path integral solution of a class of potentials related to the poschl teller potential some new exact path integral treatments are presented these are the path integrals for the symmetric top the smooth step the rosen morse and the manning rosen potential by appropriate spacetime transformations these path integral problems can be reduced to the path integral problem of the poschl teller potential

multiplicity considerations in subgroup analysis this paper deals with the general topic of subgroup analysis in late stage clinical trials with emphasis on multiplicity considerations the discussion begins with multiplicity issues arising in the context of exploratory subgroup analysis including principled approaches to subgroup search that are applied as part of subgroup exploration exercises as well as in adaptive biomarker driven designs key considerations in confirmatory subgroup analysis based on one or more pre specified patient populations are reviewed including a survey of multiplicity adjustment methods recommended in multi population phase iii clinical trials guidelines for interpretation of significant findings in several patient populations are introduced to facilitate the decision making process and achieve consistent labeling across development programs

subgroup identification based on differential effect search a recursive partitioning method for establishing response to treatment in patient subpopulations we propose a novel recursive partitioning method for identifying subgroups of subjects with enhanced treatment effects based on a differential effect search algorithm the idea is to build a collection of subgroups by recursively partitioning a database into two subgroups at each parent group such that the treatment effect within one of the two subgroups is maximized compared with the other subgroup the process of data splitting continues until a predefined stopping condition has been satisfied the method is similar to interaction tree approaches that allow incorporation of a treatment by split interaction in the splitting criterion however unlike other tree based methods this method searches only within specific regions of the covariate space and generates multiple subgroups of potential interest we develop this method and provide guidance on key topics of interest that include generating multiple promising subgroups using different splitting criteria choosing optimal values of complexity parameters via cross validation and addressing type i error rate inflation inherent in data mining applications using a resampling based method we evaluate the operating characteristics of the procedure using a simulation study and illustrate the method with a clinical trial example copyright 2011 john wiley sons ltd

a consistency adjusted alpha adaptive strategy for sequential testing in a clinical trial with two clinically important endpoints each of which can fully characterize a treatment benefit to support an efficacy claim by itself a minimum degree of consistency in the findings is expected otherwise interpretation of study findings can be problematic clinical trial literature contains examples where lack of consistency in the findings of clinically relevant endpoints led to difficulties in interpreting study results the aim of this paper is to introduce this consistency concept at the study design stage and investigate the consequences of its implementation in the statistical analysis plan the proposed methodology allows testing of hierarchically ordered endpoints to proceed as long as a pre specified consistency criterion is met in addition while an initial allocation of the alpha level is specified for the ordered endpoints at the design stage the methodology allows the alpha level allocated to the second endpoint to be adaptive to the findings of the first endpoint in addition the methodology takes into account the correlation between the endpoints in calculating the significance level and the power of the test for the next endpoint the proposed consistency adjusted alpha adaptive strategy caaas is very general several of the well known multiplicity adjustment approaches arise as special cases of this strategy by appropriate selection of the consistency level and the form of alpha adaptation function we discuss control of the type i error rate as well as power of the proposed methodology and consider its application to clinical trial data

group sequential enrichment design incorporating subgroup selection an important component of clinical trials in drug development is the analysis of treatment efficacy in patient subgroups subpopulations because of concerns of multiplicity and of the small sample sizes often involved such analyses can present substantial statistical challenges and may lead to misleading conclusions as a confirmatory seamless phase ii iii design we propose an adaptive enrichment group sequential procedure whereby resources can be concentrated on subgroups most likely to respond to treatment stopping boundaries are defined through upper and lower spending functions the procedure is presented in terms of the efficient score enabling the analysis of normal binary or time to event data it addresses the dilution effect by eliminating populations at the first stage that appear likely to derive no therapeutic benefit it subsequently proceeds with the definitive assessment of treatment efficacy among the remaining pooled populations using a group sequential design the procedure provides strong protection of familywise type i error rate and we employ a bootstrap algorithm to obtain point and interval estimates that are adjusted for the selection bias we give examples to demonstrate how the design is used we make comparisons with adaptive two stage combination test procedures and with a group sequential test that does not account for the presence of subgroups numerical results show that the procedure has high power to detect subgroup specific effects and the use of multiple interim analysis points can lead to substantial sample size savings

a flexible strategy for testing subgroups and overall population subgroup analyses in addition to the total study population analysis are common in clinical trials however it is well recognized that findings from subgroup analyses do not provide confirmatory evidence for subgroup treatment effects without placing a priori criteria for ensuring that their findings are scientifically sound in this paper we address some of the common pitfalls of subgroup analyses subgroups analyses inherently have low power for detecting treatment effects we investigate the power interplay for a subgroup analysis and that for the total study population and list factors that impact the power of a subgroup analysis then we introduce a flexible statistical strategy for testing a pre specified sequence of hypotheses for both the overall and a subgroup the proposed method strongly controls the familywise type i error rate and enjoys higher power than other traditional methods this testing strategy allows testing for a subgroup once a pre specified degree of consistency in the efficacy findings between the subgroup and the overall study population is met in addition it accounts for the dependency between test statistics for the subgroup and the overall study population we discuss the power performance of this new method and provide significance levels for subgroup analysis finally we illustrate its application through retrospective analysis of data from three published clinical trials

