interim analysis on survival data its potential bias and how to repair it we consider interim analyses in clinical trials or observational studies with a time to event outcome variable where the survival curves are compared using the hazard ratio resulting from a proportional hazards ph model or tested with the logrank test or another two sample test we show and illustrate with an example that if the ph assumption is violated the results of interim analyses can be heavily biased this is due to the fact that the censoring pattern in interim analyses can be completely different from the final analysis we argue that when the ph assumption is violated interim analyses are only sensible if a fixed time horizon for the final analysis is specified and at the time of the interim analysis sufficient information is available over the whole time interval up to the horizon we show how the bias can then be remedied by introducing in the estimation and testing procedures an appropriate weighting that reflects the weights to be expected in the final analysis the consequences for design and analysis are discussed and some practical recommendations are given copyright 2005 john wiley sons ltd

towards possible q generalizations of the malthus and verhulst growth models abstract tsallis entropy introduced in 1988 based on the equation s q 1 i p i q q 1 has been considered to obtain new possibilities for constructing a generalized thermodynamical basis for statistical physics expanding classical boltzmann gibbs thermodynamics for non equilibrium states during the last two decades this q generalized theory has been successfully applied to a considerable amount of physically interesting complex phenomena the authors wish to present a mathematical analysis of the possible q generalizations of the malthus and the verhulst growth models based on tsallis definitions of the q logarithm and the q exponential

nonparametric and semiparametric group sequential methods for comparing accuracy of diagnostic tests summary comparison of the accuracy of two diagnostic tests using the receiver operating characteristic roc curves from two diagnostic tests has been typically conducted using fixed sample designs on the other hand the human experimentation inherent in a comparison of diagnostic modalities argues for periodic monitoring of the accruing data to address many issues related to the ethics and efficiency of the medical study to date very little research has been done on the use of sequential sampling plans for comparative roc studies even when these studies may use expensive and unsafe diagnostic procedures in this article we propose a nonparametric group sequential design plan the nonparametric sequential method adapts a nonparametric family of weighted area under the roc curve statistics wieand et al 1989 biometrika 76 585 592 and a group sequential sampling plan we illustrate the implementation of this nonparametric approach for sequentially comparing roc curves in the context of diagnostic screening for nonsmall cell lung cancer we also describe a semiparametric sequential method based on proportional hazard models we compare the statistical properties of the nonparametric approach with alternative semiparametric and parametric analyses in simulation studies the results show the nonparametric approach is robust to model misspecification and has excellent finite sample performance

group sequential methods for an ordinal logistic random effects model under misspecification interim analyses in clinical trials are planned for ethical as well as economic reasons general results have been published in the literature that allow the use of standard group sequential methodology if one uses an efficient test statistic e g when wald type statistics are used in random effects models for ordinal longitudinal data these models often assume that the random effects are normally distributed however this is not always the case we will show that when the random effects distribution is misspecified in ordinal regression models the joint distribution of the test statistics over the different interim analyses is still a multivariate normal distribution but a sandwich type correction to the covariance matrix is needed in order to obtain the correct covariance matrix the independent increment structure is also investigated a bias in estimation will occur due to the misspecification however we will also show that the treatment effect estimate will be unbiased under the null hypothesis thus maintaining the type i error extensive simulations based on a toenail dermatophyte onychomycosis trial are used to illustrate our results

an approach for deriving growth equations for quantities exhibiting cumulative growth based on stochastic interpretation abstract existing growth equations have been derived by hypothesizing the growth rate as self referencing although that hypothesis is appropriate for quantities with growth directly restricted to themselves it requires the solution of differential equations that are difficult to be derived however for quantities of growth that are indirectly related to themselves and for quantities that exhibit cumulative growth such as growth of diameter or volume of trees with lignification ordinary growth models with assumptions that include self referencing or implied catabolism terms might not be appropriate for such quantities the author proposes an approach for derivation of growth models based on stochastic and microscopic interpretation results show that ordinary growth models can be interpreted from the perspective the approach enables one to derive growth functions without solving differential equations based on that approach growth functions are derived by integrating cumulative distribution functions three growth functions are generated using reasonable probability distributions the fitness of the generated growth functions for real diameter growth data was compared with that of generalized ordinary growth models results show that the presented approach has high ability to generate growth models that fit data much better than ordinary growth models for a given number of parameters

construction of a continuous stopping boundary from an alpha spending function lan and demets 1983 biometrika 70 659 663 proposed a flexible method for monitoring accumulating data that does not require the number and times of analyses to be specified in advance yet maintains an overall type i error alpha their method amounts to discretizing a preselected continuous boundary by clumping the density of the boundary crossing time at discrete analysis times and calculating the resultant discrete time boundary values in this framework the cumulative distribution function of the continuous time stopping rule is used as an alpha spending function a key assumption that underlies this method is that future analysis times are not chosen on the basis of the current value of the statistic however clinical trials may be monitored more frequently when they are close to crossing the boundary in this situation the corresponding continuous time boundary should be used here we demonstrate how to construct a continuous stopping boundary from an alpha spending function this capability is useful in the design of clinical trials we use the beta blocker heart attack trial bhat and aids clinical trials group protocol 021 for illustration

