A consequence of using a parametric frailty model with nonparametric baseline hazard for analyzing clustered time-to-event data is that its regression coefficient estimates could be sensitive to the underlying frailty distribution. Recently, there has been a proposal for specifying both the baseline hazard and the frailty distribution nonparametrically, and estimating the unknown parameters by the maximum penalized likelihood method. Instead, in this paper, we propose the nonparametric maximum likelihood method for a general class of nonparametric frailty models, i.e. models where the frailty distribution is completely unspecified but the baseline hazard can be either parametric or nonparametric. The implementation of the estimation procedure can be based on a combination of either the Broyden-Fletcher-Goldfarb-Shanno or expectation-maximization algorithm and the constrained Newton algorithm with multiple support point inclusion. Simulation studies to investigate the performance of estimation of a regression coefficient by several different model-fitting methods were conducted. The simulation results show that our proposed regression coefficient estimator generally gives a reasonable bias reduction when the number of clusters is increased under various frailty distributions. Our proposed method is also illustrated with two data examples.
A comparative study of five periodic human strains of Brugia malayi, originating from India, China, Korea, Malaysia and Indonesia, is given. This morphological analysis is based on males; the "standard" characters (oesophagus, papillae, spicules...) appear identical. On the contrary, the cuticular ornamentation of the posterior region--which is composed of the area rugosa and of a system of bosses and constitutes a secondary non-skid copulatory apparatus--differs following the geographical origin of the strain. A key is given, based on this character. 1(2) At 800-1,200 micron from the tip of tail, numerous cuticular bosses present on the right side of the body (fig. 2 and 8 B). 2(1) At 800-1,200 micron from the tip of tail, cuticular bosses absent or scarce on the right side of the body (fig. 8 D). 3(4) At 1,800-1,200 micron from the tip of tail (fig. 4), scarce and slightly projecting cuticular bosses on the dorsal side of the body contrasting with well projecting lateral cuticular bosses (fig. 9 E and F). Anterior extremity of the area rugosa made by a few stripes of tiny bosses linked transversally (fig. 9 A). 4(3) At 1,800-2,200 micron, numerous cuticular bosses on the dorsal side of the body (figs. 5, 6 and 7). Anterior extremity of the area rugosa made by the stripes of longitudinal rods (fig. 9C). 5(6) Oblong transversally stretched cuticular bosses on the dorsal and left sides of the body, anteriorly to the area rugosa (fig. 5); big oblong bosses on the left side (fig. 9 B). Transversal wrinkles and stripes of rods absent on the dorsal side of the body. 6(5) Round cuticular bosses on the dorsal and left sides of the body anteriorly to the area rugosa (figs. 6 and 7): no big oblong bosses on the left side. Transversal wrinkles or stripes of rods present on the dorsal side of the body (fig. 9 D). Nomenclaturally, such differences could be used in defining different taxa, but it could be useful to perform "blind determination" (material without labelling), to study conveniently the morphology of microfilariae (often an excellent indication for speciation in that group of Nematodes) and, evenly, to proceed to parallel studies on isoenzymes. However, whatever could be the taxonomical conclusion, the differences observed in Brugia malayi originating from different regions appear to the sufficient to consider the existence of four distinct diseases.