# Asymptotic variance-covariance estimation for True Class Fractions (TCFs) at the cut point $(c_1, c_2)$ `asy_cov_tcf` computes the asymptotic variance-covariance matrix of full data (FULL) and bias-corrected estimators (i.e. full imputation, mean score imputation, inverse probability weighting, semiparametric efficient and K nearest neighbor) of TCFs. ```r asy_cov_tcf( obj_tcf, diag_test, dise_vec, veri_stat = NULL, rho_est = NULL, pi_est = NULL, boot = FALSE, n_boot = 250, parallel = FALSE, ncpus = ifelse(parallel, detectCores()/2, NULL) ) ``` ## Arguments - `obj_tcf`: a result of a call to `rocs.tcf`. - `diag_test`: a numeric vector containing the diagnostic test values. `NA` values of `diag_test` are not accepted. - `dise_vec`: a n * 3 binary matrix with three columns, corresponding to the three classes of the disease status. In row i, 1 in column j indicates that the i-th subject belongs to class j, with j = 1, 2, 3. A row of `NA` values indicates a non-verified subject. - `veri_stat`: a binary vector containing the verification status (1 verified, 0 not verified). - `rho_est`: a result of a call to `rho_mlogit` of `rho_knn` to fit the disease model. - `pi_est`: a result of a call to `psglm` to fit the verification model. - `boot`: a logical value. Default = `FALSE`. If set to `TRUE`, a bootstrap resampling is employed to estimate the asymptotic variance-covariance matrix of bias-corrected TCFs. - `n_boot`: the number of bootstrap replicates, used when `boot = TRUE` or for FULL estimator. Usually this will be a single positive integer. Default 250. - `parallel`: a logical value. If `TRUE`, a parallel computing is employed in the bootstrap resampling process. - `ncpus`: number of processes to be used in parallel computing. Default is half of available cores. ## Returns This function returns an estimated asymptotic variance-covariance matrix for FULL estimator and bias-corrected estimators of TCFs at a fixed cut point. ## Details For bias-corrected estimators of TCFs, the asymptotic variance-covariance matrix at a fixed cut point is estimated by using the Delta method. The function `asy_cov_tcf` implements the explicit forms presented in To Duc et al. (2016, 2020). In addition, the bootstrap procedure is also available. For FULL estimator, the asymptotic variance-covariance matrix is computed via bootstrap only. ## Examples ```r data(EOC) # FULL data estimator dise_full <- pre_data(EOC$D.full, EOC$CA125) dise_vec_full <- dise_full$dise_vec full_tcf <- rocs.tcf("full", diag_test = EOC$CA125, dise_vec = dise_vec_full, cps = c(2, 4)) full_var <- asy_cov_tcf(full_tcf, diag_test = EOC$CA125, dise_vec = dise_vec_full) # Preparing the missing disease status dise_na <- pre_data(EOC$D, EOC$CA125) dise_vec_na <- dise_na$dise_vec dise_fact_na <- dise_na$dise rho_out <- rho_mlogit(dise_fact_na ~ CA125 + CA153 + Age, data = EOC, test = TRUE) ## FI estimator fi_tcf <- rocs.tcf("fi", diag_test = EOC$CA125, dise_vec = dise_vec_na, veri_stat = EOC$V, rho_est = rho_out, cps = c(2, 4)) fi_var <- asy_cov_tcf(fi_tcf, diag_test = EOC$CA125, dise_vec = dise_vec_na, veri_stat = EOC$V, rho_est = rho_out) ## MSI estimator msi_tcf <- rocs.tcf("msi", diag_test = EOC$CA125, dise_vec = dise_vec_na, veri_stat = EOC$V, rho_est = rho_out, cps = c(2, 4)) msi_var <- asy_cov_tcf(msi_tcf, diag_test = EOC$CA125, dise_vec = dise_vec_na, veri_stat = EOC$V, rho_est = rho_out) ## IPW estimator pi_out <- psglm(V ~ CA125 + CA153 + Age, data = EOC, test = TRUE) ipw_tcf <- rocs.tcf("ipw", diag_test = EOC$CA125, dise_vec = dise_vec_na, veri_stat = EOC$V, pi_est = pi_out, cps = c(2, 4)) ipw_var <- asy_cov_tcf(ipw_tcf, diag_test = EOC$CA125, dise_vec = dise_vec_na, veri_stat = EOC$V, pi_est = pi_out) ## SPE estimator spe_tcf <- rocs.tcf("spe", diag_test = EOC$CA125, dise_vec = dise_vec_na, veri_stat = EOC$V, rho_est = rho_out, pi_est = pi_out, cps = c(2, 4)) spe_var <- asy_cov_tcf(spe_tcf, diag_test = EOC$CA125, dise_vec = dise_vec_na, veri_stat = EOC$V, rho_est = rho_out, pi_est = pi_out) ## KNN estimators x_mat <- cbind(EOC$CA125, EOC$CA153, EOC$Age) rho_1nn <- rho_knn(x_mat = x_mat, dise_vec = dise_vec_na, veri_stat = EOC$V, k = 1, type = "mahala") knn_tcf <- rocs.tcf("knn", diag_test = EOC$CA125, dise_vec = dise_vec_na, veri_stat = EOC$V, rho_est = rho_1nn, cps = c(2, 4)) knn_var <- asy_cov_tcf(knn_tcf, diag_test = EOC$CA125, dise_vec = dise_vec_na, veri_stat = EOC$V, rho_est = rho_1nn) ``` ## References To Duc, K., Chiogna, M. and Adimari, G. (2016) Bias-corrected methods for estimating the receiver operating characteristic surface of continuous diagnostic tests. **Electronic Journal of Statistics**, 10 , 3063-3113. To Duc, K., Chiogna, M. and Adimari, G. (2020) Nonparametric estimation of ROC surfaces in presence of verification bias. **REVSTAT-Statistical Journal**. 18 , 5, 697–720.

asy_cov_tcf function