gravimagnetic anomaly formulas for polyhedra of spatially linear media formulas for the gravity potential field and field gradient tensor are derived for a polyhedral target body of a spatially linear density medium the formulas also define the magnetic potential and field in the case of a medium of spatially linear magnetization this work generalizes existing solutions for the gravity field of a polyhedral target of linearly varying density the formulas are analyzed for singularities and for numerical error growth error growth with increasing target distance is found to be higher than in the corresponding uniform polyhedral case examination of the error sources reveals that some error reduction is possible on this basis new algorithms of improved error control over existing algorithms are proposed for the linear medium computational results confirm the expected improvement

the mixed model for multivariate repeated measures validity conditions and an approximate test repeated measures on multivariate responses can be analyzed according to either of two models a doubly multivariate model dmm or a multivariate mixed model mmm this paper reviews both models and gives three new results concerning the mmm the first result is primarily of theoretical interest the second and third have implications for practice first it is shown that given multivariate normality a condition called multivariate sphericity of the covariance matrix is both necessary and sufficient for the validity of the mmm analysis to test for departure from multivariate sphericity the likelihood ratio test can be employed the second result is an approximation to the null distribution of the likelihood ratio test statistic useful for moderate sample sizes third for situations satisfying multivariate normality but not multivariate sphericity a multivariate e correction factor is derived the e correction factor generalizes box s e and can be used to construct an adjusted mmm test

small sample inference for fixed effects from restricted maximum likelihood restricted maximum likelihood reml is now well established as a method for estimating the parameters of the general gaussian linear model with a structured covariance matrix in particular for mixed linear models conventionally estimates of precision and inference for fixed effects are based on their asymptotic distribution which is known to be inadequate for some small sample problems in this paper we present a scaled wald statistic together with an f approximation to its sampling distribution that is shown to perform well in a range of small sample settings the statistic uses an adjusted estimator of the covariance matrix that has reduced small sample bias this approach has the advantage that it reproduces both the statistics and f distributions in those settings where the latter is exact namely for hotelling t2 type statistics and for analysis of variance f ratios the performance of the modified statistics is assessed through simulation studies of four different reml analyses and the methods are illustrated using three examples

on the singularities of the gravity field of a homogeneous polyhedral body the study of the gravity field of arbitrary polyhedral bodies of homogeneous density has provoked a series of publications over the last decades some of the researchers represented an arbitrary three dimensional body in terms of contours obtained by the intersection of horizontal planes with the body

controlling test size while gaining the benefits of an internal pilot design summary to compensate for a power analysis based on a poor estimate of variance internal pilot designs use some fraction of the planned observations to reestimate error variance and modify the final sample size ignoring the randomness of the final sample size may bias the final variance estimate and inflate test size we propose and evaluate three different tests that control test size for an internal pilot in a general linear univariate model with fixed predictors and gaussian errors test 1 uses the first sample plus those observations guaranteed to be collected in the second sample for the final variance estimate test 2 depends mostly on the second sample for the final variance estimate test 3 uses the unadjusted variance estimate and modifies the critical value to bound test size we also examine three sample size modification rules only test 2 can control conditional test size align with a modification rule and provide simple power calculations we recommend it if the minimum second incremental sample is at least moderate perhaps 20 otherwise the bounding test appears to have the highest power in small samples reanalyzing published data highlights some advantages and disadvantages of the various tests

sample size determination for hierarchical longitudinal designs with differential attrition rates we consider the problem of sample size determination for three level mixed effects linear regression models for the analysis of clustered longitudinal data three level designs are used in many areas but in particular multicenter randomized longitudinal clinical trials in medical or health related research in this case level 1 represents measurement occasion level 2 represents subject and level 3 represents center the model we consider involves random effects of the time trends at both the subject level and the center level in the most common case we have two random effects constant and a single trend at both subject and center levels the approach presented here is general with respect to sampling proportions number of groups and attrition rates over time in addition we also develop a cost model as an aid in selecting the most parsimonious of several possible competing models i e different combinations of centers subjects within centers and measurement occasions we derive sample size requirements i e power characteristics for a test of treatment by time interaction s for designs based on either subject level or cluster level randomization the general methodology is illustrated using two characteristic examples

