non deterministic polynomial optimization problems and their approximations abstract a unified and general framework for the study of nondeterministic polynomial optimization problems npop is presented and some properties of npop s are investigated a characterization of npop s with regard to their approximability properties is given by proving necessary and sufficient conditions for two approximability schemes known approximability results are shown to fit within the general frame developed in the paper finally npop s are classified and studied with regard to the possibility or impossibility of reducing certain types of npop s to other types in a sense specified in the text

approximating the domatic number a set of vertices in a graph is a dominating set if every vertex outside the set has a neighbor in the set the domatic number problem is that of partitioning the vertices of a graph into the maximum number of disjoint dominating sets let n denote the number of vertices delta the minimum degree and delta the maximum degree we show that every graph has a domatic partition with 1 o 1 delta 1 ln n dominating sets and moreover that such a domatic partition can be found in polynomial time this implies a 1 o 1 ln n approximation algorithm for domatic number since the domatic number is always at most delta 1 we also show this to be essentially best possible namely extending the approximation hardness of set cover by combining multiprover protocols with zero knowledge techniques we show that for every epsilon 0 a 1 epsilon ln n approximation implies that np subseteq dtime n o log log n this makes domatic number the first natural maximization problem known to the authors that is provably approximable to within polylogarithmic factors but no better we also show that every graph has a domatic partition with 1 o 1 delta 1 ln delta dominating sets where the o 1 term goes to zero as delta increases this can be turned into an efficient algorithm that produces a domatic partition of omega delta ln delta sets

randomization inference for treatment effect variation summary applied researchers are increasingly interested in whether and how treatment effects vary in randomized evaluations especially variation that is not explained by observed covariates we propose a model free approach for testing for the presence of such unexplained variation to use this randomization based approach we must address the fact that the average treatment effect which is generally the object of interest in randomized experiments actually acts as a nuisance parameter in this setting we explore potential solutions and advocate for a method that guarantees valid tests in finite samples despite this nuisance we also show how this method readily extends to testing for heterogeneity beyond a given model which can be useful for assessing the sufficiency of a given scientific theory we finally apply our method to the national head start impact study which is a large scale randomized evaluation of a federal preschool programme finding that there is indeed significant unexplained treatment effect variation

bootstrap tests for distributional treatment effects in instrumental variable models this article considers the problem of assessing the distributional consequences of a treatment on some outcome variable of interest when treatment intake is possibly nonrandomized but there is a binaryinstrument available for the researcher such a scenario is common in observational studies and in randomized experiments with imperfect compliance one possible approach to this problem is to compare the counterfactual cumulative distribution functions of the outcome with and without the treatment this article shows how to estimate these distributions using instrumental variable methods and a simple bootstrap procedure is proposed to test distributional hypotheses such as equality of distributions first order and second order stochastic dominance these tests and estimators are applied to the study of the effects of veteran status on the distribution of civilian earnings the results show a negative effect of military service during the vietnam era that appears to be concentrated on the lower tail of

approximate solution of np optimization problems abstract this paper presents the main results obtained in the field of approximation algorithms in a unified framework most of these results have been revisited in order to emphasize two basic tools useful for characterizing approximation classes that is combinatorial properties of problems and approximation preserving reducibilities in particular after reviewing the most important combinatorial characterizations of the classes ptas and fptas we concentrate on the class apx and as a concluding result we show that this class coincides with the class of optimization problems which are reducible to the maximum satisfiability problem with respect to a polynomial time approximation preserving reducibility

likelihood based analysis of causal effects of job training programs using principal stratification government sponsored job training programs must be subject to evaluation to assess whether their effectiveness justifies their cost to the public the evaluation usually focuses on employment and total earnings although the effect on wages is also of interest because this effect reflects the increase in human capital due to the training program whereas the effect on total earnings may be simply reflecting the increased likelihood of employment without any effect on wage rates estimating the effects of training programs on wages is complicated by the fact that even in a randomized experiment wages are truncated or less accurately censored by nonemployment that is they are only observed and well defined for individuals who are employed in this article we develop a likelihood based approach to estimate the wage effect of the us federally funded job corps training program using principal stratification our estimands are formulated in terms of 1 the effect of the training program on wages

the power of optimization over randomization in designing experiments involving small samples random assignment typically seen as the standard in controlled trials aims to make experimental groups statistically equivalent before treatment however with a small sample which is a practical reality in many disciplines randomized groups are often too dissimilar to be useful we propose an approach based on discrete linear optimization to create groups whose discrepancy in their means and variances is several orders of magnitude smaller than with randomization we provide theoretical and computational evidence that groups created by optimization have exponentially lower discrepancy than those created by randomization and that this allows for more powerful statistical inference

inference on the quantile regression process tests based on the quantile regression process can be formulated like the classical kolmogorov smirnov and cramer von mises tests of goodness of fit employing the theory of bessel processes as in kiefer 1959 however it is frequently desirable to formulate hypotheses involving unknown nuisance parameters thereby jeopardizing the distribution free character of these tests we characterize this situation as the durbin problem since it was posed in durbin 1973 for parametric empirical processes in this paper we consider an approach to the durbin problem involving a martingale transformation of the parametric empirical process suggested by khmaladze 1981 and show that it can be adapted to a wide variety of inference problems involving the quantile regression process in particular we suggest new tests of the location shift and location scale shift models that underlie much of classical econometric inference the methods are illustrated with a reanalysis of data on unemployment durations from the pennsylvania reemployment bonus experiments the pennsylvania experiments conducted in 1988 89 were designed to test the efficacy of cash bonuses paid for early reemployment in shortening the duration of insured unemployment spells

