Fit penalized mixture cure model using the E-M algorithm with cross-validation for parameter tuning

Fits penalized parametric and semi-parametric mixture cure models (MCM) using the E-M algorithm with with k-fold cross-validation for parameter tuning. The lasso (L1), MCP and SCAD penalty are supported for the Cox MCM while only lasso is currently supported for parametric MCMs. When FDR controlled variable selection is used, the model-X knockoffs method is applied and indices of selected variables are returned.

Usage

cv_cureem(
  formula,
  data,
  subset,
  x_latency = NULL,
  model = c("cox", "weibull", "exponential"),
  penalty = c("lasso", "MCP", "SCAD"),
  penalty_factor_inc = NULL,
  penalty_factor_lat = NULL,
  fdr_control = FALSE,
  fdr = 0.2,
  grid_tuning = FALSE,
  thresh = 0.001,
  scale = TRUE,
  maxit = NULL,
  inits = NULL,
  lambda_inc_list = NULL,
  lambda_lat_list = NULL,
  nlambda_inc = NULL,
  nlambda_lat = NULL,
  gamma_inc = 3,
  gamma_lat = 3,
  lambda_min_ratio_inc = 0.1,
  lambda_min_ratio_lat = 0.1,
  n_folds = 5,
  measure_inc = c("c", "auc"),
  one_se = FALSE,
  cure_cutoff = 5,
  parallel = FALSE,
  seed = NULL,
  verbose = TRUE,
  na.action = na.omit,
  ...
)

Arguments

formula

an object of class "formula" (or one that can be coerced to that class): a symbolic description of the model to be fitted. The response must be a survival object as returned by the Surv function while the variables on the right side of the formula are the covariates that are included in the incidence portion of the model.

data

a data.frame in which to interpret the variables named in the formula or in the subset argument. Rows with missing data are omitted (only na.action = na.omit is operational) therefore users may want to impute missing data prior to calling this function.

subset

an optional expression indicating which subset of observations to be used in the fitting process, either a numeric or factor variable should be used in subset, not a character variable. All observations are included by default.

x_latency

specifies the variables to be included in the latency portion of the model and can be either a matrix of predictors, a model formula with the right hand side specifying the latency variables, or the same data.frame passed to the data parameter. Note that when using the model formula syntax for x_latency it cannot handle x_latency = ~ ..

model

type of regression model to use for the latency portion of mixture cure model. Can be "cox", "weibull", or "exponential" (default is "cox").

penalty

type of penalty function. Can be "lasso", "MCP", or "SCAD" (default is "lasso").

penalty_factor_inc

vector of binary indicators representing the penalty to apply to each incidence coefficient: 0 implies no shrinkage and 1 implies shrinkage. If not supplied, 1 is applied to all incidence variables.

penalty_factor_lat

vector of binary indicators representing the penalty to apply to each latency coefficient: 0 implies no shrinkage and 1 implies shrinkage. If not supplied, 1 is applied to all latency variables.

fdr_control

logical, if TRUE, model-X knockoffs are used for FDR-controlled variable selection and indices of selected variables are returned (default is FALSE).

fdr

numeric value in (0, 1) range specifying the target FDR level to use for variable selection when fdr_control = TRUE (default is 0.2).

grid_tuning

logical, if TRUE a 2-D grid tuning approach is used to select the optimal pair of \(\lambda_b\) and \(\lambda_{\beta}\) penalty parameters for the incidence and latency portions of the model, respectively. Otherwise the \(\lambda_b\) and \(\lambda_{\beta}\) are selected from a 1-D sequence and are equal to one another (default is FALSE).

thresh

small numeric value. The iterative process stops when the differences between successive expected penalized complete-data log-likelihoods for both incidence and latency components are less than this specified level of tolerance (default is 10^-3).

scale

logical, if TRUE the predictors are centered and scaled.

maxit

maximum number of passes over the data for each lambda. If not specified, 100 is applied when penalty = "lasso" and 1000 is applied when penalty = "MCP" or penalty = "SCAD".

inits

an optional list specifiying the initial values to be used for model fitting as follows:

itct the incidence intercept.
b_u a numeric vector for the unpenalized. incidence coefficients for the incidence portion of the model.
beta_u a numeric vector for unpenalized latency coefficients in the incidence portion of the model.
lambda a numeric value for the rate parameter when fitting either a Weibull or exponential MCM using model = "weibull" or model = "exponential".
alpha a numeric value for the shape parameter when fitting a Weibull MCM using model = "weibull".
survprob a numeric vector for the latency survival probabilities \(S_u(t_i|w_i)\) for i=1,...,N when fitting a Cox MCM model = "cox".

Penalized coefficients are initialized to zero. If inits is not specified or improperly specified, initialization is automatically provided by the function.

lambda_inc_list

a numeric vector used to search for the optimal \(\lambda_b\) tuning parameter. If not supplied, the function computes a \(\lambda_b\) sequence based on nlambda_inc and lambda_min_ratio_inc. If grid_tuning = FALSE, the same sequence should be used for both \(\lambda_b\) and \(\lambda_{\beta}\).

lambda_lat_list

a numeric vector used to search for the optimal \(\lambda_{\beta}\) tuning parameter. If not supplied, the function computes a \(\lambda_{\beta}\) sequence based on nlambda_lat and lambda_min_ratio_lat. If grid_tuning = FALSE, the same sequence should be used for both \(\lambda_b\) and \(\lambda_{\beta}\).

nlambda_inc

an integer specifying the number of values to search for the optimal \(\lambda_b\) tuning parameter; default is 10 if grid_tuning = TRUE and 50 otherwise.

