurement
newpsa <- psa[!duplicated(psa$id, fromLast = TRUE),]

# Log-transform the biomarker
newpsa$l_marker1 <- log(newpsa$marker1)

# Covariate for prediction
agep <- seq(min(newpsa$age), max(newpsa$age), length = 5)
df.pred <- data.frame(age = agep)


AROC_bnp <- AROC.bnp(formula.h =  l_marker1 ~ f(age, K = 0),
                     group = "status", 
                     tag.h = 0,
                     data = newpsa,
                     standardise = TRUE,
                     p = seq(0, 1, len = 101),
                     compute.lpml = TRUE,
                     compute.WAIC = TRUE,
                     compute.DIC = TRUE,
                     pauc = pauccontrol(compute = TRUE, value = 0.5, focus = "FPF"),
                     density = densitycontrol.aroc(compute = TRUE, grid.h = NA, newdata = df.pred),
                     mcmc = mcmccontrol(nsave = 500, nburn = 100, nskip = 1)
)

Simulated endocrine data.

Description

The endosyn data set was simulated based on the data analysed in Rodriguez-Alvarez et al. (2011a,b) and Inacio de Carvalho and Rodriguez-Alvarez (2018); and presented in Botana et al. (2007) and Tome et al. (2008). The aim of these studies was to use the body mass index (BMI) to detect patients having a higher risk of cardiovascular problems, ascertaining the possible effect of age and gender on the accuracy of this measure.

Usage

data("endosyn")

Format

A data frame with 2840 observations on the following 4 variables.

gender

patient's gender. Factor with Men and Women levels.

age

patient's age.

cvd_idf

true disease status (presence/absence of two of more cardiovascular risk factors according to the International Diabetes Federation). Numerical vector (0 = absence, 1 = presence).

bmi

patient's body mass index.

Source

Botana, M.A., Mato, J.A., Cadarso-Suarez, C., Tome, M.A., Perez-Fernandez, R., Fernandez-Mario, A., Rego-Iraeta, A., Solache, I. (2007). Overweight, obesity and central obesity prevalences in the region of Galicia in Northwest Spain. Obesity and Metabolism, 3, 106–115.

Tome, M.A., Botana, M.A., Cadarso-Suarez, C., Rego-Iraeta, A., Fernandez-Mario, A., Mato, J.A, Solache, I., Perez-Fernandez, R. (2008). Prevalence of metabolic syndrome in Galicia (NW Spain) on four alternative definitions and association with insulin resistance. Journal of Endocrinological Investigation, 32, 505–511.

References

Inacio de Carvalho, V., and Rodriguez-Alvarez, M. X. (2022). The Covariate-Adjusted ROC Curve: The Concept and Its Importance, Review of Inferential Methods, and a New Bayesian Estimator. Statistical Science, 37, 541 -561.

Rodriguez-Alvarez, M.X., Roca-Pardinas, J. and Cadarso-Suarez, C. (2011a). ROC curve and covariates: extending induced methodology to the non-parametric framework. Statistics and Computing, 21(4), 483–49.

Rodriguez- Alvarez, M.X., Roca-Pardinas, J. and Cadarso-Suarez, C. (2011b). A new flexible direct ROC regression model - Application to the detection of cardiovascular risk factors by anthropometric measures. Computational Statistics and Data Analysis, 55(12), 3257–3270.

Examples

data(endosyn)
summary(endosyn)

Markov chain Monte Carlo (MCMC) parameters

Description

This function is used to set various parameters controlling the Markov chain Monte Carlo (MCMC) parameters.

Usage

mcmccontrol(nsave = 8000, nburn = 2000, nskip = 1)

Arguments

Details

The value returned by this function is used as a control argument of the AROC.bnp, cROC.bnp, and pooledROC.dpm functions.

Value

A list with components for each of the possible arguments.

See Also

pooledROC.dpm, AROC.bnp and cROC.bnp

Examples

library(ROCnReg)
data(psa)
# Select the last measurement
newpsa <- psa[!duplicated(psa$id, fromLast = TRUE),]

# Log-transform the biomarker
newpsa$l_marker1 <- log(newpsa$marker1)

cROC_bnp <- cROC.bnp(formula.h = l_marker1 ~ f(age, K = 0),
               formula.d = l_marker1 ~ f(age, K = 0),
               group = "status", 
               tag.h = 0,
               data = newpsa,
               standardise = TRUE, 
               p = seq(0, 1, len = 101),
               compute.lpml = TRUE, 
               compute.WAIC = TRUE,
               compute.DIC = TRUE, 
               pauc = pauccontrol(compute = TRUE, value = 0.5, focus = "FPF"),
               density = densitycontrol(compute = TRUE, grid.h = NA, grid.d = NA),
               mcmc = mcmccontrol(nsave = 500, nburn = 100, nskip = 1))

Partial area under the covariate-adjusted/covariate-specific/pooled ROC curve

Description

Used to set various parameters controlling the estimation of the partial area under the covariate-adjusted/covariate-specific/pooled ROC curve .

Usage

pauccontrol(compute = FALSE, focus = c("FPF", "TPF"), value = 1)

Arguments

Details

The value returned by this function is used as a control argument of the AROC.bnp, AROC.sp, AROC.kernel, cROC.bnp, cROC.sp, cROC.kernel functions.

Value

A list with components for each of the possible arguments.