proof verification and the hardness of approximation problems we show that every language in np has a probablistic verifier that checks membership proofs for it using logarithmic number of random bits and by examining a constant number of bits in the proof if a string is in the language then there exists a proof such that the verifier accepts with probability 1 i e for every choice of its random string for strings not in the language the verifier rejects every provided proof with probability at least 1 2 our result builds upon and improves a recent result of arora and safra 1998 whose verifiers examine a nonconstant number of bits in the proof though this number is a very slowly growing function of the input length as a consequence we prove that no max snp hard problem has a polynomial time approximation scheme unless np p the class max snp was defined by papadimitriou and yannakakis 1991 and hard problems for this class include vertex cover maximum satisfiability maximum cut metric tsp steiner trees and shortest superstring we also improve upon the clique hardness results of feige et al 1996 and arora and safra 1998 and show that there exists a positive e such that approximating the maximum clique size in an n vertex graph to within a factor of n e is np hard

general approximation algorithms for some arithmetical combinatorial problems abstract a general approximation technique for a large class of np hard optimization problems which involve arithmetic calculations is given this technique guarantees a worst case relative error smaller than e in time which is polynomial both in the size of the problem instance and 1 e it is also shown that problems in that class which are not approximable by this technique are not approximable in polynomial time at all provided p np and hence this technique is the most general approximation technique applicable to this class

general forms of finite population central limit theorems with applications to causal inference abstractfrequentists inference often delivers point estimators associated with confidence intervals or sets for parameters of interest constructing the confidence intervals or sets requires understanding the sampling distributions of the point estimators which in many but not all cases are related to asymptotic normal distributions ensured by central limit theorems although previous literature has established various forms of central limit theorems for statistical inference in super population models we still need general and convenient forms of central limit theorems for some randomization based causal analyses of experimental data where the parameters of interests are functions of a finite population and randomness comes solely from the treatment assignment we use central limit theorems for sample surveys and rank statistics to establish general forms of the finite population central limit theorems that are particularly useful for proving asymptotic distributions of randomization tests under th

a sharp lower bound for the geometric genus and zariski multiplicity question it is well known that the geometric genus and multiplicity are two important invariants for isolated singularities in this paper we give a sharp lower estimate of the geometric genus in terms of the multiplicity for isolated hypersurface singularities in 1971 zariski asked whether the multiplicity of an isolated hypersurface singularity depends only on its embedded topological type this problem remains unsettled in this paper we answer positively zariski s multiplicity question for isolated hypersurface singularity if milnor number or geometric genus is small

using regression models to analyze randomized trials asymptotically valid hypothesis tests despite incorrectly specified models regression models are often used to test for cause effect relationships from data collected in randomized trials or experiments this practice has deservedly come under heavy scrutiny because commonly used models such as linear and logistic regression will often not capture the actual relationships between variables and incorrectly specified models potentially lead to incorrect conclusions in this article we focus on hypothesis tests of whether the treatment given in a randomized trial has any effect on the mean of the primary outcome within strata of baseline variables such as age sex and health status our primary concern is ensuring that such hypothesis tests have correct type i error for large samples our main result is that for a surprisingly large class of commonly used regression models standard regression based hypothesis tests but using robust variance estimators are guaranteed to have correct type i error for large samples even when the models are incorrectly specified to the best of our knowledge this robustness of such model based hypothesis tests to incorrectly specified models was previously unknown for poisson regression models and for other commonly used models we consider our results have practical implications for understanding the reliability of commonly used model based tests for analyzing randomized trials

semiparametric exponential families for heavy tailed data we propose a semiparametric method for fitting the tail of a heavy tailed population given a relatively small sample from that population and a larger sample from a related background population we model the tail of the small sample as an exponential tilt of the better observed large sample tail using a robust sufficient statistic motivated by extreme value theory in particular our method induces an estimator of the small population mean and we give theoretical and empirical evidence that this estimator outperforms methods that do not use the background sample we demonstrate substantial efficiency gains over competing methods in simulation and on data from a large controlled experiment conducted by facebook

a threshold of ln n for approximating set cover given a collection f of subsets of s 1 n setcover is the problem of selecting as few as possiblesubsets from f such that their union covers s and maxk cover is the problem of selecting k subsets from f such that their union has maximum cardinality both these problems arenp hard we prove that 1 o 1 ln n is a threshold below which setcover cannot be approximated efficiently unless np has slightlysuperpolynomial time algorithms this closes the gap up to low orderterms between the ratio of approximation achievable by the greedyalogorithm which is 1 o 1 lnn and provious results of lund and yanakakis that showed hardness ofapproximation within a ratio of log 2 n 2s0 72 ln n for max k cover we show an approximationthreshold of 1 1 e up tolow order terms under assumption that p np

lasso adjustments of treatment effect estimates in randomized experiments we provide a principled way for investigators to analyze randomized experiments when the number of covariates is large investigators often use linear multivariate regression to analyze randomized experiments instead of simply reporting the difference of means between treatment and control groups their aim is to reduce the variance of the estimated treatment effect by adjusting for covariates if there are a large number of covariates relative to the number of observations regression may perform poorly because of overfitting in such cases the least absolute shrinkage and selection operator lasso may be helpful we study the resulting lasso based treatment effect estimator under the neyman rubin model of randomized experiments we present theoretical conditions that guarantee that the estimator is more efficient than the simple difference of means estimator and we provide a conservative estimator of the asymptotic variance which can yield tighter confidence intervals than the difference of means estimator simulation and data examples show that lasso based adjustment can be advantageous even when the number of covariates is less than the number of observations specifically a variant using lasso for selection and ordinary least squares ols for estimation performs particularly well and it chooses a smoothing parameter based on combined performance of lasso and ols

repeated randomization and matching in multi arm trials cluster randomized trials with relatively few clusters have been widely used in recent years for evaluation of health care strategies the balance match weighted bmw design introduced in xu and kalbfleisch 2010 biometrics 66 813 823 applies the optimal full matching with constraints technique to a prospective randomized design with the aim of minimizing the mean squared error mse of the treatment effect estimator this is accomplished through consideration of m independent randomizations of the experimental units and then selecting the one which provides the most balance evaluated by matching on the estimated propensity scores often in practice clinical trials may involve more than two treatment arms and multiple treatment options need to be evaluated therefore we consider extensions of the bmw propensity score matching method to allow for studies with three or more arms in this article we propose three approaches to extend the bmw design to clinical trials with more than two arms and evaluate the property of the extended design in simulation studies

