contrasting treatment specific survival using double robust estimators in settings where a randomized trial is infeasible observational data are frequently used to compare treatment specific survival the average causal effect ace can be used to make inference regarding treatment policies on patient populations and a valid ace estimator must account for imbalances with respect to treatment specific covariate distributions one method through which the ace on survival can be estimated involves appropriately averaging over cox regression based fitted survival functions a second available method balances the treatment specific covariate distributions through inverse probability of treatment weighting and then contrasts weighted nonparametric survival function estimators because both methods have their advantages and disadvantages we propose methods that essentially combine both estimators the proposed methods are double robust in the sense that they are consistent if at least one of the two working regression models i e logistic model for treatment and cox model for death hazard is correct the proposed methods involve estimating the ace with respect to restricted mean survival time defined as the area under the survival curve up to some prespecified time point we derive and evaluate asymptotic results through simulation we apply the proposed methods to estimate the ace of donation after cardiac death kidney transplantation with the use of data obtained from multiple centers in the netherlands

randomization inference and sensitivity analysis for composite null hypotheses with binary outcomes in matched observational studies abstractwe present methods for conducting hypothesis testing and sensitivity analyses for composite null hypotheses in matched observational studies when outcomes are binary causal estimands discussed include the causal risk difference causal risk ratio and the effect ratio we show that inference under the assumption of no unmeasured confounding can be performed by solving an integer linear program while inference allowing for unmeasured confounding of a given strength requires solving an integer quadratic program through simulation studies and data examples we demonstrate that our formulation allows these problems to be solved in an expedient manner even for large datasets and for large strata we further exhibit that through our formulation one can assess the impact of various assumptions about the potential outcomes on the performed inference r scripts are provided that implement our methods supplementary materials for this article are available online

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 a new class of sample average treatment effects we derive new variance formulas for inference on a general class of estimands of causal average treatment effects in a randomized control trial we generalize the seminal work of robins and show th

asymptotic control and stabilization of nonlinear oscillators with non isolated equilibria let h r be a c1 function on a real hilbert space h and let 0 be a positive damping parameter for any control function e r r which tends to zero as t we study the asymptotic behavior of the trajectories of the damped nonlinear oscillator hbfc x t x t x t e t x t 0 we show that if e t does not tend to zero too rapidly as t then the term e t x t asymptotically acts as a tikhonov regularization which forces the trajectories to converge to a particular equilibrium indeed in the main result of this paper it is established that when is convex and s argmin under the key assumption that e is a slow control i e 0 e t dt then each trajectory of the hbfc system strongly converges as t to the element of minimal norm of the closed convex set s as an application we consider the damped wave equation with neumann boundary condition formula

augmented weighted estimators dealing with practical positivity violation to causal inferences in a random coefficient model the inverse probability of treatment weighted iptw estimator can be used to make causal inferences under two assumptions 1 no unobserved confounders ignorability and 2 positive probability of treatment and of control at every level of the confounders positivity but is vulnerable to bias if by chance the proportion of the sample assigned to treatment or proportion of control is zero at certain levels of the confounders we propose to deal with this sampling zero problem also known as practical violation of the positivity assumption in a setting where the observed confounder is cluster identity i e treatment assignment is ignorable within clusters specifically based on a random coefficient model assumed for the potential outcome we augment the iptw estimating function with the estimated potential outcomes of treatment or of control for clusters that have no observation of treatment or control if the cluster specific potential outcomes are estimated correctly the augmented estimating function can be shown to converge in expectation to zero and therefore yield consistent causal estimates the proposed method can be implemented in the existing software and it performs well in simulated data as well as with real world data from a teacher preparation evaluation study

fundamental solutions of penny shaped and half infinite plane cracks embedded in an infinite space of one dimensional hexagonal quasi crystal under thermal loading this paper presents fundamental solutions for an infinite space of one dimensional hexagonal quasi crystal medium which contains a penny shaped or half infinite plane crack subjected to two identi

flexible regression approach to propensity score analysis and its relationship with matching and weighting in propensity score analysis the frequently used regression adjustment involves regressing the outcome on the estimated propensity score and treatment indicator this approach can be highly efficient when model assumptions are valid but can lead to biased results when the assumptions are violated we extend the simple regression adjustment to a varying coefficient regression model that allows for nonlinear association between outcome and propensity score we discuss its connection with some propensity score matching and weighting methods and show that the proposed analytical framework can shed light on the intrinsic connection among some mainstream propensity score approaches stratification regression kernel matching and inverse probability weighting and handle commonly used causal estimands we derive analytic point and variance estimators that properly take into account the sampling variability in the estimated propensity score extensive simulations show that the proposed approach possesses desired finite sample properties and demonstrates competitive performance in comparison with other methods estimating the same causal estimand the proposed methodology is illustrated with a study on right heart catheterization

