analytic smoothing for equipercentile equating under the common item nonequivalent populations design a cubic spline method for smoothing equipercentile equating relationships under the common item nonequivalent populations design is described statistical techniques based on bootstrap estimation are presented that are designed to aid in choosing an equating method degree of smoothing these include a asymptotic significance tests that compare no equating and linear equating to equipercentile equating b a scheme for estimating total equating error and for dividing total estimated error into systematic and random components the smoothing technique and statistical procedures are explored and illustrated using data from forms of a professional certification test

bootstrapping a regression equation some empirical results abstract the bootstrap like the jackknife is a technique for estimating standard errors the idea is to use monte carlo simulation based on a nonparametric estimate of the underlying error distribution the main object of this article is to present the bootstrap in the context of an econometric equation describing the demand for energy by industry as it turns out the conventional asymptotic formulas for estimating standard errors are too optimistic by factors of nearly three when applied to a particular finite sample problem in a simpler context this finding can be given a mathematical proof

modified replacement randomization we propose a slight modification of replacement randomization by imposing an imbalance criterion within a restricted range of patient entry numbers the bound on imbalance is stated most simply in terms of the maximum allowable difference in treatment assignments at the end of the trial for clinical trials in which two therapies are compared we study by means of computer simulations the dependence of both selection bias and the mean number of randomization list replacements on the parameters defining the procedure

adaptive methods for linear programming decoding detectability of failures of linear programming lp decoding and the potential for improvement by adding new constraints motivate the use of an adaptive approach in selecting the constraints for the underlying lp problem in this paper we make a first step in studying this method and show that by starting from a simple lp problem and adaptively adding the necessary constraints the complexity of lp decoding can be significantly reduced in particular we observe that with adaptive lp decoding the sizes of the lp problems that need to be solved become practically independent of the density of the parity check matrix we further show that adaptively adding extra constraints such as constraints based on redundant parity checks can provide large gains in the performance

some asymptotic theory for the bootstrap efron s bootstrap method of distribution approximation is shown to be asymptotically valid in a large number of situations including t statistics the empirical and quantile processes and von mises functionals some counter examples are also given to show that the approximation does not always succeed

adaptive cut generation algorithm for improved linear programming decoding of binary linear codes linear programming lp decoding approximates maximum likelihood ml decoding of a linear block code by relaxing the equivalent ml integer programming ip problem into a more easily solved lp problem the lp problem is defined by a set of box constraints together with a set of linear inequalities called parity inequalities that are derived from the constraints represented by the rows of a parity check matrix of the code and can be added iteratively and adaptively in this paper we first derive a new necessary condition and a new sufficient condition for a violated parity inequality constraint or cut at a point in the unit hypercube then we propose a new and effective algorithm to generate parity inequalities derived from certain additional redundant parity check rpc constraints that can eliminate pseudocodewords produced by the lp decoder often significantly improving the decoder error rate performance the cut generating algorithm is based upon a specific transformation of an initial parity check matrix of the linear block code we also design two variations of the proposed decoder to make it more efficient when it is combined with the new cut generating algorithm simulation results for several low density parity check ldpc codes demonstrate that the proposed decoding algorithms significantly narrow the performance gap between lp decoding and ml decoding

subsampling for heteroskedastic time series abstract in this article a general theory for the construction of confidence intervals or regions in the context of heteroskedastic dependent data is presented the basic idea is to approximate the sampling distribution of a statistic based on the values of the statistic computed over smaller subsets of the data this method was first proposed by politis and romano 1994b for stationary observations we extend their results to heteroskedastic observations and prove a general asymptotic validity result under minimal conditions in contrast the usual bootstrap and moving blocks bootstrap are typically valid only for asymptotically linear statistics and their justification requires a case by case analysis our general asymptotic results are applied to a regression setting with dependent heteroskedastic errors

better bootstrap confidence intervals abstract we consider the problem of setting approximate confidence intervals for a single parameter in a multiparameter family the standard approximate intervals based on maximum likelihood theory can be quite misleading in practice tricks based on transformations bias corrections and so forth are often used to improve their accuracy the bootstrap confidence intervals discussed in this article automatically incorporate such tricks without requiring the statistician to think them through for each new application at the price of a considerable increase in computational effort the new intervals incorporate an improvement over previously suggested methods which results in second order correctness in a wide variety of problems in addition to parametric families bootstrap intervals are also developed for nonparametric situations

