limited range multilinear extrapolation with applications to the bilinear hilbert transform we prove a limited range off diagonal extrapolation theorem that generalizes a number of results in the theory of rubio de francia extrapolation and use this to prove a limited range multilinear extrapolation theorem we give two applications of this result to the bilinear hilbert transform first we give sufficient conditions on a pair of weights w 1 w 2 for the bilinear hilbert transform to satisfy weighted norm inequalities of the form begin aligned bh l p 1 left w 1 p 1 right times l p 2 left w 2 p 2 right longrightarrow l p w p end aligned where w w 1w 2 and frac 1 p frac 1 p 1 frac 1 p 2 frac 3 2 this improves the recent results of culiuc et al by increasing the families of weights for which this inequality holds and by pushing the lower bound on p from 1 down to frac 2 3 the critical index from the unweighted theory of the bilinear hilbert transform second as an easy consequence of our method we obtain that the bilinear hilbert transform satisfies some vector valued inequalities with muckenhoupt weights this reproves and generalizes some of the vector valued estimates obtained by benea and muscalu in the unweighted case we also generalize recent results of carando et al on marcinkiewicz zygmund estimates for multilinear calderon zygmund operators

characteristic cycles of perverse sheaves and milnor fibers we show that the milnor monodromy and the milnor numbers naturally appear in the characteristic cycles of irreducible perverse sheaves having the singularities of a complex hypersurface we find also a new numerical constraint on the milnor monodromy for general hypersurface singularities

multilinear calderon zygmund theory abstract a systematic treatment of multilinear calderon zygmund operators is presented the theory developed includes strong type and endpoint weak type estimates interpolation a multilinear t 1 theorem and a variety of results regarding multilinear multiplier operators

the asymptotic properties of nonparametric tests for comparing survival distributions the asymptotic distribution under alternative hypotheses is derived for a class of statistics used to test the equality of two survival distributions in the presence of arbitrary and possibly unequal right censoring the test statistics include equivalents to the log rank statistic the modified wilcoxon statistic and the class of rank invariant test procedures introduced by peto peto when there are equal censoring distributions and the hazard functions are proportional the sample size formula for the f test used to compare exponential samples is shown to be valid for the log rank test in certain situations the power of the log rank test falls as the amount of censoring decreases

sample size formula for proportional hazards modelling of competing risks to test the effect of a therapeutic or prognostic factor on the occurrence of a particular cause of failure in the presence of other causes the interest has shifted in some studies from the modelling of the cause specific hazard to that of the subdistribution hazard we present approximate sample size formulas for the proportional hazards modelling of competing risk subdistribution considering either independent or correlated covariates the validity of these approximate formulas is investigated through numerical simulations two illustrations are provided a randomized clinical trial and a prospective prognostic study

minimum sample size for developing a multivariable prediction model part ii binary and time to event outcomes when designing a study to develop a new prediction model with binary or time to event outcomes researchers should ensure their sample size is adequate in terms of the number of participants n and outcome events e relative to the number of predictor parameters p considered for inclusion we propose that the minimum values of n and e and subsequently the minimum number of events per predictor parameter epp should be calculated to meet the following three criteria i small optimism in predictor effect estimates as defined by a global shrinkage factor of 0 9 ii small absolute difference of 0 05 in the model s apparent and adjusted nagelkerke s r2 and iii precise estimation of the overall risk in the population criteria i and ii aim to reduce overfitting conditional on a chosen p and require prespecification of the model s anticipated cox snell r2 which we show can be obtained from previous studies the values of n and e that meet all three criteria provides the minimum sample size required for model development upon application of our approach a new diagnostic model for chagas disease requires an epp of at least 4 8 and a new prognostic model for recurrent venous thromboembolism requires an epp of at least 23 this reinforces why rules of thumb eg 10 epp should be avoided researchers might additionally ensure the sample size gives precise estimates of key predictor effects this is especially important when key categorical predictors have few events in some categories as this may substantially increase the numbers required

on calderon s conjecture with constants cfi p1 p2 depending only on fi p1 p2 and p p1p2 p1 p2 hold the flrst result of this type is proved in 4 and the purpose of the current paper is to extend the range of exponents p1 and p2 for which 2 is known in particular the case p1 2 p2 1 is solved to the a rmative this was