using mixed integer programming for matching in an observational study of kidney failure after surgery this article presents a new method for optimal matching in observational studies based on mixed integer programming unlike widely used matching methods based on network algorithms which attempt to achieve covariate balance by minimizing the total sum of distances between treated units and matched controls this new method achieves covariate balance directly either by minimizing both the total sum of distances and a weighted sum of specific measures of covariate imbalance or by minimizing the total sum of distances while constraining the measures of imbalance to be less than or equal to certain tolerances the inclusion of these extra terms in the objective function or the use of these additional constraints explicitly optimizes or constrains the criteria that will be used to evaluate the quality of the match for example the method minimizes or constrains differences in univariate moments such as means variances and skewness differences in multivariate moments such as correlations between covari

rerandomization and regression adjustment author s li x ding p abstract 2019 royal statistical society randomization is a basis for the statistical inference of treatment effects without strong assumptions on the outcome generating process appropriately using covariates further yields more precise estimators in randomized experiments r a fisher suggested blocking on discrete covariates in the design stage or conducting analysis of covariance in the analysis stage we can embed blocking in a wider class of experimental design called rerandomization and extend the classical analysis of covariance to more general regression adjustment rerandomization trumps complete randomization in the design stage and regression adjustment trumps the simple difference in means estimator in the analysis stage it is then intuitive to use both rerandomization and regression adjustment under the randomization inference framework we establish a unified theory allowing the designer and analyser to have access to different sets of covariates we find that asymptotically for any given estimator with or without regression adjustment rerandomization never hurts either the sampling precision or the estimated precision and for any given design with or without rerandomization our regression adjusted estimator never hurts the estimated precision therefore combining rerandomization and regression adjustment yields better coverage properties and thus improves statistical inference to quantify these statements theoretically we discuss optimal regression adjusted estimators in terms of the sampling precision and the estimated precision and then measure the additional gains of the designer and the analyser we finally suggest the use of rerandomization in the design and regression adjustment in the analysis followed by the huber white robust standard error

optimum biased coin designs for sequential treatment allocation with covariate information randomized optimum designs of biased coin type are compared with other strategies for the sequential allocation of two or more treatments in a clinical trial the emphasis is on the variance of estimated treatment contrasts this variance which depends on the design strategy employed may be interpreted as the number of patients on whom information is lost simulations provide clear plots of the evolution of this loss during the course of the clinical trial

rerandomization to balance tiers of covariates when conducting a randomized experiment if an allocation yields treatment groups that differ meaningfully with respect to relevant covariates groups should be rerandomized the process involves specifying an explicit criterion for whether an allocation is acceptable based on a measure of covariate balance and rerandomizing units until an acceptable allocation is obtained here we illustrate how rerandomization could have improved the design of an already conducted randomized experiment on vocabulary and mathematics training programs then provide a rerandomization procedure for covariates that vary in importance and finally offer other extensions for rerandomization including methods addressing computational efficiency when covariates vary in a priori importance better balance should be required for more important covariates rerandomization based on mahalanobis distance preserves the joint distribution of covariates but balances all covariates equally here we propose rerandomizing based on ma

p values maximized over a confidence set for the nuisance parameter abstract for testing problems of the form h 0 v v 0 with unknown nuisance parameter various methods are used to deal with the simplest approach is exemplified by the t test where the unknown variance is replaced by the sample variance and the t distribution accounts for estimation of the variance in other problems such as the 2 2 contingency table one conditions on a sufficient statistic for 0 and proceeds as in fisher s exact test because neither of these standard methods is appropriate for all situations this article suggests a new method for handling the unknown this new method is a simple modification of the formal definition of a p value that involves taking a maximum over the nuisance parameter space of a p value obtained for the case when is known the suggested modification is to restrict the maximization to a confidence set for the nuisance parameter after giving a brief justification we give various examples to show how this new method gives improved results for 2 2 tabl

inference under covariate adaptive randomization this article studies inference for the average treatment effect in randomized controlled trials with covariate adaptive randomization here by covariate adaptive randomization we mean randomizati

completeness in approximation classes we introduce a formal framework for studying approximation properties of np optimization npo problems the classes we consider are those appearing in the literature namely the class of approximable problems within a constant e apx the class of problems having a polynomial time approximation scheme pas and the class of problems having a fully polynomial time approximation scheme fpas we define natural approximation preserving reductions and obtain completeness results for these classes a complete problem in a class can not have stronger approximation properties unless p np we also show that the degree structure of npo allows intermediate degrees that is if p np there are problems which are neither complete nor belong to a lower class

on the hardness of approximating minimization problems we prove results indicating that it is hard to compute efficiently good approximate solutions to the graph coloring set covering and other related minimization problems specifically there is an e 0 such that graph coloring cannot be approximated with ratio n e unless p np set covering cannot be approximated with ratio c log n for any c 1 4 unless np is contained in dtime n poly log n similar results follow for related problems such as clique cover fractional chromatic number dominating set and others

randomization matching and propensity scores in the design and analysis of experimental studies with measured baseline covariates in many experimental situations researchers have information on a number of covariates prior to randomization this information can be used to balance treatment assignment with respect to these covariates as well as in the analysis of the outcome data in this paper we investigate the use of propensity scores in both of these roles we also introduce a randomization procedure in which the balance of all measured covariates is approximately indexed by the variance of the empirical propensity scores and randomization is restricted to those permutations with the least variable propensity scores this procedure is compared with recently proposed methods in terms of resulting covariate balance and estimation efficiency properties of the estimators resulting from each procedure are compared with estimates which incorporate the propensity score in the analysis stage simulation results show that analytical adjustment for the propensity score yields results on par with those obtained through restricted randomization procedures and can be used in conjunction with such procedures to further improve inferential efficiency

matching on the fly sequential allocation with higher power and efficiency we propose a dynamic allocation procedure that increases power and efficiency when measuring an average treatment effect in fixed sample randomized trials with sequential allocation subjects arrive iteratively and are either randomized or paired via a matching criterion to a previously randomized subject and administered the alternate treatment we develop estimators for the average treatment effect that combine information from both the matched pairs and unmatched subjects as well as an exact test simulations illustrate the method s higher efficiency and power over several competing allocation procedures in both simulations and in data from a clinical trial