asymptotic results of stochastic decomposition for two stage stochastic quadratic programming this paper presents the stochastic decomposition sd algorithms for two classes of stochastic programming problems 1 two stage stochastic quadratic linear programming sqlp in which a quadrati

adaptive linear programming decoding of nonbinary linear codes over prime fields in this work we consider adaptive linear programming alp decoding of linear codes over prime fields i e the finite fields mathbb f p of size p where p is a prime when used over a p ary input memoryless channel in particular we provide a general construction of valid inequalities using no auxiliary variables for the codeword polytope or the convex hull of the so called constant weight embedding of a single parity check spc code over any prime field the construction is based on sets of vectors called building block classes that are assembled to form the left hand side of an inequality according to several rules in the case of almost doubly symmetric valid classes we prove that the resulting inequalities are all facet defining while we conjecture this to be true if and only if the class is valid and symmetric valid symmetric classes impose certain symmetry conditions on the elements of the vectors from the class while valid doubly symmetric classes impose further technical symmetry conditions for p 3 there is only a single valid symmetric class and we prove that the resulting inequalities together with the so called simplex constraints give a complete and irredundant description of the codeword polytope of the embedded spc code for p 5 we show that there are additional facets beyond those from the proposed construction as an example for p 7 we provide additional inequalities that all define facets of the embedded codeword polytope the resulting overall set of linear in equalities is conjectured to be irredundant and complete such sets of linear in equalities have not appeared in the literature before have a strong theoretical interest and we use them to develop an efficient relaxed alp decoder for general non spc linear codes over prime fields the key ingredient is an efficient separation algorithm based on the principle of dynamic programming furthermore we construct a decoder for linear codes over arbitrary fields mathbb f q with q p m and m 1 by a factor graph representation that reduces to several instances of the case m 1 which results in general in a relaxation of the original decoding polytope finally we present an efficient cut generating algorithm to search for redundant parity checks to further improve the performance towards maximum likelihood decoding for short to medium block lengths numerical experiments confirm that our new decoder is very efficient compared to a static lp decoder for various field sizes check node degrees and block lengths

using linear programming to decode binary linear codes a new method is given for performing approximate maximum likelihood ml decoding of an arbitrary binary linear code based on observations received from any discrete memoryless symmetric channel the decoding algorithm is based on a linear programming lp relaxation that is defined by a factor graph or parity check representation of the code the resulting lp decoder generalizes our previous work on turbo like codes a precise combinatorial characterization of when the lp decoder succeeds is provided based on pseudocodewords associated with the factor graph our definition of a pseudocodeword unifies other such notions known for iterative algorithms including stopping sets irreducible closed walks trellis cycles deviation sets and graph covers the fractional distance d sub frac of a code is introduced which is a lower bound on the classical distance it is shown that the efficient lp decoder will correct up to spl lceil d sub frac 2 spl rceil 1 errors and that there are codes with d sub frac spl omega n sup 1 spl epsi an efficient algorithm to compute the fractional distance is presented experimental evidence shows a similar performance on low density parity check ldpc codes between lp decoding and the min sum and sum product algorithms methods for tightening the lp relaxation to improve performance are also provided

accurate bootstrap confidence limits for the cumulative hazard and survivor functions under random censoring abstract in clinical trials and other settings confidence intervals for survival probabilities at a fixed point in time are often required typically such intervals are constructed on the cumulative hazard scale the quantiles required for computing these intervals are generally obtained using a first order normal theory approximation the appropriateness of such intervals for small sample sizes or under moderate to heavy censoring is suspect we demonstrate that significant improvement is possible for randomly censored data using the nonparametric bootstrap in particular using edgeworth expansion it is shown that the percentile t and bc a bootstrap methodologies based on the studentized nelson aalen estimator yield second order correct confidence limits for the cumulative hazard function at a fixed time point an explicit formula for the correction term of the edgeworth expansion is found easy to compute estimators for the components of the correction term are also found and lead to simple analytic

bootstrapping general empirical measures it is proved that the bootstrapped central limit theorem for empirical processes indexed by a class of functions mathscr f and based on a probability measure p holds a s if and only if mathscr f in mathrm clt p and int f 2 dp infty where f sup f in mathscr f f and it holds in probability if and only if mathscr f in mathrm clt p thus for a large class of statistics no local uniformity of the clt about p is needed for the bootstrap to work consistency of the bootstrap the bootstrapped law of large numbers is also characterized these results are proved under certain weak measurability assumptions on mathscr f

