For right-censored data, fit a regularized Cox cure rate model through elastic-net penalty following Masud et al. (2018), and Zou and Hastie (2005). For right-censored data with uncertain event status, fit the regularized Cox cure model proposed by Wang et al. (2020). Without regularization, the model reduces to the regular Cox cure rate model (Kuk and Chen, 1992; Sy and Taylor, 2000)
cox_cure_net( surv_formula, cure_formula, time, event, data, subset, contrasts = NULL, surv_lambda = NULL, surv_alpha = 1, surv_nlambda = 10, surv_lambda_min_ratio = 0.1, surv_l1_penalty_factor = NULL, cure_lambda = NULL, cure_alpha = 1, cure_nlambda = 10, cure_lambda_min_ratio = 0.1, cure_l1_penalty_factor = NULL, cv_nfolds = 0, surv_start = NULL, cure_start = NULL, surv_offset = NULL, cure_offset = NULL, surv_standardize = TRUE, cure_standardize = TRUE, em_max_iter = 200, em_rel_tol = 1e-05, surv_max_iter = 10, surv_rel_tol = 1e-05, cure_max_iter = 10, cure_rel_tol = 1e-05, tail_completion = c("zero", "exp", "zero-tau"), tail_tau = NULL, pmin = 1e-05, early_stop = TRUE, verbose = FALSE, ... ) cox_cure_net.fit( surv_x, cure_x, time, event, cure_intercept = TRUE, surv_lambda = NULL, surv_alpha = 1, surv_nlambda = 10, surv_lambda_min_ratio = 0.1, surv_l1_penalty_factor = NULL, cure_lambda = NULL, cure_alpha = 1, cure_nlambda = 10, cure_lambda_min_ratio = 0.1, cure_l1_penalty_factor = NULL, cv_nfolds = 0, surv_start = NULL, cure_start = NULL, surv_offset = NULL, cure_offset = NULL, surv_standardize = TRUE, cure_standardize = TRUE, em_max_iter = 200, em_rel_tol = 1e-05, surv_max_iter = 10, surv_rel_tol = 1e-05, cure_max_iter = 10, cure_rel_tol = 1e-05, tail_completion = c("zero", "exp", "zero-tau"), tail_tau = NULL, pmin = 1e-05, early_stop = TRUE, verbose = FALSE, ... )
| surv_formula | A formula object starting with |
|---|---|
| cure_formula | A formula object starting with |
| time | A numeric vector for the observed survival times. |
| event | A numeric vector for the event indicators. |
| data | An optional data frame, list, or environment that contains the
covariates and response variables ( |
| subset | An optional logical vector specifying a subset of observations to be used in the fitting process. |
| contrasts | An optional list, whose entries are values (numeric
matrices or character strings naming functions) to be used as
replacement values for the contrasts replacement function and whose
names are the names of columns of data containing factors. See
|
| surv_lambda, cure_lambda | A numeric vector consists of nonnegative values representing the tuning parameter sequence for the survival model part or the incidence model part. |
| surv_alpha, cure_alpha | A number between 0 and 1 for tuning the elastic net penalty for the survival model part or the incidence model part. If it is one, the elastic penalty will reduce to the well-known lasso penalty. If it is zero, the ridge penalty will be used. |
| surv_nlambda, cure_nlambda | A positive number specifying the number of
|
| surv_lambda_min_ratio, cure_lambda_min_ratio | The ratio of the minimum
|
| surv_l1_penalty_factor, cure_l1_penalty_factor | A numeric vector that
consists of nonnegative penalty factors (or weights) on L1-norm for the
coefficient estimate vector in the survival model part or the incidence
model part. The penalty is applied to the coefficient estimate divided
by the specified weights. The specified weights are re-scaled
internally so that their summation equals the length of coefficients.
