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

adaptive wavelet schemes for elliptic problems implementation and numerical experiments recently an adaptive wavelet scheme could be proved to be asymptotically optimal for a wide class of elliptic operator equations in the sense that the error achieved by an adaptive approximate solution behaves asymptotically like the smallest possible error that can be realized by any linear combination of the corresponding number of wavelets on one hand the results are purely asymptotic on the other hand the analysis suggests new algorithmic ingredients for which no prototypes seem to exist yet it is therefore the objective of this paper to develop suitable data structures for the new algorithmic components and to obtain a quantitative validation of the theoretical results we briefly review first the main theoretical facts describe the main ingredients of the algorithm highlight the essential data structures and illustrate the results by one and two dimensional numerical examples including comparisons with an adaptive finite element scheme

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

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

instrumental variables estimates of the effect of subsidized training on the quantiles of trainee earnings this paper reports estimates of the effects of jtpa training programs on the distribution of earnings the estimation uses a new instrumental variable iv method that measures program impacts on quantiles the quantile treatment effects qte estimator reduces to quantile regression when selection for treatment is exogenously determined qte can be computed as the solution to a convex linear programming problem although this requires first step estimation of a nuisance function we develop distribution theory for the case where the first step is estimated nonparametrically for women the empirical results show that the jtpa program had the largest proportional impact at low quantiles perhaps surprisingly however jtpa training raised the quantiles of earnings for men only in the upper half of the trainee earnings distribution

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

wavelet methods for pdes some recent developments abstract this paper is concerned with recent developments of wavelet schemes for the numerical treatment of operator equations with special emphasis on two issues adaptive solution concepts and nontrivial domain geometries after describing a general multiresolution framework the key features of wavelet bases are highlighted namely locality norm equivalences and cancellation properties assuming first that wavelet bases with these properties are available on the relevant problem domains the relevance of these features for a wide class of stationary problems is explained in subsequent sections the main issues are preconditioning and the efficient adaptive application of wavelet representations of the involved operators we indicate then how these ingredients combined with concepts from nonlinear or best n term approximation culminate in an adaptive wavelet scheme for elliptic selfadjoint problems covering boundary value problems as well as boundary integral equations these schemes can be shown to exhibit convergence rates that are in a certain sense asymptotically optimal we conclude this section with some brief remarks on data structures and implementation interrelations with regularity in a certain scale of besov spaces and strategies of extending such schemes to unsymmetric or indefinite problems we address then the adaptive evaluation of nonlinear functionals of wavelet expansions as a central task arising in connection with nonlinear problems wavelet bases on nontrivial domains are discussed next the main issues are the development of fourier free construction principles and criteria for the validity of norm equivalences finally we indicate possible combinations of wavelet concepts with conventional discretizations such as finite element or finite volume schemes in connection with convection dominated and hyperbolic problems

explicit runge kutta methods for initial value problems with oscillating solutions new pairs of embedded runge kutta methods specially adapted to the numerical solution of first order systems of differential equations which are assumed to possess oscillating solutions are obtained these pairs have been derived taking into account not only the usual properties of accuracy stability and reliability of the local error estimator to adjust the stepsize of the underlying formulas but also the dispersion and dissipation orders of the advancing formula as defined by van der houwen and sommeijer 1989 three nine stage embedded pairs of runge kutta methods with algebraic orders 7 and 5 and higher orders of dispersion and or dissipation are selected among the members of a family of pairs depending on several free parameters some numerical results are presented to show the efficiency of the new methods

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

decomposing treatment effect variation understanding and characterizing treatment effect variation in randomized experiments has become essential for going beyond the black box of the average treatment effect nonetheless traditional

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

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

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

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

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

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

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

causal estimands and confidence intervals associated with wilcoxon mann whitney tests in randomized experiments although the p value from a wilcoxon mann whitney test is used often with randomized experiments it is rarely accompanied with a causal effect estimate and its confidence interval the natural parameter for the wilcoxon mann whitney test is the mann whitney parameter which measures the probability that a randomly selected individual in the treatment arm will have a larger response than a randomly selected individual in the control arm plus an adjustment for ties we show that the mann whitney parameter may be framed as a causal parameter and show that it is not equal to a closely related and nonidentifiable causal effect the probability that a randomly selected individual will have a larger response under treatment than under control plus an adjustment for ties we review the paradox first expressed by hand that the parameter may imply that the treatment is worse or better than control while the mann whitney parameter shows the opposite unlike the mann whitney parameter is nonidentifiable from a randomized experiment we review some nonparametric assumptions that rule out hand s paradox through bounds on and use bootstrap methods to make inferences on those bounds we explore the relationship of the proportional odds parameter to hand s paradox showing that the paradox may occur for proportional odds parameters between 1 9 and 9 thus large effects are needed to ensure that if treatment appears better by the mann whitney parameter then treatment improves responses in most individuals we demonstrate these issues using a vaccine trial

an adaptive wavelet method for nonlinear differential algebraic equations abstract a multi resolution collocation method with specially designed spline wavelet is presented to numerically solve a system of nonlinear differential algebraic equations of index one in circuit simulation as the wavelet collocation method is applied to such system we will have to solve large nonlinear algebraic equations for the algebraic equations we give a modified newton type iterative method which bases on two wavelet adaptive techniques we prove the convergence the stability and the complexity on the adaptive method reported here the new approach could effectively reduce computational costs and storage requirements to large extent numerical experiments are given to further illustrate the method s principle

a randomization based perspective on analysis of variance a test statistic robust to treatment effect heterogeneity summary fisher randomization tests for neyman s null hypothesis of no average treatment effect are considered in a finite population setting associated with completely randomized experiments involving more than two treatments the consequences of using the f statistic to conduct such a test are examined and we argue that under treatment effect heterogeneity use of the f statistic in the fisher randomization test can severely inflate the type i error under neyman s null hypothesis we propose to use an alternative test statistic derive its asymptotic distributions under fisher s and neyman s null hypotheses and demonstrate its advantages through simulations

randomization tests for weak null hypotheses in randomized experiments the fisher randomization test frt is appropriate for any test statistic under a sharp null hypothesis that can recover all missing potential outcomes however it is often sought after to test a

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

adaptive wavelet methods for elliptic operator equations convergence rates this paper is concerned with the construction and analysis of wavelet based adaptive algorithms for the numerical solution of elliptic equations these algorithms approximate the solution u of the equation by a linear combination of n wavelets therefore a benchmark for their performance is provided by the rate of best approximation to u by an arbitrary linear combination of n wavelets so called n term approximation which would be obtained by keeping the n largest wavelet coefficients of the real solution which of course is unknown the main result of the paper is the construction of an adaptive scheme which produces an approximation to u with error o n s in the energy norm whenever such a rate is possible by n term approximation the range of s 0 for which this holds is only limited by the approximation properties of the wavelets together with their ability to compress the elliptic operator moreover it is shown that the number of arithmetic operations needed to compute the approximate solution stays proportional to n the adaptive algorithm applies to a wide class of elliptic problems and wavelet bases the analysis in this paper puts forward new techniques for treating elliptic problems as well as the linear systems of equations that arise from the wavelet discretization