a large sample study of cox s regression model strong consistency and asymptotic normality are established for the maximum partial likelihood estimate of the regression parameter in cox s regression model estimates are also derived for the underlying cumulative hazard function and survival distribution we establish the asymptotic normality of these estimates and calculate the limiting variances

cox s regression model for counting processes a large sample study the cox regression model for censored survival data specifies that covariates have a proportional effect on the hazard function of the life time distribution of an individual in this paper we discuss how this model can be extended to a model where covariate processes have a proportional effect on the intensity process of a multivariate counting process this permits a statistical regression analysis of the intensity of a recurrent event allowing for complicated censoring patterns and time dependent covariates furthermore this formulation gives rise to proofs with very simple structure using martingale techniques for the asymptotic properties of the estimators from such a model finally an example of a statistical analysis is included

planning the size of survival time clinical trials with allowance for stratification approximate allocation designs for comparing two survival distributions are considered and compared with the exact solution the influence of incorporating covariates by stratification accrual and follow up duration on trial size and power is investigated the bernstein and lagakos design is suited for exponential survival due to its generality and allowance for stratification while the freedman design is convenient for distribution free survival in view of its simplicity with other specifications fixed the saving in sample size becomes minimal as the accrual period or post accrual follow up time units expand above three required sample size also increases dramatically when the number of strata exceeds three

robust covariate adjusted log rank statistics and corresponding sample size formula for recurrent events data recurrent events data are frequently encountered in clinical trials this article develops robust covariate adjusted log rank statistics applied to recurrent events data with arbitrary numbers of events under independent censoring and the corresponding sample size formula the proposed log rank tests are robust with respect to different data generating processes and are adjusted for predictive covariates it reduces to the kong and slud 1997 biometrika 84 847 862 setting in the case of a single event the sample size formula is derived based on the asymptotic normality of the covariate adjusted log rank statistics under certain local alternatives and a working model for baseline covariates in the recurrent event data context when the effect size is small and the baseline covariates do not contain significant information about event times it reduces to the same form as that of schoenfeld 1983 biometrics 39 499 503 for cases of a single event or independent event times within a subject we carry out simulations to study the control of type i error and the comparison of powers between several methods in finite samples the proposed sample size formula is illustrated using data from an rhdnase study

sample size calculations in the presence of competing risks recently with the growth of statistical developments for competing risks analysis some methods have been proposed to compute sample size in this context these methods differ from a modelling approach one is based on the cox regression model for the cause specific hazard while another relies on the fine and gray regression model for the subdistribution hazard of a competing risk in this work we compare these approaches derive a new sample size for comparing cumulative incidence functions when the hazards are not proportional either cause specific or subdistribution and give practical advices to choose the approach best suited for the study question

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

supremum weighted log rank test and sample size for comparing two stage adaptive treatment strategies summary in two stage adaptive treatment strategies patients receive an induction treatment followed by a maintenance therapy given that the patient responded to the induction treatment they received to test for a difference in the effects of different induction and maintenance treatment combinations a modified supremum weighted log rank test is proposed the test is applied to a dataset from a two stage randomized trial and the results are compared to those obtained using a standard weighted log rank test a sample size formula is proposed based on the limiting distribution of the supremum weighted log rank statistic the sample size formula reduces to eng and kosorok s sample size formula for a two sample supremum log rank test when there is no second randomization monte carlo studies show that the proposed test provides sample sizes that are close to those obtained by standard weighted log rank test under a proportional hazards alternative however the proposed test is more powerful than the standard weighted log rank test under non proportional hazards alternatives

the gauss green theorem for fractal boundaries v is the 1 vectorfield dual to if 1 fi dx1 dxi dxn then v f1 fn there has been considerable effort in the literature e g jk m p to extend this formula to permit integrands of less regularity by generalizing the lebesgue integral on the other hand invariably the situations in which 1 holds require fairly strong hypotheses on the boundary e g that it should have sigmafinite n 1 measure or that the gradient of the characteristic function of be a vector valued measure with finite total variation f p however there is a natural way to expand the validity of 1 to much more general boundaries while still using the ordinary lebesgue integral this is the topic of the present paper for the case of lipschitz forms the results of this paper follow readily from whitney s theory of flat chains w2 however his approach to the gauss green theorem is not widely appreciated because he focused on chains and cochains where effectively 1 is used to define the exterior derivative in hn we extend whitney s method to treat the more general holder case only the case n 2 is discussed