approximation algorithms for the maximum hamiltonian path problem with specified endpoint s abstract this paper deals with the problem of constructing hamiltonian paths of optimal weight called hpps t if the two endpoints are specified hpps if only one endpoint is specified we show that hpps t is 1 2 differential approximable and hpps is 2 3 differential approximable moreover we observe that these problems cannot be differential approximable better than 741 742 based upon these results we obtain new bounds for standard ratio a 1 2 standard approximation for m ax hpps t and a 2 3 for m ax hpps which can be improved to 2 3 for m ax hpps t a 2a all the edge weights are within an interval a 2a to 5 6 for m ax hpps a 2a and to 2 3 for m in hpps t a 2a to 3 4 for m in hpps a 2a

recursive partitioning for heterogeneous causal effects in this paper we propose methods for estimating heterogeneity in causal effects in experimental and observational studies and for conducting hypothesis tests about the magnitude of differences in treatment effects across subsets of the population we provide a data driven approach to partition the data into subpopulations that differ in the magnitude of their treatment effects the approach enables the construction of valid confidence intervals for treatment effects even with many covariates relative to the sample size and without sparsity assumptions we propose an honest approach to estimation whereby one sample is used to construct the partition and another to estimate treatment effects for each subpopulation our approach builds on regression tree methods modified to optimize for goodness of fit in treatment effects and to account for honest estimation our model selection criterion anticipates that bias will be eliminated by honest estimation and also accounts for the effect of making additional splits on the variance of treatment effect estimates within each subpopulation we address the challenge that the ground truth for a causal effect is not observed for any individual unit so that standard approaches to cross validation must be modified through a simulation study we show that for our preferred method honest estimation results in nominal coverage for 90 confidence intervals whereas coverage ranges between 74 and 84 for nonhonest approaches honest estimation requires estimating the model with a smaller sample size the cost in terms of mean squared error of treatment effects for our preferred method ranges between 7 22

reductions completeness and the hardness of approximability in computability and in complexity theory reductions are widely used for mapping sets into sets in order to prove undecidability or hardness results in the study of the approximate solvability of hard discrete optimization problems suitable kinds of reductions called approximation preserving reductions can also be used to transfer from one problem to another either positive results solution techniques or negative results non approximability results in this paper various kinds of approximation preserving reductions are surveyed and their properties discussed the role of completeness under approximation preserving reductions is also analyzed and its relationship with hardness of approximability is explained

single machine batch scheduling of linear deteriorating jobs we consider the problem of scheduling jobs in batches on a single machine where the processing time of each job is a simple increasing linear function of its waiting time i e the time between the starting time of processing the batch to which the job belongs and the starting time of processing of the job the objective is to minimize the sum of the total job flow time and the batching cost we first show that the case with a given number of batches is strongly np hard and we present a fully polynomial time approximation scheme fptas for this case we then show that the case with an arbitrary number of batches is also strongly np hard and there is no polynomial time approximation algorithm with a constant upper bound for this case unless p np

combining propensity score matching with additional adjustments for prognostic covariates abstract propensity score matching refers to a class of multivariate methods used in comparative studies to construct treated and matched control samples that have similar distributions on many covariates this matching is the observational study analog of randomization in ideal experiments but is far less complete as it can only balance the distribution of observed covariates whereas randomization balances the distribution of all covariates both observed and unobserved an important feature of propensity score matching is that it can be easily combined with model based regression adjustments or with matching on a subset of special prognostic covariates or combinations of prognostic covariates that have been identified as being especially predictive of the outcome variables we extend earlier results by developing approximations for the distributions of covariates in matched samples created with linear propensity score methods for the practically important situation where matching uses both the estimat

efficient semiparametric estimation of quantile treatment effects this paper develops estimators for quantile treatment effects under the identifying restriction that selection to treatment is based on observable characteristics identification is achieved without requiring computation of the conditional quantiles of the potential outcomes instead the identification results for the marginal quantiles lead to an estimation procedure for the quantile treatment effect parameters that has two steps nonparametric estimation of the propensity score and computation of the difference between the solutions of two separate minimization problems root n consistency asymptotic normality and achievement of the semiparametric efficiency bound are shown for that estimator a consistent estimation procedure for the variance is also presented finally the method developed here is applied to evaluation of a job training program and to a monte carlo exercise results from the empirical application indicate that the method works relatively well even for a data set with limited overlap between treated and controls in the support of covariates the monte carlo study shows that for a relatively small sample size the method produces estimates with good precision and low bias especially for middle quantiles

scheduling deteriorating jobs subject to job or machine availability constraints we consider two problems of scheduling a set of independent non preemptable and proportionally deteriorating jobs on a single machine in the first problem the machine is not continuously available for processing but the number of non availability periods the start time and end time of each period are known in advance in the second problem the machine is available all the time but for each job a ready time and a deadline are defined in both problems the criterion of schedule optimality is the maximum completion time we show that the decision version of the first the second problem is np complete in the ordinary or in the strong sense depending on the number of non availability periods the number of ready times and deadlines

on the complexity of approximating the independent set problem we show that for some positive constant c it is not feasible to approximate independent set for graphs of n nodes within a factor of n c provided maximum 2 satisfiability does not have a randomized polynomial time approximation scheme we also study reductions preserving the quality of approximations and exhibit complete problems

adjusting treatment effect estimates by post stratification in randomized experiments summary experimenters often use post stratification to adjust estimates post stratification is akin to blocking except that the number of treated units in each stratum is a random variable because stratification occurs after treatment assignment we analyse both post stratification and blocking under the neyman rubin model and compare the efficiency of these designs we derive the variances for a post stratified estimator and a simple difference in means estimator under different randomization schemes post stratification is nearly as efficient as blocking the difference in their variances is of the order of 1 n2 with a constant depending on treatment proportion post stratification is therefore a reasonable alternative to blocking when blocking is not feasible however in finite samples post stratification can increase variance if the number of strata is large and the strata are poorly chosen to examine why the estimators variances are different we extend our results by conditioning on the observed number of treated units in each stratum conditioning also provides more accurate variance estimates because it takes into account how close or far a realized random sample is from a comparable blocked experiment we then show that the practical substance of our results remains under an infinite population sampling model finally we provide an analysis of an actual experiment to illustrate our analytical results

