penAFT.cv: Cross-validation function for fitting a regularized semiparametric accelerated failure time model

Description

A function to perform cross-validation and compute the solution path for the regularized semiparametric accelerated failure time model estimator.

Usage

penAFT.cv(X, logY, delta, nlambda = 50, 
  lambda.ratio.min = 0.1, lambda = NULL, 
  penalty = NULL, alpha = 1,weight.set = NULL, 
  groups = NULL, tol.abs = 1e-8, tol.rel = 2.5e-4, 
  standardize = TRUE, nfolds = 5, cv.index = NULL, 
  admm.max.iter = 1e4,quiet = TRUE)

Value

full.fit: A model fit with the same output as a model fit using penAFT. See documentation for penAFT for more.
cv.err.linPred: A nlambda-dimensional vector of cross-validated linear predictor scores.
cv.err.obj: A nfolds $\times$nlambda matrix of cross-validated Gehan losses.
cv.index: A list of length nfolds. Each element contains the indices for subjects belonging to that particular fold.

Arguments

X: An $n \times p$ matrix of predictors. Observations should be organized by row.
logY: An $n$-dimensional vector of log-survival or log-censoring times.
delta: An $n$-dimensional binary vector indicating whether the $j$th component of logY is an observed log-survival time ($\delta_j = 1$) or a log-censoring time ($\delta_j = 0$) for $j=1, \dots, n$.
nlambda: The number of candidate tuning parameters to consider.
lambda.ratio.min: The ratio of minimum to maximum candidate tuning parameter value. As a default, we suggest 0.1, but standard model selection procedures should be applied to select $\lambda$. Must belong to (0,1).
lambda: An optional (not recommended) prespecified vector of candidate tuning parameters. Should be in descending order.
penalty: Either "EN" or "SG" for elastic net or sparse group lasso penalties.
alpha: The tuning parameter $\alpha$. See documentation.
weight.set: A list of weights. For both penalties, $w$ is an $n$-dimensional vector of nonnegative weights. For "SG" penalty, can also include $v$ -- a non-negative vector the length of the number of groups. See documentation for usage example.
groups: When using penalty "SG", a $p$-dimensional vector of integers corresponding the to group assignment of each predictor (i.e., column of X).
tol.abs: Absolute convergence tolerance.
tol.rel: Relative convergence tolerance.
standardize: Should predictors be standardized (i.e., scaled to have unit variance) for model fitting?
nfolds: The number of folds to be used for cross-validation. Default is five. Ten is recommended when sample size is especially small.
cv.index: A list of length nfolds of indices to be used for cross-validation. This is to be used if trying to perform cross-validation for both $\alpha$ and $\lambda$. Use with extreme caution: this overwrites nfolds.
admm.max.iter: Maximum number of ADMM iterations.
quiet: TRUE or FALSE variable indicating whether progress should be printed.

Details

Given $(\log y_1 , x_1, \delta_1),\dots,(\log y_n , x_n, \delta_n)$ where for subject $i$ ($i = 1, \dots, n$), $y_i$ is the minimum of the survival time and censoring time, $x_i$ is a $p$-dimensional predictor, and $\delta_i$ is the indicator of censoring, penAFT.cv performs nfolds cross-validation for selecting the tuning parameter to be used in the argument minimizing $$\frac{1}{n^2}\sum_{i=1}^n \sum_{j=1}^n \delta_i \{ \log y_i - \log y_j - (x_i - x_j)'\beta \}^{-} + \lambda g(\beta)$$ where $\{a \}^{-} := \max(-a, 0) $, $\lambda > 0$, and $g$ is either the weighted elastic net penalty (penalty = "EN") or weighted sparse group lasso penalty (penalty = "SG"). The weighted elastic net penalty is defined as $$\alpha \| w \circ \beta\|_1 + \frac{(1-\alpha)}{2}\|\beta\|_2^2$$ where $w$ is a set of non-negative weights (which can be specified in the weight.set argument). The weighted sparse group-lasso penalty we consider is $$\alpha \| w \circ \beta\|_1 + (1-\alpha)\sum_{l=1}^G v_l\|\beta_{\mathcal{G}_l}\|_2$$ where again, $w$ is a set of non-negative weights and $v_l$ are weights applied to each of the $G$ groups.

