Data arising from Cluster Randomized Trials (CRTs) and longitudinal studies are correlated and generalized estimating equations (GEE) are a popular analysis method for correlated data. Previous research has shown that analyses using GEE could result in liberal inference due to the use of the empirical sandwich covariance matrix estimator, which can yield negatively biased standard error estimates when the number of clusters or subjects is not large. Many techniques have been presented to correct this negative bias; However, use of these corrections can still result in biased standard error estimates and thus test sizes that are not consistently at their nominal level. Therefore, there is a need for an improved correction such that nominal type I error rates will consistently result. First, GEEs are becoming a popular choice for the analysis of data arising from CRTs. We study the use of recently developed corrections for empirical standard error estimation and the use of a combination of two popular corrections. In an extensive simulation study, we find that nominal type I error rates can be consistently attained when using an average of two popular corrections developed by Mancl and DeRouen (2001, Biometrics 57, 126-134) and Kauermann and Carroll (2001, Journal of the American Statistical Association 96, 1387-1396) (AVG MD KC). Use of this new correction was found to notably outperform the use of previously recommended corrections. Second, data arising from longitudinal studies are also commonly analyzed with GEE. We conduct a simulation study, finding two methods to attain nominal type I error rates more consistently than other methods in a variety of settings: First, a recently proposed method by Westgate and Burchett (2016, Statistics in Medicine 35, 3733-3744) that specifies both a covariance estimator and degrees of freedom, and second, AVG MD KC with degrees of freedom equaling the number of subjects minus the number of parameters in the marginal model. Finally, stepped wedge trials are an increasingly popular alternative to traditional parallel cluster randomized trials. Such trials often utilize a small number of clusters and numerous time intervals, and these components must be considered when choosing an analysis method. A generalized linear mixed model containing a random intercept and fixed time and intervention covariates is the most common analysis approach. However, the sole use of a random intercept applies assumptions that will be violated in practice. We show, using an extensive simulation study based on a motivating example and a more general design, alternative analysis methods are preferable for maintaining the validity of inference in small-sample stepped wedge trials with binary outcomes. First, we show the use of generalized estimating equations, with an appropriate bias correction and a degrees of freedom adjustment dependent on the study setting type, will result in nominal type I error rates. Second, we show the use of a cluster-level summary linear mixed model can also achieve nominal type I error rates for equal cluster size settings.

Os isótopos de oxigênio (18O) e carbono (13C), registrados nas testas dos foraminíferos são descritores mensuráveis (proxies) da paleotemperatura e paleoprodutividade, respectivamente, amplamente utilizados em estudos paleoceanográficos. Em amostras de sedimento marinho, o tamanho de testa dos foraminíferos é uma importante fonte de variabilidade isotópica que não limita o uso destes proxies, desde que a mesma seja conhecida e entendida. No presente estudo, analisou-se o sinal de 18O e de 13C em testas da espécie bentônica, Cibicidoides wuellerstorfi, e das espécies planctônicas, Globigerinoides ruber (branca) e Globorotalia truncatulinoides (dextral) retidas em quatro frações de tamanho de malha de peneiras (150-250, 250-300, 300-355 e >355µm). Foram utilizadas amostras de topo de dois testemunhos localizados na Margem Continental Brasileira (em torno de 2000 metros de lâmina d´água). Foi comparado o sinal isotópico das amostras com o sinal isotópico da água do mar atual da região de estudo. Foi observado que C. wuellerstorfi calcifica em equilíbrio de 18O e com um desequilíbrio positivo (0,2-0,3) de 13C com respeito á água do mar de fundo. Os valores isotópicos desta espécie não variaram (13C) ou variaram levemente (18O) com o aumento no tamanho. Os valores de 18O registrados em G. ruber (branca) não apresentaram tendência com o tamanho de testa e refletiram uma profundidade aparente de calcificação na superfície do oceano (entre 0-100 metros). Os valores de 18O registrados em G. truncatulinoides (dextral) apresentaram aumento com o tamanho de testa (até 1,22) e refletiram uma profundidade aparente de calcificação em águas da termoclina (entre 200-400 metros). O sinal de 13C aumentou com o tamanho de testa tanto em G. ruber (branca) (até 1,85), como em G. truncatulinoides (dextral) (até 1,26). Os valores de 13C referentes aos maiores tamanhos de testas estão mais próximos dos valores de 13C da água do mar da profundidade aparente de calcificação estimada para cada espécie. Este trabalho corrobora a importância da seleção do tamanho de testa dos foraminíferos em estudos paleoceanográficos para a região de estudo, especialmente no sinal de 13C em foraminíferos planctônicos. / Stable oxygen (18O) and carbon (13C) isotope composition registered in foraminifera test are widely used in paleoceanography studies as paleotemperature and paleoprodutivity proxies. On the sea floor, samples size test is an important source of stable isotope variability related to the ecology and physiology of foraminifera. This variability does not restrict the use of these proxies when the relation between size and stable isotopes is known and understood. 18O and 13C values were analyzed in four size sieves fractions (150-250, 250-300, 300-355 e >355µm) in foraminiferal tests of Cibicidoides wuellerstorfi (benthic specie), Globigerinoides ruber (white) and Globorotalia truncatulinoides (right) (planktonic species), in two core topes located on the Brazilian continental margin (around 2,000 meters). Modern seawater stable isotope composition was used for data interpretation. In general, the relation between stable isotope and size for each species was consistent in all samples. C. wuellerstorfi calcifies in equilibrium with 18O, while it is enriched (0.2-0.3) in 13C values with respect to deep sea isotopic values. No clear isotopic size-related changes were observed for this species. 18O values do not change with test size in G. ruber (white) and reflect an apparent calcification depth in superficial waters (0-100 meters). 18O enrichment with size was observed for G. truncatulinoides (right) (up to 1.22) and reflects an apparent calcification depth in thermocline waters (between 200-400 meters). 13C values were clearly enriched with size in both G. ruber (white) (up to 1.85) and G. truncatulinoides (right) (up to 1.26). 13C values of the largest size sieve fraction reflects most accurately 13C values of the estimated calcification depth for each specie. This work highlights the importance of selecting foraminifera test size for paleoceanographic studies, especially in what concerns about carbon isotopes in planktonic foraminifera.