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

algorithmic construction of efficient fractional factorial designs with large run sizes fractional factorial ff designs are widely used in practice and typically are chosen according to the minimum aberration criterion a sequential algorithm is developed for constructing efficient ff designs a construction procedure is proposed that allows a design to be constructed only from its minimum aberration projection in the sequential buildup process to efficiently identify nonisomorphic designs designs are categorized according to moment projection pattern a fast isomorphism checking procedure is developed by matching the factors using their delete one factor projections this algorithm is used to completely enumerate all 128 run designs of resolution 4 all 256 run designs of resolution 4 up to 17 factors all 512 run designs of resolution 5 all 1024 run designs of resolution 6 and all 2048 and 4 096 run designs of resolution 7 a method is proposed for constructing minimum aberration ma designs using only a partial catalog of some good designs three approaches to constructing good de

a proportional hazards model for arbitrarily censored and truncated data turnbull 1976 journal of royal statistical society series b 38 290 295 proposed a method for nonparametric estimation of the distribution function when the data are incomplete because of censoring and truncation however as noted by frydman 1994 journal of royal statistical society series b 56 71 74 turnbull s method has to be modified to accommodate both truncation and censoring this paper presents a detailed correction of turnbull s method and an extension to the regression analysis a method of fitting the proportional hazards model for arbitrarily censored and truncated data is developed the method allows partial testing for zero regression coefficients the test can be performed using the likelihood ratio test or the wald test the methodology is applied to estimate the distribution of the induction time of patients diagnosed with transfusion associated aids and to estimate the distribution of time from diabetes onset to development of diabetic nephropathy for insulin dependent diabetics

comparison of the proportions of affected relatives of cases and controls analysis and minimum sample size formula the problem of familial aggregation has frequently been approached by comparing the proportion of affected relatives of index cases with the proportion of affected relatives of index controls this type of study has an analogy with the case control design and is frequently analysed as if it was one it is however an essentially different design which may involve dependent observations we show that conventional tests for comparing proportions of affected subjects are still valid with this design we give a formula for computing the minimum sample size required to detect a given degree of aggregation measured by the intracluster correlation coefficient which can be estimated very simply by the difference in proportions of affected relatives in the two groups

efficiency of age adjusted tests in animal carcinogenicity experiments based on asymptotic relative efficiency calculations ryan 1985 biometrics 41 525 531 concludes that in the analysis of animal carcinogenicity experiments age adjusted tests of tumor rates should be routinely preferred to simple proportions tests for both lethal and nonlethal tumors we recalculate the asymptotic efficiencies of the simple proportions test relative to the log rank test for the lethal tumor case for a simplified model it is shown that the relative efficiency may be easily computed as a function of the crude tumor rate and the survival rate at the time of terminal sacrifice more generally we calculate by numerical quadrature the asymptotic relative efficiency for all models considered by ryan and using simulations examine the relevance of asymptotic efficiencies to typical sample sizes contrary to the numerical results of ryan we find for experiments with good survival and typical tumor rates that the relative efficiencies are greater than 95 usually about 99 in the nonlethal tumor case similar results follow from ryan for tumor rates and survival rates typically encountered in practice as it is often difficult to determine whether or not a tumor is lethal we conclude for equal interim mortality rates that the simple proportions test is usually adequate in evaluating animal carcinogenicity experiments

surface integrals for domains with fractal boundaries and some applications to elasticity some possible formulations of integration of differential forms over a non smooth boundary are introduced it is supposed that the differential of a form is integrable on the whole domain in applications to continuum mechanics the condition of integral convergence selects the physically interesting cases when the energy over the domain is finite conditions providing continuity of surface integrals for domains in rn with fractal boundary of non integer hausdorff dimension and non integer box dimension for both continuous and special discontinuous differential forms are described the results obtained are applied to elastic domains in r3 with fractal boundaries some discontinuous differential forms are considered when the forms are formed by the use of the elastic stress and displacement fields the proof of the uniqueness theorem of solutions to problems of elastostatics for bodies with fractal boundaries is given

sample size determination for an exponential survival model with an unrestricted covariate we derive formulae for estimating sample size and power for detecting the effect of an unrestricted covariate on survival time these are useful in designing survival studies with different patterns of recruitment and follow up when survival time is exponentially distributed we use the asymptotic covariance matrix conditional expectation and taylor s expansion techniques to develop these formulae computer simulations indicate that the asymptotic approximations used in developing the formulae are good over a range of parameter values and different patterns of recruitment and follow up that are relevant to survival studies

properties of proportional hazards score tests under misspecified regression models the effects are investigated of misspecifying a proportional hazards regression model on the associated partial likelihood score test for comparing two randomized treatments in the presence of covariates the asymptotic efficiency of the proportional hazards score test relative to the optimal partial likelihood test declines slowly as the hazard functions for the two treatments deviate from proportionality the efficiency can be very low when the hazard functions cross or differ only at large survival times misspecification of the functional form of the regression portion of a proportional hazards model introduces a quantitative treatment covariate interaction in the situations that we examine based on a binary covariate this misspecification usually results in only a minor drop in efficiency the omission of a covariate that is balanced across treatments has a negligible effect on the size of the score test but can substantially reduce power when the covariate effect is strong the loss of power from mismodeling a balanced covariate is usually small