Next, we define the cross-validation errors. Let $\mathcal{V}_1, \dots, \mathcal{V}_K$ be a random nfolds = $K$ element partition of $[n]$ (the subjects) with the cardinality of each $\mathcal{V}_k$ (the "kth fold"") approximately equal for $k = 1, \dots, K$. Let ${\hat{\beta}}_{\lambda(-\mathcal{V}_k)}$ be the solution with tuning parameter $\lambda$ using only data indexed by $[n] \setminus \{\mathcal{V}_k\}$ (i.e., outside the kth fold). Then, defining $e_i(\beta) := \log y_i - \beta'x_i$ for $i= 1, \dots, n$, we call $$ \sum_{k=1}^K \left[\frac{1}{|\mathcal{V}_k|^2} \sum_{i \in \mathcal{V}_k} \sum_{j \in \mathcal{V}_k} \delta_i \{e_i({\hat{\beta}}_{\lambda(-\mathcal{V}_k)}) - e_{j}({\hat{\beta}}_{\lambda(-\mathcal{V}_k)})\}^{-}\right], $$ the cross-validated Gehan loss at $\lambda$ in the $k$th fold, and refer to the sum over all nfolds = $K$ folds as the cross-validated Gehan loss. Similarly, letting letting $$ \tilde{e}_i({\hat{\beta}}_\lambda) = \sum_{k = 1}^K (\log y_i - x_i'{\hat{\beta}}_{\lambda(-\mathcal{V}_k)}) \mathbf{1}(i \in \mathcal{V}_k)$$ for each $i \in [n]$, we call $$\left[\sum_{i = 1}^n \sum_{j = 1}^n \delta_i \{\tilde{e}_i({\hat{\beta}}_\lambda) - \tilde{e}_j({\hat{\beta}}_\lambda)\}^{-}\right]$$ the cross-validated linear predictor score at $\lambda$.

Examples

Run this code

 # --------------------------------------
# Generate data  
# --------------------------------------
set.seed(1)
genData <- genSurvData(n = 50, p = 50, s = 10, mag = 2,  cens.quant = 0.6)
X <- genData$X
logY <- genData$logY
delta <- genData$status
p <- dim(X)[2]

# -----------------------------------------------
# Fit elastic net penalized estimator
# -----------------------------------------------
fit.en <- penAFT.cv(X = X, logY = logY, delta = delta,
                   nlambda = 10, lambda.ratio.min = 0.1,
                   penalty = "EN", nfolds = 5,
                   alpha = 1)
# ---- coefficients at tuning parameter minimizing cross-valdiation error
coef.en <- penAFT.coef(fit.en)

# ---- predict at 8th tuning parameter from full fit
Xnew <- matrix(rnorm(10*p), nrow=10)
predict.en <- penAFT.predict(fit.en, Xnew = Xnew, lambda = fit.en$full.fit$lambda[8])

# \donttest{
  # -----------------------------------------------
  # Fit sparse group penalized estimator
  # -----------------------------------------------
  groups <- rep(1:5, each = 10)
  fit.sg <- penAFT.cv(X = X, logY = logY, delta = delta,
                    nlambda = 50, lambda.ratio.min = 0.01,
                    penalty = "SG", groups = groups, nfolds = 5,
                    alpha = 0.5)
                     
  # -----------------------------------------------
  # Pass fold indices
  # -----------------------------------------------
  groups <- rep(1:5, each = 10)
  cv.index <- list()
  for(k in 1:5){
    cv.index[[k]] <- which(rep(1:5, length=50) == k)
  }
  fit.sg.cvIndex <- penAFT.cv(X = X, logY = logY, delta = delta,
                    nlambda = 50, lambda.ratio.min = 0.01,
                    penalty = "SG", groups = groups, 
                    cv.index = cv.index,
                    alpha = 0.5)
  # --- compare cv indices
  if (FALSE) fit.sg.cvIndex$cv.index  == cv.index
# }

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