See Also

AROC.bnp

Examples

library(ROCnReg)
data(psa)
# Select the last measurement
newpsa <- psa[!duplicated(psa$id, fromLast = TRUE),]

# Log-transform the biomarker
newpsa$l_marker1 <- log(newpsa$marker1)

# Covariate for prediction
agep <- seq(min(newpsa$age), max(newpsa$age), length = 5)
df.pred <- data.frame(age = agep)


cROC_sp_normal <- cROC.sp(formula.h = l_marker1 ~ age,
                          formula.d = l_marker1 ~ age,
                          group = "status", 
                          tag.h = 0,
                          data = newpsa,
                          newdata = df.pred,
                          est.cdf = "normal",
                          pauc = list(compute = TRUE, value = 0.5, focus = "FPF"),
                          p = seq(0, 1, l = 101), 
                          B = 10)

Default AROC plotting

Description

Takes a fitted AROC object produced by AROC.bnp(), AROC.sp(), or AROC.kernel() and plots the covariate-adjusted ROC curve (AROC) and associated area under the AROC (AAUC).

Usage

## S3 method for class 'AROC'
plot(x, main = NULL, ...)

Arguments

See Also

AROC.bnp, AROC.sp or AROC.kernel

Examples

library(ROCnReg)
data(psa)
# Select the last measurement
newpsa <- psa[!duplicated(psa$id, fromLast = TRUE),]

# Log-transform the biomarker
newpsa$l_marker1 <- log(newpsa$marker1)

AROC_bnp <- AROC.bnp(formula.h = l_marker1 ~ f(age, K = 0),
group = "status", tag.h = 0, data = newpsa, standardise = TRUE,
p = seq(0,1,l=101), compute.lpml = TRUE, compute.WAIC = TRUE,
compute.DIC = TRUE)

plot(AROC_bnp)

Default cROC plotting

Description

Takes a fitted cROC object produced by cROC.bnp(), cROC.sp(), or cROC.kernel() and plots the covariate-specific ROC curve (cROC) and associated area under the cROC curve. The suitable type of graphic is chosen according to the number and nature of the covariates.

Usage

## S3 method for class 'cROC'
plot(x, ask = TRUE, ...)

Arguments

See Also

cROC.bnp, cROC.sp or cROC.kernel

Examples

library(ROCnReg)
data(psa)
# Select the last measurement
newpsa <- psa[!duplicated(psa$id, fromLast = TRUE),]

# Log-transform the biomarker
newpsa$l_marker1 <- log(newpsa$marker1)

cROC_bnp <- cROC.bnp(formula.h = l_marker1 ~ f(age, K = 0),
               formula.d = l_marker1 ~ f(age, K = 0),
               group = "status", 
               tag.h = 0,
               data = newpsa,
               standardise = TRUE, 
               p = seq(0, 1, len = 101),
               compute.lpml = TRUE, 
               compute.WAIC = TRUE,
               compute.DIC = TRUE, 
               pauc = pauccontrol(compute = TRUE, value = 0.5, focus = "FPF"),
               density = densitycontrol(compute = TRUE, grid.h = NA, grid.d = NA),
               mcmc = mcmccontrol(nsave = 500, nburn = 100, nskip = 1))

plot(cROC_bnp)

Default pooledROC plotting

Description

Takes a fitted pooledROC object produced by pooledROC.BB, pooledROC.emp, pooledROC.kernel, or pooledROC.dpm and plots the pooled ROC curve and associated area under the ROC curve (AUC).

Usage

## S3 method for class 'pooledROC'
plot(x, main = NULL, ...)

Arguments

See Also

pooledROC.BB, pooledROC.emp, pooledROC.kernel or pooledROC.dpm

Examples

library(ROCnReg)
data(psa)
# Select the last measurement
newpsa <- psa[!duplicated(psa$id, fromLast = TRUE),]

# Log-transform the biomarker
newpsa$l_marker1 <- log(newpsa$marker1)

m0_emp <- pooledROC.emp(marker = "l_marker1", group = "status",
tag.h = 0, data = newpsa, p = seq(0,1,l=101), B = 500)

summary(m0_emp)

plot(m0_emp)

Bayesian bootstrap estimation of the pooled ROC curve.

Description

This function estimates the pooled ROC curve using the Bayesian bootstrap estimator proposed by Gu et al. (2008).

Usage

pooledROC.BB(marker, group, tag.h, data, 
	p = seq(0, 1, l = 101), B = 5000, ci.level = 0.95, pauc = pauccontrol(),
  	parallel = c("no", "multicore", "snow"), ncpus = 1, cl = NULL)

Arguments

Details

Estimates the pooled ROC curve (ROC) defined as

ROC(p) = 1 - F_{D}\{F_{\bar{D}}^{-1}(1-p)\},

where

F_{D}(y) = Pr(Y_{D} \leq y),

F_{\bar{D}}(y) = Pr(Y_{\bar{D}} \leq y).

The method implemented in this function makes use of the equivalence (see Gu et al., 2008)

ROC(p) = 1 - F_{D}\{F_{\bar{D}}^{-1}(1-p)\} = 1 - Pr(1 - F_{\bar{D}}(Y_D) \leq p),

and estimates both F_{\bar{D}} and the outer probability using the Bayesian bootstrap resampling distribution.