power calculations for general linear multivariate models including repeated measures applications abstract recently developed methods for power analysis expand the options available for study design we demonstrate how easily the methods can be applied by 1 reviewing their formulation and 2 describing their application in the preparation of a particular grant proposal the focus is a complex but ubiquitous setting repeated measures in a longitudinal study describing the development of the research proposal allows demonstrating the steps needed to conduct an effective power analysis discussion of the example also highlights issues that typically must be considered in designing a study first we discuss the motivation for using detailed power calculations focusing on multivariate methods in particular second we survey available methods for the general linear multivariate model glmm with gaussian errors and recommend those based on f approximations the treatment includes coverage of the multivariate and univariate approaches to repeated measures manova anova multivariate regression and

tests for gaussian repeated measures with missing data in small samples for small samples of gaussian repeated measures with missing data barton and cramer recommended using the em algorithm for estimation and reducing the degrees of freedom for an analogue of rao s f approximation to wilks test computer simulations led to the conclusion that the modified test was slightly conservative for total sample size of n 40 here we consider additional methods and smaller sample sizes nin 12 24 we describe analogues of the pillai bartlett trace hotelling lawley trace and geisser greenhouse corrected univariate tests which allow for missing data eleven sample size adjustments were examined which replace n by some function of the numbers of non missing pairs of responses in computing error degrees of freedom overall simulation results allowed concluding that an adjusted test can always control test size at or below the nominal rate even with as few as 12 observations and up to 10 per cent missing data the choice of method varies with the test statistic replacing n by the mean number of non missing responses per variable works best for the geisser greenhouse test the pillai bartlett test requires the stronger adjustment of replacing n by the harmonic mean number of non missing pairs of responses for wilks and hotelling lawley an even more aggressive adjustment based on the minimum number of non missing pairs must be used

life span and a new critical exponent for a degenerate parabolic equation abstract in this paper we consider the positive solution of the cauchy problem for the equation u t u p u u q p 1 q 1 and give a secondary critical exponent of the behavior of initial value at infinity for the existence of global and nonglobal solutions of the cauchy problem furthermore the life span of solutions are also studied

blow up phenomena for polytropic equation with inhomogeneous density and source the subject of this investigation is the blow up phenomena of the positive solutions of the mixed problem for the one dimensional polytropic filtration equation with inhomogeneous density and source it is shown that under certain conditions on the nonlinearities and data blow up will occur at some finite time note that the technique applied for the proof does not use the zel dovich kompaneets barenblatt solutions since the construction of such type of function is more complicated in our case therefore we obtain a result by multiplying on a special factor which has convenient properties in particular by choosing the parameters of the factor and using the properties of the solution we obtain the inequality which allows us to show the blow up phenomena

multivariate test power approximations for balanced linear mixed models in studies with missing data multilevel and longitudinal studies are frequently subject to missing data for example biomarker studies for oral cancer may involve multiple assays for each participant assays may fail resulting in missing data values that can be assumed to be missing completely at random catellier and muller proposed a data analytic technique to account for data missing at random in multilevel and longitudinal studies they suggested modifying the degrees of freedom for both the hotelling lawley trace f statistic and its null case reference distribution we propose parallel adjustments to approximate power for this multivariate test in studies with missing data the power approximations use a modified non central f statistic which is a function of i the expected number of complete cases ii the expected number of non missing pairs of responses or iii the trimmed sample size which is the planned sample size reduced by the anticipated proportion of missing data the accuracy of the method is assessed by comparing the theoretical results to the monte carlo simulated power for the catellier and muller multivariate test over all experimental conditions the closest approximation to the empirical power of the catellier and muller multivariate test is obtained by adjusting power calculations with the expected number of complete cases the utility of the method is demonstrated with a multivariate power analysis for a hypothetical oral cancer biomarkers study we describe how to implement the method using standard commercially available software products and give example code copyright 2015 john wiley sons ltd

