sox_cv: cross-validation for `sox`

Description

Conduct cross-validation (cv) for sox.

Usage

sox_cv(
  x,
  ID,
  time,
  time2,
  event,
  penalty,
  lambda,
  group,
  group_variable,
  own_variable,
  no_own_variable,
  penalty_weights,
  par_init,
  nfolds = 10,
  foldid = NULL,
  stepsize_init = 1,
  stepsize_shrink = 0.8,
  tol = 1e-05,
  maxit = 1000L,
  verbose = FALSE
)

Value

A list.

lambdas: A vector of lambda used for each cross-validation.
cvm: The cv error averaged across all folds for each lambda.
cvsd: The standard error of the cv error for each lambda.
cvup: The cv error plus its standard error for each lambda.
cvlo: The cv error minus its standard error for each lambda.
nzero: The number of non-zero coefficients at each lambda.
sox.fit: A fitted model for the full data at all lambdas of class "sox".
lambda.min: The lambda such that the cvm reach its minimum.
lambda.1se: The maximum of lambda such that the cvm is less than the minimum the cvup (the minmum of cvm plus its standard error).
foldid: The fold assignments used.
index: A one column matrix with the indices of lambda.min and lambda.1se.
iterations: A vector of number of iterations it takes to converge at each $\lambda$ in lambdas.

Arguments

x: Predictor matrix with dimension $nm * p$, where $n$ is the number of subjects, $m$ is the maximum observation time, and $p$ is the number of predictors. See Details.
ID: The ID of each subjects, each subject has one ID (multiple rows in x can share one ID).
time: Represents the start of each time interval.
time2: Represents the stop of each time interval.
event: Indicator of event. event = 1 when event occurs and event = 0 otherwise.
penalty: Character string, indicating whether "overlapping" or "nested" group lasso penalty is imposed.
lambda: Sequence of regularization coefficients $\lambda$'s.
group: A $G * G$ integer matrix required to describe the structure of the overlapping and nested groups. We recommend that the users generate it automatically using overlap_structure() and nested_structure(). See Examples and Details.
group_variable: A $p * G$ integer matrix required to describe the structure of the overlapping groups. We recommend that the users generate it automatically using overlap_structure(). See Examples and Details.
own_variable: A non-decreasing integer vector of length $G$ required to describe the structure of the nested groups. We recommend that the users generate it automatically using nested_structure(). See Examples and Details.
no_own_variable: An integer vector of length $G$ required to describe the structure of the nested groups. We recommend that the users generate it automatically using nested_structure(). See Examples and Details
penalty_weights: Optional, vector of length $G$ specifying the group-specific penalty weights. We recommend that the users generate it automatically using overlap_structure() or nested_structure(). If not specified, $\mathbf{1}_G$ is used.
par_init: Optional, vector of initial values of the optimization algorithm. Default initial value is zero for all $p$ variables.
nfolds: Optional, the folds of cross-validation. Default is 10.
foldid: Optional, user-specified vector indicating the cross-validation fold in which each observation should be included. Values in this vector should range from 1 to nfolds. If left unspecified, sox will randomly assign observations to folds
stepsize_init: Initial value of the stepsize of the optimization algorithm. Default is 1.
stepsize_shrink: Factor in $(0,1)$ by which the stepsize shrinks in the backtracking linesearch. Default is 0.8.
tol: Convergence criterion. Algorithm stops when the $l_2$ norm of the difference between two consecutive updates is smaller than tol.
maxit: Maximum number of iterations allowed.
verbose: Logical, whether progress is printed.

Details

For each lambda, 10 folds cross-validation (by default) is performed. The cv error is defined as follows. Suppose we perform $K$-fold cross-validation, denote $\hat{\beta}^{-k}$ by the estimate obtained from the rest of $K-1$ folds (training set). The error of the $k$-th fold (test set) is defined as $2(P-Q)$ divided by $R$, where $P$ is the log partial likelihood evaluated at $\hat{\beta}^{-k}$ using the entire dataset, Q is the log partial likelihood evaluated at $\hat{\beta}^{-k}$ using the training set, and R is the number of events in the test set. We do not use the negative log partial likelihood evaluated at $\hat{\beta}^{-k}$ using the test set because the former definition can efficiently use the risk set, and thus it is more stable when the number of events in each test set is small (think of leave-one-out). The cv error is used in parameter tuning. To account for balance in outcomes among the randomly formed test set, we divide the deviance $2(P-Q)$ by R. To get the estimated coefficients that has the minimum cv error, use sox_cv()$Estimates[, sox_cv$index["min",]]. To apply the 1-se rule, use sox_cv()$Estimates[, sox_cv$index["1se",]].

Examples

Run this code

x <- as.matrix(sim[, c("A1","A2","C1","C2","B","A1B","A2B","C1B","C2B")])
lam.seq <- exp(seq(log(1e0), log(1e-3), length.out = 20))

# Variables:
## 1: A1
## 2: A2
## 3: C1
## 4: C2
## 5: B
## 6: A1B
## 7: A2B
## 8: C1B
## 9: C2B

# Overlapping groups:
## g1: A1, A2, A1B, A2B
## g2: B, A1B, A2B, C1B, C2B
## g3: A1B, A2B
## g4: C1, C2, C1B, C2B
## g5: C1B, C2B

overlapping.groups <- list(c(1, 2, 6, 7),
                           c(5, 6, 7, 8, 9),
                           c(6, 7),
                           c(3, 4, 8, 9),
                           c(8, 9))
                           
pars.overlapping <- overlap_structure(overlapping.groups)

cv.overlapping <- sox_cv(
  x = x,
  ID = sim$Id,
  time = sim$Start,
  time2 = sim$Stop,
  event = sim$Event,
  penalty = "overlapping",
  lambda = lam.seq,
  group = pars.overlapping$groups,
  group_variable = pars.overlapping$groups_var,
  penalty_weights = pars.overlapping$group_weights,
  nfolds = 5,
  tol = 1e-4,
  maxit = 1e3,
  verbose = FALSE
)

str(cv.overlapping)

# Nested groups (misspecified, for the demonstration of the software only.)
## g1: A1, A2, C1, C2, B, A1B, A2B, C1B, C2B
## g2: A1B, A2B, A1B, A2B
## g3: C1, C2, C1B, C2B
## g4: 1
## g5: 2
## ...
## G12: 9

nested.groups <- list(1:9,
                      c(1, 2, 6, 7),
                      c(3, 4, 8, 9),
                      1, 2, 3, 4, 5, 6, 7, 8, 9)

pars.nested <- nested_structure(nested.groups)

cv.nested <- sox_cv(
  x = x,
  ID = sim$Id,
  time = sim$Start,
  time2 = sim$Stop,
  event = sim$Event,
  penalty = "nested",
  lambda = lam.seq,
  group = pars.nested$groups,
  own_variable = pars.nested$own_variables,
  no_own_variable = pars.nested$N_own_variables,
  penalty_weights = pars.nested$group_weights,
  nfolds = 5,
  tol = 1e-4,
  maxit = 1e3,
  verbose = FALSE
)

str(cv.nested)