# Discrete Collaborative Targeted Minimum-loss based Estimation This function computes the discrete Collaborative Targeted Minimum-loss based Estimator for variable selection. It includes the greedy C-TMLE algorithm (Gruber and van der Laan 2010), and scalable C-TMLE algorithm (Ju, Gruber, and Lendle et al. 2016) with a user-specified order. ```r ctmleDiscrete(Y, A, W, Wg = W, Q = NULL, preOrder = FALSE, order = NULL, patience = FALSE, Qbounds = NULL, cvQinit = FALSE, Qform = NULL, SL.library = NULL, alpha = 0.995, family = "gaussian", gbound = 0.025, like_type = "RSS", fluctuation = "logistic", verbose = FALSE, detailed = FALSE, PEN = FALSE, V = 5, folds = NULL, stopFactor = 10^6) ``` ## Arguments - `Y`: continuous or binary outcome variable - `A`: binary treatment indicator, 1 for treatment, 0 for control - `W`: vector, matrix, or dataframe containing baseline covariates for Q bar - `Wg`: vector, matrix, or dataframe containing baseline covariates for propensity score model (defaults to W if not supplied by user) - `Q`: n by 2 matrix of initial values for Q0W, Q1W in columns 1 and 2, respectively. Current version does not support SL for automatic initial estimation of Q bar - `preOrder`: boolean indicator for using scalable C-TMLE algorithm or not - `order`: the use-specified order of covariables. Only used when (preOrder = TRUE). If not supplied by user, it would automatically order covariates from W_1 to W_p - `patience`: a number to stop early when the score in the CV function does not improve after so many covariates. Used only when (preOrder = TRUE) - `Qbounds`: bound on initial Y and predicted values for Q. - `cvQinit`: if TRUE, cross-validate initial values for Q to avoid overfits - `Qform`: optional regression formula for estimating initial Q - `SL.library`: optional vector of prediction algorithms for data adaptive estimation of Q, defaults to glm, and glmnet - `alpha`: used to keep predicted initial values bounded away from (0,1) for logistic fluctuation, 0.995 (default) - `family`: family specification for working regression models, generally 'gaussian' for continuous outcomes (default), 'binomial' for binary outcomes - `gbound`: bound on P(A=1|W), defaults to 0.025 - `like_type`: 'RSS' or 'loglike'. The metric to use for forward selection and cross-validation - `fluctuation`: 'logistic' (default) or 'linear', for targeting step - `verbose`: print status messages if TRUE - `detailed`: boolean number. If it is TRUE, return more detailed results - `PEN`: boolean. If true, penalized loss is used in cross-validation step - `V`: Number of folds. Only used if folds is not specified - `folds`: The list of indices for cross-validation step. We recommend the cv-splits in C-TMLE matchs that in gn_candidate_cv - `stopFactor`: Numerical value with default 1e6. If the current empirical likelihood is stopFactor times larger than the best previous one, the construction would stop ## Returns best_k the index of estimate that selected by cross-validation est estimate of psi_0 CI IC-based 95 pvalue pvalue for the null hypothesis that Psi = 0 likelihood sum of squared residuals, based on selected estimator evaluated on all obs or, logistic loglikelihood if like_type != 'RSS' varIC empirical variance of the influence curve adjusted for estimation of g varDstar empirical variance of the influence curve var.psi variance of the estimate varIC.cv cross-validated variance of the influence curve penlikelihood.cv penalized cross-validated likelihood cv.res all cross-validation results for each fold ## Examples ```r ## Not run: N <- 1000 p = 10 Wmat <- matrix(rnorm(N * p), ncol = p) beta1 <- 4+2*Wmat[,1]+2*Wmat[,2]+2*Wmat[,5]+2*Wmat[,6]+2*Wmat[,8] beta0 <- 2+2*Wmat[,1]+2*Wmat[,2]+2*Wmat[,5]+2*Wmat[,6]+2*Wmat[,8] tauW <- 2 tau <- 2 gcoef <- matrix(c(-1,-1,rep(-(3/((p)-2)),(p)-2)),ncol=1) Wm <- as.matrix(Wmat) g <- 1/(1+exp(Wm%*%gcoef)) A <- rbinom(N, 1, prob = g) sigma <- 1 epsilon <-rnorm(N,0,sigma) Y <- beta0 + tauW*A + epsilon # Initial estimate of Q Q <- cbind(rep(mean(Y[A == 0]), N), rep(mean(Y[A == 1]), N)) # User-suplied initial estimate time_greedy <- system.time( ctmle_discrete_fit1 <- ctmleDiscrete(Y = Y, A = A, W = data.frame(Wmat), Q = Q, preOrder = FALSE) ) # If there is no input Q, then intial Q would be estimated by SL with Sl.library ctmle_discrete_fit2 <- ctmleDiscrete(Y = Y, A = A, W = data.frame(Wmat), preOrder = FALSE, detailed = TRUE) # scalable C-TMLE with pre-order option; order is user-specified, # If 'order' is not specified takes order from W1 to Wp. time_preorder <- system.time( ctmle_discrete_fit3 <- ctmleDiscrete(Y = Y, A = A, W = data.frame(Wmat), Q = Q, preOrder = TRUE, order = rev(1:p), detailed = TRUE) ) # Compare the running time time_greedy time_preorder ## End(Not run) ```

ctmleDiscrete function