operators of harmonic analysis in weighted spaces with non standard growth last years there was increasing an interest to the so called function spaces with non standard growth known also as variable exponent lebesgue spaces for weighted such spaces on homogeneous spaces we develop a certain variant of rubio de francia s extrapolation theorem this extrapolation theorem is applied to obtain the boundedness in such spaces of various operators of harmonic analysis such as maximal and singular operators potential operators fourier multipliers dominants of partial sums of trigonometric fourier series and others in weighted lebesgue spaces with variable exponent there are also given their vector valued analogues

on a hyperplane section theorem of gurjar a hyperplane section theorem by r v gurjar 3 is given a short proof power series in any number at least three of variables satsifying the condition of the theorem are explicitly constructed in the course of the proof the restrictive looking condition of the theorem is given an easy sufficient condition from the view point of the weighted projective spaces

extrapolation from a weights and applications abstract we generalize the a p extrapolation theorem of rubio de francia to a weights in the context of muckenhoupt bases our result has several important features first it can be used to prove weak endpoint inequalities starting from strong type inequalities something which is impossible using the classical result second it provides an alternative to the technique of good inequalities for proving l p norm inequalities relating operators third it yields vector valued inequalities without having to use the theory of banach space valued operators we give a number of applications to maximal functions singular integrals potential operators commutators multilinear calderon zygmund operators and multiparameter fractional integrals in particular we give new proofs which completely avoid the good inequalities of coifman s inequality relating singular integrals and the maximal operator of the fefferman stein inequality relating the maximal operator and the sharp maximal operator and the muckenhoupt wheeden inequality relating the fractional integral operator and the fractional maximal operator

uncertainty principle estimates for vector fields abstract we derive weighted norm estimates for integral operators of potential type and for their related maximal operators these operators are generalizations of the classical fractional integrals and fractional maximal functions the norm estimates are derived in the context of a space of homogeneous type the conditions required of the weight functions involve generalizations of the fefferman phong r bump condition the results improve some earlier ones of the same kind and they also extend to homogeneous spaces some estimates that were previously known to hold only in the classical euclidean setting

asymptotic relative efficiency of survival tests with covariates summary comparison of survival for two groups of patients when adjusting for covariates is a standard problem in clinical trials in this paper the general life table models developed by cox 1972 and the stratified log rank test mantel haenszel 1959 peto peto 1972 are compared an analytical expression and numerical results for the asymptotic relative efficiency of the two tests are given when there is no censoring stratification on the covariates is asymptotically as efficient as the test arising from cox s model provided i there is no treatment effect ii the covariates are balanced across treatment groups and iii the hypotheses underlying cox s model are satisfied however if the proportional hazard model does not hold for the covariates then cox s model leads to a biased estimate of the difference between the two treatments

lack of natural weighted estimates for some singular integral operators we show that the classical hormander condition or analogously the l r hormander condition for singular integral operators t is not sufficient to derive coifman s inequality rn tf x p w x dx c rn mf x p w x dx where 0 p m is the hardy littlewood maximal operator w is any a weight and c is a constant depending upon p and the a constant of w this estimate is well known to hold when t is a calderon zygmlmd operator as a consequence we deduce that the following estimate does not hold rn t f x p w x dx c rn mf x p mw x dx where 0 p 1 and where w is an arbitrary weight however by a recent result due to a lerner this inequality is satisfied whenever t is a calderon zygmund operator one of the main ingredients of the proof is a very general extrapolation theorem for a weights

the bilinear hilbert transform in umd spaces we prove l p bounds for the bilinear hilbert transform acting on functions valued in intermediate umd spaces such bounds were previously unknown for umd spaces that are not banach lattices our proof relies on bounds on embeddings from bochner spaces l p mathbb r x into outer lebesgue spaces on the time frequency scale space mathbb r 3

