# Fitting the GORMC model with interval censored data. The Generalized Odds Rate Mixture Cure model is fitted for interval censored survival data. The EM algorithm facilitated by a gamma-poisson data augmentation is applied for estimating the coefficients in both the cure rate part and the survival part. The covariance matrix has closed forms based on the Louis method. ```r GORMC(survfun = formula(data), curefun = formula(data), data = parent.frame(), r = 0, n.int = 5, order = 3, max.iter = 1000, cov.rate = 0.001) ``` ## Arguments - `survfun`: A formula for the survival part in the GORMC model, defined using the Surv function and type=``interval2". - `curefun`: The formula of predictors of the cure rate part in the GORMC model. - `data`: The interval censored sorvival data, including the left and right end points of the time intervals and the covariates for the cure rate part and the survival part. If a subject is left(right) censored, the left(right) end point of the subject should be defined as ``NA", see example. - `r`: The transformation parameter in the GORMC model, should be greater than or equal to 0. r=0 refers to the PHMC model and r=1 refers to the POMC model. The default is 0. - `n.int`: Number of interior knots of the splines. Default is 5. - `order`: Order of the spline basis functions. Default is 3, i.e. the cubic splines. - `max.iter`: The maximum number of interations for the EM algorithm. Default is 1000. - `cov.rate`: The bound for convergence of the algorithm, which defined as the difference between the log-likelihood values of two consecutive iterations smaller than this value. Default is 0.001. ## Details The formula defined for ``survfun" is based on the Surv() function, where the left and right end points of the time interval are included and the type is equal to ``interval2". The left(right) end points of left(right) censored individuals should be defined as ``NA" in the data frame before running the function.\ The transformation parameter r is a nonnegative number corresponding to a specific model in the GORMC family of models. Special cases include the PHMC model(r=0) and the POMC model(r=1). Other positive numbers can also be specified. The grid search method is suggested to find the best model in practice. That is, try a sequence of r values and choose the one with the greatest log-likelihood value. ## Returns - **ParEst**: A list includes the estimated coefficients (Eta,Beta,gl), the whole hessian matrix (Hessian), AIC, and the log-likelihood value(loglik). - **ParVcov**: The estimated covariance matrix of the coefficients Eta and Beta. ## References Zhou, J., Zhang, J. and Lu, W. (2017+). Computationally Efficient Estimation for the Generalized Odds Rate Mixture Cure Model with Interval Censored Data. ## Note The estimated hessian matrix can be very large and sometimes not invertable. In which case, we try the QR decomposition, g-inverse or even numerical methods to get the covariance matrix. Different values of hess in the ParVcov indicating the different cases. hess=0:the hessian matrix is invertable; hess=1:the QR decomposition is applied to solve the hessian matrix; hess=2:the g-inverce is applied to the hessian matrix; hess=3:the hessian matrix is obtained from numerical methods. The variance estimates may be unreliable for the cases when hess>0. ## Examples ```r data(Hemophilia) head(Hemophilia) # Set Left/Right Interval End Points as NA Hemophilia$L[Hemophilia$d1==1]<-Hemophilia$R[Hemophilia$d3==1]<-NA # Fit PHMC Model (r=0) fit<-GORMC(survfun=Surv(L,R)~Low+Medium+High,curefun=~Low+Medium+High, data=Hemophilia,r=0) summary(fit) # Fit POMC Model (r=1) # fit<-GORMC(survfun=Surv(L,R)~Low+Medium+High,curefun=~Low+Medium+High, # data=Hemophilia,r=1) # summary(fit) # Predict Cure Rate and Survival Curve for a New Individual # Specify coveriate vectors for new.z and new.x pred1<-predict(fit,new.z=c(1,0,0,0),new.x=c(0,0,0)) pred2<-predict(fit,new.z=c(1,1,0,0),new.x=c(1,0,0)) pred3<-predict(fit,new.z=c(1,0,1,0),new.x=c(0,1,0)) pred4<-predict(fit,new.z=c(1,0,0,1),new.x=c(0,0,1)) # Obtain Cure Rates pred1$CureRate pred2$CureRate pred3$CureRate pred4$CureRate # Plot the Survival Curves plot(pred1,xlab="Time",ylab="Survival Probability",ylim=c(0,1)) lines(pred2$Survival,col=2) lines(pred3$Survival,col=3) lines(pred4$Survival,col=4) legend(0,0.3,c("None","Low","Medium","High"),lty=1,col=1:4) # Not run: Grid Search r # rr<-seq(0,2,0.2) # logl<-numeric() # for(i in 1:length(rr)){ # fit<-GORMC(survfun=Surv(L,R)~Low+Medium+High,curefun=~Low+Medium+High, # data=Hemophilia,r=rr[i]) # logl[i]<-fit$ParEst$loglik # } # plot(rr,logl,type="l",xlab="r",ylab="Log-likelihood") ```

GORMC function