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

lie algebras and 3 transpositions abstract we describe a construction of an algebra over the field of order 2 starting from a conjugacy class of 3 transpositions in a group in particular we determine which simple lie algebras arise by this construction among other things this construction yields a natural embedding of the sporadic simple group fi 22 in the group e 6 2 2

sequential monitoring of survival data with the wilcoxon statistic when a spending function is used in sequential data monitoring of a clinical trial it is important to know the information fraction at the times of interim analysis in a maximum duration designed study the information fraction is unknown when data are monitored and it has to be estimated the modified wilcoxon statistic developed by peto and peto and modified by prentice is often used to compare two survival curves in a clinical trial we give guidelines for estimating the information fraction in a maximum duration trial when this statistic is employed when there is a relatively low event rate or the survival time is approximately exponential the information fraction for the peto peto prentice wilcoxon statistic is very close to that of the popular logrank statistic in other cases it would be helpful to estimate the information fraction as a function of elapsed calendar time we discuss both group sequential and continuous monitoring

a unifying family of group sequential test designs summary currently the design of group sequential clinical trials requires choosing among several distinct design categories design scales and strategies for determining stopping rules this approach can limit the design selection process so that clinical issues are not fully addressed this paper describes a family of designs that unifies previous approaches and allows continuous movement among the previous categories this unified approach facilitates the process of tailoring the design to address important clinical issues the unified family of designs is constructed from a generalization of a four boundary group sequential design in which the shape and location of each boundary can be independently specified methods for implementing the design using error spending functions are described examples illustrating the use of the design family are also presented

on the inefficiency of the adaptive design for monitoring clinical trials summary adaptive designs which allow the sample size to be modified based on sequentially computed observed treatment differences have been advocated recently for monitoring clinical trials although such methods have a great deal of appeal on the surface we show that such methods are inefficient and that one can improve uniformly on such adaptive designs using standard group sequential tests based on the sequentially computed likelihood ratio test statistic

sample size calculation for planning group sequential longitudinal trials procedures are developed in this paper for sample size calculations for planning a group sequential longitudinal trial with various correlation structures using a test statistic based on generalized estimating equations lee kim and tsiatis and group sequential boundaries based on type i error spending functions lan and demets

adaptive group sequential and decision theoretic approaches to sample size determination this paper presents two adaptive methods for sample size re estimation within a unified group sequential framework the conceptual and practical distinction between these adaptive modifications and more traditional sample size changes due to revised estimates of nuisance parameters is highlighted the motivation for the adaptive designs is discussed having established that adaptive sample size modifications can be made without inflating the type 1 error the paper concludes with a novel decision theoretic approach for determining the magnitude of the sample size modification

effective carrying capacity and analytical solution of a particular case of the richards like two species population dynamics model abstract we consider a generalized two species population dynamic model and analytically solve it for the amensalism and commensalism ecological interactions these two species models can be simplified to a one species model with a time dependent extrinsic growth factor with a one species model with an effective carrying capacity one is able to retrieve the steady state solutions of the previous one species model the equivalence obtained between the effective carrying capacity and the extrinsic growth factor is complete only for a particular case the gompertz model here we unveil important aspects of sigmoid growth curves which are relevant to growth processes and population dynamics

a one dimensional model of population growth abstract in this work a one dimensional population growth model is proposed the model based on the cooperative and competitive individual individual distance dependent interaction allows us to get a full analytical solution with this analytical approach it was possible to investigate the dynamics of the population according to some parameters as intrinsic growth rate strength of the interaction between individuals and the distance dependent interaction as a consequence of the individuals interaction a rich phase diagram to which the population has access was observed the phases observed are convergence to carrying capacity exponential growth divergence at finite time and extinction moreover it was also observed that some phases are strictly dependent on the initial condition for instance in the cooperative regime with negative intrinsic growth rate the population can diverge or become extinct according to the initial population size the phases accessible to the population can be seen as a macroscopic behavior which emerges from the interaction among the individuals the microscopic level