sample sizes for proportional hazards survival studies with arbitrary patient entry and loss to follow up distributions in proportional hazards survival studies power depends on the observed number of deaths d for a given choice of survival loss and patient entry distributions sample sizes can be determined by equating d to the expected number of deaths approximating the survival and loss distributions with piecewise exponential distributions and patient entry with a piecewise linear distribution significantly reduces the computational overhead and the expected number of deaths can be evaluated routinely the merits of this approach are illustrated by a clinical trial of chemotherapy for large bowel cancer

survival analysis with heterogeneous covariate measurement error this article is motivated by a time to event analysis where the covariate of interest was measured at the wrong time we show that the problem can be formulated as a special case of survival analysis with heterogeneous covariate measurement error and develop a general analytic framework we study the asymptotic behavior of the naive partial likelihood estimates and analytically demonstrate that under the heterogeneous measurement error structure and the assumption that all components of the covariate vector and the measurement error vector combined are mutually independent these naive estimates will shrink toward 0 and that the degree of attenuation increases as the measurement error increases we also give counterexamples for reverse attenuation when the independence conditions are violated we use our analytical results to derive a simple bias correcting estimator that performs well in simulations for small and moderate amounts of measurement error our framework can be used to provide insight into th

integration over non rectifiable curves and riemann boundary value problems abstract in this paper we introduce an alternative way of defining the curvilinear cauchy integral over non rectifiable curves in the case of complex functions of one complex variable especially the jump behavior on the boundary is considered as an application solvability conditions of the riemann boundary value problem are derived on very weak conditions to the boundary besides the complex case the consideration can be extended to the theory of douglis algebra valued functions

a simple approach to power and sample size calculations in logistic regression and cox regression models for a given regression problem it is possible to identify a suitably defined equivalent two sample problem such that the power or sample size obtained for the two sample problem also applies to the regression problem for a standard linear regression model the equivalent two sample problem is easily identified but for generalized linear models and for cox regression models the situation is more complicated an approximately equivalent two sample problem may however also be identified here in particular we show that for logistic regression and cox regression models the equivalent two sample problem is obtained by selecting two equally sized samples for which the parameters differ by a value equal to the slope times twice the standard deviation of the independent variable and further requiring that the overall expected number of events is unchanged in a simulation study we examine the validity of this approach to power calculations in logistic regression and cox regression models several different covariate distributions are considered for selected values of the overall response probability and a range of alternatives for the cox regression model we consider both constant and non constant hazard rates the results show that in general the approach is remarkably accurate even in relatively small samples some discrepancies are however found in small samples with few events and a highly skewed covariate distribution comparison with results based on alternative methods for logistic regression models with a single continuous covariate indicates that the proposed method is at least as good as its competitors the method is easy to implement and therefore provides a simple way to extend the range of problems that can be covered by the usual formulas for power and sample size determination

survival analysis 1982 1991 the second decade of the proportional hazards regression model a short review of the cox regression model and its development from publication to 1982 is given i discuss how the model is used and presented in medical journals and how improvement may be achieved finally some extensions of the model are outlined

efficiency and power of tests for multiple binary outcomes abstract global tests provide a useful tool for comparing two or more groups with respect to multiple correlated outcomes we adapt and compare the performance of tests that have been suggested for use with multiple continuous outcomes to the case of multiple binary outcomes comparisons and guidelines are based on asymptotic relative efficiencies are s and simulations these results are illustrated using an application from teratology we extend the work of lefkopoulou and ryan to include general m group comparisons alternatives where group effects may differ for each outcome a concise form for this general class of score tests is derived to compute the are s for this class of tests we devise a useful characterization of the alternative space based on multivariate polar coordinates our findings indicate that the common outcome effect tests are efficient for a remarkably large range of circumstances a simple formula applies to compute the maximum number of unaffected outcomes that can be included i

a graph aided method for planning two level experiments when certain interactions are important in planning a fractional factorial experiment prior knowledge may suggest that some interactions are potentially important and should therefore be estimated free of the main effects in this article we propose a graph aided method to solve this problem for two level experiments first we choose the defining relations for a 2 n k design according to a goodness criterion such as the minimum aberration criterion then we construct all of the nonisomorphic graphs that represent the solutions to the problem of simultaneous estimation of main effects and two factor interactions for the given defining relations in each graph a vertex represents a factor and an edge represents the interaction between the two factors for the experiment planner the job is simple draw a graph representing the specified interactions and compare it with the list of graphs obtained previously our approach is a substantial improvement over taguchi s linear graphs