sample size considerations for the evaluation of prognostic factors in survival analysis when the role of a new prognostic factor is investigated careful planning of an appropriate study is required this includes an assessment of the power of the study in terms of sample sizes an adequate analysis of the independent prognostic effect of a new factor has to be adjusted for the existing standard factors with survival time as endpoint this will usually be done with the cox proportional hazards model sample size and power formulae in survival analysis have been developed by schoenfeld for randomized treatment comparisons in the analysis of prognostic factors the covariates included are expected to be correlated with the factor of primary interest in this situation the existing sample size and power formulae are not valid and may not be applied in this paper schoenfeld s formula is first extended to the situation where a correlated factor is included in the analysis the validity of the resulting approximate asymptotic formula is investigated for its asymptotic behaviour by numerical integration and for its finite behaviour by simulation second an approximate formula for sample size and power is provided to detect an interaction between the interesting and a second correlated factor this extends the formula for independent effects finally the approach is illustrated by an example on the prognostic impact of dna ploidy and other factors in advanced ovarian cancer

extrapolation of weights revisited new proofs and sharp bounds abstract we use an appropriate factorization of the a p weights to give another proof of the extrapolation theorem of rubio de francia it provides sharp bounds in terms of the a p constant of the weights then we extend the result to more general settings including off diagonal and partial range extrapolation among the applications we prove by iteration a multivariable extrapolation theorem and give a sharp bound for calderon zygmund operators on l p w for weights in a q q p

a simple method of sample size calculation for linear and logistic regression a sample size calculation for logistic regression involves complicated formulae this paper suggests use of sample size formulae for comparing means or for comparing proportions in order to calculate the required sample size for a simple logistic regression model one can then adjust the required sample size for a multiple logistic regression model by a variance inflation factor this method requires no assumption of low response probability in the logistic model as in a previous publication one can similarly calculate the sample size for linear regression models this paper also compares the accuracy of some existing sample size software for logistic regression with computer power simulations an example illustrates the methods

the power of testing a semi parametric shared gamma frailty parameter in failure time data frailty is usually modeled as an unobserved random variable acting multiplicatively on the baseline hazard function where a shared unobserved quantity in the intensity induces a positive correlation among the observed failure times using the asymptotic properties of the nonparametric maximum likelihood estimator for the gamma frailty model we derive a power function for testing the shared frailty parameter and evaluate the effect of number of groups and number of individuals per groups on the power of the test by simulation studies the results show that choosing between 8 and 25 groups with sample sizes between 200 and 500 individuals will be enough to get a high power

sample size tables for logistic regression sample size tables are presented for epidemiologic studies which extend the use of whittemore s formula the tables are easy to use for both simple and multiple logistic regressions monte carlo simulations are performed which show three important results firstly the sample size tables are suitable for studies with either high or low event proportions secondly although the tables can be inaccurate for risk factors having double exponential distributions they are reasonably adequate for normal distributions and exponential distributions finally the power of a study varies both with the number of events and the number of individuals at risk

power sample size calculations for generalized linear models an approach for estimating power sample size is described within the framework of generalized linear models this approach is based on an asymptotic approximation to the power of the score test under contiguous alternatives and is applicable to tests of composite null hypotheses an implementation is described for the special case of logistic regression models simulation studies are presented which indicate that the asymptotic approximation to the finite sample situation is good over a range of parameter configurations

sample size formula for the proportional hazards regression model a formula is derived for determining the number of observations necessary to test the equality of two survival distributions when concomitant information is incorporated this formula should be useful in designing clinical trials with a heterogeneous patient population schoenfeld 1981 biometrika 68 316 319 derived the asymptotic power of a class of statistics used to test the equality of two survival distributions that result is extended to the case where concomitant information is available for each individual and where the proportional hazards model holds the loss of efficiency caused by ignoring concomitant variables is also computed

quasi banach valued inequalities via the helicoidal method abstract we extend the helicoidal method from 1 to the quasi banach context proving in this way multiple banach and quasi banach vector valued inequalities for paraproducts and for the bilinear hilbert transform bht as an immediate application we obtain mixed norm estimates for in the whole range of lebesgue exponents one of the novelties in the quasi banach framework that is when 0 r 1 which we expect to be useful in other contexts as well is the linearization of the operator k t f k g k r 1 r achieved by dualizing its weak l p quasinorms through l r see proposition 8 another important role is played by the sharp evaluation of the operatorial norm t i 0 f 1 f g 1 g 1 h r which is obtained by dualizing the weak l p quasinorms through l with r in the banach case the linearization of the operator and the sharp estimates for the localized operatorial norm can be both achieved through the classical generalized restricted type l 1 dualization