iterative approximate linear programming decoding of ldpc codes with linear complexity the problem of low complexity linear programming lp decoding of low density parity check ldpc codes is considered an iterative algorithm similar to min sum and belief propagation for efficient approximate solution of this problem was proposed by vontobel and koetter in this paper the convergence rate and computational complexity of this algorithm are studied using a scheduling scheme that we propose in particular we are interested in obtaining a feasible vector in the lp decoding problem that is close to optimal in the following sense the distance normalized by the block length between the minimum and the objective function value of this approximate solution can be made arbitrarily small it is shown that such a feasible vector can be obtained with a computational complexity which scales linearly with the block length combined with previous results that have shown that the lp decoder can correct some fixed fraction of errors we conclude that this error correction can be achieved with linear computational complexity this is achieved by first applying the iterative lp decoder that decodes the correct transmitted codeword up to an arbitrarily small fraction of erroneous bits and then correcting the remaining errors using some standard method these conclusions are also extended to generalized ldpc codes

comparisons of minimization and atkinson s algorithm some general points regarding efficiency in clinical trials are made reasons as to why fitting many covariates to adjust the estimate of the treatment effect may be less problematic than commonly supposed are given two methods of dynamic allocation of patients based on covariates minimization and atkinson s approach are compared and contrasted for the particular case where all covariates are binary the results of monte carlo simulations are also presented it is concluded that in the cases considered atkinson s approach is slightly more efficient than minimization although the difference is unlikely to be very important in practice both are more efficient than simple randomization although it is concluded that fitting covariates may make a more valuable and instructive contribution to inferences about treatment effects than only balancing them

optimized design of finite length separable circulant based spatially coupled codes an absorbing set based analysis in this paper we characterize the finite length performance of separable circulant based spatially coupled scb sc ldpc codes for transmission over the additive white gaussian noise channel for a general class of finite length graph based codes it is known that the existence of small absorbing sets causes a performance degradation in the error floor regime we first present the mathematical conditions for the existence of absorbing sets in binary scb sc codes this analysis enables us to find the exact number of absorbing sets as a function of the design parameters in particular our results show that the choice of the cutting vector affects the number of absorbing sets and therefore the error floor performance of the code for a fixed column weight we find provably optimal cutting vectors that result in the least number of absorbing sets furthermore we extend our analysis to nonbinary scb sc codes where we show that the choice of the cutting vector is not as critical as in the binary case we provide an algorithm which provably removes the problematic nonbinary absorbing sets from nonbinary scb sc codes by informed selection of edge labels our simulation results show the superior error floor performance of our designed binary and nonbinary scb sc codes compared with binary unstructured and nonbinary quasi cyclic sc codes available in the open literature

sieve bootstrap with variable length markov chains for stationary categorical time series a bootstrap for stationary categorical time series based on the method of sieves is studied here the data generating process is approximated by the so called variable length markov chain vlmc a flexible class of markov models that allows for parsimonious structure then the resampling is given by simulating from the fitted model it is shown that for a whole class of stationary categorical time series that is more general than vlmc the vlmc sieve has faster rate of convergence for variance estimation than the more general block bootstrap results are illustrated from a theoretical and empirical perspective for the latter a real data application about in homogeneity classification of a dna strand is also presented finally the vlmc sieve scheme enjoys an implementational advantage of using the plug in rule for bootstrapping a statistical procedure which generally is not the case for the block method

on the bootstrap and continuity correction on propose une correction de continuite utile lorsque le bootstrap est utilise pour approcher des distributions de moyennes d echantillons a valeurs sur un treillis