nlambda_lat

an integer specifying the number of values to search for the optimal \(\lambda_{\beta}\) tuning parameter; default is 10 if grid_tuning = TRUE and 50 otherwise.

gamma_inc

numeric value for the penalization parameter \(\gamma\) for variables in the incidence portion of the model when penalty = "MCP" or penalty = "SCAD" (default is 3).

gamma_lat

numeric value for the penalization parameter \(\gamma\) for variables in the latency portion of the model when penalty = "MCP" or penalty = "SCAD" (default is 3).

lambda_min_ratio_inc

numeric value in (0,1) representing the smallest value for \(\lambda_b\) as a fraction of lambda.max_inc, the data-derived entry value at which essentially all penalized variables in the incidence portion of the model have a coefficient estimate of 0 (default is 0.1).

lambda_min_ratio_lat

numeric value in (0.1) representing the smallest value for \(\lambda_{\beta}\) as a fraction of lambda.max_lat, the data-derived entry value at essentially all penalized variables in the latency portion of the model have a coefficient estimate of 0 (default is 0.1).

n_folds

an integer specifying the number of folds for the k-fold cross-valiation procedure (default is 5).

measure_inc

character string specifying the evaluation criterion used in selecting the optimal \(\lambda_b\) which can be either

"c" specifying to use the C-statistic for cure status weighting (CSW) method proposed by Asano and Hirakawa (2017) to select both \(\lambda_b\) and \(\lambda_{\beta}\)
"auc" specifying to use the AUC for cure prediction using the mean score imputation (MSI) method proposed by Asano et al. (2014) to select \(\lambda_b\) while the C-statistic with CSW is used for \(\lambda_{\beta}\).

one_se

logical, if TRUE then the one standard error rule is applied for selecting the optimal parameters. The one standard error rule selects the most parsimonious model having evaluation criterion no more than one standard error worse than that of the best evaluation criterion (default is FALSE).

cure_cutoff

numeric value representing the cutoff time value that represents subjects not experiencing the event by this time are cured. This value is used to produce a proxy for the unobserved cure status when calculating C-statistic and AUC (default is 5 representing 5 years). Users should be careful to note the time scale of their data and adjust this according to the time scale and clinical application.

parallel

logical. If TRUE, parallel processing is performed for K-fold CV using foreach and the doParallel package is required.

seed

optional integer representing the random seed. Setting the random seed fosters reproducibility of the results.

verbose

logical, if TRUE running information is printed to the console (default is FALSE).

na.action

this function requires complete data so "na.omit" is invoked. Users can impute missing data as an alternative prior to model fitting.

...

additional arguments.

Value

b0: Estimated intercept for the incidence portion of the model.
b: Estimated coefficients for the incidence portion of the model.
beta: Estimated coefficients for the latency portion of the model.
alpha: Estimated shape parameter if the Weibull model is fit.
rate: Estimated rate parameter if the Weibull or exponential model is fit.
logLik_inc: Expected penalized complete-data log-likelihood for the incidence portion of the model.
logLik_lat: Expected penalized complete-data log-likelihood for the latency portion of the model.
selected_lambda_inc: Value of \(\lambda_b\) selected using cross-validation. NULL when fdr_control is TRUE.
selected_lambda_lat: Value of \(\lambda_{\beta}\) selected using cross-validation. NULL when fdr_control is TRUE.
max_c: Maximum C-statistic achieved.
max_auc: Maximum AUC for cure prediction achieved; only output when measure_inc="auc".
selected_index_inc: Indices of selected variables for the incidence portion of the model when fdr_control=TRUE. If no variables are selected, int(0) will be returned.
selected_index_lat: Indices of selected variables for the latency portion of the model when fdr_control=TRUE. If no variables are selected, int(0) will be returned.
call: the matched call.

References

Archer, K. J., Fu, H., Mrozek, K., Nicolet, D., Mims, A. S., Uy, G. L., Stock, W., Byrd, J. C., Hiddemann, W., Braess, J., Spiekermann, K., Metzeler, K. H., Herold, T., Eisfeld, A.-K. (2024) Identifying long-term survivors and those at higher or lower risk of relapse among patients with cytogenetically normal acute myeloid leukemia using a high-dimensional mixture cure model. Journal of Hematology & Oncology, 17:28.

Examples

library(survival)
withr::local_seed(1234)
temp <- generate_cure_data(n = 200, j = 25, n_true = 5, a = 1.8)
training <- temp$training
fit.cv <- cv_cureem(Surv(Time, Censor) ~ .,
  data = training,
  x_latency = training, fdr_control = FALSE,
  grid_tuning = FALSE, nlambda_inc = 10, nlambda_lat = 10,
  n_folds = 2, seed = 23, verbose = TRUE
)
#> Fold 1 out of 2 training...
#> Fold 2 out of 2 training...
#> Selected lambda for incidence: 0.052
#> Selected lambda for latency: 0.052
#> Maximum C-statistic: 0.713333376903194
#> Fitting a final model...
fit.cv.fdr <- cv_cureem(Surv(Time, Censor) ~ .,
  data = training,
  x_latency = training, model = "weibull", penalty = "lasso",
  fdr_control = TRUE, grid_tuning = FALSE, nlambda_inc = 10,
  nlambda_lat = 10, n_folds = 2, seed = 23, verbose = TRUE)
#> Fold 1 out of 2 training...
#> Fold 2 out of 2 training...
#> Selected lambda for incidence: 0.09
#> Selected lambda for latency: 0.09
#> Maximum C-statistic: 0.691812421454886
#> Fitting a final model...

Fit penalized mixture cure model using the E-M algorithm with cross-validation for parameter tuning

Usage

Arguments

Value

References

See also

Examples

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