internal pilots for a class of linear mixed models with gaussian and compound symmetric data an internal pilot design uses interim sample size analysis without interim data analysis to adjust the final number of observations the approach helps to choose a sample size sufficiently large to achieve the statistical power desired but not too large which would waste money and time we report on recent research in cerebral vascular tortuosity curvature in three dimensions which would benefit greatly from internal pilots due to uncertainty in the parameters of the covariance matrix used for study planning unfortunately observations correlated across the four regions of the brain and small sample sizes preclude using existing methods however as in a wide range of medical imaging studies tortuosity data have no missing or mistimed data a factorial within subject design the same between subject design for all responses and a gaussian distribution with compound symmetry for such restricted models we extend exact small sample univariate methods for internal pilots to linear mixed models with any between subject design not just two groups planning a new tortuosity study illustrates how the new methods help to avoid sample sizes that are too small or too large while still controlling the type i error rate

sample size calculations for two group repeated measures experiments vonesh and schork 1986 biometrics 42 601 610 presented a statistical methodology for computing the minimum sample size required for the within subjects repeated measures design they applied hotelling s t2 analysis and demonstrated the utility of these techniques under general covariance structures in this paper we extend these procedures to the between subjects repeated measures design when there are two treatment groups under consideration the multivariate analysis of variance approach to analyzing repeated measurements is considered and this model also resolves to hotelling s t2 analysis two models are put forward to contend with those situations where the correlation structure among the repeated measures is unknown these include the autoregressive model and the assumption of compound symmetry and they are compared and contrasted throughout tables of the minimum sample sizes required for several hypotheses arising from repeated measures designs are presented

global hypothesis testing for high dimensional repeated measures outcomes high throughput technology in metabolomics genomics and proteomics gives rise to high dimension low sample size data when the number of metabolites genes or proteins exceeds the sample size for a limited class of designs the classic univariate approach for gaussian repeated measures can provide a reasonable global hypothesis test we derive new tests that not only accurately allow more variables than subjects but also give valid analyses for data with complex between subject and within subject designs our derivations capitalize on the dual of the error covariance matrix which is nonsingular when the number of variables exceeds the sample size to ensure correct statistical inference and enhance computational efficiency simulation studies demonstrate that the new tests accurately control type i error rate and have reasonable power even with a handful of subjects and a thousand outcome variables we apply the new methods to the study of metabolic consequences of vitamin b6 deficiency free software implementing the new methods applies to a wide range of designs including one group pre intervention and post intervention comparisons multiple parallel group comparisons with one way or factorial designs and the adjustment and evaluation of covariate effects copyright 2011 john wiley sons ltd

avoiding bias in mixed model inference for fixed effects analysis of a large longitudinal study of children motivated our work the results illustrate how accurate inference for fixed effects in a general linear mixed model depends on the covariance model selected for the data simulation studies have revealed biased inference for the fixed effects with an underspecified covariance structure at least in small samples one underspecification common for longitudinal data assumes a simple random intercept and conditional independence of the within subject errors i e compound symmetry we prove that the underspecification creates bias in both small and large samples indicating that recruiting more participants will not alleviate inflation of the type i error rate associated with fixed effect inference enumerations and simulations help quantify the bias and evaluate strategies for avoiding it when practical backwards selection of the covariance model starting with an unstructured pattern provides the best protection tutorial papers can guide the reader in minimizing the chances of falling into the often spurious software trap of nonconvergence in some cases the logic of the study design and the scientific context may support a structured pattern such as an autoregressive structure the sandwich estimator provides a valid alternative in sufficiently large samples authors reporting mixed model analyses should note possible biases in fixed effects inference because of the following i the covariance model selection process ii the specific covariance model chosen or iii the test approximation

