aftgee: Accelerated Failure Time with Generalized Estimating Equation

Description

A package that uses Generalized Estimating Equations (GEE) to estimate Multivariate Accelerated Failure Time Model (AFT). This package implements recently developed inference procedures for AFT models with both the rank-based approach and the least squares approach. For the rank-based approach, the package allows various weight choices and uses an induced smoothing procedure that leads to much more efficient computation than the linear programming method. With the rank-based estimator as an initial value, the generalized estimating equation approach is used as an extension of the least squares approach to the multivariate case. Additional sampling weights are incorporated to handle missing data needed as in case-cohort studies or general sampling schemes.

Author(s)

Maintainer: Sy Han Chiou schiou@smu.edu

Authors:

• Sangwook Kang

• Jun Yan

References

Chiou, S., Kim, J. and Yan, J. (2014) Marginal Semiparametric Multivariate Accelerated Failure Time Model with Generalized Estimating Equation. Life Time Data, 20(4): 599–618.

Chiou, S., Kang, S. and Yan, J. (2014) Fast Accelerated Failure Time Modeling for Case-Cohort Data. Statistics and Computing, 24(4): 559–568.

Chiou, S., Kang, S. and Yan, J. (2014) Fitting Accelerated Failure Time Model in Routine Survival Analysis with R Package aftgee. Journal of Statistical Software, 61(11): 1–23.

Huang, Y. (2002) Calibration Regression of Censored Lifetime Medical Cost. Journal of American Statistical Association, 97, 318–327.

Jin, Z. and Lin, D. Y. and Ying, Z. (2006) On Least-squares Regression with Censored Data. Biometrika, 90, 341–353.

Johnson, L. M. and Strawderman, R. L. (2009) Induced Smoothing for the Semiparametric Accelerated Failure Time Model: Asymptotic and Extensions to Clustered Data. Biometrika, 96, 577 – 590.

Zeng, D. and Lin, D. Y. (2008) Efficient Resampling Methods for Nonsmooth Estimating Functions. Biostatistics, 9, 355–363

See Also

Useful links:

• https://github.com/stc04003/aftgee

• Report bugs at https://github.com/stc04003/aftgee/issues

Quasi Information Criterion

Description

Implementation based on MES::QIC.geeglm

Usage

QIC(object)

Arguments

Examples

## Simulate data from an AFT model with possible depended response
datgen <- function(n = 100, tau = 0.3, dim = 2) {
    x1 <- rbinom(dim * n, 1, 0.5)
    x2 <- rnorm(dim * n)
    e <- c(t(exp(MASS::mvrnorm(n = n, mu = rep(0, dim), Sigma = tau + (1 - tau) * diag(dim)))))
    tt <- exp(2 + x1 + x2 + e)
    cen <- runif(n, 0, 100)
    data.frame(Time = pmin(tt, cen), status = 1 * (tt < cen),
               x1 = x1, x2 = x2, id = rep(1:n, each = dim))
}
set.seed(1); dat <- datgen(n = 50, dim = 2)
fm <- Surv(Time, status) ~ x1 + x2
fit1 <- aftgee(fm, data = dat, id = id, corstr = "ind")
fit2 <- aftgee(fm, data = dat, id = id, corstr = "ex")

QIC(fit1)
QIC(fit2)

Least-Squares Approach for Accelerated Failure Time with Generalized Estimating Equation

Description

Fits a semiparametric accelerated failure time (AFT) model with least-squares approach. Generalized estimating equation is generalized to multivariate AFT modeling to account for multivariate dependence through working correlation structures to improve efficiency.

Usage

aftgee(
  formula,
  data,
  subset,
  id = NULL,
  contrasts = NULL,
  weights = NULL,
  margin = NULL,
  corstr = c("independence", "exchangeable", "ar1", "unstructured", "userdefined",
    "fixed"),
  binit = "srrgehan",
  B = 100,
  control = aftgee.control()
)

Arguments

Value

An object of class "aftgee" representing the fit. The aftgee object is a list containing at least the following components:

coefficients

a vector of initial value and a vector of point estimates

coef.res

a vector of point estimates

var.res

estimated covariance matrix

coef.init

a vector of initial value

var.init.mat

estimated initial covariance matrix

binit

a character string specifying the initial estimator.

conv

An integer code indicating type of convergence after GEE iteration. 0 indicates successful convergence; 1 indicates that the iteration limit maxit has been reached

ini.conv

An integer code indicating type of convergence for initial value. 0 indicates successful convergence; 1 indicates that the iteration limit maxit has been reached

conv.step

An integer code indicating the step until convergence

References

Chiou, S., Kim, J. and Yan, J. (2014) Marginal Semiparametric Multivariate Accelerated Failure Time Model with Generalized Estimating Equation. Lifetime Data Analysis, 20(4): 599–618.

Jin, Z. and Lin, D. Y. and Ying, Z. (2006) On Least-squares Regression with Censored Data. Biometrika, 90, 341–353.

Examples

## Simulate data from an AFT model with possible depended response
datgen <- function(n = 100, tau = 0.3, dim = 2) {
    x1 <- rbinom(dim * n, 1, 0.5)
    x2 <- rnorm(dim * n)
    e <- c(t(exp(MASS::mvrnorm(n = n, mu = rep(0, dim), Sigma = tau + (1 - tau) * diag(dim)))))
    tt <- exp(2 + x1 + x2 + e)
    cen <- runif(n, 0, 100)
    data.frame(Time = pmin(tt, cen), status = 1 * (tt < cen),
               x1 = x1, x2 = x2, id = rep(1:n, each = dim))
}
set.seed(1); dat <- datgen(n = 50, dim = 2)
fm <- Surv(Time, status) ~ x1 + x2
fit1 <- aftgee(fm, data = dat, id = id, corstr = "ind")
fit2 <- aftgee(fm, data = dat, id = id, corstr = "ex")
summary(fit1)
summary(fit2)

confint(fit1)
confint(fit2)

Auxiliary for Controlling AFTGEE Fitting

Description

Auxiliary function as user interface for aftgee and aftsrr fitting.