asymptotic distribution of auc nris and idi based on theory of u statistics the change in area under the curve auc the integrated discrimination improvement idi and net reclassification index nri are commonly used measures of risk prediction model performance some authors have reported good validity of associated methods of estimating their standard errors se and construction of confidence intervals whereas others have questioned their performance to address these issues we unite the auc idi and three versions of the nri under the umbrella of the u statistics family we rigorously show that the asymptotic behavior of auc nris and idi fits the asymptotic distribution theory developed for u statistics we prove that the auc nris and idi are asymptotically normal unless they compare nested models under the null hypothesis in the latter case asymptotic normality and existing se estimates cannot be applied to auc nris or idi in the former case se formulas proposed in the literature are equivalent to se formulas obtained from u statistics theory if we ignore adjustment for estimated parameters we use sukhatme randles dewet condition to determine when adjustment for estimated parameters is necessary we show that adjustment is not necessary for ses of the auc and two versions of the nri when added predictor variables are significant and normally distributed the ses of the idi and three category nri should always be adjusted for estimated parameters these results allow us to define when existing formulas for se estimates can be used and when resampling methods such as the bootstrap should be used instead when comparing nested models we also use the u statistic theory to develop a new se estimate of auc copyright 2017 john wiley sons ltd

a likelihood ratio test against stochastic ordering in several populations abstract the likelihood ratio test is often used to test hypotheses involving a stochastic ordering distribution theory for the likelihood ratio test has been developed only for two stochastically ordered distributions for testing equality of distributions against a stochastic ordering in several populations this paper derives the null asymptotic distribution of the likelihood ratio test statistic which is characterized by minimization problems and has no closed form a monte carlo simulation is conducted to study the limiting distribution because the limiting distribution depends on the specific values of the unknown distributions under the null hypothesis asymptotic and bootstrap approaches are proposed to overcome practical difficulties and implement tests based on the likelihood principle asymptotic validities for these tests are established and simulations are carried out to check their performances for finite sample sizes the tests are applied to an example involving data for survival time f

low complexity lp decoding of nonbinary linear codes linear programming lp decoding of low density parity check ldpc codes has attracted much attention in the research community in the past few years lp decoding has been derived for binary and nonbinary linear codes however the most important problem with lp decoding for both binary and nonbinary linear codes is that the complexity of standard lp solvers such as the simplex algorithm remains prohibitively large for codes of moderate to large block length to address this problem two low complexity lp lclp decoding algorithms for binary linear codes have been proposed by vontobel and koetter henceforth called the basic lclp decoding algorithm and the subgradient lclp decoding algorithm in this paper we generalize these lclp decoding algorithms to nonbinary linear codes the computational complexity per iteration of the proposed nonbinary lclp decoding algorithms scales linearly with the block length of the code a modified bcjr algorithm for efficient check node calculations in the nonbinary basic lclp decoding algorithm is also proposed which has complexity linear in the check node degree several simulation results are presented for nonbinary ldpc codes defined over z4 gf 4 and gf 8 using quaternary phase shift keying and 8 phase shift keying respectively over the awgn channel it is shown that for some group structured ldpc codes the error correcting performance of the nonbinary lclp decoding algorithms is similar to or better than that of the min sum decoding algorithm

decomposition methods for large scale lp decoding when binary linear error correcting codes are used over symmetric channels a relaxed version of the maximum likelihood decoding problem can be stated as a linear program lp this lp decoder can be used to decode error correcting codes at bit error rates comparable to state of the art belief propagation bp decoders but with significantly stronger theoretical guarantees however lp decoding when implemented with standard lp solvers does not easily scale to the block lengths of modern error correcting codes in this paper we draw on decomposition methods from optimization theory specifically the alternating direction method of multipliers admm to develop efficient distributed algorithms for lp decoding the key enabling technical result is a two slice characterization of the parity polytope the polytope formed by taking the convex hull of all codewords of the single parity check code this new characterization simplifies the representation of points in the polytope using this simplification we develop an efficient algorithm for euclidean norm projection onto the parity polytope this projection is required by the admm decoder and its solution allows us to use lp decoding with all its theoretical guarantees to decode large scale error correcting codes efficiently we present numerical results for ldpc codes of lengths more than 1000 the waterfall region of lp decoding is seen to initiate at a slightly higher snr than for sum product bp however an error floor is not observed for lp decoding which is not the case for bp our implementation of lp decoding using the admm executes as fast as our baseline sum product bp decoder is fully parallelizable and can be seen to implement a type of message passing with a particularly simple schedule

bootstrap methods for testing homogeneity of variances this article describes the use of bootstrap methods for the problem of testing homogeneity of variances when means are not assumed equal or known the methods are new in this context and allow the use of normal theory test statistics such as f s 2 1 s 2 2 without the normality assumption that is crucial for validity of critical values obtained from the f distribution both asymptotic analysis and monte carlo sampling show that the new resampling procedures compare favorably with older methods in terms of test validity and power