exact statistical inference for group sequential trials summary this paper considers clinical trials comparing two treatments with dichotomous responses where the data are examined periodically for early evidence of treatment difference the existing group sequential methods for such trials are based on the large sample normal approximation to the joint distribution of the estimators of treatment difference over interim analyses we demonstrate through extensive numerical studies that for small and even moderate sized trials these approximate procedures may lead to tests with supranominal size mainly when unpooled estimators of variance are used and confidence intervals with under nominal coverage probability we then study exact methods for group sequential testing repeated interval estimation and interval estimation following sequential testing the new procedures can accommodate any treatment allocation rules an example using real data is provided in clinical trials with sequential patient entry fixed sample size designs are sometimes unjustified on ethical grounds and fully sequential procedures are often impractical one compromise is the group sequential design under which multiple looks are performed during the trial it is widely recognized that the use of the standard single stage f1xedsample size test or confidence interval at each look would lead to an overall type i error or noncoverage rate significantly higher than the nominal level a number of methods have been proposed to determine proper critical values for maintaining a prespecified overall significance level in sequential testing to construct repeated confidence intervals with a given overall coverage probability and to obtain a valid confidence interval following a sequential trial an excellent review on the existing methodology was given by jennison and turnbull 1989 in this paper we are mainly concerned with the commonly used group sequential trials comparing two treatments a and b with relatively quick dichotomous responses success and failure the data are reviewed periodically with cumulative mk subjects receiving treatment a and nk subjects receiving treatment b by the kth look let random variables xk and yk denote the numbers of successes out of the msk and nk subjects respectively we assume that the maximum number of looks k is fixed in advance though some of the procedures developed here can be extended to the case where k is not fixed in advance

sequential rank tests with repeated measurements in clinical trials abstract for comparing responses in two groups of subjects observed repeatedly we propose a group sequential procedure based on linear rank statistics the asymptotic normality of the sequentially computed linear rank statistics is obtained and construction of the group sequential boundaries is based on this distribution theory by virtue of this asymptotic approximation the proposed procedure can be applied to interim analyses with either continuous or discrete repeated measurements even for staggered patient entry simulation results suggest the theory is approximately correct it can also be useful for testing the equality of two changes and rates of change as well as the equality of two means of the responses this procedure is illustrated with a real example

point and interval estimation following a sequential clinical trial summary the usual methodology employed in analysis after a sequential clinical trial is based on orderings of the possible samples resulting from the design however this approach lacks flexibility for use in wider applications in this paper two estimation techniques not based on orderings are considered and modified to obtain improved accuracy a bias adjusted maximum likelihood estimate together with a new and general method for setting confidence limits are discussed the realistic scenario of group sequential monitoring is assumed and methods for exact estimation are given accuracy of the methodology after a triangular test and an o brien fleming test are demonstrated through simulation

application of bayesian statistics to decision making during a clinical trial we describe the application of bayesian methods to the monitoring and analysis of a trial of treatment for patients with advanced colorectal carcinoma we discuss the choice of prior distribution and justify the use of a truncated normal distribution with a probability mass at zero difference the stopping rule based on the trials of the posterior distribution and a chosen range of equivalence yields an upper boundary very close to the pocock group sequential boundary the bayes stopping rule is quite sensitive to the amount of probability mass at zero in the prior distribution

the dual test safeguarding p value combination tests for adaptive designs many modern adaptive designs apply an analysis where p values from different stages are weighted together to an overall hypothesis test one merit of this combination approach is that the design can be made very flexible however combination tests violate the sufficiency and conditionality principles as a consequence combination tests may lead to absurd conclusions such as proving a positive effect while the average effect is negative we explore the possibility of modifying the test so that such illogical conclusions are no longer possible the dual test requires both the weighted combination test and a nave test ignoring the adaptations to be statistically significant the result is that the flexibility and type i error level control of the combination test are preserved while the nave test adds a safeguard against unconvincing results the dual test is by construction at least as conservative as the combination test however many design changes will not lead to any power loss a typical situation where the combination approach can be used is two stage sample size reestimation ssr for this case we give a complete specification of all sample size modifications for which the two tests are equally powerful we also study the overall power loss for some suggested ssr rules rules based on conditional power generally lead to ignorable power loss while a decision analytic approach exhibits clear discrepancies between the two tests

increasing the sample size when the unblinded interim result is promising increasing the sample size based on unblinded interim result may inflate the type i error rate and appropriate statistical adjustments may be needed to control the type i error rate at the nominal level we briefly review the existing approaches which allow early stopping due to futility or change the test statistic by using different weights or adjust the critical value for final test or enforce rules for sample size recalculation the implication of early stopping due to futility and a simple modification to the weighted z statistic approach are discussed in this paper we show that increasing the sample size when the unblinded interim result is promising will not inflate the type i error rate and therefore no statistical adjustment is necessary the unblinded interim result is considered promising if the conditional power is greater than 50 per cent or equivalently the sample size increment needed to achieve a desired power does not exceed an upper bound the actual sample size increment may be determined by important factors such as budget size of the eligible patient population and competition in the market the 50 per cent conditional power approach is extended to a group sequential trial with one interim analysis where a decision may be made at the interim analysis to stop the trial early due to a convincing treatment benefit or to increase the sample size if the interim result is not as good as expected the type i error rate will not be inflated if the sample size may be increased only when the conditional power is greater than 50 per cent if there are two or more interim analyses in a group sequential trial our simulation study shows that the type i error rate is also well controlled