construction of orthogonal two level designs of user specified resolution where n 2 k fractional two level factorial designs are often used in the early stages of an investigation to screen for important factors traditionally 2 n k fractional factorial designs of resolution iii iv or v have been used for this purpose when the investigator is able to specify the set of nonnegligible factorial effects it is sometimes possible to obtain an orthogonal design with fewer runs than a standard textbook design by searching within a wider class of designs called parallel flats designs the run sizes in this class of designs do not necessarily need to be powers of 2 we discuss an algorithm for constructing orthogonal parallel flats designs to meet user specifications several examples illustrate the use of the algorithm

sample size calculation for cluster randomization trials with a time to event endpoint cluster randomization trials randomize groups called clusters of subjects called subunits between intervention arms and observations are collected from each subject in this case subunits within each cluster share common frailties so that the observations from subunits of each cluster tend to be correlated oftentimes the outcome of a cluster randomization trial is a time to event endpoint with censoring in this article we propose a closed form sample size formula for weighted rank tests to compare the marginal survival distributions between intervention arms under cluster randomization with possibly variable cluster sizes extensive simulation studies are conducted to evaluate the performance of our sample size formula under various design settings real study examples are taken to demonstrate our method

checking the cox model with cumulative sums of martingale based residuals summary this paper presents a new class of graphical and numerical methods for checking the adequacy of the cox regression model the procedures are derived from cumulative sums of martingale based residuals over follow up time and or covariate values the distributions of these stochastic processes under the assumed model can be approximated by zero mean gaussian processes each observed process can then be compared both visually and analytically with a number of simulated realizations from the approximate null distribution these comparisons enable the data analyst to assess objectively how unusual the observed residual patterns are special attention is given to checking the functional form of a covariate the form of the link function and the validity of the proportional hazards assumption an omnibus test consistent against any model misspecification is also studied the proposed techniques are illustrated with two real data sets

statistical considerations when using a composite endpoint for comparing treatment groups when comparing two treatment groups in a time to event analysis it is common to use a composite event consisting of two or more distinct outcomes the goal of this paper is to develop a statistical methodology to derive efficiency guidelines for deciding whether to expand a study primary endpoint from e1 for example non fatal myocardial infarction and cardiovascular death to the composite of e1 and e2 for example non fatal myocardial infarction cardiovascular death or revascularisation we investigate this problem by considering the asymptotic relative efficiency of a log rank test for comparing treatment groups with respect to a primary relevant endpoint e1 versus the composite primary endpoint say e of e1 and e2 where e2 is some additional endpoint

a comparison of estimated proportional hazards models and regression trees we present examples of the usage of regression trees for censored response via two real world datasets one a rheumatoid arthritis survival study and the other a hip replacement study and draw comparisons with the results of cox proportional hazards modelling the two methods pursue different goals motivation of the tree techniques is the desire to extract meaningful prognostic groups while the proportional hazards model enables assessment of the impact of risk factors the methods are thus complementary for the arthritis study the two techniques corroborate one another although the flavour of the conclusions derived differ for the hip replacement study however the regression tree approach reveals structure that would not emerge from a routine proportional hazards analysis we also discuss the treatment of data analytic issues such as the handling of missing values and influence in the presence of non uniform censoring

functional form diagnostics for cox s proportional hazards model summary we propose a new type of residual and an easily computed functional form test for the cox proportional hazards model the proposed test is a modification of the omnibus test for testing the overall fit of a parametric regression model developed by stute gonzalez manteiga and presedo quindimil 1998 journal of the american statistical association93 141 149 and is based on what we call censoring consistent residuals in addition we develop residual plots that can be used to identify the correct functional forms of covariates we compare our test with the functional form test of lin wei and ying 1993 biometrika80 557 572 in a simulation study the practical application of the proposed residuals and functional form test is illustrated using both a simulated data set and a real data set

covariate measurement errors and parameter estimation in a failure time regression model summary estimation in cox s failure time regression model is considered when the regression vector is subject to measurement error a hazard function model is induced for the failure rate given the measured covariate and a partial likelihood function is derived for the relative risk parameters this partial likelihood function may involve the baseline hazard function as well as the regression parameter but useful inference techniques arise for testing whether the regression parameter equals zero and for more general inferences in important special cases explicit consideration is given to testing equality of survival curves when group membership is subject to misclassification and to relative risk estimation with normally distributed covariates approaches to regression estimation using the overall likelihood function and a marginal likelihood function based on failure time ranks are also indicated illustration of the possible effect of covariate errors on relative risk estimation is provided