Regarding the area under the curve, we note that

AUC = \int_{0}^{1}ROC(p)dp = 1 - E\{U_D\},

where U_D = 1 - F_{\bar{D}}(Y_D). In our implementation, the expectation is computed using the Bayesian bootstrap (using the same weights as those used to estimate the pooled ROC). As far as the partial area under the curve is concerned, when focus = "FPF" and assuming an upper bound u_1 for the FPF, what it is computed is

pAUC_{FPF}(u_1)=\int_0^{u_1} ROC(p)dp = u_1 - E\{U_{D,u_1}\},

where U_{D,u_1} = \min\{u_1, 1 - F_{\bar{D}}(Y_D)\}. Again, the expectation is computed using the Bayesian bootstrap. The returned value is the normalised pAUC, pAUC_{FPF}(u_1)/u_1 so that it ranges from u_1/2 (useless test) to 1 (perfect marker). Conversely, when focus = "TPF", and assuming a lower bound for the TPF of u_2, the partial area corresponding to TPFs lying in the interval (u_2,1) is computed as

pAUC_{TPF}(u_2)=\int_{u_2}^{1}ROC_{TNF}(p)dp,

where ROC_{TNF}(p) is a 270^\circ rotation of the ROC curve, and it can be expressed as ROC_{TNF}(p) = F_{\bar{D}}\{F_{D}^{-1}(1-p)\}. Thus

pAUC_{TPF}(u_2)=\int_{u_2}^{1}ROC_{TNF}(p)dp = E\{U_{\bar{D}, u_2} - u_2\},

where U_{\bar{D}, u_2} = \max\{u_2, 1 - F_{D}(Y_{\bar{D}})\}, and the expectation is computed using the Bayesian bootstrap. The returned value is the normalised pAUC, pAUC_{TPF}(u_2)/(1-u_2), so that it ranges from (1-u_2)/2 (useless test) to 1 (perfect test).

Value

As a result, the function provides a list with the following components:

References

Gu, J., Ghosal, S., and Roy, A. (2008). Bayesian bootstrap estimation of ROC curve. Statistics in Medicine, 27, 5407–5420.

See Also

AROC.bnp, AROC.sp, AROC.kernel, pooledROC.BB, pooledROC.emp, pooledROC.kernel, pooledROC.dpm, cROC.bnp, cROC.sp or AROC.kernel.

Examples

library(ROCnReg)
data(psa)
# Select the last measurement
newpsa <- psa[!duplicated(psa$id, fromLast = TRUE),]

# Log-transform the biomarker
newpsa$l_marker1 <- log(newpsa$marker1)

m0_BB <- pooledROC.BB(marker = "l_marker1", group = "status",
tag.h = 0, data = newpsa, p = seq(0,1,l=101), B = 5000)

summary(m0_BB)

plot(m0_BB)

Nonparametric Bayesian inference of the pooled ROC curve

Description

This function estimates the pooled ROC curve using a Dirichlet process mixture of normals model as proposed by Erkanli et al. (2006).

Usage

pooledROC.dpm(marker, group, tag.h, data, 
  standardise = TRUE, p = seq(0, 1, l = 101), ci.level = 0.95, 
  compute.lpml = FALSE, compute.WAIC = FALSE, compute.DIC = FALSE, 
  pauc = pauccontrol(), density = densitycontrol(), 
  prior.h = priorcontrol.dpm(), prior.d = priorcontrol.dpm(),
  mcmc = mcmccontrol(),
  parallel = c("no", "multicore", "snow"), ncpus = 1, cl = NULL)

Arguments

Details

Estimates the pooled ROC curve (ROC) defined as

ROC(p) = 1 - F_{D}\{F_{\bar{D}}^{-1}(1-p)\},

where

F_{D}(y) = Pr(Y_{D} \leq y),

F_{\bar{D}}(y) = Pr(Y_{\bar{D}} \leq y).

The method implemented in this function estimates F_{D}(\cdot) and F_{\bar{D}}(\cdot) by means of a Dirichlet process mixture of normals model. More precisely, and letting \{y_{\bar{D}i}\}_{i=1}^{n_{\bar{D}}} and \{y_{Dj}\}_{j=1}^{n_{D}} be two independent random samples from the nondiseased and diseased populations, respectively, the model postulated for the distribution function is as follows

F_{\bar{D}}(y_{\bar{D}i}) = \sum_{l=1}^{L_{\bar{D}}}\omega_{l\bar{D}}\Phi(y_{\bar{D}i}\mid\mu_{l\bar{D}},\sigma_{l\bar{D}}^2),

F_{D}(y_{Dj}) = \sum_{l=1}^{L_{D}}\omega_{lD}\Phi(y_{Dj}\mid\mu_{lD},\sigma_{lD}^2),

where L_{d} is pre-specified is a pre-specified upper bound on the number of mixture components (d \in \{D, \bar{D}\}). The \omega_{ld}'s result from a truncated version of the stick-breaking construction (\omega_{1d} = v_{1d}; \omega_{ld} = v_{ld}\prod_{r<l}(1-v_{dr}), l=2,\ldots,L_{d}; v_{d1},\ldots,v_{L_{d}-1}\sim Beta (1,\alpha_{d}); v_{Ld} = 1, \alpha_d \sim \Gamma(a_{\alpha_d},b_{\alpha_d})), \beta_{ld}\sim N(m_{0d},S_{0d}), and \sigma_{ld}^{-2}\sim\Gamma(a_{d},b_{d}).