monitoring a general class of two sample survival statistics with applications summary this paper considers a general class of statistics for testing the equality of two survival distributions in clinical trials with sequential monitoring the tests can be expressed as lebesgue stieltjes integrals of a weight function with respect to the difference between two survival distributions prominent members of this class include the two sample difference in kaplan meier estimates the test of medians brookmeyer crowley 1982 a truncated version of efron s 1967 test and the pepe fleming statistic pepe fleming 1989 1991 statistics in this class are shown to converge to a gaussian process indexed by information time under both null and local alternatives even if different statistics are used at different information times properly standardised statistics in a subclass converge to gaussian processes with independent increments so that the usual group sequential techniques for monitoring a clinical trial can be applied the design of a trial comparing two treatments with respect to mother to newborn transmission of hiv is used to illustrate practical aspects of monitoring

nonparametric rank based methods for group sequential monitoring of paired censored survival data summary this research gives methods for nonparametric sequential monitoring of paired censored survival data in the two sample problem using paired weighted log rank statistics with adjustments for dependence in survival and censoring outcomes the joint asymptotic closed form distribution of these sequentially monitored statistics has a dependent increments structure simulations validating operating characteristics of the proposed methods highlight power and size consequences of ignoring even mildly correlated data a motivating example is presented via the early treatment diabetic retinopathy study

discrete sequential boundaries for clinical trials summary pocock 1977 o brien fleming 1979 and slud wei 1982 have proposed different methods to construct discrete sequential boundaries for clinical trials these methods require that the total number of decision times be specified in advance in the present paper we propose a more flexible way to construct discrete sequential boundaries the method is based on the choice of a function a t which characterizes the rate at which the error level ac is spent the boundary at a decision time is determined by a t and by past and current decision times but does not depend on the future decision times or the total number of decision times

adaptive group sequential designs for clinical trials combining the advantages of adaptive and of classical group sequential approaches a general method is presented integrating the concept of adaptive interim analyses into classical group sequential testing this allows the researcher to represent every group sequential plan as an adaptive trial design and to make design changes during the course of the trial after every interim analysis in the same way as with adaptive designs the concept of adaptive trial designing is thereby generalized to a large variety of possible sequential plans

group sequential tests for delayed responses with discussion summary group sequential methods are used routinely to monitor clinical trials and to provide early stopping when there is evidence of a treatment effect a lack of an effect or concerns about patient safety in many studies the response of clinical interest is measured some time after the start of treatment and there are subjects at each interim analysis who have been treated but are yet to respond we formulate a new form of group sequential test which gives a proper treatment of these pipeline subjects these tests can be applied even when the continued accrual of data after the decision to stop the trial is unexpected we illustrate our methods through a series of examples we define error spending versions of these new designs which handle unpredictable group sizes and provide an information monitoring framework that can accommodate nuisance parameters such as an unknown response variance by studying optimal versions of our new designs we show how the benefits of lower expected sample size that are normally achieved by a group sequential test are reduced when there is a delay in response the loss of efficiency for larger delays can be ameliorated by incorporating data on a correlated short term end point fitting a joint model for the two end points but still making inferences on the original longer term end point we derive p values and confidence intervals on termination of our new tests

estimation of conditional and marginal odds ratios using the prognostic score introduced by hansen in 2008 the prognostic score pgs has been presented as the prognostic analogue of the propensity score pps pps based methods are intended to estimate marginal effects most previous studies evaluated the performance of existing pgs based methods adjustment stratification and matching using the pgs in situations in which the theoretical conditional and marginal effects are equal i e collapsible situations to support the use of pgs framework as an alternative to the pps framework applied researchers must have reliable information about the type of treatment effect estimated by each method we propose four new pgs based methods each developed to estimate a specific type of treatment effect we evaluated the ability of existing and new pgs based methods to estimate the conditional treatment effect cte the marginal average treatment effect on the whole population ate and the marginal average treatment effect on the treated population att when the odds ratio a non collapsible estimator is the measure of interest the performance of pgs based methods was assessed by monte carlo simulations and compared with pps based methods and multivariate regression analysis existing pgs based methods did not allow for estimating the ate and showed unacceptable performance when the proportion of exposed subjects was large when estimating marginal effects pps based methods were too conservative whereas the new pgs based methods performed better with low prevalence of exposure and had coverages closer to the nominal value when estimating cte the new pgs based methods performed as well as traditional multivariate regression copyright 2016 john wiley sons ltd

