mixture.2states: Horizontal Mixture Model for Two Competing Events

Description

The 2-state mixture model which includes an initial state (X=1) and two absorbing states in competition (X=2 and X=3). Parameters are estimated by (weighted) Likelihood maximization.

Usage

mixture.2states(times, sequences, weights=NULL, dist, cuts.12=NULL, cuts.13=NULL,
 ini.dist.12=NULL, ini.dist.13=NULL, cov.12=NULL, init.cov.12=NULL,
 names.12=NULL, cov.13=NULL, init.cov.13=NULL, names.13=NULL,
 cov.p=NULL, init.cov.p=NULL, names.p=NULL, init.intercept.p=NULL,
 conf.int=TRUE, silent=TRUE, precision=10^(-6))

Value

object: The character string indicating the estimated model: "mixture.2states (mixture model with two competing events)".
dist: A character vector with two arguments describing respectively the distributions of duration time for transitions 1->2 and 1->3.
cuts.12: A numeric vector indicating the timepoints in days for the piecewise exponential distribution related to the time from X=1 to X=2.
cuts.13: A numeric vector indicating the timepoints in days for the piecewise exponential distribution related to the time from X=1 to X=3.
covariates: A numeric vector indicating the numbers of covariates respectively related to the time to the event X=2, the time to the event X=3, the long-term probability P(X=2).
table: A data frame containing the estimated parameters of the model (Estimate). When the option conf.int=TRUE is specified, this data frame includes three additional columns: the Standard Errors of parameters (Std.Error), the value of the Wald statistic (t.value), and the p-value for the Wald test (Pr(>|t|)).
cov.matrix: A data frame corresponding to variance-covariance matrix of the parameters.
LogLik: A numeric value corresponding to the (weighted) log-likelihood of the model.
AIC: A numeric value corresponding to the Akaike Information Criterion of the model.

Arguments

times: A numeric vector with the observed times in days from baseline to the last observation.
sequences: A numeric vector with the sequence of observed states. Three possible values are allowed: 1 (the individual is right-censored in X=1), 12 (the individual transits to X=2) and 13 (the individual transits to X=3).
weights: A numeric vector with the weights for correcting the contribution of each individual. When the vector is completed, the IPW estimator is implemented. Default is NULL which means that no weighting is applied.
dist: A character vector with two arguments describing respectively the distributions of duration time for transitions 1->2 and 1->3. Arguments allowed are "E" for Exponential distribution, "PE" for the piecewise exponential distribution, "W" for Weibull distribution or "WG" for Generalized Weibull distribution. When the user choose "PE", the arguments "cut.XX" have also to be defined.
cuts.12: A numeric vector indicating the timepoints in days for the piecewise exponential distribution related to the time from X=1 to X=2. Only internal timepoints are allowed: timepoints cannot be 0 or Inf. Default is NULL which means that the distribution is not piecewise. Piecewise model is only allowed for exponential distribution.
cuts.13: A numeric vector indicating the timepoints in days for the piecewise exponential distribution related to the time from X=1 to X=3. Only internal timepoints are allowed: timepoints cannot be 0 or Inf. Default is NULL which means that the distribution is not piecewise. Piecewise model is only allowed for exponential distribution.
ini.dist.12: A numeric vector of initial values for the distribution from X=1 to X=2. The logarithm of the parameters have to be declared. Default value is 1.
ini.dist.13: A numeric vector of initial values for the distribution from X=1 to X=3. The logarithm of the parameters have to be declared. Default value is 1.
cov.12: A matrix (or data frame) with the explicative time-fixed variable(s) related to the time from X=1 to X=2.
init.cov.12: A numeric vector of initial values for regression coefficients (logarithm of the cause-specific hazards ratios) associated to cov.12. Default initial value is 0.
names.12: An optional character vector with name of explicative variables associated to cov.12.
cov.13: A numeric matrix (or data frame) with the explicative time-fixed variable(s) related to the time from X=1 to X=3.
init.cov.13: A numeric vector of initial values for regression coefficients (logarithm of the cause-specific hazards ratios) associated to cov.13. Default initial value is 0.
names.13: An optional character vector with name of explicative variables associated to cov.13.
cov.p: A matrix (or data frame) with the explicative time-fixed variable(s) related to the probability P(X=2), which is regressing according to a logistic function.
init.cov.p: A numeric vector of initial values for regression coefficients (logarithm of the cause-specific hazards ratios) associated to cov.p. Default initial value is 0.
names.p: An optional character vector with name of explicative variables associated to cov.p.
init.intercept.p: A numeric value to iniate the intercept of the logit of P(X=2). Default value is 0.
conf.int: A logical value specifying if the pointwise confidence intervals for parameters and the variance-covariance matrix should be returned. Default is TRUE.
silent: A logical value specifying if the log-likelihood value should be returned at each iteration. Default is TRUE, which corresponds to silent mode (no display).
precision: A numeric positive value indicating the required precision for the log-likelihood maximization between each iteration. Default is \(10^{-6}\).

