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sox: Structured Learning in Time-Dependent Cox Models

Description

Efficient procedures for fitting and cross-validating the structurally-regularized time-dependent Cox models.

Author(s)

Maintainer: Yi Lianyi.lian@mail.mcgill.ca

Authors:

• Guanbo Wang

• Archer Y. Yang

• Mireille E. Schnitzer

• Robert W. Platt

• Rui Wang

• Marc Dorais

• Sylvie Perreault

Other contributors:

• Julien Mairal [contributor]

• Yuansi Chen [contributor]

Automatically generate objects used to describe the structure of the nested group lasso penalty.

Description

Automatically generate objects used to describe the structure of the nested group lasso penalty. The output is then used bysox() andsox_cv().

Usage

nested_structure(group_list)

Arguments

Value

A list of objects describing the group structure.

Examples

# p = 9 Variables:## 1: A1## 2: A2## 3: C1## 4: C2## 5: B## 6: A1B## 7: A2B## 8: C1B## 9: C2B# G = 12 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: 9nested.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)str(pars.nested)

Automatically generate objects used to describe the structure of the overlapping group lasso penalty

Description

Automatically generate objects used to describe the structure of the overlapping group lasso penalty The output is then used bysox() andsox_cv().

Usage

overlap_structure(group_list)

Arguments

Value

A list of objects describing the group structure.

Examples

# p = 9 Variables:## 1: A1## 2: A2## 3: C1## 4: C2## 5: B## 6: A1B## 7: A2B## 8: C1B## 9: C2B# G = 5 Overlapping groups:## g1: A1, A2, A1B, A2B## g2: B, A1B, A2B, C1B, C2B## g3: A1B, A2B## g4: C1, C2, C1B, C2B## g5: C1B, C2Boverlapping.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)str(pars.overlapping)

Solution path plot forsox()

Description

Plot the solution path generated bysox().

Usage

## S3 method for class 'sox'plot(x, type = "l", log = "x", ...)

Arguments

Value

Produces a coefficient profile plot of the coefficient paths for a fittedsox model.

See Also

sox,sox_cv.

Examples

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))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)fit.overlapping <- sox(  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$group_weights,  tol = 1e-4,  maxit = 1e3,  verbose = FALSE)plot(fit.overlapping)              cv.overlapping <- sox_cv(  x = x,  ID = sim$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)plot(cv.overlapping$sox.fit)

Plots forsox_cv

Description

Plot the solution path or cross-validation curves produced bysox_cv().

Usage

## S3 method for class 'sox_cv'plot(x, type = "cv-curve", ...)

Arguments

Value

The "solution-path" plot produces a coefficient profile plot of the coefficient paths for a fittedsox model. The "cv-curve" plot is thecvm (red dot) for each lambda with its standard error (vertical bar). The two vertical dashed lines corresponds to thelambda.min andlambda.1se

See Also

sox,sox_cv.

Examples

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))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)plot(cv.overlapping)plot(cv.overlapping, type = "solution-path")

A simulated demo datasetsim

Description

A simulated demo datasetsim

Usage

data(sim)

Format

A simulated data frame that is used to illustrate the use of the sox package. The max follow-up time for each subject is set to be 5. The total number of subject is 50.

Id

The ID of each subject.

Event

During the time fromStart toStop, if the subject experience the event. We use the functionpermalgorithm in theR packagePermAlgo to generate the Event.

Start

Start time.

Stop

Stop time.

Fup

The total follow-up time for the subject.

Covariates

A1, A2, C1, C2, B, A1B, A2B, C1B, C2B. The dataset contains 5 variables (9 columns after one-hot encoding). Variable A is a e 3-level categorical variable, which results in 2 binary variables (A1 and A2), the same with the variable C. B is a continuous variable. The interaction term AB and CB are also two 3-level categorical variables. The code for generating the covariates is given below.