a review of methods for futility stopping based on conditional power conditional power cp is the probability that the final study result will be statistically significant given the data observed thus far and a specific assumption about the pattern of the data to be observed in the remainder of the study such as assuming the original design effect or the effect estimated from the current data or under the null hypothesis in many clinical trials a cp computation at a pre specified point in the study such as mid way is used as the basis for early termination for futility when there is little evidence of a beneficial effect brownian motion can be used to describe the distribution of the interim z test value the corresponding b value and the cp values under a specific assumption about the future data a stopping boundary on the cp value specifies an equivalent boundary on the b value from which the probability of stopping for futility can then be computed based on the planned study design sample size and duration and the assumed true effect size this yields expressions for the total type i and ii error probabilities as the probability of stopping increases the probability of a type i error alpha decreases from the nominal desired level e g 0 05 while the probability of a type ii error beta increases from the level specified in the study design thus a stopping boundary on the b value can be determined such that the inflation in type ii error probability is controlled at a desired level an iterative procedure is also described that determines a stopping boundary on the b value and a final test critical z value with specified type i and ii error probabilities the implementation in conjunction with a group sequential analysis for effectiveness is also described

implementation of group sequential logrank tests in a maximum duration trial summary to control the type i error probability in a group sequential procedure using the logrank test it is important to know the information times fractions at the times of interim analyses conducted for purposes of data monitoring for the logrank test the information time at an interim analysis is the fraction of the total number of events to be accrued in the entire trial in a maximum information trial design the trial is concluded when a prespecified total number of events has been accrued for such a design therefore the information time at each interim analysis is known however many trials are designed to accrue data over a fixed duration of follow up on a specified number of patients this is termed a maximum duration trial design under such a design the total number of events to be accrued is unknown at the time of an interim analysis for a maximum duration trial design therefore these information times need to be estimated a common practice is to assume that a fixed fraction of information will be accrued between any two consecutive interim analyses and then employ a pocock or o brien fleming boundary in this article we describe an estimate of the information time based on the fraction of total patient exposure which tends to be slightly negatively biased i e conservative if survival is exponentially distributed we then present a numerical exploration of the robustness of this estimate when nonexponential survival applies we also show that the lan demets 1983 biometrika 70 659 663 procedure for constructing group sequential boundaries with the desired level of type i error control can be computed using the estimated information fraction even though it may be biased finally we discuss the implications of employing a biased estimate of study information for a group sequential procedure

sample size determination in group sequential clinical trials with two co primary endpoints we discuss sample size determination in group sequential designs with two endpoints as co primary we derive the power and sample size within two decision making frameworks one is to claim the test intervention s benefit relative to control when superiority is achieved for the two endpoints at the same interim timepoint of the trial the other is when superiority is achieved for the two endpoints at any interim timepoint not necessarily simultaneously we evaluate the behaviors of sample size and power with varying design elements and provide a real example to illustrate the proposed sample size methods in addition we discuss sample size recalculation based on observed data and evaluate the impact on the power and type i error rate copyright 2014 john wiley sons ltd

confidence intervals for secondary parameters following a sequential test in sequential studies formal interim analyses are usually restricted to a consideration of a single null hypothesis concerning a single parameter of interest valid frequentist methods of hypothesis testing and of point and interval estimation for the primary parameter have already been devised for use at the end of such a study however the completed data set may warrant a more detailed analysis involving the estimation of parameters corresponding to effects that were not used to determine when to stop and yet correlated with those that were this paper describes methods for setting confidence intervals for secondary parameters in a way which provides the correct coverage probability in repeated frequentist realizations of the sequential design used the method assumes that information accumulates on the primary and secondary parameters at proportional rates this requirement will be valid in many potential applications but only in limited situations in survival analysis

exact confidence intervals following a group sequential trial a comparison of methods summary with interest in designing group sequential clinical trials increasing methodology for analysing the data arising in such trials is needed we discuss several approaches to constructing confidence intervals for the parameter of interest in a group sequential trial after the trial has ended we compare these met1fods to one already in the literature and point out interesting differences

an improved approximation for calculation of confidence intervals after a sequential clinical trial a method for the calculation of confidence intervals following a sequential clinical trial is proposed which is more accurate than methods based solely on continuous monitoring assumptions but is not as computationally laborious as previous methods suggested for group sequential trials the calculation takes account of the excess of the sample path over the boundary at the final inspection of the trial which may be substantial after group sequential monitoring and is ignored when continuous monitoring is assumed accuracy of such confidence intervals after a triangular test is investigated by simulation of coverage probabilities individual confidence limit probabilities and p value curves

miscellanea properties of spending function boundaries clinical trials are monitored periodically for safety and efficacy resulting in several looks at interim data if no account is taken of this the type i error rate may be substantially higher than planned one of the most popular methods of generating interim boundaries that result in an overall type i error rate of is the spending function approach this paper proves several properties of these boundaries including a continuity like property for looks occurring close to each other and a monotonicity property when additional looks are taken how past monitoring affects future boundaries is also studied