matrix elements for powers of x dependent operators for the hyperbolic poschl teller potentials analytical matrix elements of the position operator xn n 0 and of the first and second order differential operators are derived using the eigenfunctions of two hyperbolic poschl teller potentials these general relations are written in closed form and calculated from the properties of the integral powers of hypergeometric series additionally an explicit expression is derived for the successive derivatives of the beta function convergence and reliability of these results will be discussed and an application to the calculation of the rovibrational energy spectrum of co will be considered

testing each hypothesis marginally at alpha while still controlling fwer how and when this paper proposes a multiple testing procedure that allows one to reject each individual hypothesis at a prespecified level while still controlling the familywise error rate at in the strong sense typically rejecting a hypothesis when its marginal p value is in a multiple hypothesis testing setting will lead to an inflation of familywise error rate however this inflation can be avoided if a particular consistency criterion is prespecified and incorporated in the testing algorithm the criterion is equivalent to requiring that all p values be smaller than or equal to a particular threshold in the one sided hypothesis testing setting extensions to the two sided hypothesis testing setting and extensions to situations where the criterion can be chosen per user s preference are also presented copyright 2012 john wiley sons ltd

key multiplicity issues in clinical drug development much progress has been made over the past decade with the development of novel methods for addressing increasingly more complex multiplicity problems arising in confirmatory phase iii clinical trials this includes traditional problems with a single source of multiplicity for example analysis of multiple endpoints or dose placebo contrasts in addition more advanced problems with several sources of multiplicity have attracted attention in clinical drug development these problems include two or more families of objectives such as multiple endpoints evaluated at multiple dose levels or in multiple patient populations this paper provides a review of concepts that play a central role in defining and solving multiplicity problems error rate definitions and introduces main classes of multiple testing procedures widely used in clinical trials nonparametric semiparametric and parametric procedures the paper also presents recent advances in multiplicity research including gatekeeping procedures for clinical trials with multiple sets of objectives the concepts and methods introduced in the paper are illustrated using several case studies on the basis of real clinical trials software implementation of commonly used multiple testing and gatekeeping procedures is discussed copyright 2012 john wiley sons ltd

an sl 4 r lie algebraic approach to the bargmann functions and its application to the second poschl teller equation the sl 4 r lie algebraic treatment of the winger su 2 matrices 1959 is extended by analytical continuation to the bargmann su 1 1 matrices 1947 corresponding to the positive discrete series irreducible representations it is then used to obtain an sl 4 r dynamical potential algebra for the negative energy solutions of the second poschl teller equations 1933

three dimensional hierarchical finite element free vibration analysis of annular sector plates an annular sector solid hierarchical finite element is presented and applied to three dimensional free vibration analysis of annular sector plates the element s displacements are expressed in terms of a fixed number of linear polynomial shape functions plus a variable number of shape functions which are forms of the shifted legendre orthogonal polynomial the linear polynomial shape functions are used to describe the element s nodal displacements and the higher order shape functions are used to provide additional freedom to the edges faces and interior of the element results of frequency calculations are found for annular sector plates with two straight edges simply supported like a diaphragm condition and comparisons are made with those obtained by the finite prism method the manner of convergence of the solution as a function of the numbers of hierarchical modes is also investigated furthermore contributions to original three dimensional frequencies are made for annular sector plates with other boundary conditions along the two straight edges

algebraic treatment of second poschl teller morse rosen and eckart equations the method of an earlier paper see ibid vol 20 p 4075 1987 is applied to the non compact case to solve a family of second poschl teller morse rosen and eckart equations with quantised coupling constants both discrete and continuous spectra bound state and scattering wavefunctions transmission coefficients are found from the matrix elements of group representations

lie algebras for systems with mixed spectra i the scattering poschl teller potential starting from an n body quantum space we consider the lie algebraic framework where the poschl teller hamiltonian 1 2 2 c sech2 s csch2 is the single sp 2 r casimir operator the spectrum of this system is mixed it contains a finite number of negative energy bound states and a positive energy continuum of free states it is identified with the clebsch gordan series of the d d representation coupling the wave functions are the sp 2 r clebsch gordan coefficients of that coupling in the parabolic basis using only lie algebraic techniques we find the asymptotic behavior of these wave functions for the special pure trough potential s 0 we derive thus the transmission and reflection amplitudes of the scattering matrix