The area under the curve is

AUC=\int_{0}^{1}ROC(p)dp.

When the upper bound on the number of mixture components is 1, i.e., L_d = 1 (d \in \{D, \bar{D}\}), there is a closed-form expression for the AUC (binormal model), which is used in the package. In contrast, when L_D > 1 or L_{\bar{D}} > 1, the AUC is computed using results presented in Erkanli et al. (2006). With regard to the partial area under the curve, when focus = "FPF" and assuming an upper bound u_1 for the FPF, what it is computed is

pAUC_{FPF}(u_1)=\int_0^{u_1} ROC(p)dp.

As for the AUC, when L_d = 1 (d \in \{D, \bar{D}\}), there is a closed-form expression for the pAUC_{FPF} (Hillis and Metz, 2012), and when L_D > 1 or L_{\bar{D}} > 1 the integral is approximated numerically using Simpson's rule. The returned value is the normalised pAUC, pAUC_{FPF}(u_1)/u_1 so that it ranges from u_1/2 (useless test) to 1 (perfect marker). Conversely, when focus = "TPF", and assuming a lower bound for the TPF of u_2, the partial area corresponding to TPFs lying in the interval (u_2,1) is computed as

pAUC_{TPF}(u_2)=\int_{u_2}^{1}ROC_{TNF}(p)dp,

where ROC_{TNF}(p) is a 270^\circ rotation of the ROC curve, and it can be expressed as ROC_{TNF}(p) = F_{\bar{D}}\{F_{D}^{-1}(1-p)\}. Again, when L_d = 1 (d \in \{D, \bar{D}\}), there is a closed-form expression for the pAUC_{TNF} (Hillis and Metz, 2012), and when L_D > 1 or L_{\bar{D}} > 1 the integral is approximated numerically using Simpson's rule. The returned value is the normalised pAUC, pAUC_{TPF}(u_2)/(1-u_2), so that it ranges from (1-u_2)/2 (useless test) to 1 (perfect test).

It is worth referring that with respect to the computation of the DIC, when L=1, it is computed as in Spiegelhalter et al. (2002), and when L>1, DIC3 as described in Celeux et al. (2006) is computed. Also, for the computation of the conditional predictive ordinates (CPO) we follow the stable version proposed by Gelman et al. (2014).

Value

As a result, the function provides a list with the following components:

References

Erkanli, A., Sung M., Jane Costello, E., and Angold, A. (2006). Bayesian semi-parametric ROC analysis. Statistics in Medicine, 25, 3905–3928.

Geisser, S. and Eddy, W.F. (1979) A Predictive Approach to Model Selection, Journal of the American Statistical Association, 74, 153–160.

Gelman, A., Carlin, J.B., Stern, H.S., Dunson, D.B., Vehtari, A., and Rubin, D.B. (2014). Bayesian Data Analysis, 3rd ed. CRC Press: Boca Raton, FL.

Gelman, A., Hwang, J., and Vehtari, A. (2014). Understanding predictive information criteria for Bayesian models. Statistics and Computing, 24, 997–1010.

Hillis, S. L. and Metz, C.E. (2012). An Analytic Expression for the Binormal Partial Area under the ROC Curve. Academic Radiology, 19, 1491–1498.

Watanabe, S. (2010). Asymptotic Equivalence of Bayes Cross Validation and Widely Applicable Information Criterion in Singular Learning Theory. Journal of Machine Learning Research, 11, 3571–3594.

See Also

AROC.bnp, AROC.sp, AROC.kernel, pooledROC.BB, pooledROC.emp, pooledROC.kernel, pooledROC.dpm, cROC.bnp, cROC.sp or AROC.kernel.

Examples

library(ROCnReg)
data(psa)
# Select the last measurement
newpsa <- psa[!duplicated(psa$id, fromLast = TRUE),]

# Log-transform the biomarker
newpsa$l_marker1 <- log(newpsa$marker1)

m0_dpm <- pooledROC.dpm(marker = "l_marker1", group = "status",
            tag.h = 0, data = newpsa, standardise = TRUE, 
            p = seq(0,1,l=101), compute.WAIC = TRUE, compute.lpml = TRUE, 
            compute.DIC = TRUE, 
            prior.h = priorcontrol.dpm(m0 = 0, S0 = 10, a = 2, b = 0.5, alpha = 1, 
            L =10),
            prior.d = priorcontrol.dpm(m0 = 0, S0 = 10, a = 2, b = 0.5, alpha = 1, 
            L =10),
            mcmc = mcmccontrol(nsave = 400, nburn = 100, nskip = 1))

summary(m0_dpm)

plot(m0_dpm)

  

Empirical estimation of the pooled ROC curve.

Description

This function estimates the pooled ROC curve using the empirical estimator proposed by Hsieh and Turnbull (1996).

Usage

pooledROC.emp(marker, group, tag.h, data, 
	p = seq(0, 1, l = 101), B = 1000, ci.level = 0.95, 
    method = c("ncoutcome", "coutcome"), pauc = pauccontrol(),
  	parallel = c("no", "multicore", "snow"), ncpus = 1, cl = NULL)

Arguments

Details

Estimates the pooled ROC curve (ROC) defined as

ROC(p) = 1 - F_{D}\{F_{\bar{D}}^{-1}(1-p)\},

where

F_{D}(y) = Pr(Y_{D} \leq y),

F_{\bar{D}}(y) = Pr(Y_{\bar{D}} \leq y).