the versatility of function indexed weighted log rank statistics abstract two sample weighted log rank statistics are used in the presence of right censoring to test whether failure times from two populations have different survival distributions kosorok has showed that large families of these statistics form stochastic processes indexed by weight functions and that these function indexed statistics can be used to construct versatile test procedures simultaneously sensitive to a wide array of both ordered hazards and stochastic ordering alternatives the complexity of the asymptotic distribution of these statistics precludes obtaining p values through analytical means in this article we develop a monte carlo method for accurately obtaining these p values and we evaluate the moderate sample size properties of this method and compare the power of function indexed statistics with previously developed weighted log rank tests these statistics are also examined in a data analysis of the beta blocker heart attack trial bhat the results of this article demonstrate that

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

graphical methods for assessing violations of the proportional hazards assumption in cox regression a major assumption of the cox proportional hazards model is that the effect of a given covariate does not change over time if this assumption is violated the simple cox model is invalid and more sophisticated analyses are required this paper describes eight graphical methods for detecting violations of the proportional hazards assumption and demonstrates each on three published datasets with a single binary covariate i discuss the relative merits of these methods smoothed plots of the scaled schoenfeld residuals are recommended for assessing ph violations because they provide precise usable information about the time dependence of the covariate effects

effects of mismodelling and mismeasuring explanatory variables on tests of their association with a response variable we consider three commonly used statistical tests for assessing the association between an explanatory variable and a measured binary or survival time response variable and investigate the loss in efficiency from mismodelling or mismeasuring the explanatory variable with respect to mismodelling we examine the consequences of using an incorrect dose metameter in a test for trend of mismodelling a continuous explanatory variable and of discretizing a continuous explanatory variable we also examine the consequences of classification errors for a discrete explanatory variable and of measurement errors for a continuous explanatory variable for all three statistical tests the asymptotic relative efficiency are corresponding to each type of mis specification equals the square of the correlation between the correct and fitted form of the explanatory variable this result is evaluated numerically for the different types of mis specification to provide insight into the selection of tests the interpretation of results and the design of studies where the correct explanatory variable cannot be measured exactly

the robust inference for the cox proportional hazards model abstract we derive the asymptotic distribution of the maximum partial likelihood estimator for the vector of regression coefficients under a possibly misspecified cox proportional hazards model as in the parametric setting this estimator converges to a well defined constant vector in addition the random vector n 1 2 is asymptotically normal with mean 0 and with a covariance matrix that can be consistently estimated the newly proposed robust covariance matrix estimator is similar to the so called sandwich variance estimators that have been extensively studied for parametric cases for many misspecified cox models the asymptotic limit or part of it can be interpreted meaningfully in those circumstances valid statistical inferences about the corresponding covariate effects can be drawn based on the aforementioned asymptotic theory of and the related results for the score statistics extensive studies demonstrate that the proposed robust tests and interval estimation procedures

a clinical trial design using the concept of proportional time using the generalized gamma ratio distribution traditional methods of sample size and power calculations in clinical trials with a time to event end point are based on the logrank test and its variations cox proportional hazards ph assumption or comparison of means of 2 exponential distributions of these sample size calculation based on ph assumption is likely the most common and allows adjusting for the effect of one or more covariates however when designing a trial there are situations when the assumption of ph may not be appropriate additionally when it is known that there is a rapid decline in the survival curve for a control group such as from previously conducted observational studies a design based on the ph assumption may confer only a minor statistical improvement for the treatment group that is neither clinically nor practically meaningful for such scenarios a clinical trial design that focuses on improvement in patient longevity is proposed based on the concept of proportional time using the generalized gamma ratio distribution simulations are conducted to evaluate the performance of the proportional time method and to identify the situations in which such a design will be beneficial as compared to the standard design using a ph assumption piecewise exponential hazards assumption and specific cases of a cure rate model a practical example in which hemorrhagic stroke patients are randomized to 1 of 2 arms in a putative clinical trial demonstrates the usefulness of this approach by drastically reducing the number of patients needed for study enrollment

sample size calculations for ordered categorical data many clinical trials yield data on an ordered categorical scale such as very good good moderate poor under the assumption of proportional odds such data can be analysed using techniques of logistic regression in simple comparisons of two treatments this approach becomes equivalent to the mann whitney test in this paper sample size formulae consistent with an eventual logistic regression analysis are derived the influence on efficiency of the number and breadth of categories will be examined effects of misclassification and of stratification are discussed and examples of the calculations are given

