Estimate parameters and hazard function via EM algorithm.

Description

Estimate the vector of parameters for baseline covariates \beta and baseline cumulative hazard function \Lambda(\cdot) using the expectation-maximization algorithm. \Lambda(t) is estimated as a step function with jumps only at the observed failure times. Typically, it would only be used in a call to trans.m or Simu.

Usage

EM_est(Y, X, delta, alpha, Q = 60, EM_itmax = 250)

Arguments

Value

a list containing

References

Abramowitz, M., and Stegun, I.A. (1972). Handbook of Mathematical Functions (9th ed.). Dover Publications, New York.

Evans, M. and Swartz, T. (2000). Approximating Integrals via Monte Carlo and Deterministic Methods. Oxford University Press.

Liu, Q. and Pierce, D.A. (1994). A note on Gauss-Hermite quadrature. Biometrika 81: 624-629.

Examples

  gen_data = generate_data(200, 1, 0.5, c(-0.5, 1))
  delta = gen_data$delta
  Y = gen_data$Y
  X = gen_data$X
  EM_est(Y, X, delta, alpha = 1)$beta_new - c(-0.5, 1)

Generate data for simulation.

Description

Generate observed event times, covariates and other data used for simulations in the paper.

Usage

generate_data(n, alpha, rho, beta_true, now_repeat = 1)

Arguments

Details

The survival function for t of the ith observation takes the form

S_{i}(t| X_{i}) = \exp\left\{-H \{\Lambda(t) \exp ( \beta^T X_{i} ) \}\right\}.

The failure time T_i can be generated by

T_i = \left\{\begin{array}{l l} \exp\{ \frac{U^{-\alpha}-1}{\alpha\rho\exp\{\beta^TX_i \}} \}-1& \alpha > 0, \\ \exp\{ \frac{-log(U)}{\rho\exp\{\beta^TX_i \}} \}-1, & \alpha = 0. \end{array}\right\}

Value

a list containing

References

Abramowitz, M., and Stegun, I.A. (1972). Handbook of Mathematical Functions (9th ed.). Dover Publications, New York. +- Evans, M. and Swartz, T. (2000). Approximating Integrals via Monte Carlo and Deterministic Methods. Oxford University Press.

Liu, Q. and Pierce, D.A. (1994). A note on Gauss-Hermite quadrature. Biometrika 81: 624-629.

Examples

 generate_data(200,0.5,1,c(0.5,-1))

Adaptive LASSO for Semiparametric Transformation Models.

Description

Select the important variables in semiparametric transformation models for right censored data using adaptive lasso.

Usage

trans_alasso(Z, Y, delta_i, r, lamb_vec, solu = TRUE)

Arguments

Details

The initial value of the coefficient \beta used as the adapting weights is EM estimator, which is computed by the function EM_est. The tuning parameter \lambda is data-dependent and we select it using generalized crossvalidation. There may be some errors for small \lambda, in which case the \lambda and the number of adaptive lasso iteration are recorded in the skip_para.

Value

a list containing

References

Xiaoxi, L. , & Donglin, Z. . (2013). Variable selection in semiparametric transformation models for right-censored data. Biometrika(4), 859-876.

Examples

 if(!requireNamespace("MASS", quietly = TRUE))
 {stop("package MASS needed for this example. Please install it.")}

 gen_lasdat = function(n,r,rho,beta_true,a,b,seed=66,std = FALSE)
{

  set.seed(seed)
  beta_len = length(beta_true)
  beta_len = beta_len
  sigm = matrix(0, nrow = beta_len, ncol = beta_len)
  for(i in 1:(beta_len-1))
  {
    diag(sigm[1:(beta_len+1-i),i:beta_len]) = rho^(i-1)
  }
  sigm[1,beta_len] = rho^(beta_len-1)
  sigm[lower.tri(sigm)] = t(sigm)[lower.tri(sigm)]

  Z = MASS::mvrnorm(n, mu = rep(0, beta_len), Sigma = sigm)
  beta_Z.true = c(Z %*% beta_true)
  U = runif(n)
  if(r>0)
  {
    t = ((U^(-r)-1)/(a*r*exp(beta_Z.true)))^(1/b)
  }else if(r == 0)
  {
    t = (-log(U)/(a*exp(beta_Z.true)))^(1/b)
    #t = (exp(-log(U)/(0.5 * exp(beta_Z.true))) - 1)
  }
  C = runif(n,0,8)
  Y = pmin(C,t)
  delta_i = ifelse( C >= t, 1, 0)
  if(std)
  {
    Z = apply(Z,2,normalize)
  }
  return(list(Z = Z, Y = Y, delta_i = delta_i,censor = mean(1-delta_i)))
}

now_rep=1
dat = gen_lasdat(100,1,0.5,c(0.3,0.5,0.7,0,0,0,0,0,0,0),2,5,seed= 6+60*now_rep,std = FALSE)
Z = dat$Z
Y = dat$Y
delta_i = dat$delta_i

tra_ala = trans_alasso(Z,Y,delta_i,lamb_vec = c(5,7),r=1)
tra_ala$GCV_res
tra_ala$beta_res
tra_ala$lamb_res

Regression Analysis of Right-censored Data using Semiparametric Transformation Models.

Description

This function is used to conduct the regression analysis of right-censored data using semiparametric transformation models. It calculates the estimators, standard errors and p values. A plot of estimated baseline cumulative hazard function and confidence intervals can be produced.

Usage

trans_m(
  X,
  delta,
  Y,
  plot.Lamb = TRUE,
  alpha = seq(0.1, 1.1, by = 0.1),
  trsmodel = TRUE,
  EM_itmax = 250,
  show_res = TRUE
)

Arguments

Details

If \alpha is unknown, we firse set \alpha=alpha. Then, for each \alpha, we estimate the parameters and record the value of observed log-likelihood function. The \alpha that maximizes the observed log-likelihood function and the corresponding \hat\beta and \hat\Lambda(\cdot) are chosen as the best estimators. Nonparametric maximum likelihood estimators are developed for the regression parameters and cumulative intensity functions of these models based on censored data.

Value

a list containing

References

Cheng, S.C., Wei, L.J., and Ying, Z. (1995). Analysis of transformation models with censored data. Biometrika 82, 835-845.

Zeng, D. and Lin, D.Y. (2007). Maximum likelihood estimation in semiparametric regression models with censored data. J. R. Statist. Soc. B 69, 507-564.

Abramowitz, M., and Stegun, I.A. (1972). Handbook of Mathematical Functions (9th ed.). Dover Publications, New York.

Evans, M. and Swartz, T. (2000). Approximating Integrals via Monte Carlo and Deterministic Methods. Oxford University Press.

Liu, Q. and Pierce, D.A. (1994). A note on Gauss-Hermite quadrature. Biometrika 81, 624-629.

Louis, T. (1982). Finding the Observed Information Matrix when Using the EM Algorithm. Journal of the Royal Statistical Society. Series B (Methodological), 44(2), 226-233.

See Also

EM_est

Examples

 gen_data = generate_data(200, 1, 0.5, c(-0.5,1))
 delta = gen_data$delta
 Y = gen_data$Y
 X = gen_data$X
 res.trans = trans_m(X, delta,  Y, plot.Lamb = TRUE, show_res = FALSE)