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

effect analysis and causation in linear structural equation models this paper considers total and direct effects in linear structural equation models adopting a causal perspective that is implicit in much of the literature on the subject the paper concludes that in many instances the effects do not admit the interpretations imparted in the literature drawing a distinction between concomitants and factors the paper concludes that a concomitant has neither total nor direct effects on other variables when a variable is a factor and one or more intervening variables are concomitants the notion of a direct effect is not causally meaningful even when the notion of a direct effect is meaningful the usual estimate of this quantity may be inappropriate the total effect is usually interpreted as an equilibrium multiplier in the case where there are simultaneity relations among the dependent variables in tghe model the results in the literature for the total effects of dependent variables on other dependent variables are not equilibrium multipliers and thus the usual interpretation is incorrect to remedy some of these deficiencies a new effect the total effect of a factorx on an outcomey holding a set of variablesf constant is defined when defined the total and direct effects are a special case of this new effect and the total effect of a dependent variable on a dependent variable is an equilibrium multiplier

constructing a confidence interval for the fraction who benefit from treatment using randomized trial data the fraction who benefit from treatment is the proportion of patients whose potential outcome under treatment is better than that under control inference on this parameter is challenging since it is only partially identifiable even in our context of a randomized trial we propose a new method for constructing a confidence interval for the fraction when the outcome is ordinal or binary our confidence interval procedure is pointwise consistent it does not require any assumptions about the joint distribution of the potential outcomes although it has the flexibility to incorporate various user defined assumptions our method is based on a stochastic optimization technique involving a second order asymptotic approximation that to the best of our knowledge has not been applied to biomedical studies this approximation leads to statistics that are solutions to quadratic programs which can be computed efficiently using optimization tools in simulation our method attains the nominal coverage probability or higher and can have narrower average width than competitor methods we apply it to a trial of a new intervention for stroke

substantial gains in bias reduction from matching with a variable number of controls summary in observational studies that match several controls to each treated subject substantially greater bias reduction is possible if the number of controls is not fixed but rather is allowed to vary from one matched set to another in certain cases matching with a fixed number of controls may remove only 50 of the bias in a covariate whereas matching with a variable number of controls may remove 90 of the bias even though both control groups have the same number of controls in total an example of matching in a study of surgical mortality is discussed in detail

two stage instrumental variable methods for estimating the causal odds ratio analysis of bias we present closed form expressions of asymptotic bias for the causal odds ratio from two estimation approaches of instrumental variable logistic regression i the two stage predictor substitution 2sps method and ii the two stage residual inclusion 2sri approach under the 2sps approach the first stage model yields the predicted value of treatment as a function of an instrument and covariates and in the second stage model for the outcome this predicted value replaces the observed value of treatment as a covariate under the 2sri approach the first stage is the same but the residual term of the first stage regression is included in the second stage regression retaining the observed treatment as a covariate our bias assessment is for a different context from that of terza j health econ 2008 27 3 531 543 who focused on the causal odds ratio conditional on the unmeasured confounder whereas we focus on the causal odds ratio among compliers under the principal stratification framework our closed form bias results show that the 2sps logistic regression generates asymptotically biased estimates of this causal odds ratio when there is no unmeasured confounding and that this bias increases with increasing unmeasured confounding the 2sri logistic regression is asymptotically unbiased when there is no unmeasured confounding but when there is unmeasured confounding there is bias and it increases with increasing unmeasured confounding the closed form bias results provide guidance for using these iv logistic regression methods our simulation results are consistent with our closed form analytic results under different combinations of parameter settings

balancing covariates via propensity score weighting abstractcovariate balance is crucial for unconfounded descriptive or causal comparisons however lack of balance is common in observational studies this article considers weighting strategies for balancing covariates we define a general class of weights the balancing weights that balance the weighted distributions of the covariates between treatment groups these weights incorporate the propensity score to weight each group to an analyst selected target population this class unifies existing weighting methods including commonly used weights such as inverse probability weights as special cases general large sample results on nonparametric estimation based on these weights are derived we further propose a new weighting scheme the overlap weights in which each unit s weight is proportional to the probability of that unit being assigned to the opposite group the overlap weights are bounded and minimize the asymptotic variance of the weighted average treatment effect among the class of balancing wei

