# U-shaped Hazard Function Estimation `Uhaz` computes the nonparametric maximum likelihood esimate (NPMLE) of a U-shaped hazard function from exact or interval-censored data, or a mix of the two types of data. ```r Uhaz(data, w = 1, deg = 1, maxit = 100, tol = 1e-06, verb = 0) ``` ## Arguments - `data`: vector or matrix, or an object of class `icendata`. - `w`: weights or multiplicities of the observations. - `deg`: nonnegative real number for spline degree (i.e., p in the formula below). - `maxit`: maximum number of iterations. - `tol`: tolerance level for stopping the algorithm. It is used as the threshold on the increase of the log-likelihood after each iteration. - `verb`: verbosity level for printing intermediate results in each iteration. ## Returns An object of class `Uhaz`, which is a list with components: - **convergence**: = `TRUE`, converged successfully; = `FALSE`, maximum number of iterations reached. - **grad**: gradient values at the knots. - **numiter**: number of iterations used. - **ll**: log-likelihood value of the NPMLE `h`. - **h**: NPMLE of the U-shaped hazard function, an object of class `uh`. ## Details If `data` is a vector, it contains only exact observations, with weights given in `w`. If `data` is a matrix with two columns, it contains interval-censored observations, with the two columns storing their left and right end-points, respectively. If the left and right end-points are equal, then the observation is exact. Weights are provided by `w`. If `data` is a matrix with three columns, it contains interval-censored observations, with the first two columns storing their left and right end-points, respectively. The weight of each observation is the third-column value multiplied by the corresponding weight value in `w`. The algorithm used for the computing the NPMLE of a hazard function under the U-shape restriction is is proposed by Wang and Fani (2015). Such a hazard function is given by A U-shaped hazard function is given by $$ h(t) = \alpha + \sum_{j = 1}^k \nu_j(\tau_j - t)_+^p + \sum_{j = 1}^{m} \mu_j (t-\eta_j)_+^p,h(t) = alpha + sum_{j=1}^k nu_j (tau_j - t)_+^p+ sum_{j=1}^m mu_j (t - eta_j)_+^p, $$ where $alpha, nu_j, mu_j \ge 0$, $tau_1 < ... < tau_k <= eta_1 < ... < eta_m,$ and $p >= 0$ is the the spline degree which determines the smoothness of the U-shaped hazard. As p increases, the family of hazard functions becomes increasingly smoother, but at the time, smaller. When p = 0, the hazard function is U-shaped, as studied by Bray et al. (1967). When p = 1, the hazard function is convex, as studied by Jankowski and Wellner (2009a,b). Note that `deg` (i.e., p in the above mathematical display) can take on any nonnegative real value. ## Examples ```r ## Interval-censored observations data(ap) (r = Uhaz(ap, deg=0)) plot(r, ylim=c(0,.3), col=1) for(i in 1:6) plot(Uhaz(ap, deg=i/2), add=TRUE, col=i+1) legend(15, 0.01, paste0("deg = ", 0:6/2), lwd=2, col=1:7, xjust=1, yjust=0) ## Exact observations data(nzmort) x = with(nzmort, nzmort[ethnic=="maori",])[,1:2] # Maori mortality (h0 = Uhaz(x[,1]+0.5, x[,2], deg=0)$h) # U-shaped hazard (h1 = Uhaz(x[,1]+0.5, x[,2], deg=1)$h) # convex hazard (h2 <- Uhaz(x[,1]+0.5, x[,2], deg=2)$h) # smooth U-shaped hazard plot(h0, pch=2) # plot hazard functions plot(h1, add=TRUE, col="green3", pch=1) plot(h2, add=TRUE, col="red3", pch=19) age = 0:max(x[,1]) # plot densities count = integer(length(age)) count[x[,"age"]+1] = x[,"deaths"] barplot(count/sum(count), space=0, col="lightgrey") axis(1, pos=NA, at=0:10*10) plot(h0, fn="d", add=TRUE, pch=2) plot(h1, fn="d", add=TRUE, col="green3", pch=1) plot(h2, fn="d", add=TRUE, col="red3", pch=19) plot(h0, fn="s", pch=2) # plot survival functions plot(h1, fn="s", add=TRUE, col="green3", pch=1) plot(h2, fn="s", add=TRUE, col="red3", pch=19) ## Exact and right-censored observations data(gastric) plot(h0<-Uhaz(gastric, deg=0)$h) # plot hazard functions plot(h1<-Uhaz(gastric, deg=1)$h, add=TRUE, col="green3") plot(h2<-Uhaz(gastric, deg=2)$h, add=TRUE, col="red3") plot(npsurv(gastric), fn="s", col="grey") # plot survival functions plot(h0, fn="s", add=TRUE) plot(h1, fn="s", add=TRUE, col="green3") plot(h2, fn="s", add=TRUE, col="red3") ``` ## References Bray, T. A., Crawford, G. B., and Proschan, F. (1967). **Maximum Likelihood Estimation of a U-shaped Failure Rate Function**. Defense Technical Information Center. Jankowski, H. K. and Wellner, J. A. (2009a). Computation of nonparametric convex hazard estimators via profile methods. **Journal of Nonparametric Statistics**, 21 , 505-518. Jankowski, H. K. and Wellner, J. A. (2009b). Nonparametric estimation of a convex bathtub-shaped hazard function. **Bernoulli**, 15 , 1010-1035. Wang, Y. and Fani, S. (2018). Nonparametric maximum likelihood computation of a U-shaped hazard function. **Statistics and Computing**, 28 , 187-200. ## See Also `icendata`, `nzmort`. ## Author(s) Yong Wang

Uhaz function