performance bounds for nonbinary linear block codes over memoryless symmetric channels the performance of nonbinary linear block codes is studied in this paper via the derivation of new upper bounds on the block error probability under maximum likelihood ml decoding the transmission of these codes is assumed to take place over a memoryless and symmetric channel the new bounds which are based on the gallager bounds and their variations are applied to the gallager ensembles of nonbinary and regular low density parity check ldpc codes these upper bounds are also compared with sphere packing lower bounds this study indicates that the new upper bounds are useful for the performance evaluation of coded communication systems which incorporate nonbinary coding techniques

on the bootstrap and confidence intervals we derive an explicit formula for the first term in an unconditional edgeworth type expansion of coverage probability for the nonparametric bootstrap technique applied to a very broad class of studentized statistics the class includes sample mean k sample mean sample correlation coefficient maximum likelihood estimators expressible as functions of vector means etc we suggest that the bootstrap is really an empiric one term edgeworth inversion with the bootstrap simulations implicitly estimating the first term in an edgeworth expansion this view of the bootstrap is reinforced by our discussion of the iterated bootstrap which inverts an edgeworth expansion to arbitrary order by simulating simulations

nonparametric estimation of species richness two procedures the jackknife and the bootstrap are discussed as methods for estimating the number of species by the sampling of quadrats explicit formulas for both procedures are presented and evaluated under a model with a random distribution of individuals the jackknife and bootstrap are shown to reduce the bias although they underestimate the actual number of species if there is a large number of rare species and the number of quadrats sampled is small when a small number of quadrats is sampled the jackknife is shown to give better estimates when the number of quadrats is large the jackknife tends to overestimate the number of species and the bootstrap performs better

estimating the asymptotic covariance matrix for quantile regression models a monte carlo study abstract this monte carlo study examines several estimation procedures of the asymptotic covariance matrix in the quantile and censored quantile regression models design matrix bootstrap error bootstrapping order statistic sigma bootstrap homoskedastic kernel and heteroskedastic kernel the monte carlo samples are drawn from two alternative data sets 1 a the unaltered current population survey cps for 1987 and 2 b this cps data with independence between error term and regressors imposed this special setup allows one to evaluate the estimators under various realistic scenarios the results favor the design bootstrap for the general case but also support the order statistic when the error term is independent of the regressors

second order correctness of the poisson bootstrap abstract rao pathak and koltchinskii 1997 have recently studied a sequential approach to resampling in which resampling is carried out sequentially one by one with replacement each time until the bootstrap sample contains m approximately 1 e 1 n approximately 632n distinct observations from the original sample in our previous work we have established that the main empirical characteristics of the sequential bootstrap go through in the sense of being within a distance of order o n 3 4 from those of the usual bootstrap however the theoretical justification of the second order correctness of the sequential bootstrap is somewhat difficult it is the main topic of this investigation among other things we accomplish it by approximating our sequential scheme by a resampling scheme based on the poisson distribution with mean u 1 and censored at x 0

a separation algorithm for improved lp decoding of linear block codes maximum likelihood ml decoding is the optimal decoding algorithm for arbitrary linear block codes and can be written as an integer programming ip problem feldman relaxed this ip problem and presented linear programming lp based decoding in this paper we propose a new separation algorithm to improve the error correcting performance of lp decoding for binary linear block codes we use an ip formulation with indicator variables that help in detecting the violated parity checks we derive gomory cuts from the ip and use them in our separation algorithm an efficient method of finding cuts induced by redundant parity checks rpc is also proposed under certain circumstances we can guarantee that these rpc cuts are valid and cut off the fractional optimal solutions of lp decoding it is demonstrated on three ldpc codes and two bch codes that our separation algorithm performs significantly better than lp decoding and belief propagation bp decoding

mathematical programming decoding of binary linear codes theory and algorithms mathematical programming is a branch of applied mathematics and has recently been used to derive new decoding approaches challenging established but often heuristic algorithms based on iterative message passing concepts from mathematical programming used in the context of decoding include linear integer and nonlinear programming network flows notions of duality as well as matroid and polyhedral theory this paper reviews and categorizes decoding methods based on mathematical programming approaches for binary linear codes over binary input memoryless symmetric channels