See Also

PermAlgo

Examples

# generate Bgen_con=function(m){ X=rnorm(m/5) XX=NULL for (i in 1:length(X)) {   if (length(XX)<m){   X.rep=rep(X[i],round(runif(1,5,10),0))   XX=c(XX,X.rep)   } } return(XX[1:m])}# generate A and Cgen_cat=function(m){ X=sample.int(3, m/5,replace = TRUE) XX=NULL for (i in 1:length(X)) {   if (length(XX)<m){     X.rep=rep(X[i],round(runif(1,5,10),0))     XX=c(XX,X.rep)   } } return(XX[1:m])}# generate covariate for one subjectgen_X=function(m){ A=gen_cat(m);B=gen_con(m);C=gen_cat(m) A1=ifelse(A==1,1,0);A2=ifelse(A==2,1,0) C1=ifelse(C==1,1,0);C2=ifelse(C==2,1,0) A1B=A1*B;A2B=A2*B C1B=C1*B;C2B=C2*B return(as.matrix(cbind(A1,A2,C1,C2,B,A1B,A2B,C1B,C2B)))}# generate covariate for all subjectgen_X_n=function(m,n){ Xn=NULL for (i in 1:n) {   X=gen_X(m)   Xn=rbind(Xn,X) } return(Xn)} n=50;m=5 covariates=gen_X_n(m,n) # generate outcomes # library(PermAlgo) # data <- permalgorithm(n, m, covariates,  #                       XmatNames = c("A1","A2","C1","C2","B","A1B","A2B","C1B","C2B"), #                       #change according to scenario 1/2 #                       betas = c(rep(log(3),2),rep(0,2), log(4), rep(log(3),2),rep(0,2)), #                       groupByD=FALSE ) # fit.original = coxph(Surv(Start, Stop, Event) ~ . ,data[,-c(1,3)])

(Time-dependent) Cox model with structured variable selection

Description

Fit a (time-dependent) Cox model with overlapping (including nested) group lasso penalty. The regularization path is computed at a grid of values for the regularization parameter lambda.

Usage

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

Arguments

Details

The predictor matrix should be of dimensionnm * p. Each row records the values of covariates for one subject at one time, for example, the values at the day fromtime (Start) totime2 (Stop). An example datasetsim is provided. The dataset has the format produced by theR packagePermAlgo. The specification of the argumentsgroup,group_variable,own_variable andno_own_variable for the grouping structure can be found inhttps://thoth.inrialpes.fr/people/mairal/spams/doc-R/html/doc_spams006.html#sec26 andhttps://thoth.inrialpes.fr/people/mairal/spams/doc-R/html/doc_spams006.html#sec27.

In the Examples below,p=9,G=5, the group structure is:

g_1 = \{A_{1}, A_{2}, A_{1}B, A_{2}B\},

g_2 = \{B, A_{1}B, A_{2}B, C_{1}B, C_{2}B\},

g_3 = \{A_{1}B, A_{2}B\},

g_4 = \{C_1, C_2, C_{1}B, C_{2}B\},

g_5 = \{C_{1}B, C_{2}B\}.

whereg_3 is a subset ofg_1 andg_2, andg_5 is a subset ofg_2 andg_4.

Value

A list with the following three elements.

Examples

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, C2Boverlapping.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)fit.overlapping <- sox(  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$group_weights,  tol = 1e-4,  maxit = 1e3,  verbose = FALSE)str(fit.overlapping)# Nested groups (misspecified, for the demonstration of the software only.)## g1: A1, A2, C1, C2, B, A1B, A2B, C1B, C2B## g2: A1B, A2B, A1B, A2B## g3: C1, C2, C1B, C2B## g4: 1## g5: 2## ...## G12: 9nested.groups <- list(1:9,                      c(1, 2, 6, 7),                      c(3, 4, 8, 9),                      1, 2, 3, 4, 5, 6, 7, 8, 9)pars.nested <- nested_structure(nested.groups)fit.nested <- sox(  x = x,  ID = sim$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,  tol = 1e-4,  maxit = 1e3,  verbose = FALSE)str(fit.nested)

cross-validation forsox

Description

Conduct cross-validation (cv) forsox.

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)

Arguments

Details

For each lambda, 10 folds cross-validation (by default) is performed. The cv error is defined as follows. Suppose we performK-fold cross-validation, denote\hat{\beta}^{-k} by the estimate obtained from the rest ofK-1 folds (training set). The error of thek-th fold (test set) is defined as2(P-Q) divided byR, whereP 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 deviance2(P-Q) by R. To get the estimated coefficients that has the minimum cv error, usesox_cv()$Estimates[, sox_cv$index["min",]]. To apply the 1-se rule, usesox_cv()$Estimates[, sox_cv$index["1se",]].

Value

A list.

See Also

sox,plot.sox_cv.

Examples

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, C2Boverlapping.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$group_weights,  nfolds = 5,  tol = 1e-4,  maxit = 1e3,  verbose = FALSE)str(cv.overlapping)# Nested groups (misspecified, for the demonstration of the software only.)## g1: A1, A2, C1, C2, B, A1B, A2B, C1B, C2B## g2: A1B, A2B, A1B, A2B## g3: C1, C2, C1B, C2B## g4: 1## g5: 2## ...## G12: 9nested.groups <- list(1:9,                      c(1, 2, 6, 7),                      c(3, 4, 8, 9),                      1, 2, 3, 4, 5, 6, 7, 8, 9)pars.nested <- nested_structure(nested.groups)cv.nested <- sox_cv(  x = x,  ID = sim$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)