SurvRegCensCov (version 1.1)

SurvRegCens: Weibull Survival Regression Model with a censored covariate

Description

Computes estimators for the shape and scale parameter of the Weibull distribution, as well as for the vector of regression parameters in a parametric survival model with potentially right-censored time-to-event endpoint distributed according to a Weibull distribution. The regression allows for one potentially interval-censored and an arbitrary number of non-censored covariates.

Usage

SurvRegCens(time, event, CovariateNonCens = NULL, CovariateCens, 
            Density, initial, conf.level = 0.95, intlimit = 10^-10, 
            namCens = "VarCens", trace = 0, reltol = 10^-8)

Arguments

time

Vector of event times.

event

Censoring indicator for the time-to-event endpoint (0: censored, 1: event).

CovariateNonCens

data.frame containing the non-censored covariates. If no non-censored covariates are to be included in the model, set to NULL.

CovariateCens

Information on censored covariate. Format as for censdata in ParamSampleCens.

Density

Density function of the censored covariate.

initial

Initial values for the parameters to be optimized over, ordered according to Scale parameter, Shape parameter, regression parameters (i.e. $\beta$) linked to the non-censored covariates, regression parameter (i.e. $\beta$) linked to the censored covariate

conf.level

Confidence level of confidence intervals.

intlimit

In computation of integrals, values of the function to be integrated below intlimit are set to 0. This makes integration results more accurate and speeds up integration. If the data is such that the absolute values of the underlying baseline

namCens

Name of censored variables, to tidy outputs.

trace

trace argument in optim, indicates whether to show optimization progress.

reltol

reltol argument in optim. By changing this one can modify the relative tolerance in maximization of the likelihood function.

Value

• coeff
  Estimators, confidence intervals, $p$-values for the for the null hypothesis: {Estimators is equal to 0}, and this for each of the parameters of the Weibull survival regression model.
• percent.cens
  Percentage of censored observations in the censored covariate.
• loglik
  Log-likelihood function value at the estimators.
• info.converg
  Convergence information provided by the function optim.
• info.converg.message
  Message provided by optim.

Details

The time-to-event distributed according to a Weibull distribution, i.e. time-to-event $\sim$ Weibull$(\lambda,\gamma)$, has conditional density given by, $$f_{Y_i}(t|\mathbf{x}_i,\boldsymbol{\beta}) =\gamma \lambda t^{\gamma-1} \exp\left(\mathbf{x}_i\boldsymbol{\beta}\right)\exp\left( - \lambda t^{\gamma} \exp\left(\mathbf{x}_i\boldsymbol{\beta}\right) \right),$$ conditional hazard function given by, $$h_i(t|\mathbf{x}_i,\boldsymbol{\beta})= \lambda \gamma t^{\gamma-1} \exp\left( \mathbf{x}_i\boldsymbol{\beta} \right),$$ and conditional survival function given by, $$S_i(t|\mathbf{x}_i,\boldsymbol{\beta}) = \exp\left(-\lambda t^{\gamma} \exp\left(\mathbf{x}_i\boldsymbol{\beta}\right)\right),$$ where $\mathbf{x}_i$ collects the values of each covariate for observation $i$ and $\boldsymbol{\beta}$ represents the regression parameters.

References

Hubeaux, S. (2013). Parametric Surival Regression Model with left- and/or interval-censored covariate. Technical report, Biostatistics Oncology, F. Hoffmann-La Roche Ltd. Hubeaux, S. and Rufibach, K. (2014). SurvRegCensCov: Weibull Regression for a Right-Censored Endpoint with a Censored Covariate. Preprint, http://arxiv.org/abs/1402.0432. Sattar, A., Sinha, S. K. and Morris, N. J. (2012). A Parametric Survival Model When a Covariate is Subject to Left-Censoring. Biometrics & Biostatistics, S3(2).

Examples

library(survival)

## --------------------------------------------------------------
## 1 censored-covariate and 2 non-censored covariates 
## no censoring, to compare result with survival::survreg
## modify prop.cens to introduce censoring of covariate
## --------------------------------------------------------------

set.seed(158)
size <- 100
lambda <- exp(-2)
gamma <- 1.5

## vector of regression parameters: the last entry is the one for the censored covariate
ParamBeta <- c(0.3, -0.2, 0.25) 
true <- c(lambda, gamma, ParamBeta)

## simulate from a Weibull regression model
Censdata <- rnorm(size, mean = 5, sd = 0.5)
min.value <- min(Censdata)
prop.cens <- 0
LOD <- qnorm(prop.cens, mean = 5, sd = 0.5)
Censdata[Censdata <= LOD] <- LOD
index.noncens <- which(Censdata > LOD)
InfoCens <- matrix(nrow = size, ncol = 1)
InfoCens[index.noncens] <- Censdata[index.noncens]
CovariateCens <- cbind(InfoCens, Censdata)

NonCensdata <- data.frame("var1" = rnorm(size, mean = 4, sd = 0.5), 
    "var2" = rnorm(size, mean = 4, sd = 0.5))
index.cens <- which(Censdata <= LOD)
index.cens <- index.cens[index.cens >= 250]
InfoCens[index.cens] <- min.value
time <- TimeSampleWeibull(covariate_noncens = NonCensdata, covariate_cens = Censdata, 
                          lambda = lambda, gamma = gamma, beta=ParamBeta) 

IndicTime <- matrix(1, nrow = size, ncol = 1)
TimeCensVector <- rweibull(size, 1.5, exp(3))
IndicTime[time >= TimeCensVector] <- 0
index.cens.time <- which(IndicTime >= TimeCensVector)
time[index.cens.time] <- TimeCensVector[index.cens.time]

## specify the true density for the censored covariate:
DensityCens <- function(value){return(dnorm(value, mean = 5 , sd = 0.5))}

## use Weibull regression where each censored covariate value is set 
## to LOD ("naive" method)
naive <- survreg(Surv(time, IndicTime) ~ NonCensdata[,1] + 
                    NonCensdata[,2] + Censdata, dist = "weibull")
initial <- as.vector(ConvertWeibull(naive)$vars[, 1])

## use new method that takes into account the left-censoring of one covariate
cens1 <- SurvRegCens(time = time, event = IndicTime, CovariateNonCens = NonCensdata, 
                     CovariateCens = CovariateCens, Density = DensityCens,
                     initial = initial, namCens = "biomarker")

## compare estimates
tab <- data.frame(cbind(true, initial, cens1$coeff[, 1]))
colnames(tab) <- c("true", "naive", "Weibull MLE")
tab

## compare confidence intervals
ConvertWeibull(naive)$HR[, 2:3]
cens1$coeff[, 7:8]



## --------------------------------------------------------------
## model without the non-censored covariates
## --------------------------------------------------------------
naive2 <- survreg(Surv(time, IndicTime) ~ Censdata, dist = "weibull")
initial2 <- as.vector(ConvertWeibull(naive2)$vars[, 1])

## use new method that takes into account the left-censoring of one covariate
cens2 <- SurvRegCens(time = time, event = IndicTime, CovariateNonCens = NULL, 
                     CovariateCens = CovariateCens, Density = DensityCens,
                     initial = initial2, namCens = "biomarker")

## compare estimates
tab <- data.frame(cbind(true[c(1, 2, 5)], initial2, cens2$coeff[, 1]))
colnames(tab) <- c("true", "naive", "Weibull MLE")
tab

## compare confidence intervals
ConvertWeibull(naive2)$HR[, 2:3]
cens2$coeff[, 7:8]