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 formula for clustered survival data using weighted log rank statistics summary we present a simple sample size formula for weighted log rank statistics applied to clustered survival data with variable cluster sizes and arbitrary treatment assignments within clusters this formula is based on the asymptotic normality of weighted log rank statistics under certain local alternatives in the clustered data context we also provide consistent variance estimators the derived sample size formula reduces to schoenfeld s 1983 formula for cases of no clustering or independence within clusters simulation results verify control of the type i error and accuracy of the sample size formula use of the sample size formula in an event driven clinical trial design is illustrated using data from the early treatment diabetic retinopathy study

a monte carlo study of some two sample rank tests with censored data abstract the powers of several nonparametric tests for the two sample problem with censored data are compared by simulation the tests studied include gehan s efron s and peto s and prentice s generalized wilcoxon tests along with the logrank test the test with the greatest power changes with the sample sizes censoring mechanism and distribution of the random variables the test that performed the best overall was the peto prentice generalized wilcoxon statistic with an asymptotic variance estimate

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

sample size determination for comparing more than two survival distributions we examine the asymptotic properties of the tarone and ware and harrington and fleming classes of test statistics under alternative hypotheses when there are comparisons between more than two survival distributions in the presence of arbitrary right censoring when we assume equal censoring distributions across treatment groups and proportional hazards we derive the sample size formula for testing the equality of k 2 survival distributions using the logrank test this work extends schoenfeld s derivation for comparing two survival distributions and also generalizes the results of makuch and simon we also derive the sample size formula for testing monotone dose response using tarone s trend test we then investigate the practicality of the formula in various situations by presenting empirical power with use of monte carlo simulations in addition with stratification present we derive the sample size formula for the stratified logrank test which is an extension of palta and amini

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

generalized martingale residual processes for goodness of fit inference in cox s type regression models in the paper a general class of stochastic processes based on the sums of weighted martingale transform residuals for goodness of fit inference in general cox s type regression models is studied their form makes the inference robust to covariate outliers a weak convergence result for such processes is obtained giving the possibility of establishing the randomness of their graphs together with the construction of the formal x 2 type goodness of fit tests by using the khmaladze innovation approach a modified version of the initial class of processes is also defined weak convergence results for the processes are derived this leads to the main application which concerns the formal construction of the kolmogorov smirnov and cramer von mises type goodness of fit tests this is done within the general situation considered

nonparametric analysis of truncated survival data with application to aids summary one source of data for the induction distribution of aids arises from persons infected by the aids virus from contaminated blood transfusions analyses of these data are complicated because the number of individuals infected by transfusion is unknown information is available only for those who are infected and develop aids within a certain chronologic time interval the statistical problem is one of making inferences about a stochastic process of infection and disease for which realizations are right truncated in chronologic time by considering the process in reverse time we transform the problem to one of analysing survival data that are left truncated in internal time we develop nonparametric methods for estimating and comparing the identifiable aspects of the induction distributions of several groups an important feature of the natural history and population dynamics of aids is the induction period between infection with the aids virus and the onset of clinical aids this time is sometimes referred to as the latency period or incubation period follow up studies of individuals at risk of being infected will provide direct observations from the induction distribution but most of these studies will not be completed for several years one alternative source of information arises from persons infected with the aids virus from a contaminated blood transfusion of persons infected in this way only those who develop aids by a certain date can be identified the total number who are infected is not known a similar sampling distribution arises in the study of pediatric aids children who contract aids as a result of being infected in utero or at birth can be identified but the total number infected in this way is unknown the statistical problem is one of making inferences about a stochastic process of infection and subsequent disease in which realizations are right truncated in chronologic time by considering the process in reverse time the problem can be transformed to one of survival data that are left truncated in internal time we use this relationship to develop nonparametric methods for estimating and comparing the identifiable aspects of the induction distributions for

sample size in cluster randomized trials with time to event as the primary endpoint in cluster randomized trials groups of individuals clusters are randomized to the treatments or interventions to be compared in many of those trials the primary objective is to compare the time for an event to occur between randomized groups and the shared frailty model well fits clustered time to event data members of the same cluster tend to be more similar than members of different clusters causing correlations as correlations affect the power of a trial to detect intervention effects the clustered design has to be considered in planning the sample size in this publication we derive a sample size formula for clustered time to event data with constant marginal baseline hazards and correlation within clusters induced by a shared frailty term the sample size formula is easy to apply and can be interpreted as an extension of the widely used schoenfeld s formula accounting for the clustered design of the trial simulations confirm the validity of the formula and its use also for non constant marginal baseline hazards findings are illustrated on a cluster randomized trial investigating methods of disseminating quality improvement to addiction treatment centers in the usa copyright 2012 john wiley sons ltd

