# Horizontal Mixture Model for Two Competing Events 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. ```r 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)) ``` ## 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 $10^{-6}$. ## 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. ## Returns - **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. ## 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. . ## Author(s) Yohann Foucher ## Examples ```r # 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 ```

mixture.2states function