critical exponents and non extinction for a fast diffusive polytropic filtration equation with nonlinear boundary sources in this paper we discuss the critical exponents and non extinction property for a nonlinear boundary value problem of a fast diffusive polytropic filtration equation

closed form formula of magnetic gradient tensor for a homogeneous polyhedral magnetic target a tetrahedral grid example abstracta closed form formula is developed for the full magnetic gradient tensor of a polyhedral body with a homogeneous magnetization vector it is based on the direct derivative technique on the closed form of the magnetic field these analytical expressions are implemented into an easy to use c package which simultaneously calculates the magnetic potential the magnetic field and the full magnetic gradient tensor for magnetic targets modern unstructured tetrahedral grids are adopted to represent the polyhedral body so that our code can deal with arbitrarily complicated magnetic targets a prismatic body is tested to verify the accuracies of our closed form formula excellent agreements are obtained between our closed form solutions and solutions of a prismatic magnetic body with differences up to machine precision a pipeline model is used to demonstrate its capability to deal with complicated magnetic targets this c code is freely available to the magnetic exploration community

analytical computation of the full gravity tensor of a homogeneous arbitrarily shaped polyhedral source using line integrals the analytical computation of the full gravity tensor from a polyhedral source of homogeneous density is presented with emphasis on its algorithmic implementation the theoretical development is based on the subsequent transition of the general expressions from volume to surface and from surface to line integrals defined along the closed polygons building each polyhedral face however the accurate numerical computation of the obtained transcendental expressions is linked with the relative position of the computation point and its corresponding projections on the plane of each face and on the line of each segment with respect to the polygons defining each face depending on this geometric setup the application of the divergence theorem of gauss leads to the appearance of additional correction terms valid only for these boundary conditions and crucial for the correct numerical evaluation of the polyhedral related gravity quantities at those locations of the computation point a program in fortran is su

comparison of gravimetric formulas for uniform polyhedra the gravitational anomaly of a polyhedral target of uniform density is amenable to analytical solution and has received wide attention we analyze a number of apparently dissimilar solutions for consistency and comment on their suitability for numeric computation consistency is determined by evaluating a method over a special contour for which all methods must agree numerical performance is assessed through operation counts and algorithm classification into vertex line or surface type each type having a known accuracy bound for a given floating point precision we verify performance predictions through model calculations in particular we demonstrate the effectiveness of revised vertex and line algorithms and of a new analytical surface method our scheme of analysis and classification has general application it provides criteria by which existing and novel solutions may be examined for consistency and numerical performance

sample size for a two group comparison of repeated binary measurements using gee controlled clinical trials often randomize subjects to two treatment groups and repeatedly evaluate them at baseline and intervals across a treatment period of fixed duration a popular primary objective in these trials is to compare the change rates in the repeated measurements between treatment groups repeated measurements usually involve missing data and a serial correlation within each subject the generalized estimating equation gee method has been widely used to fit the time trend in repeated measurements because of its robustness to random missing and misspecification of the true correlation structure in this paper we propose a closed form sample size formula for comparing the change rates of binary repeated measurements using gee for a two group comparison the sample size formula is derived incorporating missing patterns such as independent missing and monotone missing and correlation structures such as ar 1 model we also propose an algorithm to generate correlated binary data with arbitrary marginal means and a markov dependency and use it in simulation studies copyright 2005 john wiley sons ltd

existence results for doubly nonlinear higher order parabolic equations on unbounded domains on demontre un resultat d existence pour le probleme aux valeurs limites et initiales u q 1 sign u t a 1 u f sur 0 t q u satisfait des conditions de dirichlet sur 0 t u u 0 sur pour t 0 a 1 u 1 m m d d u p 1 sign d u pour p q 1 m n 1 et r n un ensemble ouvert

support properties of solutions to a degenerate equation with absorption and variable density abstract in this paper we study the properties of solutions to a degenerate parabolic equation with variable density and absorption we first obtain a critical exponent which distinguishes the localization of solutions from the positivity of them when positivity prevails we obtain the other critical exponent with respect to the decay of the variable density which separates the global existence of interfaces from the disappearance of them moreover the long time behavior of interfaces is characterized

