Package 'transmdl'

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

Y	observed event times
X	design matrix
delta	censoring indicator. If $Y_i$ is censored, delta=0. If not, delta=1.
alpha	parameter in transformation function
Q	number of nodes and weights in Gaussian quadrature. Defaults to 60.
EM_itmax	maximum iteration of EM algorithm. Defaults to 250.

Value

a list containing

beta_new			estimator of $\beta$
Lamb_Y			estimator of $\Lambda(Y)$
lamb_Y			estimator of $\lambda(Y)$
lamb_Ydot			estimator of $\lambda(Y')$
Y_eq_Yhat			a matrix used in trans.m and Simu
Y_geq_Yhat			a matrix used in trans.m and Simu

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

n	number of subjects
alpha	parameter in transformation function
rho	parameter in baseline cumulative hazard function $\Lambda(t) = \rho log(1+t)$ assumed in simulation
beta_true	parameter $\beta$
now_repeat	number of duplication of simulation

Details

The survival function for $t$ of the $i$th 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

X			design matrix
beta_X			$X\cdot\beta^T$
Y			observed event time

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

Z	the baseline covariates
Y	observed event times
delta_i	censoring indicator. If $Y$ is censored, delta_i=0. If not, delta_i=1.
r	parameter in transformation function
lamb_vec	the grad of the tuning parameter $\lambda$
solu	determines whether the solution path will be plotted. The default is TRUE.

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

beta_res			the estimated $\beta$ with the selected tuning parameter $\lambda$
GCV_res			the value of GCV with the selected tuning parameter $\lambda$
lamb_res			the selected tuning parameter $\lambda$
beta_all			estimated $\beta$ with all tuning parameters
CSV_all			value of GCV with all tuning parameters
skip_para			a list containing the $\lambda$ and the number of adaptive lasso iteration when adaptive lasso doesn't work.

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

X	design matrix
delta	censoring indicator. If $Y_i$ is censored, delta=0. If not, delta=1.
Y	observed event times
plot.Lamb	If TRUE, plot the estimated baseline cumulative hazard function and confidence intervals. The default is TRUE.
alpha	parameter in transformation function. Generally, $\alpha$ can not be observed in medical applications. In that situation, alpha indicates the scale of choosing $\alpha$. The default is $(0.1,0.2,...,1.1)$. If $\alpha$ is known, alpha indicates the true value of $\alpha$.
trsmodel	logical value indicating whether to implement transformation models. The default is TRUE.
EM_itmax	maximum iteration of EM algorithm. Defaults to 250.
show_res	show results after trans_m.

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

beta.est			estimators of $\beta$
SE.beta			standard errors of the estimated $\beta$
SE.Ydot			standard errors of the estimated $\Lambda(Y')$
Ydot			vector of sorted event times with duplicate values removed
Lamb.est			estimated baseline cumulative hazard
lamb.est			estimated jump sizes of baseline cumulative hazard function
choose.alpha			the chosen $\alpha$
Lamb.upper			upper confidence limits for the estimated baseline cumulative hazard function
Lamb.lower			lower confidence limits for the estimated baseline cumulative hazard function
p.beta			P values of estimated $\beta$
p.Lamb			P values of estimated baseline cumulative hazard
p.beta

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)

---