power analysis for multivariable cox regression models in power analysis for multivariable cox regression models variance of the estimated log hazard ratio for the treatment effect is usually approximated by inverting the expected null information matrix because in many typical power analysis settings assumed true values of the hazard ratios are not necessarily close to unity the accuracy of this approximation is not theoretically guaranteed to address this problem the null variance expression in power calculations can be replaced with one of the alternative expressions derived under the assumed true value of the hazard ratio for the treatment effect this approach is explored analytically and by simulations in the present paper we consider several alternative variance expressions and compare their performance to that of the traditional null variance expression theoretical analysis and simulations demonstrate that whereas the null variance expression performs well in many nonnull settings it can also be very inaccurate substantially underestimating or overestimating the true variance in a wide range of realistic scenarios particularly those where the numbers of treated and control subjects are very different and the true hazard ratio is not close to one the alternative variance expressions have much better theoretical properties confirmed in simulations the most accurate of these expressions has a relatively simple form it is the sum of inverse expected event counts under treatment and under control scaled up by a variance inflation factor

power and sample size determination for group comparison of patient reported outcomes using polytomous rasch models the analysis of patient reported outcomes or other psychological traits can be realized using the rasch measurement model when the objective of a study is to compare groups of individuals it is important before the study to define a sample size such that the group comparison test will attain a given power the raschpower procedure rp allows doing so with dichotomous items the rp is extended to polytomous items several computational issues were identified and adaptations have been proposed the performance of this new version of rp is assessed using simulations this adaptation of rp allows obtaining a good estimate of the expected power of a test to compare groups of patients in a large number of practical situations a stata module as well as its implementation online is proposed to perform the rp two versions of the rp for polytomous items are proposed deterministic and stochastic versions these two versions produce similar results in all of the tested cases we recommend the use of the deterministic version when the measure is obtained using small questionnaires or items with a few number of response categories and the stochastic version elsewhere so as to optimize computing time

extrapolation for multilinear muckenhoupt classes and applications abstract in this paper we solve a long standing problem about the multivariable rubio de francia extrapolation theorem for the multilinear muckenhoupt classes a p which were extensively studied by lerner et al and which are the natural ones for the class of multilinear calderon zygmund operators as well as for some bilinear rough singular integral operators furthermore we go beyond the classes a p and extrapolate within the classes a p r which appear naturally associated to the weighted norm inequalities for multilinear sparse forms which control fundamental operators such as the bilinear hilbert transform we give several applications which can be easily obtained using extrapolation first weighted norm inequalities scalar and vector valued for the bilinear rough singular integral operators are established second for the bilinear hilbert transform one can extrapolate from the recent result of culiuc et al who considered the banach range extend the estimates to the quasi banach range and furthermore prove trivially vector valued inequalities we also extend recent results of carando et al on marcinkiewicz zygmund estimates for multilinear calderon zygmund operators finally our last application gives new weighted estimates scalar and vector valued for the commutators of multilinear calderon zygmund operators bilinear rough singular integral operators and for the bilinear hilbert transform with bmo functions using ideas from benyi et al

sample size calculations for evaluating mediation this paper presents sample size calculations for testing the mediation of the effect of a primary predictor by an intermediate variable this problem is related to validating surrogate markers and to testing the effect of a primary predictor in the presence of confounders for those problems proposals for sample size calculation exist in the literature and can be adapted to the problem of mediation methods based on the variance inflation factor in linear regression provide exact sample size calculations for the linear model and approximations for the logistic poisson and cox models we propose another procedure based on simulation of the underlying data structure with applications to the logistic and cox models for the poisson model a new analytic method is also proposed the behavior of the different proposals is investigated by means of simulation studies copyright 2008 john wiley sons ltd

sample size for logistic regression with small response probability abstract the fisher information matrix for the estimated parameters in a multiple logistic regression can be approximated by the augmented hessian matrix of the moment generating function for the covariates the approximation is valid when the probability of response is small with its use one can obtain a simple closed form estimate of the asymptotic covariance matrix of the maximum likelihood parameter estimates and thus approximate sample sizes needed to test hypotheses about the parameters the method is developed for selected distributions of a single covariate and for a class of exponential type distributions of several covariates it is illustrated with an example concerning risk factors for coronary heart disease