homogeneous diffusion in ℝ with power like nonlinear diffusivity we study the nonnegative solutions of the initial value problem ut ur ux p 1ux x u x 0 l1 where p 0 r p 0 the local velocity of propagation of the solutions is identified as v vx vx p 1 where v cu with r p 1 p and c r p r p 1 is the nonlinear potential our main result is the a priori estimate vx vx p 1 x frac 1 2p r t

properties of internal pilots with the univariate approach to repeated measures uncertainty surrounding the error covariance matrix often presents the biggest barrier to achieving accurate power analysis in the univariate approach to repeated measures analysis of variance unirep a poor choice gives either an overpowered study which wastes resources or an underpowered study with little chance of success internal pilot designs were introduced to resolve such uncertainty about error variance for t tests in earlier papers we extended the use of internal pilots to any univariate linear model with fixed predictors and independent gaussian errors here we further extend our exact and approximate results to unirep analysis for a fixed treatment effect the inaccuracy in a power calculation depends only on the ratio of the true variance to the value used for planning the greater complexity of repeated measures requires generalizing misspecification of error variance to the misspecification of the eigenvalues of the error covariance we recommend approximating the misspecification in terms of the first and second moments of the eigenvalues for both fixed sample and internal pilot designs we also describe an unadjusted approach for internal pilots with repeated measures simulations illustrate the fact that both positive and negative properties in the univariate setting extend to repeated measures analysis in particular internal pilots allow maintaining power or reducing expected sample size when the covariance matrix used for planning differs from the true value however an unadjusted approach can inflate test size at least with small to moderate sample sizes hence new adjusted methods must be developed for small samples at this time we caution against using an internal pilot design with repeated measures without first conducting simulations to document the amount of test size inflation possible for the conditions of interest

gravimagnetic similarity in anomaly formulas for uniform polyhedra gravitational and magnetic anomalies of an arbitrary target body are linked through poisson s differential relation for a uniform polyhedral target poisson s relation reduces to an algebraic link between gravity and magnetic anomaly formulas the derivation is given in tensor form it identifies for each target facet edge a vector function in terms of which the gravitational and magnetic potential and field anomaly formulas are similarly expressed as appropriately weighted linear combinations this similarity unifies the theory of uniform polyhedral anomalies it benefits analysis and construction of software that naturally embraces all anomalies in a single code the analysis is exemplified by a discussion of singularities and by the adaptation of three gravity field algorithms to the remaining gravitational and magnetic cases while retaining the respective computational advantages of the former gravity field algorithms

critical exponents for a fast diffusive polytropic filtration equation with nonlinear boundary flux abstract in this paper we deal with a fast diffusive polytropic filtration equation u t x u m x p 2 u m x 1 p 1 1 m in r 0 subject to a nonlinear boundary flux u m x p 2 u m x 0 t u q 0 t t 0 we first get the behavior of the solution at infinity and establish the critical global existence exponent and critical fujita exponent for the fast diffusive polytropic filtration equation furthermore give the blow up set and upper bound of the blow up rate for the nonglobal solutions

sample size estimation for gee method for comparing slopes in repeated measurements data sample size calculation is an important component at the design stage of clinical trials controlled clinical trials often use a repeated measurement design in which individuals are randomly assigned to treatment groups and followed up for measurements at intervals across a treatment period of fixed duration in studies with repeated measurements one of the popular primary interests is the comparison of the rates of change in a response variable between groups statistical models for calculating sample sizes for repeated measurement designs often fail to take into account the impact of missing data correctly in this paper we propose to use the generalized estimating equation gee method in comparing the rates of change in repeated measurements and introduce closed form formulae for sample size and power that can be calculated using a scientific calculator since the sample size formula is based on asymptotic theory we investigate the performance of the estimated sample size in practical settings through simulations