asymptotic theory of rerandomization in treatment control experiments although complete randomization ensures covariate balance on average the chance of observing significant differences between treatment and control covariate distributions increases with many covariates rerandomization discards randomizations that do not satisfy a predetermined covariate balance criterion generally resulting in better covariate balance and more precise estimates of causal effects previous theory has derived finite sample theory for rerandomization under the assumptions of equal treatment group sizes gaussian covariate and outcome distributions or additive causal effects but not for the general sampling distribution of the difference in means estimator for the average causal effect we develop asymptotic theory for rerandomization without these assumptions which reveals a non gaussian asymptotic distribution for this estimator specifically a linear combination of a gaussian random variable and truncated gaussian random variables this distribution follows because rerandomization affects only the projection of potential outcomes onto the covariate space but does not affect the corresponding orthogonal residuals we demonstrate that compared with complete randomization rerandomization reduces the asymptotic quantile ranges of the difference in means estimator moreover our work constructs accurate large sample confidence intervals for the average causal effect

rerandomization to improve covariate balance in experiments randomized experiments are the gold standard for estimating causal effects yet often in practice chance imbalances exist in covariate distributions between treatment groups if covariate data are available before units are exposed to treatments these chance imbalances can be mitigated by first checking covariate balance before the physical experiment takes place provided a precise definition of imbalance has been specified in advance unbalanced randomizations can be discarded followed by a rerandomization and this process can continue until a randomization yielding balance according to the definition is achieved by improving covariate balance rerandomization provides more precise and trustworthy estimates of treatment effects

model assisted design of experiments in the presence of network correlated outcomes summaryin this paper we consider how to assign treatment in a randomized experiment in which the correlation among the outcomes is informed by a network available pre intervention working within the potential outcome causal framework we develop a class of models that posit such a correlation structure among the outcomes we use these models to develop restricted randomization strategies for allocating treatment optimally by minimizing the mean squared error of the estimated average treatment effect analytical decompositions of the mean squared error due both to the model and to the randomization distribution provide insights into aspects of the optimal designs in particular the analysis suggests new notions of balance based on specific network quantities in addition to classical covariate balance the resulting balanced optimal restricted randomization strategies are still design unbiased when the model used to derive them does not hold we illustrate how the proposed treatment allocation strategies improve on allocations that ignore the network structure

a calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves any graph manifold can be obtained by plumbing according to some plumbing graph i a calculus for plumbing which includes normal forms for such graphs is developed this is applied to answer several questions about the topology of normal complex surface singularities and analytic families of complex curves for instance it is shown that the topology of the minimal resolution of a normal complex surface singularity is determined by the link of the singularity and even by its fundamental group if the singularity is not a cyclic quotient singularity or a cusp singularity in this paper we describe a calculus for plumbed manifolds which lets one algorithmically determine when the oriented 3 manifolds m tx and a ftj ob tained by plumbing according to two graphs tx and t2 are homeomorphic 3 mani folds are oriented 3 manifolds throughout this paper and homeomorphisms of 3 manifolds are orientation preserving we then apply the calculus to answer several questions about the topology of isolated singularities of complex surfaces and one parameter families of complex curves these results are described below since the class of 3 manifolds obtainable by plumbing is precisely the class of graph manifolds which were classified with minor exceptions by waldhausen 24 a calculus for plumbing is in some sense implicit in waldhausen s work moreover it has been known for some time that the calculus can be put in a form like the one given here but the details have never appeared in the literature a related calculus for plumbing trees has been worked by bonahon and siebenmann 1 in order to classify their algebraic knots we describe the calculus in greater generality than is needed for the present applications since this involves minimal extra work and the calculus is needed elsewhere 4 14 and 15 in particular in an appendix we describe two generalizations of it the calculus consists of a collection of moves one can do to a plumbing graph t without altering the plumbed manifold m t to see these moves are sufficient we describe how they can be used to reduce any graph to a normal form which is

optimum biased coin designs for sequential clinical trials with prognostic factors patients in a clinical trial arrive sequentially and are assigned to one of t treatments this assignment should maintain a balance between the numbers receiving each treatment yet should be sufficiently random to avoid any suspicion of conscious or unconscious cheating to balance these requirements efron 1971 introduced biased coin designs for the comparison of two treatments in which allocation of the treatment is determined probabilistically but with a bias towards the underrepresented treatment one disadvantage of efron s scheme is that it does not include balance over covariates or prognostic factors which may affect the response of the patient to the treatment biased coin schemes which do force balance over both treatments and prognostic factors are given by pocock simon 1975 and efron 1980 the properties of the designs have been elucidated by numerical studies and they are now being increasingly used in clinical trials reviews of the literature and discussions of the practical implications of the designs are given by pocock 1979 and simon 1979 however the designs suffer from the disadvantage that they rely on arbitrary functions to achieve the desired balance the procedures thus lack a firm theoretical framework an alternative approach in the presence of prognostic factors begg iglewicz 1980 uses optimum design theory to suggest a deterministic design criterion which is then modified for computational convenience in this paper i use optimum design theory to provide a procedure of the biased coin type for an arbitrary number of treatments in the presence or absence of prognostic factors this has the theoretical advantage of obviating dependence on a series of arbitrary functions the necessary optimum design theory is presented in 2 and in 3 applied to biased coin experiments the simplest case that of two treatments in the absence of prognostic factors is studied in 4 the extension to three or more treatments is in 5 followed in 6 by the allowance for prognostic factors although in the biased coin designs allocation of treatments is made probabilistically the results of the paper can be used for the sequential construction of da optimum designs in which randomization is not introduced by the experimenter all the designs mentioned so far are sequential but not adaptive in the useful distinction made by pericchi 1981 they are data dependent but not outcomedependent each allocation depends on the previous allocations and prognostic factors