bagged one to one matching for efficient and robust treatment effect estimation observational studies present challenges due to bias from imbalance in baseline confounders one to one matching oom a popular cohort construction technique for observational studies reduces bias and provides a compelling basis for inference but generally leads to at least some loss of efficiency due to the exclusion of potentially informative subjects we introduce the bagged one to one matching boom estimator which combines the bias reducing properties of oom with the variance reducing properties of bootstrap aggregation bagging we describe the boom algorithm in detail provide r code for its implementation and investigate its performance in simulation studies and a case study in the simulation studies under different types of model misspecification we compare the boom estimator s performance in terms of mean squared error bias variance accuracy of standard error estimation and coverage of nominal 95 confidence intervals to that of oom and to that of ordinary least squares estimation inverse probability weighting and targeted maximum likelihood estimation all on the full unmatched cohort in our simulations the boom estimator achieves as much bias reduction as the estimator based on oom while having much lower variance in all of the settings examined in the simulations the boom s mean squared error is comparable to or better than that of the comparison methods in the case study boom yields estimates similar to those from the established methods with narrower 95 confidence intervals

asymptotic behavior of coupled dynamical systems with multiscale aspects abstract we study the asymptotic behavior as time variable t goes to of nonautonomous dynamical systems involving multiscale features as a benchmark case given h a general hilbert space h r and h r two closed convex functions and a function of t which tends to as t goes to we consider the differential inclusion x t x t t x t 0 this system models the emergence of various collective behaviors in game theory as well as the asymptotic control of coupled systems we show several results ranging from weak ergodic to strong convergence of the trajectories as a key ingredient we assume that for every p belonging to the range of n c 0 t p t c p t d t where is the fenchel conjugate of c is the support function of c argmin and n c x is the normal cone to c at x as a by product we revisit the system x t t x t x t 0 where t tends to zero as t goes to and 0 t d t whose asymptotic behavior can be derived from the preceding one by time rescaling applications are given in game theory optimal control variational problems and pdes

on the nature and discovery of structure abstract extending principles of experimentation we discuss conditions under which nonexperimental data allow consistent estimation of effects of the kind revealed by experimentation and relevant to decisions we show how implications of these conditions are often overlooked and how failure to distinguish between factors and concomitants makes almost anything said about a model ambiguous if not wrong the effects to be estimated dictate the factors to be included consistency and efficiency determine the concomitants whose effects are not to be estimated concomitants may affect but must not be affected by the factors effects of excluded variables on an included variable may cause inconsistency if the included variable is a factor can only reduce inconsistency if the included variable is a concomitant exclusion of a variable because it is highly correlated with another may sometimes be legitimate if both variables are concomitants never if either is a factor a condition for consistent estimatio

building a stronger instrument in an observational study of perinatal care for premature infants an instrument is a random nudge toward acceptance of a treatment that affects outcomes only to the extent that it affects acceptance of the treatment nonetheless in settings in which treatment assignment is mostly deliberate and not random there may exist some essentially random nudges to accept treatment so that use of an instrument might extract bits of random treatment assignment from a setting that is otherwise quite biased in its treatment assignments an instrument is weak if the random nudges barely influence treatment assignment or strong if the nudges are often decisive in influencing treatment assignment although ideally an ostensibly random instrument is perfectly random and not biased it is not possible to be certain of this thus a typical concern is that even the instrument might be biased to some degree it is known from theoretical arguments that weak instruments are invariably sensitive to extremely small biases for this reason strong instruments are preferred the strength of an

optimal multilevel matching in clustered observational studies a case study of the effectiveness of private schools under a large scale voucher system abstracta distinctive feature of a clustered observational study is its multilevel or nested data structure arising from the assignment of treatment in a nonrandom manner to groups or clusters of units or individuals examples are ubiquitous in the health and social sciences including patients in hospitals employees in firms and students in schools what is the optimal matching strategy in a clustered observational study at first thought one might start by matching clusters of individuals and then within matched clusters continue by matching individuals but as we discuss in this article the optimal strategy is the opposite in typical applications where the intracluster correlation is not one it is best to first match individuals and once all possible combinations of matched individuals are known then match clusters in this article we use dynamic and integer programming to implement this strategy and extend optimal matching methods to hierarchical and multilevel settings among other match