elastic and piezoelectric fields due to polyhedral inclusions abstract we derive explicit closed form expressions describing elastic and piezoelectric deformations due to polyhedral inclusions in uniform half space and bi materials our analysis is based on the linear elasticity theory and green s function method the method involves evaluation of volume and surface integrals of harmonic and bi harmonic potentials in case of polyhedra such integrals are expressed through algebraic functions our results generalize numerous studies on this subject and they allow to obtain fully analytical solutions for a number of physical and engineering problems in the limiting case of an infinite space our relations have an essentially more compact form than relations obtained by other authors we present solutions to classical mindlin and cherruti problems we describe the elastic relaxation of a misfitting polygonal quantum dot in bi materials assuming isotropic and vertically isotropic properties it is explained how to analyze non hydrostatic and non uniform inclusions we also study piezoelectric fields induced by inclusions in materials with cubic and hexagonal lattices among other results we have found that a cubic inclusion in an isotropic material reproduces fields of quantum dots in gaas 0 0 1 and gaas 1 1 1 depending on the orientation of the cube this suggests that one can qualitatively model crystals with different lattices by choosing an appropriate inclusion shape

positivity versus localization in degenerate diffusion equations on considere le probleme de cauchy u t u m f u 0 dans s r n o u x o u 0 x x r n ou m 1 n 1 et les fonctions f et u 0 satisfont les hypotheses 1 f c 1 o f o 0 2 u 0 est une fonction continue non negative definie sur r n a support borne

on solutions of some doubly nonlinear degenerate parabolic equations with absorption abstract the existence uniqueness regularity and behavior of solutions to the initial boundary value problem for the equation u t i 1 n x i u x i p 2 u x i u 0 with zero dirichlet boundary condition and with nonnegative initial condition are studied in a bounded domain in r n where 0 and p 1

approximate power for repeated measures anova lacking sphericity abstract violation of sphericity of covariance across repeated measures inflates type i error rates in univariate repeated measures analysis of variance hence the use of geisser greenhouse or huynh feldt is now common to provide improved type i error rate with nonsphericity no method has been available to compute power a convenient method is suggested for approximating power and test size under nonsphericity new approximations are suggested for a a weighted sum of independent noncentral 2 s b the trace of a noncentral wishart or pseudo wishart matrix c the expected values of and and d the noncentral test statistic whether corrected or uncorrected the new approximations are extensions of the work of box 1954a b and satterthwaite 1941 the method performed well when evaluated against published and new simulations

statistical tests with accurate size and power for balanced linear mixed models the convenience of linear mixed models for gaussian data has led to their widespread use unfortunately standard mixed model tests often have greatly inflated test size in small samples many applications with correlated outcomes in medical imaging and other fields have simple properties which do not require the generality of a mixed model alternately stating the special cases as a general linear multivariate model allows analysing them with either the univariate or multivariate approach to repeated measures unirep multirep even in small samples an appropriate unirep or multirep test always controls test size and has a good power approximation in sharp contrast to mixed model tests hence mixed model tests should never be used when one of the unirep tests uncorrected huynh feldt geisser greenhouse box conservative or multirep tests wilks hotelling lawley roy s pillai bartlett apply convenient methods give exact power for the uncorrected and box conservative tests simulations demonstrate that new power approximations for all four unirep tests eliminate most inaccuracy in existing methods in turn free software implements the approximations to give a better choice of sample size two repeated measures power analyses illustrate the methods the examples highlight the advantages of examining the entire response surface of power as a function of sample size mean differences and variability

quadratic forms in random variables theory and applications textbook for a one semester graduate course for students specializing in mathematical statistics or in multivariate analysis or reference for theoretical as well as applied statisticians confines its discussion to quadratic forms and second degree polynomials in real normal random vectors and matr