Usage

aftgee.control(
  maxiter = 50,
  reltol = 0.001,
  trace = FALSE,
  seIni = FALSE,
  parallel = FALSE,
  parCl = parallel::detectCores()/2,
  gp.pwr = -999
)

Arguments

Details

When trace is TRUE, output for each iteration is printed to the screen.

Value

A list with the arguments as components.

See Also

aftgee

Accelerated Failure Time with Smooth Rank Regression

Description

Fits a semiparametric accelerated failure time (AFT) model with rank-based approach. General weights, additional sampling weights and fast sandwich variance estimations are also incorporated. Estimating equations are solved with Barzilar-Borwein spectral method implemented as BBsolve in package BB.

Usage

aftsrr(
  formula,
  data,
  subset,
  id = NULL,
  contrasts = NULL,
  weights = NULL,
  B = 100,
  rankWeights = c("gehan", "logrank", "PW", "GP", "userdefined"),
  eqType = c("is", "ns", "mis", "mns"),
  se = c("NULL", "bootstrap", "MB", "ZLCF", "ZLMB", "sHCF", "sHMB", "ISCF", "ISMB"),
  control = list()
)

Arguments

Details

When se = "bootstrap" or se = "MB", the variance-covariance matrix is estimated through a bootstrap fashion. Bootstrap samples that failed to converge are removed when computing the empirical variance matrix. When bootstrap is not called, we assume the variance-covariance matrix has a sandwich form

\Sigma = A^{-1}V(A^{-1})^T,

where V is the asymptotic variance of the estimating function and A is the slope matrix. In this package, we provide seveal methods to estimate the variance-covariance matrix via this sandwich form, depending on how V and A are estimated. Specifically, the asymptotic variance, V, can be estimated by either a closed-form formulation (CF) or through bootstrap the estimating equations (MB). On the other hand, the methods to estimate the slope matrix A are the inducing smoothing approach (IS), Zeng and Lin's approach (ZL), and the smoothed Huang's approach (sH).

Value

aftsrr returns an object of class "aftsrr" representing the fit. An object of class "aftsrr" is a list containing at least the following components:

beta

A vector of beta estimates

covmat

A list of covariance estimates

convergence

An integer code indicating type of convergence.

0

indicates successful convergence.

1

indicates that the iteration limit maxit has been reached.

2

indicates failure due to stagnation.

3

indicates error in function evaluation.

4

is failure due to exceeding 100 step length reductions in line-search.

5

indicates lack of improvement in objective function.

bhist

When variance = "MB", bhist gives the bootstrap samples.

References

Chiou, S., Kang, S. and Yan, J. (2014) Fast Accelerated Failure Time Modeling for Case-Cohort Data. Statistics and Computing, 24(4): 559–568.

Chiou, S., Kang, S. and Yan, J. (2014) Fitting Accelerated Failure Time Model in Routine Survival Analysis with R Package Aftgee. Journal of Statistical Software, 61(11): 1–23.

Huang, Y. (2002) Calibration Regression of Censored Lifetime Medical Cost. Journal of American Statistical Association, 97, 318–327.

Johnson, L. M. and Strawderman, R. L. (2009) Induced Smoothing for the Semiparametric Accelerated Failure Time Model: Asymptotic and Extensions to Clustered Data. Biometrika, 96, 577 – 590.

Varadhan, R. and Gilbert, P. (2009) BB: An R Package for Solving a Large System of Nonlinear Equations and for Optimizing a High-Dimensional Nonlinear Objective Function. Journal of Statistical Software, 32(4): 1–26

Zeng, D. and Lin, D. Y. (2008) Efficient Resampling Methods for Nonsmooth Estimating Functions. Biostatistics, 9, 355–363

Examples

## Simulate data from an AFT model
datgen <- function(n = 100) {
    x1 <- rbinom(n, 1, 0.5)
    x2 <- rnorm(n)
    e <- rnorm(n)
    tt <- exp(2 + x1 + x2 + e)
    cen <- runif(n, 0, 100)
    data.frame(Time = pmin(tt, cen), status = 1 * (tt < cen),
               x1 = x1, x2 = x2, id = 1:n)
}
set.seed(1); dat <- datgen(n = 50)
summary(aftsrr(Surv(Time, status) ~ x1 + x2, data = dat, se = c("ISMB", "ZLMB"), B = 10))

## Data set with sampling weights
data(nwtco, package = "survival")
subinx <- sample(1:nrow(nwtco), 668, replace = FALSE)
nwtco$subcohort <- 0
nwtco$subcohort[subinx] <- 1
pn <- mean(nwtco$subcohort)
nwtco$hi <- nwtco$rel + ( 1 - nwtco$rel) * nwtco$subcohort / pn
nwtco$age12 <- nwtco$age / 12
nwtco$study <- factor(nwtco$study)
nwtco$histol <- factor(nwtco$histol)
sub <- nwtco[subinx,]
fit <- aftsrr(Surv(edrel, rel) ~ histol + age12 + study, id = seqno,
              weights = hi, data = sub, B = 10, se = c("ISMB", "ZLMB"),
              subset = stage == 4)
summary(fit)
confint(fit)

Surv function imported from survival

Description

This function is imported from the survival package. See Surv.

is.Surv function imported from survival

Description

This function is imported from the survival package. See is.Surv.