blocked regular fractional factorial designs with maximum estimation capacity in this paper the problem of constructing optimal blocked regular fractional factorial designs is considered the concept of minimum aberration due to fries and hunter is a well accepted criterion for selecting good unblocked fractional factorial designs cheng steinberg and sun showed that a minimum aberration design of resolution three or higher maximizes the number of two factor interactions which are not aliases of main effects and also tends to distribute these interactions over the alias sets very uniformly we extend this to construct block designs in which i no main effect is aliased with any other main effect not confounded with blocks ii the number of two factor interactions that are neither aliased with main effects nor confounded with blocks is as large as possible and iii these interactions are distributed over the alias sets as uniformly as possible such designs perform well under the criterion of maximum estimation capacity a criterion of model robustness which has a direct statistical meaning some general results on the construction of blocked regular fractional factorial designs with maximum estimation capacity are obtained by using a finite projective geometric approach

nonparametric estimation of duration of accrual and total study length for clinical trials nonparametric estimates for all possible values of accrual duration and total study length required to achieve a specified power and level of significance are given assuming a proportional hazards model comparing two treatment groups based on data from a previous representative clinical trial the nonparametric maximum likelihood estimate of total study length and its asymptotic distribution are derived as functions of accrual duration an example using this technique is given for designing a clinical trial comparing two adjuvant chemotherapy regimens for large bowel cancer

minimum aberration two level fractional factorial split plot designs it is often impractical to perform the experimental runs of a fractional factorial in a completely random order in these cases restrictions on the randomization of the experimental trials are imposed and the design is said to have a split plot structure we rank these fractional factorial split plot designs similarly to fractional factorials using the aberration criterion to find the minimum aberration design we introduce an algorithm that constructs the set of all nonisomorphic two level fractional factorial split plot designs more efficiently than existing methods the algorithm can be easily modified to efficiently produce sets of all nonisomorphic fractional factorial designs fractional factorial designs in which the number of levels is a power of a prime and fractional factorial split plot designs in which the number of levels is a power of a prime

sample size evaluation for a multiply matched case control study using the score test from a conditional logistic discrete cox ph regression model the conditional logistic regression model breslow ne covariance adjustment of relative risk estimates in matched studies biometrics 1982 38 661 672 provides a convenient method for the assessment of qualitative or quantitative covariate effects on risk in a study with matched sets each containing a possibly different numbers of cases and controls the conditional logistic likelihood is identical to the stratified cox proportional hazards model likelihood with an adjustment for ties regression models and life tables with discussion j roy statist soc b 1972 34 187 220 this likelihood also applies to a nested case control study with multiply matched cases and controls selected from those at risk at selected event times herein the distribution of the score test for the effect of a covariate in the model is used to derive simple equations to describe the power of the test to detect a coefficient log odds ratio or log hazard ratio or the number of cases or matched sets and controls required to provide a desired level of power additional expressions are derived for a quantitative covariate as a function of the difference in the assumed mean covariate values among cases and controls and for a qualitative covariate in terms of the difference in the probabilities of exposure for cases and controls examples are presented for a nested case control study and a multiply matched case control study

selecting defining contrasts and confounded effects in p n m factorial experiment fractional factorial designs presented in standard tables are not always suitable when some factors are more likely than others to interact if the fraction is small compared to the total number of treatments the selection of defining contrasts to generate a design meeting specified requirements is not straightforward a systematic procedure is given for selecting defining contrasts and confounded effects in p n m factorial designs where p 2 is prime

a sample size formula for the supremum log rank statistic un avantage du test du log rank supremum sur le test du log rank standard est une sensibilite accrue a une plus large variete d alternatives stochastiques ordonnees dans ce papier nous etablissons une formule de calcul de la taille de l echantillon pour des etudes utilisant le test du log rank supremum l idee est de fonder la puissance sur l alternative de hasards proportionnels si bien que le log rank supremum aura la meme puissance que le log rank standard dans la situation ou le log rank standard est optimal cela a pour consequence une legere augmentation de la taille de l echantillon par rapport a celle requise pour le log rank standard par exemple cette augmentation est de 5 733 pour un test bilateral avec une erreur de type i de 0 05 et une puissance de 0 80 cette legere augmentation est compensee par les gains significatifs de puissance que permet le test du log rank supremum pour une large famille d alternatives a hazards non proportionnels une petite etude de simulation est utilisee a titre d illustration ces resultats devraient faciliter une utilisation plus large du test du log rank supremum dans les essais cliniques