confidence intervals for causal parameters consider an unbiased follow up study designed to investigate the causal effect of a dichotomous exposure on a dichotomous disease outcome under a deterministic outcome model a standard 95 per cent binomial confidence interval may fail to cover the causal parameter of interest at the nominal rate when we take the causal parameter to be a parameter associated with the observed study population regardless of whether the observed study population was sampled from a larger superpopulation i propose new interval estimators that in this setting improve upon the performance of the standard binomial confidence interval

semiparametric estimation of treatment effects given base line covariates on an outcome measured after a post randomization event occurs summary we consider estimation from a double blind randomized trial of treatment effect within levels of base line covariates on an outcome that is measured after a post treatment event e has occurred in the subpopulation pe e that would experience event e regardless of treatment specifically we consider estimation of the parameters y indexing models for the outcome mean conditional on treatment and base line covariates in the subpopulation p e such parameters are not identified from randomized trial data but become identified if additionally it is assumed that the subpopulation p of subjects that would experience event e under the second treatment but not under the first s empty and a parametric model for the conditional probability that a subject experiences event e if assigned to the first treatment given that the subject would experience the event if assigned to the second treatment his or her outcome under the second treatment and his or her pretreatment covariates we develop a class of estimating equations whose solutions comprise up to asymptotic equivalence all consistent and asymptotically normal estimators of 7y under these two assumptions in addition we derive a locally semiparametric efficient estimator of y we apply our methods to estimate the effect on mean viral load of vaccine versus placebo after infection with human immunodeficiency virus the event e in a placebo controlled randomized acquired immune deficiency syndrome vaccine trial

on the propensity score weighting analysis with survival outcome estimands estimation and inference propensity score analysis is widely used in observational studies to adjust for confounding and estimate the causal effect of a treatment on the outcome when the outcome is survival time there are special considerations on the definition of the causal estimand point and variance estimation that have not been thoroughly studied in the literature we investigate propensity score analysis of survival data with a class of weighting methods we consider the following estimands in the two sample context average survival time restricted average survival time survival probability survival quantile and the marginal hazard ratio we propose a unified analytic framework to obtain the point and variance estimators the proposed methodology properly adjusts for the sampling variability in the estimated propensity scores extensive simulations show that the point and variance estimators possess desired finite sample properties and demonstrate better numerical performance than some existing weighting and matching methods commonly used in the literature the proposed methodology is illustrated with data from a breast cancer study

sharp nonparametric bounds and randomization inference for treatment effects on an ordinal outcome in clinical research investigators are interested in inferring the average causal effect of a treatment however the causal parameter that can be used to derive the average causal effect is not well defined for ordinal outcomes although some definitions have been proposed they are limited in that they are not identical to the well defined causal risk for a binary outcome which is the simplest ordinal outcome in this paper we propose the use of a causal parameter for an ordinal outcome defined as the proportion that a potential outcome under one treatment condition would not be smaller than that under the other condition for a binary outcome this proportion is identical to the causal risk unfortunately the proposed causal parameter cannot be identified even under randomization therefore we present a numerical method to calculate the sharp nonparametric bounds within a sample reflecting the impact of confounding when the assumption of independent potential outcomes is included the causal parameter can be identified when randomization is in play then we present exact tests and the associated confidence intervals for the relative treatment effect using the randomization based approach which are an extension of the existing methods for a binary outcome our methodologies are illustrated using data from an emetic prevention clinical trial

response to comment on randomization inference for treatment effects on a binary outcome we thank professor yasutaka chiba 1 for commenting on rigdon and hudgens rh 2 chiba 1 described a certain exact confidence interval reported in rh as somewhat unnatural chiba also presented an alternative approach to constructing confidence intervals 3 in this response we i provide a simple explanation why the confidence interval in rh appeared unnatural and ii explain the relationship between the rh 2 and chiba 3 confidence intervals essentially the two approaches are equivalent except rh entails inverting one two sided test whereas chiba inverts two one sided tests we present a more computationally efficient method rlh for computing the rh intervals based on chiba s principal stratification formulation of the problem we also propose a third method based on blaker 4 which inverts a single two sided test but forms a confidence interval that is at least as narrow as inverting two one sided tests simulation results show the rlh intervals tend to be as narrow or narrower than the chiba and blaker intervals on average