longitudinal clinical trials with adaptive choice of follow up time in longitudinal studies comparing two treatments with a maximum follow up time there may be interest in examining treatment effects for intermediate follow up times one motivation may be to identify the time period with greatest treatment difference when there is a non monotone treatment effect over time another motivation may be to make the trial more efficient in terms of time to reach a decision on whether a new treatment is efficacious or not here we test the composite null hypothesis of no difference at any follow up time versus the alternative that there is a difference at at least one follow up time the methods are applicable when a few measurements are taken over time such as in early longitudinal trials or in ancillary studies suppose the test statistic z t k will be used to test the hypothesis of no treatment effect at a fixed follow up time t k in this context a common approach is to perform a pilot study on n1 subjects and evaluate the treatment effect at the fixed time points t1 t k and choose t as the value of t k for which z t k is maximized having chosen t a second trial can be designed in a setting with group sequential testing we consider several adaptive alternatives to this approach that treat the pilot and second trial as a seamless combined entity and evaluate type i error and power characteristics the adaptive designs we consider typically have improved power over the common separate trial approach

p value adjustment in sequential clinical trials many randomized clinical trials utilize a group sequential monitoring procedure for assessing treatment efficacy during the course of the trial for a group sequential trial in which the null hypothesis is ultimately rejected the nominal p value generally overstates the statistical significance of the result and some adjustment is required several orderings of the sample space have been proposed for performing this adjustment each with unique operating characteristics in this article we compare four proposed methods with respect to the degree to which each captures the strength of evidence against the null hypothesis implied by the data we conclude that the ordering of the sample space induced by the observed z score has the most desirable operating characteristics

sequential monitoring for comparison of changes in a response variable in clinical studies the spending function approach proposed by lan and demets 1983 biometrika 70 659 663 for sequential monitoring of clinical trials is applied to situations where comparison of changes in a continuous response variable between two groups is the primary concern death loss to follow up and missed visits could cause follow up measurements to be right censored or missing for some participants furthermore the probability of being censored may be dependent on the parameter value of the response variable informative censoring we propose to compare treatment effects by comparing areas under the expected response change curves between the two groups when the response curves are linear as a function of time in both groups this comparison is equivalent to comparing the rates of change in the response variable covariances of the sequential test statistics are derived conditions for having independent increments are presented for studies designed to evaluate long term treatment effects spending functions obtained by shifting the usual spending functions kim and demets 1987 biometrika 74 149 154 to the right and then rescaling to the remaining interval are also proposed such a shifted spending function is applied to the monitoring plan for the lung health study anthonisen 1989 american review of respiratory diseases 140 871 872

an improved method for deriving optimal one sided group sequential tests summary we derive one sided group sequential tests for normal responses which minimize expected sample size minimization is at a single value of the normal mean or integrated with respect to a normal density the methods employed are much faster and also numerically more stable than those of jennison 1987 they provide solutions for cases with as many as 200 groups and for small numbers of groups facilitate optimization over the choice of group sizes we present a new method for constructing near optimal tests when group sizes are unpredictable based on interpolating the error spending function of an optimal test with a large number of groups

spending functions and continuous monitoring boundaries clinical trials are monitored periodically for safety efficacy and futility the spending function is a popular tool for efficacy monitoring because it does not require pre specification of the number or timing of interim analyses but there are infinitely many spending functions so some guidance on how to choose one is helpful we consider spending functions that are generated from different continuous monitoring boundaries for brownian motion we use properties of the continuous monitoring boundaries to derive properties of the associated spending function copyright 2012 john wiley sons ltd

type i error in sample size re estimations based on observed treatment difference sample size re estimation based on an observed difference can ensure an adequate power and potentially save a large amount of time and resources in clinical trials one of the concerns for such an approach is that it may inflate the type i error however such a possible inflation has not been mathematically quantified in this paper the mathematical mechanism of this inflation is explored for two sample normal tests a conditional type i error function based on normal data is derived this function not only provides the quantification but also gives mathematical mechanisms of possible inflation in the type i error due to the sample size re estimation theoretically based on their decision rules certain upper and lower bounds people can calculate this function and exactly visualize the changes in type i error computer simulations are performed to ensure the results if there are no bounds for the adjustment the inflation is evident if proper adjusting rules are used the inflation can be well controlled in some cases the type i error can even be reduced the trade off is to give up some unrealistic power we investigated several scenarios in which the mechanisms to change the type i error are different our simulations show that similar results may apply to other distributions copyright 2001 john wiley sons ltd