The method implemented in this function estimates F_{D}(\cdot) and F_{\bar{D}}(\cdot) by means of the empirical dsitributions. More precisely, and letting \{y_{\bar{D}i}\}_{i=1}^{n_{\bar{D}}} and \{y_{Dj}\}_{j=1}^{n_{D}} be two independent random samples from the nondiseased and diseased populations, respectively, the distribution functions in each group take the form

\widehat{F}_{D}(y)=\frac{1}{n_D}\sum_{j=1}^{n_D}I(y_{Dj}\leq y),

\widehat{F}_{\bar{D}}(y)=\frac{1}{n_{\bar{D}}}\sum_{i=1}^{n_{\bar{D}}}I(y_{\bar{D}i}\leq y).

The area under the curve is

AUC=\int_{0}^{1}ROC(p)dp

and is estimated empirically by means of the Mann-Whitney U-statistic. With regard to the partial area under the curve, when focus = "FPF" and assuming an upper bound u_1 for the FPF, what it is computed is

pAUC_{FPF}(u_1)=\int_0^{u_1} ROC(p)dp,

where again is estimated empirically. The returned value is the normalised pAUC, pAUC_{FPF}(u_1)/u_1 so that it ranges from u_1/2 (useless test) to 1 (perfect marker). Conversely, when focus = "TPF", and assuming a lower bound for the TPF of u_2, the partial area corresponding to TPFs lying in the interval (u_2,1) is computed as

pAUC_{TPF}(u_2)=\int_{u_2}^{1}ROC_{TNF}(p)dp,

where ROC_{TNF}(p) is a 270^\circ rotation of the ROC curve, and it can be expressed as ROC_{TNF}(p) = F_{\bar{D}}\{F_{D}^{-1}(1-p)\}. Again, ROC_{TNF}(p) is estimated empirically. The returned value is the normalised pAUC, pAUC_{TPF}(u_2)/(1-u_2), so that it ranges from (1-u_2)/2 (useless test) to 1 (perfect test).

Value

As a result, the function provides a list with the following components:

References

Hsieh, F., and Turnbull, B.W. (1996). Nonparametric and semiparametric estimation of the receiver operating characteristic curve, The Annals of Statistics, 24, 25–40.

See Also

AROC.bnp, AROC.sp, AROC.kernel, pooledROC.BB, pooledROC.emp, pooledROC.kernel, pooledROC.dpm, cROC.bnp, cROC.sp or AROC.kernel.

Examples

library(ROCnReg)
data(psa)
# Select the last measurement
newpsa <- psa[!duplicated(psa$id, fromLast = TRUE),]

# Log-transform the biomarker
newpsa$l_marker1 <- log(newpsa$marker1)

m0_emp <- pooledROC.emp(marker = "l_marker1", group = "status",
tag.h = 0, data = newpsa, p = seq(0,1,l=101), B = 10,
method = "coutcome", pauc = pauccontrol(compute = TRUE, value = 0.5, focus = "FPF"))

summary(m0_emp)

plot(m0_emp)

Kernel-based estimation of the pooled ROC curve.

Description

This function estimates the pooled ROC curve using the kernel-based density estimator proposed by Zhou et al. (1997).

Usage

pooledROC.kernel(marker, group, tag.h, data, 
	p = seq(0, 1, l = 101), 
	bw = c("SRT", "UCV"), B = 1000, ci.level = 0.95, 
	method = c("ncoutcome", "coutcome"), pauc = pauccontrol(),
  parallel = c("no", "multicore", "snow"), ncpus = 1, cl = NULL)

Arguments

Details

Estimates the pooled ROC curve (ROC) defined as

ROC(p) = 1 - F_{D}\{F_{\bar{D}}^{-1}(1-p)\},

where

F_{D}(y) = Pr(Y_{D} \leq y),

F_{\bar{D}}(y) = Pr(Y_{\bar{D}} \leq y).

The method implemented in this function estimates F_{D}(\cdot) and F_{\bar{D}}(\cdot) by means of kernel methods. More precisely, and letting \{y_{\bar{D}i}\}_{i=1}^{n_{\bar{D}}} and \{y_{Dj}\}_{j=1}^{n_{D}} be two independent random samples from the nondiseased and diseased populations, respectively, the distribution functions in each group take the form

F_{D}(y)=\frac{1}{n_{D}}\sum_{j=1}^{n_D}\Phi\left(\frac{y-y_{Dj}}{h_D}\right).

F_{\bar{D}}(y)=\frac{1}{n_{\bar{D}}}\sum_{i=1}^{n_D}\Phi\left(\frac{y-y_{\bar{D}i}}{h_{\bar{D}}}\right).

where \Phi(y) stands for the standard normal distribution function evaluated at y. For the bandwidth h_d, d \in \{D,\bar{D}\} which controls the amount of smoothing, two options are available. When bw = "SRT",

h_{d}=0.9\min\{SD(\mathbf{y}_d),IQR(\mathbf{y}_d)/1.34\}n_{d}^{-0.2},

where SD(\mathbf{y}_d) and IQR(\mathbf{y}_d) are the standard deviation and interquantile range, respectively, of \mathbf{y}_d=(y_{d1},\ldots,y_{dn_{d}}). In turn, when bw = "UCV", the bandwidth is selected via unbiased cross-validation, for further details we refer to bw.ucv from the base package stats.