testing for imbalance of covariates in controlled experiments results of a controlled experiment are often adjusted for covariates found by a preliminary test to differ significantly between the treatment and control groups the resulting test s true significance level is lower than the nominal level greater power can be achieved by always adjusting for a covariate that is highly correlated with the response regardless of its distribution between groups

the central role of the propensity score in observational studies for causal effects abstract the results of observational studies are often disputed because of nonrandom treatment assignment for example patients at greater risk may be overrepresented in some treatment group this paper discusses the central role of propensity scores and balancing scores in the analysis of observational studies the propensity score is the estimated conditional probability of assignment to a particular treatment given a vector of observed covariates both large and small sample theory show that adjustment for the scalar propensity score is sufficient to remove bias due to all observed covariates applications include matched sampling on the univariate propensity score which is equal percent bias reducing under more general conditions than required for discriminant matching multivariate adjustment by subclassification on balancing scores where the same subclasses are used to estimate treatment effects for all outcome variables and in all subpopulations and visual representation of multivariate adjustment by a two dimensional plot author

finding approximate and constrained motifs in graphs one of the most relevant topics in the analysis of biological networks is the identification of functional motifs inside a network a recent approach introduced in literature called graph motif represents the network as a vertex colored graph and the motif m as a multiset of colors an occurrence of a motif m in a vertex colored graph g is a connected induced subgraph of g whose vertex set is colored exactly as m in this paper we investigate three different variants of the graph motif problem the first two variants minimum adding motif min add graph motif and minimum substitution motif min sub graph motif deal with approximate occurrences of a motif in the graph while the third variant constrained graph motif cgm constrains the motif to contain a given set of vertices we investigate the computational and parameterized complexity of the three problems we show that min add graph motifand min sub graph motifare both np hard even when m is a set and the graph is a tree with maximum degree 4 in which each color appears at most twice then we show that min sub graph motifis fixed parameter tractable when parameterized by the size of m finally we consider the parameterized complexity of the cgmproblem we give a fixed parameter algorithm for graphs of bounded treewidth and show that the problem is w 2 hard when parameterized by m even if the input graph has diameter 2

multivariate matching methods that are monotonic imbalance bounding we introduce a new monotonic imbalance bounding mib class of matching methods for causal inference with a surprisingly large number of attractive statistical properties mib generalizes and extends in several new directions the only existing class equal percent bias reducing epbr which is designed to satisfy weaker properties and only in expectation we also offer strategies to obtain specific members of the mib class and analyze in more detail a member of this class called coarsened exact matching whose properties we analyze from this new perspective we offer a variety of analytical results and numerical simulations that demonstrate how members of the mib class can dramatically improve inferences relative to epbr based matching methods

using standard tools from finite population sampling to improve causal inference for complex experiments abstractthis article considers causal inference for treatment contrasts from a randomized experiment using potential outcomes in a finite population setting adopting a neymanian repeated sampling approach that integrates such causal inference with finite population survey sampling an inferential framework is developed for general mechanisms of assigning experimental units to multiple treatments this framework extends classical methods by allowing the possibility of randomization restrictions and unequal replications novel conditions that are milder than strict additivity of treatment effects yet permit unbiased estimation of the finite population sampling variance of any treatment contrast estimator are derived the consequences of departures from such conditions are also studied under the criterion of minimax bias and a new justification for using the neymanian conservative sampling variance estimator in experiments is provided the proposed approach can readily be extended to the case of treatm

inapproximability of dominating set on power law graphs we prove the first logarithmic lower bounds for the approximability of the minimum dominating set problem for the case of connected power law graphs for being a size parameter and the power law exponent we give also a best up to now upper approximation bound for this problem in the case of the parameters 2 we develop also a new functional method for proving lower approximation bounds and display a sharp approximation phase transition area between approximability and inapproximability of the underlying problems our results depend on a method which could be also of independent interest

product partition and related problems of scheduling and systems reliability computational complexity and approximation abstract problem product partition differs from the np complete problem partition in that the addition operation is replaced by the multiplication operation furthermore it differs from the np complete problem subset product in that it does not contain the product value b in its input we prove that problem product partition and several of its modifications are np complete in the strong sense our results imply the strong np hardness of a number of scheduling problems with start time dependent job processing times and a problem of designing a reliable system with a series parallel structure it should be noticed that the strong np hardness of the considered optimization problems does not preclude the existence of a fully polynomial time approximation scheme fptas for them we present a simple fptas for one of these problems

on mitigating the analytical limitations of finely stratified experiments although attractive from a theoretical perspective finely stratified experiments such as paired designs suffer from certain analytical limitations that are not present in block randomized experiments with multiple treated and control individuals in each block in short when using a weighted difference in means to estimate the sample average treatment effect the traditional variance estimator in a paired experiment is conservative unless the pairwise average treatment effects are constant across pairs however in more coarsely stratified experiments the corresponding variance estimator is unbiased if treatment effects are constant within blocks even if they vary across blocks using insights from classical least squares theory we present an improved variance estimator that is appropriate in finely stratified experiments the variance estimator remains conservative in expectation but is asymptotically no more conservative than the classical estimator and can be considerably less conservative the magnitude of the improvement depends on the extent to which effect heterogeneity can be explained by observed covariates aided by this estimator a new test for the null hypothesis of a constant treatment effect is proposed these findings extend to some but not all superpopulation models depending on whether the covariates are viewed as fixed across samples

randomization based inference within principal strata in randomized studies treatment comparisons conditional on intermediate postrandomization outcomes using standard analytic methods do not have a causal interpretation an alternative approach entails treatment comparisons within principal strata defined by the potential outcomes for the intermediate outcome that would be observed under each treatment assignment in this article we develop methods for randomization based inference within principal strata we compare our proposed methods with existing large sample methods as well as traditional intent to treat approaches this research is motivated by hiv prevention studies where few infections are expected and inference is desired within the always infected principal stratum that is all individuals who would become infected regardless of randomization assignment