a method for sequential analysis of survival data with nonproportional hazards two tests are proposed for comparing the survival curves of patients randomised between an experimental treatment and a control treatment when it is anticipated that the two survival curves may not satisfy the assumption of proportional hazards the tests are particularly useful for the situation in which the survival curves are coincident or cross over early in the follow up period and then diverge the tests compare the probabilities of survival for longer than some fixed time since randomisation for the two groups of patients both methods take account of the right censored observations and both are associated with methods for estimating and setting confidence limits for treatment differences the first method is a mathematically direct approach based on the derivation of the efficient score statistic and fisher s information the second method is simpler being based on kaplan meier estimates and their variances conventional methods of sample size determination require the assumption of proportional hazards here a sequential approach is used as it is difficult to set the sample size in advance without strong assumptions about the relationship between the two survival curves simulation results giving information on the size and power of the proposed tests are provided and the tests are applied to data from a clinical trial in breast cancer

sequential implementation of monte carlo tests with uniformly bounded resampling risk this paper introduces an open ended sequential algorithm for computing the p value of a test using monte carlo simulation it guarantees that the resampling risk the probability of a different decision than the one based on the theoretical p value is uniformly bounded by an arbitrarily small constant previously suggested sequential or nonsequential algorithms using a bounded sample size do not have this property although the algorithm is open ended the expected number of steps is finite except when the p value is on the threshold between rejecting and not rejecting the algorithm is suitable as standard for implementing tests that require re sampling it can also be used in other situations to check whether a test is conservative iteratively to implement double bootstrap tests and to determine the sample size required for a certain power an r package implementing the sequential algorithm is available online

design and monitoring of multi arm multi stage clinical trials two arm group sequential designs have been widely used for over 40 years especially for studies with mortality endpoints the natural generalization of such designs to trials with multiple treatment arms and a common control mams designs has however been implemented rarely while the statistical methodology for this extension is clear the main limitation has been an efficient way to perform the computations past efforts were hampered by algorithms that were computationally explosive with the increasing interest in adaptive designs platform designs and other innovative designs that involve multiple comparisons over multiple stages the importance of mams designs is growing rapidly this article provides break through algorithms that can compute mams boundaries rapidly thereby making such designs practical for designs with efficacy only boundaries the computational effort increases linearly with number of arms and number of stages for designs with both efficacy and futility boundaries the computational effort doubles with successive increases in number of stages

first exit densities of brownian motion through one sided moving boundaries let a a e be a sequence of curved boundaries which tend to infinity as a increases let t a inf t 0 w t geqq psi a t where w t denotes the standard brownian motion under regularity conditions on the boundaries uniform approximations for the first exit densities of ta are derived the consequences for upper and lower class functions are discussed the approximations for the first exit densities of brownian motion with drift which are also derived lead to uniform approximations for the power functions of sequential tests the quality of the approximations is demonstrated by some figures

group sequential distribution free methods for the analysis of multivariate observations many studies involve the collection of multivariate observations such as repeated measures on two groups of subjects who are recruited over time i e with staggered entry of subjects various marginal distribution free multivariate methods have been proposed for the analyses of such multivariate observations where some measures may be missing at random using the multivariate u statistic of wei and johnson 1985 biometrika 72 359 364 we describe the group sequential analysis of such a study where the multivariate observations are observed sequentially both within and among subjects we describe a multivariate generalization of the hodges and lehmann 1963 annals of mathematical statistics 34 598 611 estimator of a location shift that can be obtained via the multivariate u statistic with the mann whitney wilcoxon kernel we then describe large sample group sequential interval estimators and tests based on an aggregate estimate of the location shift combined over all of the repeated measures we also describe how the same steps could be employed to perform a group sequential analysis based on any one of the variety of marginal multivariate methods that have been proposed these methods are applied to a real life example

two sample repeated significance tests based on the modified wilcoxon statistic abstract the asymptotic distribution theory of sequentially computed modified wilcoxon scores is developed for two sample survival data with random staggered entry and random loss to follow up the asymptotic covariance indicates generally dependent modified wilcoxon increments contradicting the authors reading of jones and whitehead 1979 a repeated significance testing procedure is presented for testing the equality of two survival distributions based on the asymptotic theory the early stopping properties of this procedure are illustrated by a prostate cancer example

simultaneous group sequential analysis of rank based and weighted kaplan meier tests for paired censored survival data this research sequentially monitors paired survival differences using a new class of nonparametric tests based on functionals of standardized paired weighted log rank pwlr and standardized paired weighted kaplan meier pwkm tests during a trial these tests may alternately assume the role of the more extreme statistic by monitoring pemax the maximum between the absolute values of the standardized pwlr and pwkm one combines advantages of rank based rb and non rb paired testing paradigms simulations show that monitoring treatment differences using pemax maintains type i error and is nearly as powerful as using the more advantageous of the two tests in proportional hazards ph as well as non ph situations hence pemax preserves power more robustly than individually monitored pwlr and pwkm while maintaining a reasonably simple approach to design and analysis of results an example from the early treatment diabetic retinopathy study etdrs is given