The area under the curve is

AUC=\int_{0}^{1}ROC(p)dp

and is computed numerically (using Simpson's rule). With regard to the partial area under the curve, when focus = "FPF" and assuming an upper bound u_1 for the FPF, what it is computed is

pAUC_{FPF}(u_1)=\int_0^{u_1} ROC(p)dp,

where again the integral is approximated numerically (Simpson's rule). The returned value is the normalised pAUC, pAUC_{FPF}(u_1)/u_1 so that it ranges from u_1/2 (useless test) to 1 (perfect marker). Conversely, when focus = "TPF", and assuming a lower bound for the TPF of u_2, the partial area corresponding to TPFs lying in the interval (u_2,1) is computed as

pAUC_{TPF}(u_2)=\int_{u_2}^{1}ROC_{TNF}(p)dp,

where ROC_{TNF}(p) is a 270^\circ rotation of the ROC curve, and it can be expressed as ROC_{TNF}(p) = F_{\bar{D}}\{F_{D}^{-1}(1-p)\}. Again, the computation of the integral is done via Simpson's rule. The returned value is the normalised pAUC, pAUC_{TPF}(u_2)/(1-u_2), so that it ranges from (1-u_2)/2 (useless test) to 1 (perfect test).

Value

As a result, the function provides a list with the following components:

References

Zou, K.H., Hall, W.J., Shapiro, D.E. (1997) Smooth non-parametric receiver operating characteristic (ROC) curves for continuous diagnostic tests. Statistics in Medicine, 16, 2143–2156.

See Also

AROC.bnp, AROC.sp, AROC.kernel, pooledROC.BB, pooledROC.emp, pooledROC.kernel, pooledROC.dpm, cROC.bnp, cROC.sp or AROC.kernel.

Examples

library(ROCnReg)
data(psa)
# Select the last measurement
newpsa <- psa[!duplicated(psa$id, fromLast = TRUE),]

# Log-transform the biomarker
newpsa$l_marker1 <- log(newpsa$marker1)

m0_kernel <- pooledROC.kernel(marker = "l_marker1", group = "status",
tag.h = 0, data = newpsa, p = seq(0,1,l=101), bw = "SRT",
B = 500, method = "coutcome", 
pauc = pauccontrol(compute = TRUE, value = 0.5, focus = "FPF"))

summary(m0_kernel)

plot(m0_kernel)

Posterior predictive checks.

Description

Implements posterior predictive checks for objects of AROC or cROC as produced by AROC.bnp or cROC.bnp.

Usage

predictive.checks(object, 
	statistics = c("min", "max", "kurtosis", "skewness"), 
	ndensity = 512, devnew = TRUE)

Arguments

Details

Compares a selected test statistic computed based on the observed diagnostic test outcome (either in the nondiseased group, AROC object, or in both the nondiseased and diseased groups, cROC object) against the same test statistics computed based on simulated data from the posterior predictive distribution of the diagnostic test outcome obtained using a single-weights linear dependent Dirichlet process mixture of normals model. The following graphics are depicted: (1) histograms of the desired statistics computed from simulated datasets (nsave of them) from the posterior predictive distribution of the diagnostic test outcome. In these plots, the estimated statistics from the observed diagnostic test outcome are also depicted. (2) Kernel density estimates computed computed from simulated datasets (nsave of them) from the posterior predictive distribution of the diagnostic test outcome. In these plots, the kernel density estimate of the observed diagnostic test outcome is also depicted. In the case of an object of class AROC, the abovementioned graphics are depicted only for the diagnostic test outcome in the nondiseased group. However, for an object of class cROC, the graphics are depicted, separately, for both the nondiseased and diseased groups. For a detailed discussion about predictive checks, see Gabry et al. (2019).

Value

As a result, the function provides a list with the following components:

References

Gabry, J., Simpson, D., Vehtari, A., Betancourt, M., and Gelman, A. (2019). Visualization in Bayesian workflow. Journal of the Royal Statistical Society, Series A, 182, 1–14.

Inacio de Carvalho, V., and Rodriguez-Alvarez, M. X. (2022). The Covariate-Adjusted ROC Curve: The Concept and Its Importance, Review of Inferential Methods, and a New Bayesian Estimator. Statistical Science, 37, 541 -561.

See Also

AROC.bnp, cROC.bnp

Examples

library(ROCnReg)
data(psa)
# Select the last measurement
newpsa <- psa[!duplicated(psa$id, fromLast = TRUE),]

# Log-transform the biomarker
newpsa$l_marker1 <- log(newpsa$marker1)

AROC_bnp <- AROC.bnp(formula.h = l_marker1 ~ f(age, K = 0),
group = "status", tag.h = 0, data = newpsa, standardise = TRUE,
p = seq(0,1,l=101), compute.lpml = TRUE, compute.WAIC = TRUE)

predictive.checks(AROC_bnp, statistics = "skewness")

Print method for AROC objects

Description

Default print method for objects fitted with AROC.bnp(), AROC.sp(), and AROC.kernel() functions.

Usage

## S3 method for class 'AROC'
print(x, ...)

Arguments

Details

A short summary is printed including the area under the covariate-adjusted ROC curve (AAUC), and if required, the partial area under the covariate-adjusted ROC curve (pAAUC).