a versatile test for equality of two survival functions based on weighted differences of kaplan meier curves with censored event time observations the logrank test is the most popular tool for testing the equality of two underlying survival distributions although this test is asymptotically distribution free it may not be powerful when the proportional hazards assumption is violated various other novel testing procedures have been proposed which generally are derived by assuming a class of specific alternative hypotheses with respect to the hazard functions the test considered by pepe and fleming 1989 is based on a linear combination of weighted differences of the two kaplan meier curves over time and is a natural tool to assess the difference of two survival functions directly in this article we take a similar approach but choose weights that are proportional to the observed standardized difference of the estimated survival curves at each time point the new proposal automatically makes weighting adjustments empirically the new test statistic is aimed at a one sided general alternative hypothesis and is distributed with a short right tail under the null hypothesis but with a heavy tail under the alternative the results from extensive numerical studies demonstrate that the new procedure performs well under various general alternatives with a caution of a minor inflation of the type i error rate when the sample size is small or the number of observed events is small the survival data from a recent cancer comparative study are utilized for illustrating the implementation of the process copyright 2015 john wiley sons ltd

procedures for comparing samples with multiple endpoints summary five procedures are considered for the comparison of two or more multivariate samples these procedures include a newly proposed nonparametric rank sum test and a generalized least squares test also considered are the following tests ordinary least squares hotelling s p and a bonferroni per experiment error rate approach applications are envisaged in which each variable represents a qualitatively different measure of response to treatment the null hypothesis of no treatment difference is tested with power directed towards alternatives in which at least one treatment is uniformly better than the others in all simulations the nonparametric procedure provided relatively good power and accurate control over the size of the test and is recommended for general use alternatively the generalized least squares procedure may also be useful with normally distributed data in moderate or large samples a convenient expression for this procedure is obtained and its asymptotic relative efficiency with respect to the ordinary least squares test is evaluated clinical trials are often conducted for the purpose of evaluating the relative efficacy of two or more modes of therapy when the therapies are administered to independent groups and efficacy is measured on the basis of a single response variable appropriate parametric and nonparametric one way analysis of variance anova procedures and their properties are well known often however efficacy is measured by more than one variable although univariate methods for assessing each characteristic individually are useful in this setting there is often the additional need for a single overall objective probability statement that addresses the question of whether or not the experimental therapy is efficacious this need is particularly acute when the medical question is controversial and the sample size is small although a large literature exists on the comparison of multivariate samples the procedures tend to be infrequently used in practice as pointed out by meier 1975 p 523 the standard analysis for the comparison of two multivariate samples which is based on hotelling s t2 statistic addresses somewhat the wrong question and consequently has very poor power for the alternatives of primary interest specifically the t2 procedure addresses the question are one or more of the treatments different making no distinction between variables that change favorably and variables that change unfavorably a second approach would be to assign per experiment error rates to each of the univariate tests by using bonferroni s inequality i e by multiplying each univariate p value by the number of variables studied the desired overall probability statement is then given by the minimum of the per experiment univariate p values although this approach may be useful in some settings it may lack power for alternatives in which most or all measures of efficacy are improved this will be of particular concern when the number k of endpoints studied is large relative to sample size in fact statistical significance will not be possible in some instances this problem will be exacerbated when the measures of efficacy are highly

sample size reestimation for clinical trials with censored survival data a flexible design with updating of sample size in clinical trials is proposed for censored survival data patients enter trials serially and are subject to random loss to follow up the statistical inference is based on a weighted average of the linear rank statistics where the weight function at each look depends on prior observed data a stopping rule is devised to allow early termination and acceptance of the null hypothesis when the experimental treatment offers no advantage or is inferior to the control the null hypothesis that two survival distributions are equal may be rejected only at the last step when the weight function is used up the overall type i error rate is preserved independent increments for the sequential linear rank statistic is key to deriving asymptotic properties of the test statistic some extensive simulations have been carried out to compare the operating characteristics of the method under different scenarios and for a comparison with the usual log rank test with fixed sam

sample size calculation for the weighted rank statistics with paired survival data this paper introduces a sample size calculation method for the weighted rank test statistics with paired two sample survival data our sample size formula requires specification of joint survival and censoring distributions for modelling the distribution of paired survival variables we may use a paired exponential survival distribution that is specified by the marginal hazard rates and a measure of dependency also in most trials randomizing paired subjects the subjects of each pair are accrued and censored at the same time over an accrual period and an additional follow up period so that the paired subjects have a common censoring time under these practical settings the design parameters include type i and type ii error probabilities marginal hazard rates under the alternative hypothesis correlation coefficient accrual period or accrual rate and follow up period if pilot data are available we may estimate the survival distributions from them but we specify the censoring distribution based on the specified accrual trend and the follow up period planned for the new study through simulations the formula is shown to provide accurate sample sizes under practical settings real studies are taken to demonstrate the proposed method

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