Author

Yohann Foucher <Yohann.Foucher@univ-poitiers.fr>

Details

Hazard functions available are:

Exponential distribution	\(\lambda(t)=1/\sigma\)
Weibull distribution	\(\lambda(t)=\nu(\frac{1}{\sigma})^{\nu}t^{\nu-1}\)
Generalized Weibull distribution	\(\lambda(t)=\frac{1}{\theta}\left(1+\left(\frac{t}{\sigma}\right)^{\nu}\right)^{\frac{1}{\theta}-1} \nu\left(\frac{1}{\sigma}\right)^{\nu} t^{\nu-1}\)

with \(\sigma\), \(\nu\),and \(\theta>0\). The parameter \(\sigma\) varies for each interval when the distribution is piecewise Exponential. We advise to initialize the logarithm of these parameters in ini.dist.12, ini.dist.13 and ini.dist.23.

To estimate the marginal effect of a binary exposure, the weights may be equal to 1/p, where p is the estimated probability that the individual belongs to his or her own observed group of exposure. The probabilities p are often estimated by a logistic regression in which the dependent binary variable is the exposure. The possible confounding factors are the explanatory variables of this logistic model.

References

Trebern-Launay et al. Horizontal mixture model for competing risks: a method used in waitlisted renal transplant candidates. European Journal of Epidemiology. 33(3):275-286, 2018. <doi: 10.1007/s10654-017-0322-3>.

Examples

Run this code

# import the observed data
# X=1 corresponds to initial state with a functioning graft,
# X=2 to acute rejection episode (transient state), 
# X=3 to return to dialysis, X=4 to death with a functioning graft

data(dataDIVAT1)

# A subgroup analysis to reduce the time needed for this example

dataDIVAT1$id<-c(1:nrow(dataDIVAT1))
set.seed(2)
d2<-dataDIVAT1[dataDIVAT1$id %in% sample(dataDIVAT1$id, 300, replace = FALSE),]

# Data-management: two competing events
# the patient death is now X=2
# the return in dialysis is now X=3

d2$time<-NA
d2$time[d2$trajectory==1]<-d2$time1[d2$trajectory==1]

d2$time[d2$trajectory==12]<-d2$time2[d2$trajectory==12]
d2$trajectory[d2$trajectory==12]<-1

d2$time[d2$trajectory==13]<-d2$time1[d2$trajectory==13]

d2$time[d2$trajectory==123]<-d2$time2[d2$trajectory==123]
d2$trajectory[d2$trajectory==123]<-13

d2$time[d2$trajectory==14]<-d2$time1[d2$trajectory==14]

d2$time[d2$trajectory==124]<-d2$time2[d2$trajectory==124]
d2$trajectory[d2$trajectory==124]<-14

d2$trajectory[d2$trajectory==14]<-12

table(d2$trajectory)

# Univariable horizontal mixture model one binary explicative variable
# z is 1 if delayed graft function and 0 otherwise

mm2.test <- mixture.2states(times=d2$time, sequences=d2$trajectory, weights=NULL,
  dist=c("E","W"), cuts.12=NULL, cuts.13=NULL, 
  ini.dist.12=c(9.28), ini.dist.13=c(9.92, -0.23), 
  cov.12=d2$z, init.cov.12=0.84, names.12="beta_12",
  cov.13=d2$z, init.cov.13=0.76, names.13="beta_13",
  cov.p=NULL, init.cov.p=NULL, names.p=NULL, init.intercept.p=-0.75,
  conf.int=TRUE, silent=FALSE)
  
mm2.test$table

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