# EM algorithm for fitting generalized odds-rate model with specified link function (i.e., alpha value) under interval-censored data Fits the generalized odds-rate model based on penalized B-splines to interval censored data via an EM algorithm. ```r EM_fit(g0,b0,d1,d2,d3,Li,Ri,Z,nsub,alpha,qn,order,t.seq,tol=1e-5,itmax=500,lamu=1e5) ``` ## Arguments - `g0`: initial estimate of the spline coefficients; should be of length qn+order+1. - `b0`: initial estimate of regression coefficients; should be of length dim(Z)[2]. - `d1`: vector indicating whether an observation is left-censored (1) or not (0). - `d2`: vector indicating whether an observation is interval-censored (1) or not (0). - `d3`: vector indicating whether an observation is right-censored (1) or not (0). - `Li`: the left endpoint of the observed interval; if an observation is left-censored, its corresponding entry should be 0. - `Ri`: the right endpoint of the observed interval; if an observation is right-censored, its corresponding entry should be Inf. - `Z`: design matrix of predictor variables (in columns); should be specified without an intercept term. - `nsub`: size of observed dataset. - `alpha`: parameter of link function; alpha=0 for the PH model and alpha=1 for the PO model. - `qn`: the number of interior knots to be used; should not exceed square root of sample size. - `order`: the order of the basis functions; order=3 for cubic spline. - `tol`: the convergence criterion of the EM algorithm. - `t.seq`: an increasing sequence of points at which the cumulative baseline hazard function is evaluated. - `itmax`: maximum iterations of EM procedure. - `lamu`: upper limit of smoothing parameter. ## Details The above function fits the generalized odds-rate model (with specified value of alpha) to interval censored data via an EM algorithm using penalized monotone B-splines. ## Returns - **b**: estimates of the regression coefficients. - **g**: estimates of the spline coefficients. - **se**: the standard deviation of b. - **base**: estimated cumulative baseline hazard function evaluated at the points t.seq. - **lambda**: final value of smooth parameter. - **flag**: the indicator whether the procedure converged; 0 if converged. ## References Lu, M., Liu, Y., Li, C. and Sun, J. (2019+). An efficient penalized estimation approach for a semi-parametric linear transformation model with interval-censored data. arXiv:1912.11703. ## Examples ```r set.seed(1) case <- 2 nsub <- 35 # Generate interval-censored data under PH model dat <- dataPA(nsub,case,alpha=0) rp <- c(mean(dat$d1),mean(dat$d2),mean(dat$d3)) rp # [1] 0.2571429 0.3428571 0.4000000 t.seq <- seq(0.01,4,0.01) # number of interior knots to be used qn <- ceiling(nsub^(1/3))-2 order <- 3 d1 <- dat$d1 d2 <- dat$d2 d3 <- dat$d3 Ri <- dat$Ri Li <- dat$Li Z <- dat$Z p <- ncol(Z) b0 <- rep(0,p) g0 <- sort(runif(qn+order+1,-1,1)) # Fit data under PH model fit <- EM_fit(g0,b0,d1,d2,d3,Li,Ri,Z,nsub,alpha=0,qn,order,t.seq,tol=1e-2,itmax=100,lamu=1e5) cbind(fit$b,fit$se) # [,1] [,2] #[1,] -1.0655212 0.5021835 #[2,] 0.7649178 0.3185045 ```

EMfit function