sensitivity of two sample permutation inferences in observational studies abstract in observational studies subjects are not randomly assigned to treatment or control so they may differ in their chances of receiving the treatment a simple method is developed and demonstrated for displaying the sensitivity of conventional two group permutation inferences to departures from random assignment of treatments the unmatched case discussed here differs in certain technical and computational details from the matched case discussed previously however the underlying model and the method for quantifying departures from randomization are the same the method may be applied to wilcoxon s rank sum test the gehan and log rank tests for censored outcomes mantel s test for scored categories and fisher s exact test for binary responses the method embeds the usual randomization reference distribution in a one parameter family of departures involving an unobserved covariate that would have been controlled by adjustments had it been observed as this parameter is varied the sensitivity

assessing the performance of the generalized propensity score for estimating the effect of quantitative or continuous exposures on binary outcomes propensity score methods are increasingly being used to estimate the effects of treatments and exposures when using observational data the propensity score was initially developed for use with binary exposures the generalized propensity score gps is an extension of the propensity score for use with quantitative or continuous exposures eg dose or quantity of medication income or years of education we used monte carlo simulations to examine the performance of different methods of using the gps to estimate the effect of continuous exposures on binary outcomes we examined covariate adjustment using the gps and weighting using weights based on the inverse of the gps we examined both the use of ordinary least squares to estimate the propensity function and the use of the covariate balancing propensity score algorithm the use of methods based on the gps was compared with the use of g computation all methods resulted in essentially unbiased estimation of the population dose response function however gps based weighting tended to result in estimates that displayed greater variability and had higher mean squared error when the magnitude of confounding was strong of the methods based on the gps covariate adjustment using the gps tended to result in estimates with lower variability and mean squared error when the magnitude of confounding was strong we illustrate the application of these methods by estimating the effect of average neighborhood income on the probability of death within 1 year of hospitalization for an acute myocardial infarction

three dimensional elastic problems of three dimensional quasicrystals in this paper the three dimensional initial value problem for three dimensional quasicrystals is considered for special polynomial form of initial conditions and nonhomogeneous terms it is proved that the solution can be presented in a similar form an analytic method for finding the exact solution of the problem with this polynomial form of initial conditions and nonhomogeneous terms is presented

causal inference without counterfactuals abstract a popular approach to the framing and answering of causal questions relies on the idea of counterfactuals outcomes that would have been observed had the world developed differently for example if the patient had received a different treatment by definition one can never observe such quantities nor assess empirically the validity of any modeling assumptions made about them even though one s conclusions may be sensitive to these assumptions here i argue that for making inference about the likely effects of applied causes counterfactual arguments are unnecessary and potentially misleading an alternative approach based on bayesian decision analysis is presented properties of counterfactuals are relevant to inference about the likely causes of observed effects but close attention then must be given to the nature and context of the query as well as to what conclusions can and cannot be supported empirically in particular even in the absence of statistical uncertainty such inferences m

asymptotic behavior of nonautonomous monotone and subgradient evolution equations in a hilbert setting h we study the asymptotic behavior of the trajectories of nonautonomous evolution equations dot x t a t x t ni 0 where for each t geq 0 a t h rightrightarrows h denotes a maximal monotone operator we provide general conditions guaranteeing the weak ergodic convergence of each trajectory x cdot to a zero of a limit maximal monotone operator a infty as the time variable t tends to infty the crucial point is to use the brezis haraux function or equivalently the fitzpatrick function to express at which rate the excess of mathrm gph a infty over mathrm gph a t tends to zero this approach gives a sharp and unifying view of this subject in the case of operators a t partial phi t which are subdifferentials of proper closed convex functions phi t we show convergence results for the trajectories then we specialize our results to multiscale evolution equations and obtain asymptotic properties of hierarchical minimization and selection of viscosity solutions illustrations are given in the field of coupled systems and partial differential equations

sharp bounds on causal effects using a surrogate endpoint this paper considers a problem of evaluating the causal effect of a treatment x on a true endpoint y using a surrogate endpoint s in the presence of unmeasured confounders between s and y such confounders render the causal effect of x on y unidentifiable from the causal effect of x on s and the joint probability of s and y to evaluate the causal effect of x on y in such a situation this paper derives closed form formulas for the sharp bounds on the causal effect of x on y based on both the causal effect of x on s and the joint probability of s and y under various assumptions in addition we show that it is not always necessary to observe y to test the null causal effect of x on y under the monotonicity assumption between x and s these bounds enable clinical practitioners and researchers to assess the causal effect of a treatment on a true endpoint using a surrogate endpoint with minimum computational effort