a group sequential holm procedure with multiple primary endpoints we propose a group sequential holm procedure when there are multiple primary endpoints this method addresses multiplicities arising from multiple primary endpoints and from multiple analyses in a group sequential design it has been shown to be a closed testing procedure and preserves the familywise error rate in the strong sense when multiple endpoints are the only concern without an interim analysis the method simplifies to the weighted holm procedure the proposed method is more powerful than the parallel group sequential method and avoids the need to anticipate the testing order as in the fixed sequence testing scheme

group sequential designs for monitoring survival probabilities in the present paper we derive the asymptotic joint distribution of the nelson aalen estimators for the cumulative hazard function or the kaplan meier estimators for the survival function calculated at different possibly random calendar time points of a survival study this result is used to develop group sequential designs for clinical trials in which patients are enrolled over an extended period and the survival probability at a given failure time point is the parameter of primary interest the proposed designs allow monitoring the survival probability or probabilities at interim looks so that the study can be terminated when there is sufficient evidence for a positive or negative result a detailed illustration is provided with a problem from the national wilms tumor study group which motivated this research

sequential methods for parametric survival models summary the asymptotic sequential distribution is derived for score tests and maximum likelihood estimates in parametric survival models in the presence of nuisance parameters where individuals enter over time and the data are subject to right censoring it is shown that these sequential tests and estimates have an independent increments structure this allows the use of standard group sequential methods for early stopping

group sequential analysis incorporating covariate information abstract in this article we survey existing results concerning the joint distribution of the sequence of estimates of the parameter vector when a model is fitted to accumulating data and provide a unified theory that explains the independent increments structure commonly seen in group sequential test statistics our theory covers normal linear models including the case of correlated observations and asymptotic results extend to generalized linear models and the proportional hazards regression model for survival data the asymptotic results are derived using standard methods for the nonsequential case and they hold as long as these nonsequential techniques are applicable at each individual analysis in all cases the joint distribution of the sequence of parameter estimates has the same form exactly or asymptotically as that of the sequence of means of an increasing number of independent identically distributed normal variables thus our results provide the formal basis for extending the scope of s

sequential monitoring of clinical trials the role of information and brownian motion sequential monitoring has been a topic of major interest in clinical trials methodology over the past two decades this paper presents a unified conceptual framework for sequential monitoring that covers a wide variety of monitoring procedures in a wide variety of clinical trial settings the central elements of this framework consist of a suitable concept of statistical information and a scheme for using this concept as a basis for summarizing the accumulating results of a trial in a standardized form through a stochastic process that can be shown to approximate classical brownian motion the ideas are developed in a simple step by step fashion and illustrated by several practical examples

exact confidence bounds following adaptive group sequential tests summary we provide a method for obtaining confidence intervals point estimates and p values for the primary effect size parameter at the end of a two arm group sequential clinical trial in which adaptive changes have been implemented along the way the method is based on applying the adaptive hypothesis testing procedure of muller and schafer 2001 to a sequence of dual tests derived from the stage wise adjusted confidence interval of tsiatis rosner and mehta 1984 in the non adaptive setting this confidence interval is known to provide exact coverage in the adaptive setting exact coverage is guaranteed provided the adaptation takes place at the penultimate stage in general however all that can be claimed theoretically is that the coverage is guaranteed to be conservative nevertheless extensive simulation experiments supported by an empirical characterization of the conditional error function demonstrate convincingly that for all practical purposes the coverage is exact and the point estimate is median unbiased no procedure has previously been available for producing confidence intervals and point estimates with these desirable properties in an adaptive group sequential setting the methodology is illustrated by an application to a clinical trial of deep brain stimulation for parkinson s disease

data collapse scaling functions and analytical solutions of generalized growth models we consider a nontrivial one species population dynamics model with finite and infinite carrying capacities time dependent intrinsic and extrinsic growth rates are considered in these models through the model per capita growth rate we obtain a heuristic general procedure to generate scaling functions to collapse data into a simple linear behavior even if an extrinsic growth rate is included with this data collapse all the models studied become independent from the parameters and initial condition analytical solutions are found when time dependent coefficients are considered these solutions allow us to perceive nontrivial transitions between species extinction and survival and to calculate the transition s critical exponents considering an extrinsic growth rate as a cancer treatment we show that the relevant quantity depends not only on the intensity of the treatment but also on when the cancerous cell growth is maximum

a computationally simpler algorithm for the umvue of a normal mean following a group sequential trial when using data collected in a group sequential clinical trial the sample mean is no longer the uniform minimum variance unbiased estimator umvue of the mean of a normal distribution emerson 1993 computers and biomedical research 26 68 73 described an algorithm for computing the umvue in this setting this algorithm although computationally expensive used only the basic software necessary for deriving group sequential boundaries in this paper we present an improved algorithm that results in greatly decreased computation times