quantifiers and approximation we investigate the relationship between logical expressibility of np optimization problems and their approximation properties first such attempt was made by papadimitrou and yannakakis 1988 who defined the class of npo problems max np we show that many important optimization problems do not belong to max np and that in fact there are problems in p which are not in max np the problems that we consider fit naturally in a new complexity class that we call max 1 we prove that several natural optimization problems are complete for max 1 under approximation preserving reductions all these complete problems are not approximable unless p np this motivates the definition of subclasses of max 1 that only contain problems which are presumably eaiser with respect to approximation in particular the class that we call rmax 2 contains approximable problems and problems like max clique that are not known to be nonapproximable we prove the max clique and several other optimization problems are complete for rmax 2 all the complete problems in rmax 2 share the interesting property that they either are nonapproximable or are approximable to any degree of accuracy

approximation algorithms for the m dimensional 0 1 knapsack problem worst case and probabilistic analyses abstract we describe a polynomial approximation scheme for an m constraint 0 1 integer programming problem m fixed based on the use of the dual simplex algorithm for linear programming we also analyse the asymptotic properties of a particular random model

lower bounds on time accuracy trade offs for the 0 1 knapsack problem the 0 1 knapsack optimization problem is well known to be np hard the best known approximation algorithms for this problem are approximation schemes which exhibit trade offs between the running time of the algorithm and the guaranteed accuracy of the solution this paper establishes lower bounds on the time required to find approximate solutions of guaranteed accuracy for a class of knapsack algorithms which must use oracles to perform feasibility tests and dominance tests

a physicist s approach to number partitioning the statistical physics approach to the number partioning problem a classical np hard problem is both simple and rewarding very basic notions and methods from statistical mechanics are enough to obtain analytical results for the phase boundary that separates the easy to solve from the hard to solve phase of the npp as well as for the probability distributions of the optimal and sub optimal solutions in addition it can be shown that solving a number portioning problem of size n to some extent corresponds to locating the minimum in an unsorted list of o 2n numbers considering this correspondence it is not surprising that known heuristics for the partitioning problem are not signi4cantly better than simple random search

effects attributable to treatment inference in experiments and observational studies with a discrete pivot summary in randomisation and permutation inference pivotal arguments remove the hypothesised treatment effect thereby basing inferences on the null distribution in which the treatment has no effect this is common for instance with additive treatment effects the current paper uses attributable effects to expand substantially the scope of pivotal arguments attributable effects are defined for three cases namely the 2 x 2 contingency table displacement effects and the mann whitney wilcoxon statistic and in each case removing an appropriate attributable effect restores the familiar null randomisation distribution of the associated statistic yielding exact inferences the procedure extends immediately for use in sensitivity analysis in nonrandomised observational studies

causal inference using potential outcomes design modeling decisions causal effects are defined as comparisons of potential outcomes under different treatments on a common set of units observed values of the potential outcomes are revealed by the assignment mechanism a probabilistic model for the treatment each unit receives as a function of covariates and potential outcomes fisher made tremendous contributions to causal inference through his work on the design of randomized experiments but the potential outcomes perspective applies to other complex experiments and nonrandomized studies as well as noted by kempthorne in his 1976 discussion of savage s fisher lecture fisher never bridged his work on experimental design and his work on parametric modeling a bridge that appears nearly automatic with an appropriate view of the potential outcomes framework where the potential outcomes and covariates are given a bayesian distribution to complete the model specification also this framework crisply separates scientific inference for causal effects and decisions based on s

hardness and inapproximability of convex recoloring problems abstract given a graph with an arbitrary vertex coloring the convex recoloring problem cr consists in recoloring the minimum number of vertices so that each color induces a connected subgraph we focus on the complexity and inapproximability of this problem on k colored graphs for fixed k 2 we prove a strong complexity result showing that for each k 2 cr is already np hard on k colored grids and therefore also on planar graphs with maximum degree 4 for each k 2 we prove that for a positive constant c there is no c ln n approximation algorithm for k colored n vertex bipartite graphs unless p np we also prove that cr parameterized by the number of color changes is w 2 hard for 2 colored q q 4 graphs a class that includes cographs and p 4 sparse graphs we present linear time algorithms for fixed q the same complexity and inapproximability results are obtained for two relaxations of the problem where only one fixed color or any color is required to induce a connected subgraph respectively

interval estimation for treatment effects using propensity score matching in causal studies without random assignment of treatment causal effects can be estimated using matched treated and control samples where matches are obtained using estimated propensity scores propensity score matching can reduce bias in treatment effect estimators in cases where the matched samples have overlapping covariate distributions despite its application in many applied problems there is no universally employed approach to interval estimation when using propensity score matching in this article we present and evaluate approaches to interval estimation when using propensity score matching

a new u statistic with superior design sensitivity in matched observational studies summary in an observational or nonrandomized study of treatment effects a sensitivity analysis indicates the magnitude of bias from unmeasured covariates that would need to be present to alter the conclusions of a naive analysis that presumes adjustments for observed covariates suffice to remove all bias the power of sensitivity analysis is the probability that it will reject a false hypothesis about treatment effects allowing for a departure from random assignment of a specified magnitude in particular if this specified magnitude is no departure then this is the same as the power of a randomization test in a randomized experiment a new family of u statistics is proposed that includes wilcoxon s signed rank statistic but also includes other statistics with substantially higher power when a sensitivity analysis is performed in an observational study wilcoxon s statistic has high power to detect small effects in large randomized experiments that is it often has good pitman efficiency but small effects are invariably sensitive to small unobserved biases members of this family of u statistics that emphasize medium to large effects can have substantially higher power in a sensitivity analysis for example in one situation with 250 pair differences that are normal with expectation 1 2 and variance 1 the power of a sensitivity analysis that uses wilcoxon s statistic is 0 08 while the power of another member of the family of u statistics is 0 66 the topic is examined by performing a sensitivity analysis in three observational studies using an asymptotic measure called the design sensitivity and by simulating power in finite samples the three examples are drawn from epidemiology clinical medicine and genetic toxicology

