RISCA (version 1.0.7)

markov.3states: 3-State Time-Inhomogeneous Markov Model

Description

The 3-state Markov model includes an initial state (X=1), a transient state (X=2) and an absorbing state (X=3). Usually, X=1 corresponds to disease-free or remission, X=2 to relapse, and X=3 to death. In this illness-death model, the possible transitions are: 1->2, 1->3 and 2->3.

Usage

markov.3states(times1, times2, sequences, weights=NULL, dist, 
cuts.12=NULL, cuts.13=NULL, cuts.23=NULL, 
ini.dist.12=NULL, ini.dist.13=NULL, ini.dist.23=NULL,
cov.12=NULL, init.cov.12=NULL, names.12=NULL,
cov.13=NULL, init.cov.13=NULL, names.13=NULL,
cov.23=NULL, init.cov.23=NULL, names.23=NULL, 
conf.int=TRUE, silent=TRUE, precision=10^(-6))

Value

object

The character string indicating the estimated model: "markov.3states (3-state time-inhomogeneous markov model)".

dist

A character vector with two arguments describing respectively the distributions of duration time for transitions 1->2, 1->3 and 2->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.

cuts.23

A numeric vector indicating the timepoints in days for the piecewise exponential distribution related to the time from X=2 to X=3.

covariates

A numeric vector indicating the numbers of covariates respectively related to the transition 1->2, 1->3 and 2->3.

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 related p-value (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

times1

A numeric vector with the observed times (in days) from baseline to the first transition (in X=2 or X=3) or to the right-censoring (in X=1 at the last follow-up).

times2

A numeric vector with the observed times (in days) from baseline to the second transition (in X=3) or to the right censoring (in X=2 at the last follow-up). NA for individuals right-censored in X=1 or individuals who directly transited from X=1 to X=3.

sequences

A numeric vector with the sequences of observed states. Four possible values are allowed: 1 (individual right-censored in X=1), 12 (individual right-censored in X=2), 13 (individual who directly transited from X=1 to X=3), 123 (individual who transited from X=1 to X=3 through X=2).

weights

A numeric vector with the weights for correcting the contribution of each individual. Default is NULL which means that no weighting is applied.

dist

A character vector with three arguments describing respectively the distributions of duration time for transitions 1->2, 1->3 and 2->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.

cuts.23

A numeric vector indicating the timepoints in days for the piecewise exponential distribution related to the time from X=2 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.

ini.dist.23

A numeric vector of initial values for the distribution from X=2 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.23

A numeric matrix (or data frame) with the explicative time-fixed variable(s) related to the time from X=2 to X=3.

init.cov.23

A numeric vector of initial values for regression coefficients (logarithm of the cause-specific hazards ratios) associated to cov.23. Default initial value is 0.

names.23

An optional character vector with name of explicative variables associated to cov.23.

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 <[e-mail]>

Florence Gillaizeau <[e-mail]>

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

Huszti et al. Relative survival multistate Markov model. Stat Med. 10;31(3):269-86, 2012. <DOI: 10.1002/sim.4392>

Gillaizeau et al. Inverse Probability Weighting to control confounding in an illness-death model for interval-censored data. Stat Med. 37(8):1245-1258, 2018. <doi: 10.1002/sim.7550>.

Examples

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

data(dataDIVAT1)

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

dataDIVAT1$id<-c(1:nrow(dataDIVAT1))
set.seed(2)
d3<-dataDIVAT1[dataDIVAT1$id %in% sample(dataDIVAT1$id, 300, replace = FALSE),]

# Individuals with trajectory 13 and 123 are 
# censored at the time of transition into state X=3

d3$trajectory[d3$trajectory==13]<-1
d3$trajectory[d3$trajectory==123]<-12
d3$trajectory[d3$trajectory==14]<-13
d3$trajectory[d3$trajectory==124]<-123

# 3-state parametric Markov model including one explicative variable 
# (z is the delayed graft function) on the transition 1->2. We only reduced
# the precision and the number of iteration to save time in this example,
# prefer the default values.

markov.3states(times1=d3$time1, times2=d3$time2, sequences=d3$trajectory, weights=NULL,
  dist=c("E","E","E"), ini.dist.12=c(9.93),
  ini.dist.13=c(11.54), ini.dist.23=c(10.21),
  cov.12=d3$z, init.cov.12=c(-0.13), names.12=c("beta12_z"),
  conf.int=TRUE,  silent=FALSE, precision=0.001)