solvable potentials associated with su 1 1 algebras a systematic study we consider a specific differential realization of the su 1 1 algebra and use it to explore such algebraic structures associated with shape invariant potentials our approach combines elements of various methods of solving the schrodinger equation such as supersymmetric quantum mechanics or the factorization method algebraic techniques and special function theory in fact it amounts to reformulating transformations mapping the schrodinger equation into the differential equation of orthogonal polynomials in group theoretical terms our systematic study recovers a number of earlier results in a natural unified way and also leads to new findings the procedure presented here implicitly contains a similar treatment of the compact su 2 algebra as well possible generalizations of this approach involving different realizations of the su 1 1 algebra other algebraic structures and larger classes of potentials are also outlined

the potential group approach and hypergeometric differential equations this paper proposes a generalized realization of the potential groups so 2 1 and so 2 2 to describe the confluent hypergeometric and the hypergeometric equations respectively it implies that the classes of schrodinger equations with solvable potentials whose analytical solutions are related to the confluent hypergeometric and the hypergeometric functions can be realized in terms of the above group structure

traditional multiplicity adjustment methods in clinical trials this tutorial discusses important statistical problems arising in clinical trials with multiple clinical objectives based on different clinical variables evaluation of several doses or regiments of a new treatment analysis of multiple patient subgroups etc simultaneous assessment of several objectives in a single trial gives rise to multiplicity if unaddressed problems of multiplicity can undermine integrity of statistical inferences the tutorial reviews key concepts in multiple hypothesis testing and introduces main classes of methods for addressing multiplicity in a clinical trial setting general guidelines for the development of relevant and efficient multiple testing procedures are presented on the basis of application specific clinical and statistical information case studies with common multiplicity problems are used to motivate and illustrate the statistical methods presented in the tutorial and software implementation of the multiplicity adjustment methods is discussed copyright 2013 john wiley sons ltd

validity of the hochberg procedure revisited for clinical trial applications there is much interest in using the hochberg procedure hp for statistical tests on primary endpoints of confirmatory clinical trials the procedure is simple to use and enjoys more power than the bonferroni and the holm procedures however the hp is not assumption free like the other two procedures it controls the familywise type i error rate when test statistics used for statistical tests are independent or if dependent satisfy a conditionally independent formulation otherwise its properties for dependent tests at present are not fully understood consequently its use for confirmatory trials especially for their primary endpoints remains worrisome confirmatory trials are typically designed with 1 2 primary endpoints therefore a question was raised at the food and drug administration as to whether the hp is a valid test for the simple case of performing treatment to control comparisons on two primary endpoints when their test statistics are not independent confirmatory trials for statistical tests normally use simple test statistics such as the normal z student s t and chi square the literature does include some work on the hp for dependent cases covering these test statistics but concerns remain regarding its use for confirmatory trials for which endpoint tests are mostly of the dependent kind the purpose of this paper is therefore to revisit this procedure and provide sufficient details for better understanding of its performance for dependent cases related to the aforementioned question published 2015 this article is a u s government work and is in the public domain in the usa

inconsistencies in the description of a quantum system with a finite number of bound states by a compact dynamical group in a recent paper aldaya v and guerrero j 2005 j phys a math gen 38 6939 it is shown that the bound states of the modified poschl teller potential should be described by the non compact dynamical group su 1 1 instead of the usual compact group su 2 here we prove that su 2 cannot be the dynamical group for a potential with a finite number of bound states on the basis of the modified poschl teller potential and the morse potential this suggests that a quantum system with a continuum spectrum and a finite number of bound states should be described both in the continuum and the discrete spectrum by a non compact dynamical group instead of a compact one

extension of adaptive alpha allocation methods for strong control of the family wise error rate with recent advancements in clinical trial design and the availability of rigorous statistical methods that provide strong control of the family wise type i error rate for multiple testing of hypotheses it is now common for sponsors to design clinical trials with prospectively specified multiple testing of hypotheses of both primary and secondary endpoints and with the intent to obtain labeling claims for secondary endpoints one of these recent advancements in multiple testing techniques is the adaptive alpha allocation approach 4a proposed by li and mehrotra statistics in medicine 2008 27 5377 5391 which groups the hypotheses into two families on the basis of perceived trial power and allows the significance level for the second family to be set adaptively on the basis of the largest observed p value in the first family we introduce a class of flexible functions that generalize the 4a procedure and can lead to relatively more powerful test statistics in the case when the test statistics are correlated we introduce well defined functions to calculate the significance level for the second family the numerical computation for our methods is straightforward making application in practice easy copyright 2012 john wiley sons ltd

matrix elements of u and p for the modified poschl teller potential closed analytical expressions in terms of a single sum are obtained for the matrix elements of the momentum and the natural variable u tanh x in the basis of the modified poschl teller mpt bound eigenstates these matrix elements are first expressed in terms of franck condon factors which thereafter are substituted for analytic expressions expansions of the variables p and u in terms of creation and annihilation operators associated with the mpt bound eigenfunctions are also presented