variance identification and efficiency analysis in randomized experiments under the matched pair design in his 1923 landmark article neyman introduced randomization based inference to estimate average treatment effects from experiments under the completely randomized design under this framework neyman considered the statistical estimation of the sample average treatment effect and derived the variance of the standard estimator using the treatment assignment mechanism as the sole basis of inference in this paper i extend neyman s analysis to randomized experiments under the matched pair design where experimental units are paired based on their pre treatment characteristics and the randomization of treatment is subsequently conducted within each matched pair i study the variance identification for the standard estimator of average treatment effects and analyze the relative efficiency of the matched pair design over the completely randomized design i also show how to empirically evaluate the relative efficiency of the two designs using experimental data obtained under the matched pair design my randomization based analysis differs from previous studies in that it avoids modeling and other assumptions as much as possible finally the analytical results are illustrated with numerical and empirical examples

optimal matching for observational studies abstract matching is a common method of adjustment in observational studies currently matched samples are constructed using greedy heuristics or stepwise procedures that produce in general suboptimal matchings with respect to a particular criterion a matched sample is suboptimal if it could be improved by changing the controls assigned to specific treated units that is if it could be improved with the data at hand here optimal matched samples are obtained using network flow theory in addition to providing optimal matched pair samples this approach yields optimal constructions for several statistical matching problems that have not been studied previously including the construction of matched samples with multiple controls with a variable number of controls and the construction of balanced matched samples that combine features of pair matching and frequency matching computational efficiency is discussed extensive use is made of ideas from two essentially disjoint literatures namely st

scheduling linear deteriorating jobs with an availability constraint on a single machine we consider a single machine scheduling problem in which the processing time of a job is a simple linear increasing function of its starting time and the machine is subject to an availability constraint we consider the non resumable case the objectives are to minimize the makespan and the total completion time we show that both problems are np hard and present pseudopolynomial time optimal algorithms to solve them furthermore for the makespan problem we present an optimal approximation algorithm for the on line case and a fully polynomial time approximation scheme for the off line case for the total completion time problem we provide a heuristic and evaluate its efficiency by computational experiments

optimal a priori balance in the design of controlled experiments summary we develop a unified theory of designs for controlled experiments that balance baseline covariates a priori before treatment and before randomization using the framework of minimax variance and a new method called kernel allocation we show that any notion of a priori balance must go hand in hand with a notion of structure since with no structure on the dependence of outcomes on baseline covariates complete randomization no special covariate balance is always minimax optimal restricting the structure of dependence either parametrically or non parametrically gives rise to certain covariate imbalance metrics and optimal designs this recovers many popular imbalance metrics and designs previously developed ad hoc including randomized block designs pairwise matched allocation and rerandomization we develop a new design method called kernel allocation based on the optimal design when structure is expressed by using kernels which can be parametric or non parametric relying on modern optimization methods kernel allocation which ensures nearly perfect covariate balance without biasing estimates under model misspecification offers sizable advantages in precision and power as demonstrated in a range of real and synthetic examples we provide strong theoretical guarantees on variance consistency and rates of convergence and develop special algorithms for design and hypothesis testing

max np completeness made easy we introduce a simple technique to obtain reductions between optimization constraint satisfaction problems with respect to an approximation preserving reducibility more powerful than the commonly used l reducibility the technique applies the probabilistic method to reduce the size of disjunctions as a first application we prove the max np completeness of max 3sat without using the pcp theorem thus solving an open question posed in khanna et al 1994 successively we show that the planar restrictions of several optimization constraint satisfaction problems admit linear time approximation schemes thus improving the results of khanna and motwani 1996

a unified approach to approximation algorithms for bottleneck problems in this paper a powerful and yet simple technique for devising approximation algorithms for a wide variety of np complete problems in routing location and communication network design is investigated each of the algorithms presented here delivers an approximate solution guaranteed to be within a constant factor of the optimal solution in addition for several of these problems we can show that unless p np there does not exist a polynomial time algorithm that has a better performance guarantee

reduced sensitivity to hidden bias at upper quantiles in observational studies with dilated treatment effects summary when a treatment has a dilated effect with larger effects when responses are higher there can be much less sensitivity to bias at upper quantiles than at lower quantiles i e small plausible hidden biases might explain the ostensible effect of the treatment for many subjects and yet only quite large hidden biases could explain the effect on a few subjects having dramatically elevated responses an example concerning kidney function of cadmium workers is discussed in detail in that example the treatment effect is far from additive it is plausibly zero at the lower quartile of responses to control and it is large and fairly insensitive to bias at the upper quartile

improving massive experiments with threshold blocking inferences from randomized experiments can be improved by blocking assigning treatment in fixed proportions within groups of similar units however the use of the method is limited by the difficulty in deriving these groups current blocking methods are restricted to special cases or run in exponential time are not sensitive to clustering of data points and are often heuristic providing an unsatisfactory solution in many common instances we present an algorithm that implements a widely applicable class of blocking threshold blocking that solves these problems given a minimum required group size and a distance metric we study the blocking problem of minimizing the maximum distance between any two units within the same group we prove this is a nondeterministic polynomial time hard problem and derive an approximation algorithm that yields a blocking where the maximum distance is guaranteed to be at most four times the optimal value this algorithm runs in o n log n time with o n space complexity this makes it to our knowledge the first blocking method with an ensured level of performance that works in massive experiments whereas many commonly used algorithms form pairs of units our algorithm constructs the groups flexibly for any chosen minimum size this facilitates complex experiments with several treatment arms and clustered data a simulation study demonstrates the efficiency and efficacy of the algorithm tens of millions of units can be blocked using a desktop computer in a few minutes

efficient approximation of min set cover by moderately exponential algorithms we study the approximation of min set cover combining ideas and results from polynomial approximation and from exact computation with non trivial worst case complexity upper bounds for np hard problems we design approximation algorithms for min set cover achieving ratios that cannot be achieved in polynomial time unless problems in np could be solved by slightly super polynomial algorithms with worst case complexity much lower though super polynomial than those of an exact computation