If |
| cv_nfolds | An non-negative integer specifying number of folds in
cross-validation (CV). The default value is |
| surv_start | An optional numeric vector representing the
starting values for the survival model component or the incidence model
component. If |
| cure_start | An optional numeric vector representing the
starting values for the survival model component or the incidence model
component. If |
| surv_offset | An optional numeric vector representing the
offset term in the survival model compoent or the incidence model
component. The function will internally try to find values of the
specified variable in the |
| cure_offset | An optional numeric vector representing the
offset term in the survival model compoent or the incidence model
component. The function will internally try to find values of the
specified variable in the |
| surv_standardize, cure_standardize | A logical value specifying whether
to standardize the covariates for the survival model part or the
incidence model part. If |
| em_max_iter | A positive integer specifying the maximum iteration
number of the EM algorithm. The default value is |
| em_rel_tol | A positive number specifying the tolerance that determines
the convergence of the EM algorithm in terms of the convergence of the
covariate coefficient estimates. The tolerance is compared with the
relative change between estimates from two consecutive iterations, which
is measured by ratio of the L1-norm of their difference to the sum of
their L1-norm. The default value is |
| surv_max_iter, cure_max_iter | A positive integer specifying the maximum
iteration number of the M-step routine related to the survival model
component or the incidence model component. The default value is
|
| surv_rel_tol | A positive number specifying the tolerance
that determines the convergence of the M-step related to the survival
model component or the incidence model component in terms of the
convergence of the covariate coefficient estimates. The tolerance is
compared with the relative change between estimates from two consecutive
iterations, which is measured by ratio of the L1-norm of their
difference to the sum of their L1-norm. The default value is
|
| cure_rel_tol | A positive number specifying the tolerance
that determines the convergence of the M-step related to the survival
model component or the incidence model component in terms of the
convergence of the covariate coefficient estimates. The tolerance is
compared with the relative change between estimates from two consecutive
iterations, which is measured by ratio of the L1-norm of their
difference to the sum of their L1-norm. The default value is
|
| tail_completion | A character string specifying the tail completion
method for conditional survival function. The available methods are
|
| tail_tau | A numeric number specifying the time of zero-tail
completion. It will be used only if |
| pmin | A numeric number specifying the minimum value of probabilities
for sake of numerical stability. The default value is |
| early_stop | A logical value specifying whether to stop the iteration
once the negative log-likelihood unexpectedly increases, which may
suggest convergence on likelihood, or indicate numerical issues or
implementation bugs. The default value is |
| verbose | A logical value. If |
| ... | Other arguments for future usage. A warning will be thrown if any invalid argument is specified. |
| surv_x | A numeric matrix for the design matrix of the survival model component. |
| cure_x | A numeric matrix for the design matrix of the cure rate model
component. The design matrix should exclude an intercept term unless we
want to fit a model only including the intercept term. In that case, we
need further set |
| cure_intercept | A logical value specifying whether to add an intercept
term to the cure rate model component. If |
cox_cure_net object for regular Cox cure rate model or
cox_cure_net_uncer object for Cox cure rate model with uncertain
events.
The model estimation procedure follows expectation maximization (EM) algorithm. Variable selection procedure through regularization by elastic net penalty is developed based on cyclic coordinate descent and majorization-minimization (MM) algorithm.
Kuk, A. Y. C., & Chen, C. (1992). A mixture model combining logistic regression with proportional hazards regression. Biometrika, 79(3), 531--541.
Masud, A., Tu, W., & Yu, Z. (2018). Variable selection for mixture and promotion time cure rate models. Statistical methods in medical research, 27(7), 2185--2199.
Peng, Y. (2003). Estimating baseline distribution in proportional hazards cure models. Computational Statistics & Data Analysis, 42(1-2), 187--201.
Sy, J. P., & Taylor, J. M. (2000). Estimation in a Cox proportional hazards cure model. Biometrics, 56(1), 227--236.
Wang, W., Luo, C., Aseltine, R. H., Wang, F., Yan, J., & Chen, K. (2020). Suicide Risk Modeling with Uncertain Diagnostic Records. arXiv preprint arXiv:2009.02597.
Zou, H., & Hastie, T. (2005). Regularization and variable selection via the elastic net. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 67(2), 301--320.
cox_cure for regular Cox cure rate model.