a consistent bootstrap procedure for the maximum score estimator abstract in this paper we propose a new model based smoothed bootstrap procedure for making inference on the maximum score estimator of manski 1975 1985 and prove its consistency we provide a set of sufficient conditions for the consistency of any bootstrap procedure in this problem we compare the finite sample performance of different bootstrap procedures through simulation studies the results indicate that our proposed smoothed bootstrap outperforms other bootstrap schemes including the m out of n bootstrap additionally we prove a convergence theorem for triangular arrays of random variables arising from binary choice models which may be of independent interest

iterative linear programming decoding of nonbinary ldpc codes with linear complexity the problem of low complexity linear programming lp decoding of nonbinary low density parity check ldpc codes is considered and an iterative lp decoding algorithm is presented results that were previously derived for binary ldpc codes are extended to the nonbinary case both simple and generalized nonbinary ldpc codes are considered it is shown how the algorithm can be implemented efficiently using a finite field fast fourier transform then the convergence rate of the algorithm is analyzed the complexity of the algorithm scales linearly in the block length and it can approximate up to an arbitrarily small relative error the objective function of the exact lp solution when applied to a typical code from an appropriate nonbinary ldpc code ensemble the algorithm can correct a constant fraction of errors in linear in the block length computational complexity computer experiments with the new iterative lp decoding algorithm show that in the error floor region it can have better performance compared to belief propagation decoding with similar computational requirements

coverage probabilities of bootstrap confidence intervals for quantiles an asymptotic expansion of length 2 is established for the coverage probabilities of confidence intervals for the underlying q quantile which are derived by bootstrapping the sample q quantile the corresponding level error turns out to be of order o n 1 2 which is unexpectedly low a confidence interval of even more practical use is derived by using backward critical points the resulting confidence interval is of the same length as the one derived by ordinary bootstrap but it is distribution free and has higher coverage probability

the moving blocks bootstrap and robust inference for linear least squares and quantile regressions abstract this paper presents the moving blocks bootstrap mbb as a new inference procedure for linear regression estimation which is robust to heteroskedasticity and autocorrelation of unknown forms the mbb covariance estimator is shown to provide heteroskedasticity and autocorrelation consistent hac standard errors for least squares ls and quantile regression qr coefficient estimators the mbb covariance estimator is shown to be asymptotically equivalent to the bartlett kernel estimator suggested by newey and west 1987 and the asymptotically optimal choice of the blocksize is discussed a monte carlo study is included showing that the mbb fares well in comparison to standard hac inference procedures considering strong mixing data generating processes the paper extends existing asymptotic results for the qr estimator the analogy to the ls case is stressed the paper analyzes both the cases of stochastic and of nonstochastic regressors and suggests two new grenander like conditions for the latter case the use of the mbb approach is illustrated for a practical example using a standard econometric package

bootstrap procedures under some non i i d models it is shown in this article that the classical i i d bootstrap remains a valid procedure for estimating the sampling distributions of certain symmetric estimators of location as long as the random observations are independently drawn from distributions with essentially a common location this may be viewed as a robust property of the classical i i d bootstrap also included is a study of the second order properties of a different bootstrap procedure proposed by wu in the context of heteroscedasticity in regression

randomization with a posteriori constraints description and properties the use of randomization for assigning patients to treatment groups in clinical trials is firmly acknowledged as providing the best quality results two standard methods are used in order to achieve well balanced groups with respect to prognostic factors i e factors influencing the disease outcome stratification and minimization stratification is recommended when the number of strata is not too high otherwise minimization is preferred however minimization may compromise blinding since the search for balance is performed a priori and furthermore use of the technique has been questioned by the european agency for the evaluation of medicinal products we have developed a new procedure for adaptive randomization which we have named randomization with a posteriori constraints by using a search for balance a posteriori this procedure ensures that patient groups are similar with respect to prognostic factors while being less vulnerable to selection bias the aim of this work was to describe the new method and to compare it using simulations with stratification and minimization in the case of trials with few prognostic factors the recourse to minimization or randomization with a posteriori constraints does not appear to be useful in such a context stratification has suitable properties and its simplicity of implementation encourages its use however when the number of prognostic factors is higher randomization with a posteriori constraints is less predictable than minimization and the chance of imbalance is lower than for stratification in conclusion randomization with a posteriori constraints with an adequate threshold seems to be a good compromise between minimization and stratification copyright 2007 john wiley sons ltd