Mixed-effects models have become a popular approach for the analysis of grouped data that arise in many areas as diverse as clinical trials, epidemiology, and sociology. Examples of grouped data include longitudinal data, repeated measures, and multilevel data. In the case of linear mixed-effects (LME) models, the likelihood function can be expressed in closed form, with efficient computational algorithms having been proposed for maximum likelihood and restricted maximum likelihood estimation. For nonlinear mixed-effects (NLME) models and generalized linear mixed models (GLMMs), however, the likelihood function does not have a closed form. Different likelihood approximations, with varying degrees of accuracy and computational complexity, have been proposed for these models. This article describes algorithms for one such approximation, the adaptive Gaussian quadrature (AGQ), for GLMMs which scale up efficiently to multilevel models with arbitrary number of levels. The proposed algorithms greatly reduce the computational complexity and the memory usage for approximating the multilevel GLMM likelihood, when compared to a direct application of a single-level AGQ approximation algorithm to the multilevel case. The accuracy of the associated estimates is evaluated and compared to that of estimates obtained from other approximations via simulation studies.
Coesite, the high-pressure polymorph of SiO(2), hitherto known only as a synthetic compound, is identified as an abundant mineral in sheared Coconino sandstone at Meteor Crater, Arizona. This natural occurrence has important bearing on the recognition of meteorite impact craters in quartz-bearing geologic formations.
The Ries basin is a shallow, nearly circular depression about 17 miles in diameter that lies between the Swabian and Franconian plateaus of southern Germany. Great masses of breccia and a system of thrust sheets associated with the Ries have been studied by German geologists for about a century. E. Werner and Otto Stutzer suggested that the Ries was an impact crater, but the consensus of the principal investigators has been that it was formed by some sort of volcanic explosion. The only direct evidence of magmatic activity at the Ries is the presence of glass in scattered patches of a breccia called suevite. Some of the glass has long been recognized as sintered fragments of old crystalline rocks. We have found that glasses of various composition coexist in single specimens of suevite. In addition, coesite, a high‐pressure polymorph of SiO2, and lechatelierite, SiO2 glass, occur in the sintered rocks in the suevite. The presence of the same phases in sintered rock fragments at Meteor Crater, Arizona, and the coexistence of glasses of different composition suggest that the glassy components of suevite are of impact rather than volcanic origin.
Shock effects in quartz, plagioclase, biotite, amphibole, and some accessory minerals have been observed in rocks subjected to various degrees of meta morphism by meteoritic impact. The shock features described are unique; they are never observed in rocks from normal geologic environments. Such features are described: 1) Multiple sets of closely spaced planar microstructures occur in quartz, plagioclase, and other rock-forming minerals. Those characteristic of shock consist of alternating platelets, with a range of reduced mean index of re fraction and birefringence; they con sist of platelets that have been partially or completely transformed to an amor phous phase. 2) Quartz and plagioclase are selec tively and completely transformed to silica glass and plagioclase glass in the solid state, whereas the associated mafic minerals remained crystalline. There is no reaction between adjacent minerals. 3) High-pressure polymorphs occur, such as coesite or stishovite. Coesite oc Curs exclusively within silica glass; it has not been observed as a reaction or breakdown product. 4) Nickel-iron spherules occur in the fused glass or impactites. 5) The occurrence of droplets of ilmenite, rutile, pseudobrookite, and baddaleyite in impactites indicates a temperature of formation exceeding 150 degrees C. 6) Dense glass occurs, similar in composition to bulk rock, in which iron oxide, such as fine particles of mag netite, is completely dissolved. All these features are characteristic of a process involving the rapid rise and fall of extremiiely high pressures and temperatures. Minerals and mineral as semblages experiencing such high strain rates and sudden changes of pressures and temperatures react and change in dependently to the bulk chemical com position, under nonequilibrium condi tions. Many aspects of shock features re quire careful study. Kink bands in biotite and deformation lamellae in quartz occur in tectonically deformed rocks. These features should be studied with great care in order to determine whether reduction in mean index of refraction and total birefringence along the planar structures have resulted from vitrification or phase transition; their presence is additional evidence in favor of a shock mechanism. Vitreous phases or glasses formed by shock also have many unique prop erties; they have not been studied by such methods as thermoluminescence, electron spin resonance, low-angle x ray diffraction, or infrared spectroscopy. Shock-fused glass of high density needs to be studied in detail in carefully con trolled laboratory conditions. Experimental shock-wave studies of the equation-of-state of single minerals and mineral assemblages, under care fully controlled conditions, must pre cede estimates of peak pressures and peak and residual temperatures of shocked natural mineral assemblages. Detailed petrographic and mineralogic studies, however, have provided useful and definitive criteria for characteriza tion of impact events. Such data should be of paramount importance in the study of samples brought back f...
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