gravimetric analysis of uniform polyhedra analytical formulas for the gravity anomaly of a uniform polyhedral body are subject to numerical error that increases with distance from the target while the anomaly decreases this leads to a limited range of target distances in which the formulas are operational beyond which the calculations are dominated by rounding error we analyze the sources of error and propose a combination of numerical and analytical procedures that exhibit advantages over existing methods namely 1 errors that diminish with distance 2 enhanced operating range and 3 algorithmic simplicity the latter is achieved by avoiding the need to transform coordinates and the need to discriminate between projected observation points that lie inside on or outside a target facet boundary our error analysis is verified in computations based on a published code and on a code implementing our methods the former requires a numerical precision of one part in 1016 double precision in problems of geophysical interest whereas our c

adjusting power for a baseline covariate in linear models the analysis of covariance provides a common approach to adjusting for a baseline covariate in medical research with gaussian errors adding random covariates does not change either the theory or the computations of general linear model data analysis however adding random covariates does change the theory and computation of power analysis many data analysts fail to fully account for this complication in planning a study we present our results in five parts i a review of published results helps document the importance of the problem and the limitations of available methods ii a taxonomy for general linear multivariate models and hypotheses allows identifying a particular problem iii we describe how random covariates introduce the need to consider quantiles and conditional values of power iv we provide new exact and approximate methods for power analysis of a range of multivariate models with a gaussian baseline covariate for both small and large samples the new results apply to the hotelling lawley test and the four tests in the univariate approach to repeated measures unadjusted huynh feldt geisser greenhouse box the techniques allow rapid calculation and an interactive graphical approach to sample size choice v calculating power for a clinical trial of a treatment for increasing bone density illustrates the new methods we particularly recommend using quantile power with a new satterthwaite style approximation

exact test size and power of a gaussian error linear model for an internal pilot study wittes and brittain recommended using an internal pilot study to adjust sample size the approach involves five steps in testing a general linear hypothesis for a general linear univariate model with gaussian errors first specify the design hypothesis desired test size power a smallest clinically meaningful effect and a speculated error variance second conduct a power analysis to choose provisionally a planned sample size third collect a specified proportion of the planned sample as the internal pilot sample and estimate the variance but do not test the hypothesis fourth update the power analysis with the variance estimate to adjust the total sample size fifth finish the study and test the hypothesis with all data we describe methods for computing exact test size and power under this scenario our analytic results agree with simulations of wittes and brittain furthermore our exact results apply to any general linear univariate model with fixed predictors which is much more general than the two sample t test considered by wittes and brittain in addition our results allow for examination of the impact on test size of internal pilot studies for more complicated designs in the framework of the general linear model we examine the impact of i small samples ii allowing the planned sample size to decrease iii the choice of internal pilot sample size and iv the maximum allowable size of the second sample all affect test size power and expected total sample size we present a number of examples including one that uses an internal pilot study in a three group analysis of variance

on blow up of solution of nonhomogeneous polytropic equation with source abstract in this paper we discuss the blow up property of the positive solutions of the mixed problem for a one dimensional polytropic filtration equation with source

power calculation for overall hypothesis testing with high dimensional commensurate outcomes the complexity of system biology means that any metabolic genetic or proteomic pathway typically includes so many components e g molecules that statistical methods specialized for overall testing of high dimensional and commensurate outcomes are required while many overall tests have been proposed very few have power and sample size methods we develop accurate power and sample size methods and software to facilitate study planning for high dimensional pathway analysis with an account of any complex correlation structure between high dimensional outcomes the new methods allow power calculation even when the sample size is less than the number of variables we derive the exact finite sample and approximate non null distributions of the univariate approach to repeated measures test statistic as well as power equivalent scenarios useful to generalize our numerical evaluations extensive simulations of group comparisons support the accuracy of the approximations even when the ratio of number of variables to sample size is large we derive a minimum set of constants and parameters sufficient and practical for power calculation using the new methods and specifying the minimum set to determine power for a study of metabolic consequences of vitamin b6 deficiency helps illustrate the practical value of the new results free software implementing the power and sample size methods applies to a wide range of designs including one group pre intervention and post intervention comparisons multiple parallel group comparisons with one way or factorial designs and the adjustment and evaluation of covariate effects