See Also

AROC.bnp, AROC.sp or AROC.kernel.

Examples

library(ROCnReg)
data(psa)
# Select the last measurement
newpsa <- psa[!duplicated(psa$id, fromLast = TRUE),]

# Log-transform the biomarker
newpsa$l_marker1 <- log(newpsa$marker1)

AROC_bnp <- AROC.bnp(formula.h = l_marker1 ~ f(age, K = 0),
group = "status", tag.h = 0, data = newpsa, standardise = TRUE,
p = seq(0,1,l=101), compute.lpml = TRUE, compute.WAIC = TRUE)

AROC_bnp

Print method for cROC objects

Description

Default print method for objects fitted with cROC.bnp(), cROC.sp() and cROC.kernel() functions.

Usage

## S3 method for class 'cROC'
print(x, ...)

Arguments

Details

A short summary is printed.

See Also

cROC.bnp, cROC.sp or cROC.kernel.

Examples

library(ROCnReg)
data(psa)
# Select the last measurement
newpsa <- psa[!duplicated(psa$id, fromLast = TRUE),]

# Log-transform the biomarker
newpsa$l_marker1 <- log(newpsa$marker1)

cROC_bnp <- cROC.bnp(formula.h = l_marker1 ~ f(age, K = 0),
               formula.d = l_marker1 ~ f(age, K = 0),
               group = "status", 
               tag.h = 0,
               data = newpsa,
               standardise = TRUE, 
               p = seq(0, 1, len = 101),
               compute.lpml = TRUE, 
               compute.WAIC = TRUE,
               compute.DIC = TRUE, 
               pauc = pauccontrol(compute = TRUE, value = 0.5, focus = "FPF"),
               density = densitycontrol(compute = TRUE, grid.h = NA, grid.d = NA),
               mcmc = mcmccontrol(nsave = 500, nburn = 100, nskip = 1))

cROC_bnp

Print method for pooledROC objects

Description

Default print method for objects fitted with pooledROC.BB, pooledROC.emp, pooledROC.kernel or pooledROC.dpm functions.

Usage

## S3 method for class 'pooledROC'
print(x, ...)

Arguments

Details

A short summary is printed including the area under the pooled ROC curve (AUC).

See Also

pooledROC.BB, pooledROC.emp, pooledROC.kernel or pooledROC.dpm.

Examples

library(ROCnReg)
data(psa)
# Select the last measurement
newpsa <- psa[!duplicated(psa$id, fromLast = TRUE),]

# Log-transform the biomarker
newpsa$l_marker1 <- log(newpsa$marker1)

m0_emp <- pooledROC.emp(marker = "l_marker1", group = "status",
tag.h = 0, data = newpsa, p = seq(0,1,l=101), B = 500)

summary(m0_emp)

plot(m0_emp)

Prior information for the AROC.bnp and cROC.bnp

Description

This function is used to set various parameters controlling the prior information to be used in the AROC.bnp and cROC.bnp functions.

Usage

priorcontrol.bnp(m0 = NA, S0 = NA, nu = NA, Psi = NA, a = 2, b = NA, 
	alpha = 1, L = 10)

Arguments

Value

A list with components for each of the possible arguments.

See Also

AROC.bnp and cROC.bnp

Prior information for the pooledROC.dpm

Description

This function is used to set various parameters controlling the prior information to be used in the pooledROC.dpm function.

Usage

priorcontrol.dpm(m0 = NA, S0 = NA, a = 2, b = NA, alpha = 1, L = 10)

Arguments

Value

A list with components for each of the possible arguments.

See Also

pooledROC.dpm

Prostate specific antigen (PSA) biomarker study.

Description

The dataset contains 71 prostate cases and 71 controls who participated in a lung cancer prevention trial (CARET, Beta-carotene and retinol trial). For details, see Etzioni et al. (1999) and Pepe (2003).

Usage

data("psa")

Format

A data frame with 683 observations on the following 6 variables.

id

Patient identifier.

marker1

Total prostate specific antigen (PSA).

marker2

Free prostate specific antigen (PSA)

status

Presence/absence of prostate cancer. The non-cancer patients are controls matched to cases on age and number of serum samples available for analysis (see Details).

age

Patient age at blood draw (serum sample).

t

Time (years) relative to prostate cancer diagnosis.

Details

The CARET enrolled 12000 men, aged between 50 and 65 years, at high risk of lung cancer. For each subject on the study, serum samples were drawn at baseline and at two-year intervals after that. The data presented here represent a subsample of the original sample, and it was reported by Etzioni et al. (1999). It contains 71 cases of prostate cancer that occurred during the study. All these cases had, at least, three and up to eight serum samples. As far as controls are concerned, they were selected from the participants of the CARET study verifying that they had not been diagnosed with prostate cancer by the time of the original study, and the selection was done by matching to cases on date of birth and number of serum samples available for analysis.

Source

The dataset can be downloaded from https://research.fredhutch.org/diagnostic-biomarkers-center/en/datasets.html.

References

Pepe, M. S. (2003). The Statistical Evaluation of Medical Tests for Classification and Prediction. Oxford Statistical Science Series. Oxford University Press, New York.