library(intsurv) ### regularized Cox cure rate model ================================== ## simulate a toy right-censored data with a cure fraction set.seed(123) n_obs <- 100 p <- 10 x_mat <- matrix(rnorm(n_obs * p), nrow = n_obs, ncol = p) colnames(x_mat) <- paste0("x", seq_len(p)) surv_beta <- c(rep(0, p - 5), rep(1, 5)) cure_beta <- c(rep(1, 2), rep(0, p - 2)) dat <- simData4cure(nSubject = n_obs, lambda_censor = 0.01, max_censor = 10, survMat = x_mat, survCoef = surv_beta, cureCoef = cure_beta, b0 = 0.5, p1 = 1, p2 = 1, p3 = 1) ## model-fitting from given design matrices fit1 <- cox_cure_net.fit(x_mat, x_mat, dat$obs_time, dat$obs_event, surv_nlambda = 10, cure_nlambda = 10, surv_alpha = 0.8, cure_alpha = 0.8) ## model-fitting from given model formula fm <- paste(paste0("x", seq_len(p)), collapse = " + ") surv_fm <- as.formula(sprintf("~ %s", fm)) cure_fm <- surv_fm fit2 <- cox_cure_net(surv_fm, cure_fm, data = dat, time = obs_time, event = obs_event, surv_alpha = 0.5, cure_alpha = 0.5) ## summary of BIC's BIC(fit1)#> df surv_df cure_df BIC #> 1 2 0 2 571.6569 #> 2 2 0 2 567.3181 #> 3 3 0 3 566.3957 #> 4 4 0 4 567.0484 #> 5 6 0 6 572.1122 #> 6 7 0 7 572.3148 #> 7 9 0 9 578.0111 #> 8 10 0 10 579.5357 #> 9 10 0 10 577.3777 #> 10 11 0 11 580.3645 #> 11 3 1 2 574.8266 #> 12 3 1 2 570.4866 #> 13 4 1 3 569.5839 #> 14 5 1 4 570.2510 #> 15 6 1 5 570.7119 #> 16 8 1 7 575.5220 #> 17 10 1 9 581.2231 #> 18 11 1 10 582.7494 #> 19 11 1 10 580.5933 #> 20 12 1 11 583.5942 #> 21 4 2 2 571.2981 #> 22 4 2 2 566.9017 #> 23 5 2 3 565.9767 #> 24 6 2 4 566.6088 #> 25 7 2 5 567.0865 #> 26 9 2 7 571.9303 #> 27 11 2 9 577.6616 #> 28 12 2 10 579.1972 #> 29 12 2 10 577.0534 #> 30 13 2 11 580.0550 #> 31 6 4 2 572.0628 #> 32 6 4 2 567.2647 #> 33 7 4 3 566.1255 #> 34 8 4 4 566.5304 #> 35 9 4 5 567.1024 #> 36 11 4 7 572.0521 #> 37 12 4 8 573.2681 #> 38 14 4 10 579.5170 #> 39 14 4 10 577.4542 #> 40 15 4 11 580.4828 #> 41 7 5 2 561.6637 #> 42 7 5 2 556.4483 #> 43 8 5 3 555.4625 #> 44 9 5 4 555.9295 #> 45 9 5 4 552.1100 #> 46 12 5 7 561.6186 #> 47 13 5 8 562.8272 #> 48 15 5 10 569.1044 #> 49 15 5 10 567.0317 #> 50 16 5 11 570.0577 #> 51 8 6 2 553.9714 #> 52 8 6 2 549.1187 #> 53 9 6 3 548.2548 #> 54 10 6 4 548.7672 #> 55 10 6 4 544.9904 #> 56 13 6 7 554.6001 #> 57 15 6 9 560.3878 #> 58 16 6 10 562.0681 #> 59 16 6 10 559.9757 #> 60 17 6 11 562.9819 #> 61 8 6 2 546.0186 #> 62 9 7 2 545.7555 #> 63 10 7 3 544.8851 #> 64 11 7 4 545.4099 #> 65 11 7 4 541.6443 #> 66 14 7 7 551.3362 #> 67 16 7 9 557.1108 #> 68 17 7 10 558.8130 #> 69 17 7 10 556.7136 #> 70 18 7 11 559.7027 #> 71 10 8 2 549.1734 #> 72 10 8 2 544.4072 #> 73 11 8 3 543.5357 #> 74 12 8 4 544.0616 #> 75 12 8 4 540.3017 #> 76 14 8 6 545.4479 #> 77 17 8 9 555.8321 #> 78 18 8 10 557.5624 #> 79 18 8 10 555.4684 #> 80 19 8 11 558.4443 #> 81 12 10 2 553.4890 #> 82 12 10 2 548.5615 #> 83 13 10 3 547.6633 #> 84 14 10 4 548.1603 #> 85 14 10 4 544.3829 #> 86 16 10 6 549.5268 #> 87 19 10 9 559.9157 #> 88 20 10 10 561.6565 #> 89 20 10 10 559.5554 #> 90 21 10 11 562.5128 #> 91 12 10 2 549.7380 #> 92 12 10 2 545.0480 #> 93 13 10 3 544.0637 #> 94 14 10 4 544.5302 #> 95 14 10 4 540.7353 #> 96 16 10 6 545.8863 #> 97 19 10 9 556.3017 #> 98 20 10 10 558.0478 #> 99 21 10 11 560.5439 #> 100 21 10 11 558.8899#> df surv_df cure_df BIC #> 1 2 0 2 571.7958 #> 2 2 0 2 568.4844 #> 3 3 0 3 568.4511 #> 4 4 0 4 569.2552 #> 5 6 0 6 574.2389 #> 6 7 0 7 574.6851 #> 7 8 0 8 575.6070 #> 8 10 0 10 581.4934 #> 9 10 0 10 578.9973 #> 10 11 0 11 581.7157 #> 11 3 1 2 575.2438 #> 12 3 1 2 571.9320 #> 13 4 1 3 571.9106 #> 14 5 1 4 572.7245 #> 15 7 1 6 577.7091 #> 16 8 1 7 578.1582 #> 17 9 1 8 579.0867 #> 18 11 1 10 584.9727 #> 19 11 1 10 582.4786 #> 20 12 1 11 585.2040 #> 21 4 2 2 573.4404 #> 22 4 2 2 570.0984 #> 23 5 2 3 570.0641 #> 24 6 2 4 570.8632 #> 25 8 2 6 575.8681 #> 26 9 2 7 576.3247 #> 27 11 2 9 581.8671 #> 28 12 2 10 583.1623 #> 29 12 2 10 580.6766 #> 30 13 2 11 583.4063 #> 31 6 4 2 576.5390 #> 32 6 4 2 573.0304 #> 33 7 4 3 572.8699 #> 34 8 4 4 573.5255 #> 35 10 4 6 578.5651 #> 36 11 4 7 579.1056 #> 37 13 4 9 584.6872 #> 38 14 4 10 586.0446 #> 39 14 4 10 583.6066 #> 40 15 4 11 586.3663 #> 41 7 5 2 569.3109 #> 42 7 5 2 565.4851 #> 43 8 5 3 565.3667 #> 44 9 5 4 566.0206 #> 45 11 5 6 571.2470 #> 46 12 5 7 571.7412 #> 47 14 5 9 577.3128 #> 48 15 5 10 578.7157 #> 49 15 5 10 576.2806 #> 50 16 5 11 579.0345 #> 51 8 6 2 562.1769 #> 52 8 6 2 558.4644 #> 53 9 6 3 558.4153 #> 54 10 6 4 559.0812 #> 55 11 6 5 559.7931 #> 56 13 6 7 564.9129 #> 57 15 6 9 570.4631 #> 58 16 6 10 571.8919 #> 59 16 6 10 569.4491 #> 60 17 6 11 572.1968 #> 61 8 6 2 553.2210 #> 62 8 6 2 549.5238 #> 63 9 6 3 549.5231 #> 64 10 6 4 550.2024 #> 65 11 6 5 550.9437 #> 66 13 6 7 556.1250 #> 67 15 6 9 561.6600 #> 68 16 6 10 563.1094 #> 69 16 6 10 560.6603 #> 70 17 6 11 563.3988 #> 71 10 8 2 555.7105 #> 72 10 8 2 551.8308 #> 73 11 8 3 551.8037 #> 74 12 8 4 552.4865 #> 75 13 8 5 553.2438 #> 76 15 8 7 558.4816 #> 77 17 8 9 564.0158 #> 78 18 8 10 565.4922 #> 79 18 8 10 563.0476 #> 80 19 8 11 565.7790 #> 81 12 10 2 558.9853 #> 82 12 10 2 555.1329 #> 83 13 10 3 555.0467 #> 84 14 10 4 555.7139 #> 85 15 10 5 556.4813 #> 86 17 10 7 561.7638 #> 87 19 10 9 567.2879 #> 88 20 10 10 568.7817 #> 89 20 10 10 566.3330 #> 90 21 10 11 569.0486 #> 91 12 10 2 554.2454 #> 92 12 10 2 550.4021 #> 93 13 10 3 550.3402 #> 94 14 10 4 551.0016 #> 95 15 10 5 551.7540 #> 96 17 10 7 557.0672 #> 97 19 10 9 562.5825 #> 98 20 10 10 564.0952 #> 99 20 10 10 561.6475 #> 100 21 10 11 564.3541#> $surv #> x1 x2 x3 x4 x5 x6 #> 0.00000000 0.04350424 -0.01435817 0.00000000 -0.05024160 0.60281005 #> x7 x8 x9 x10 #> 0.53291918 0.70067227 0.69438086 0.39645481 #> #> $cure #> (Intercept) x1 x2 x3 x4 x5 #> 0.18536867 0.57611279 0.28251292 0.00000000 0.00000000 0.00000000 #> x6 x7 x8 x9 x10 #> 0.00000000 0.07310365 0.00000000 0.00000000 0.00000000 #>#> $surv #> x1 x2 x3 x4 x5 x6 x7 #> 0.00000000 0.04206453 0.00000000 0.00000000 0.00000000 0.37514690 0.34846484 #> x8 x9 x10 #> 0.51954260 0.51481914 0.23175666 #> #> $cure #> (Intercept) x1 x2 x3 x4 x5 #> 0.16781197 0.26825151 0.08013352 0.00000000 0.00000000 0.00000000 #> x6 x7 x8 x9 x10 #> 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 #>### regularized Cox cure model with uncertain event status =========== ## simulate a toy data set.seed(123) n_obs <- 100 p <- 10 x_mat <- matrix(rnorm(n_obs * p), nrow = n_obs, ncol = p) colnames(x_mat) <- paste0("x", seq_len(p)) surv_beta <- c(rep(0, p - 5), rep(1, 5)) cure_beta <- c(rep(1, 2), rep(0, p - 2)) dat <- simData4cure(nSubject = n_obs, lambda_censor = 0.01, max_censor = 10, survMat = x_mat, survCoef = surv_beta, cureCoef = cure_beta, b0 = 0.5, p1 = 0.95, p2 = 0.95, p3 = 0.95) ## model-fitting from given design matrices fit1 <- cox_cure_net.fit(x_mat, x_mat, dat$obs_time, dat$obs_event, surv_nlambda = 5, cure_nlambda = 5, surv_alpha = 0.8, cure_alpha = 0.8) ## model-fitting from given model formula fm <- paste(paste0("x", seq_len(p)), collapse = " + ") surv_fm <- as.formula(sprintf("~ %s", fm)) cure_fm <- surv_fm fit2 <- cox_cure_net(surv_fm, cure_fm, data = dat, time = obs_time, event = obs_event, surv_nlambda = 5, cure_nlambda = 5, surv_alpha = 0.5, cure_alpha = 0.5) ## summary of BIC's BIC(fit1)#> df surv_df cure_df BIC #> 1 2 0 2 660.0951 #> 2 3 0 3 649.6661 #> 3 8 0 8 662.4689 #> 4 8 0 8 653.7398 #> 5 9 0 9 654.4038 #> 6 4 2 2 653.2886 #> 7 5 2 3 642.7620 #> 8 10 2 8 655.9078 #> 9 10 2 8 647.3588 #> 10 11 2 9 648.0941 #> 11 5 4 1 649.1912 #> 12 7 4 3 642.3998 #> 13 11 4 7 651.2044 #> 14 12 4 8 647.3566 #> 15 13 4 9 648.1374 #> 16 9 8 1 654.9063 #> 17 11 8 3 648.6869 #> 18 14 8 6 653.0220 #> 19 16 8 8 653.6878 #> 20 17 8 9 654.4016 #> 21 10 9 1 652.2151 #> 22 12 9 3 646.4180 #> 23 15 9 6 650.8193 #> 24 16 8 8 646.8662 #> 25 17 8 9 647.5568#> df surv_df cure_df BIC #> 1 2 0 2 660.1474 #> 2 3 0 3 652.6713 #> 3 7 0 7 661.2005 #> 4 8 0 8 656.0703 #> 5 8 0 8 651.2207 #> 6 4 2 2 656.1159 #> 7 5 2 3 648.6202 #> 8 10 2 8 661.5912 #> 9 10 2 8 652.5115 #> 10 11 2 9 652.3423 #> 11 4 3 1 646.9408 #> 12 6 3 3 643.6201 #> 13 11 3 8 656.7878 #> 14 11 3 8 647.7554 #> 15 12 3 9 647.6006 #> 16 9 8 1 658.2755 #> 17 11 8 3 654.7547 #> 18 15 8 7 663.5134 #> 19 16 8 8 659.1908 #> 20 16 7 9 654.4754 #> 21 10 9 1 654.1403 #> 22 12 9 3 651.1082 #> 23 16 9 7 659.9199 #> 24 17 9 8 655.5378 #> 25 18 9 9 655.3806#> $surv #> x1 x2 x3 x4 x5 x6 x7 #> 0.04679663 0.00000000 0.00000000 0.00000000 0.00000000 0.47929140 0.01572914 #> x8 x9 x10 #> 0.41052238 0.00000000 0.00000000 #> #> $cure #> (Intercept) x1 x2 x3 x4 x5 #> -0.1996682 0.3369252 0.2668648 0.0000000 0.0000000 0.0000000 #> x6 x7 x8 x9 x10 #> 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 #>#> $surv #> x1 x2 x3 x4 x5 x6 x7 #> 0.02924518 0.00000000 0.00000000 0.00000000 0.00000000 0.40564512 0.00000000 #> x8 x9 x10 #> 0.36028994 0.00000000 0.00000000 #> #> $cure #> (Intercept) x1 x2 x3 x4 x5 #> -0.2051408 0.2676769 0.2131741 0.0000000 0.0000000 0.0000000 #> x6 x7 x8 x9 x10 #> 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 #>