Modified Ring specimens are of the shape of discs having a hole inside and flattened ends. These specimens are used for determination of Mode I fracture toughness. Finite element program, named ABAQUS, is used for numerical modeling for finding stress intensity factors. Varying disc geometries were used for the experiments and numerical modeling in which size of the flat ends, radius of the hole inside, and external radius of the specimen were varied. Experiments were done by using pink Ankara andesite. Effects of internal hole radius, external disc radius and size of the flat ends on both stress intensity factor and fracture toughness were studied. In order to compare the results, fracture tests with semi-circular specimens under three point bending (SCB) were also performed. From a similar previous study, fracture toughness values of gray andesite were recalculated and compared to the fracture toughness values of pink andesite for varying geometrical factors. Size effect studies were performed as well for varying diameter of core specimens.Fracture toughness values of andesite were found to increase with increasing specimen size. Fracture toughness of 100 mm specimens was determined as 1.11&plusmn / 0.07 MPa&amp / #8730 / m, whereas fracture toughness of 75 mm specimens was 0.96&plusmn / 0.08 MPa&amp / #8730 / m. 100 mm or larger diameter specimens were suggested for the fracture toughness determination with the modified ring tests.

Os isótopos de oxigênio (18O) e carbono (13C), registrados nas testas dos foraminíferos são descritores mensuráveis (proxies) da paleotemperatura e paleoprodutividade, respectivamente, amplamente utilizados em estudos paleoceanográficos. Em amostras de sedimento marinho, o tamanho de testa dos foraminíferos é uma importante fonte de variabilidade isotópica que não limita o uso destes proxies, desde que a mesma seja conhecida e entendida. No presente estudo, analisou-se o sinal de 18O e de 13C em testas da espécie bentônica, Cibicidoides wuellerstorfi, e das espécies planctônicas, Globigerinoides ruber (branca) e Globorotalia truncatulinoides (dextral) retidas em quatro frações de tamanho de malha de peneiras (150-250, 250-300, 300-355 e >355µm). Foram utilizadas amostras de topo de dois testemunhos localizados na Margem Continental Brasileira (em torno de 2000 metros de lâmina d´água). Foi comparado o sinal isotópico das amostras com o sinal isotópico da água do mar atual da região de estudo. Foi observado que C. wuellerstorfi calcifica em equilíbrio de 18O e com um desequilíbrio positivo (0,2-0,3) de 13C com respeito á água do mar de fundo. Os valores isotópicos desta espécie não variaram (13C) ou variaram levemente (18O) com o aumento no tamanho. Os valores de 18O registrados em G. ruber (branca) não apresentaram tendência com o tamanho de testa e refletiram uma profundidade aparente de calcificação na superfície do oceano (entre 0-100 metros). Os valores de 18O registrados em G. truncatulinoides (dextral) apresentaram aumento com o tamanho de testa (até 1,22) e refletiram uma profundidade aparente de calcificação em águas da termoclina (entre 200-400 metros). O sinal de 13C aumentou com o tamanho de testa tanto em G. ruber (branca) (até 1,85), como em G. truncatulinoides (dextral) (até 1,26). Os valores de 13C referentes aos maiores tamanhos de testas estão mais próximos dos valores de 13C da água do mar da profundidade aparente de calcificação estimada para cada espécie. Este trabalho corrobora a importância da seleção do tamanho de testa dos foraminíferos em estudos paleoceanográficos para a região de estudo, especialmente no sinal de 13C em foraminíferos planctônicos. / Stable oxygen (18O) and carbon (13C) isotope composition registered in foraminifera test are widely used in paleoceanography studies as paleotemperature and paleoprodutivity proxies. On the sea floor, samples size test is an important source of stable isotope variability related to the ecology and physiology of foraminifera. This variability does not restrict the use of these proxies when the relation between size and stable isotopes is known and understood. 18O and 13C values were analyzed in four size sieves fractions (150-250, 250-300, 300-355 e >355µm) in foraminiferal tests of Cibicidoides wuellerstorfi (benthic specie), Globigerinoides ruber (white) and Globorotalia truncatulinoides (right) (planktonic species), in two core topes located on the Brazilian continental margin (around 2,000 meters). Modern seawater stable isotope composition was used for data interpretation. In general, the relation between stable isotope and size for each species was consistent in all samples. C. wuellerstorfi calcifies in equilibrium with 18O, while it is enriched (0.2-0.3) in 13C values with respect to deep sea isotopic values. No clear isotopic size-related changes were observed for this species. 18O values do not change with test size in G. ruber (white) and reflect an apparent calcification depth in superficial waters (0-100 meters). 18O enrichment with size was observed for G. truncatulinoides (right) (up to 1.22) and reflects an apparent calcification depth in thermocline waters (between 200-400 meters). 13C values were clearly enriched with size in both G. ruber (white) (up to 1.85) and G. truncatulinoides (right) (up to 1.26). 13C values of the largest size sieve fraction reflects most accurately 13C values of the estimated calcification depth for each specie. This work highlights the importance of selecting foraminifera test size for paleoceanographic studies, especially in what concerns about carbon isotopes in planktonic foraminifera.