Etzioni, R., Pepe, M. S., Longton, G., Hu. C., and Goodman, G. (1999). Incorporating the time dimension in receiver operating characteristic curves: A case study of prostate cancer. Medical Decision Making, 19(3), 242-251.

Examples

data(psa)
summary(psa)

Summary method for AROC objects

Description

Default summary method for objects fitted with AROC.bnp(), AROC.sp(), or AROC.kernel() functions.

Usage

## S3 method for class 'AROC'
summary(object, ...)

Arguments

Details

The information printed depends on the method. In all cases, the call to the function, the method, the area under the covariate-adjusted ROC curve (AAUC), the partial area under the covariate-adjusted ROC curve (if required) (AAUC), and the sample sizes are printed. For the semiparametric approach (AROC.sp()), the estimated coefficients (and 95% confidence intervals, if required) of the model for the healthy population are printed. In addition, the function provides the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC). For the nonparametric Bayesian approach (AROC.bnp()), and if required, the function provides the log pseudo marginal likelihood (LPML), the widely applicable information criterion (WAIC) and/or the deviance information criterion (DIC). For the kernel-based approach (AROC.kernel()), information regarding the selected bandwidth and the type of kernel estimator (for both regression and variance functions) is printed.

See Also

AROC.bnp, AROC.sp or AROC.kernel.

Examples

library(ROCnReg)
data(psa)
# Select the last measurement
newpsa <- psa[!duplicated(psa$id, fromLast = TRUE),]

# Log-transform the biomarker
newpsa$l_marker1 <- log(newpsa$marker1)

m0 <- AROC.bnp(formula.h = l_marker1 ~ f(age, K = 0),
group = "status", tag.h = 0, data = newpsa, standardise = TRUE,
p = seq(0,1,l=101), compute.lpml = TRUE, compute.WAIC = TRUE)

summary(m0)

Summary method for cROC objects

Description

Default summary method for objects fitted with cROC.bnp(), cROC.sp(), or cROC.kernel() functions.

Usage

## S3 method for class 'cROC'
summary(object, ...)

Arguments

Details

The information printed depends on the method. In all cases, the call to the function, the method, and the sample sizes are printed. For the semiparametric approach (cROC.sp()), the estimated coefficients (and 95% confidence intervals, if required) of the model for the healthy population, the diseased population and the conditional ROC curve, are printed. In addition, the function provides the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC). For the nonparametric Bayesian approach (cROC.bnp()), and if required, the function provides the log pseudo marginal likelihood (LPML), the widely applicable information criterion (WAIC) and/or the deviance information criterion (DIC) (for both healthy and diseased populations). For the kernel-based approach (cROC.kernel()), information regarding the selected bandwidths and the type of kernel estimator(for both healthy and diseased populations and for both regression and variance functions) is printed.

References

Rodriguez-Alvarez, M.X., Tahoces, P.G., Cadarso-Suarez, C. and Lado, M.J. (2011). Comparative study of ROC regression techniques. Applications for the computer-aided diagnostic system in breast cancer detection. Computational Statistics and Data Analysis, 55, 888–902.

See Also

cROC.bnp, cROC.sp or cROC.kernel.

Examples

library(ROCnReg)
data(psa)
# Select the last measurement
newpsa <- psa[!duplicated(psa$id, fromLast = TRUE),]

# Log-transform the biomarker
newpsa$l_marker1 <- log(newpsa$marker1)

cROC_bnp <- cROC.bnp(formula.h = l_marker1 ~ f(age, K = 0),
               formula.d = l_marker1 ~ f(age, K = 0),
               group = "status", 
               tag.h = 0,
               data = newpsa,
               standardise = TRUE, 
               p = seq(0, 1, len = 101),
               compute.lpml = TRUE, 
               compute.WAIC = TRUE,
               compute.DIC = TRUE, 
               pauc = pauccontrol(compute = TRUE, value = 0.5, focus = "FPF"),
               density = densitycontrol(compute = TRUE, grid.h = NA, grid.d = NA),
               mcmc = mcmccontrol(nsave = 500, nburn = 100, nskip = 1))

summary(cROC_bnp)

Summary method for pooledROC objects

Description

Default summary method for objects fitted with pooledROC.BB, pooledROC.emp, pooledROC.kernel, or pooledROC.dpm functions.

Usage

## S3 method for class 'pooledROC'
summary(object, ...)

Arguments

Details

A short summary is printed including the call to the function, the method, samples sizes, the area under the pooled ROC curve (AUC), and if required, the partial area under the pooled ROC curve. For the nonparametric Bayesian approach (pooledROC.dpm()), and if required, the function provides the log pseudo marginal likelihood (LPML), the widely applicable information criterion (WAIC) and/or the deviance information criterion (DIC). For the kernel-based approach (pooledROC.dpm()), information regarding the selected bandwidths and the density bandwidth selection method is presented.

See Also

pooledROC.BB, pooledROC.emp, pooledROC.kernel or pooledROC.dpm.

Examples

library(ROCnReg)
data(psa)
# Select the last measurement
newpsa <- psa[!duplicated(psa$id, fromLast = TRUE),]

# Log-transform the biomarker
newpsa$l_marker1 <- log(newpsa$marker1)

m0_emp <- pooledROC.emp(marker = "l_marker1", group = "status",
            tag.h = 0, data = newpsa, p = seq(0,1,l=101), B = 500)

summary(m0_emp)

plot(m0_emp)