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

causal inference in outcome dependent two phase sampling designs summary we consider estimation of the causal effect of a treatment on an outcome from observational data collected in two phases in the first phase a simple random sample of individuals is drawn from a population on these individuals information is obtained on treatment outcome and a few low dimensional covariates these individuals are then stratified according to these factors in the second phase a random subsample of individuals is drawn from each stratum with known stratum specific selection probabilities on these individuals a rich set of covariates is collected in this setting we introduce five estimators simple inverse weighted simple doubly robust enriched inverse weighted enriched doubly robust locally efficient we evaluate the finite sample performance of these estimators in a simulation study we also use our methodology to estimate the causal effect of trauma care on in hospital mortality by using data from the national study of cost and outcomes of trauma

bayesian inference for a generalized population attributable fraction the impact of early vitamin a levels on chronic lung disease in very low birthweight infants in this paper the population attributable fraction is studied using the potential responses framework of rubin s causal model this framework facilitates definition of a general measure of population attributable effect which can accommodate many valued and multivariate exposures as well as many valued responses inferential issues are considered from the bayesian perspective finite population inference is emphasized with inference in the case of a fully observed population given particular attention the key inferential issue concerns computation of the posterior distribution of unobserved potential responses given observed responses exposures and covariates a dependency on model parameters about which observed data are uninformative is highlighted and this reflects the unobservable nature of causal effects in an application to a small cohort study of respiratory problems in very low birthweight infants posterior inferences were found to be insensitive to assumptions concerning the joint distribution of potential response variables but sensitive to the assumption of weak ignorability a weaker form of the more familiar assumption of no confounding by omitted covariates in a model based set up the weak ignorability assumption is identified with setting a model parameter to zero and consequently uncertainty concerning this assumption can in principle be handled via the prior distribution for the model parameters

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

generalized cornfield conditions for the risk difference a central question in causal inference with observational studies is the sensitivity of conclusions to unmeasured confounding the classical cornfield condition allows us to assess whether an unmeasured binary confounder can explain away the observed relative risk of the exposure on the outcome it states that for an unmeasured confounder to explain away an observed relative risk the association between the unmeasured confounder and the exposure and the association between the unmeasured confounder and the outcome must both be larger than the observed relative risk in this paper we extend the classical cornfield condition in three directions first we consider analogous conditions for the risk difference and allow for a categorical not just a binary unmeasured confounder second we provide more stringent thresholds that the maximum of the above mentioned associations must satisfy rather than weaker conditions that both must satisfy third we show that all the earlier results on cornfield conditions hold under weaker assumptions than previously used we illustrate the potential applications by real examples where our new conditions give more information than the classical ones

a characterization of optimal designs for observational studies summary an empirical investigation of the effects of a treatment is an observational study if it involves the comparison of treated and control groups that were not formed by randomization in such studies treated and control groups may differ systematically with respect to pretreatment measures or covariates and addressing these pre treatment differences is a central concern matching and subclassification are two standard methods of adjusting for observed pretreatment differences they may be used alone e g cochran 1968 or in conjunction with analytical or model based adjustments e g rubin 1973 1979 holford 1978 rosenbaum and rubin 1984 section 3 3 and rosenbaum 1987 1988a in particular rubin s simulation studies suggest that model based adjustments applied to matched or subclassified samples are more robust than model based adjustments applied to unmatched samples the purpose of this paper is to identify the structure of an optimal subclassification i e one in which the treated and control subjects in the same subclass are on average as similar as possible with respect to observed covariates it turns out that this structure is simple and intuitive while adjustments may control imbalances in observed covariates they cannot

exact confidence intervals for the average causal effect on a binary outcome based on the physical randomization of completely randomized experiments in a recent article in statistics in medicine rigdon and hudgens propose two approaches to obtaining exact confidence intervals for the average causal effect on a binary outcome they construct the first confidence interval by combining with the bonferroni adjustment the prediction sets for treatment effects among treatment and control groups and the second one by inverting a series of randomization tests with sample size n their second approach requires performing o n4 randomization tests we demonstrate that the physical randomization also justifies other ways to constructing exact confidence intervals that are more computationally efficient by exploiting recent advances in hypergeometric confidence intervals and the stochastic order information of randomization tests we propose approaches that either do not need to invoke monte carlo or require performing at most o n2 randomization tests we provide technical details and r code in the supporting information

evaluation of subset matching methods and forms of covariate balance this paper conducts a monte carlo simulation study to evaluate the performance of multivariate matching methods that select a subset of treatment and control observations the matching methods studied are the widely used nearest neighbor matching with propensity score calipers and the more recently proposed methods optimal matching of an optimally chosen subset and optimal cardinality matching the main findings are i covariate balance as measured by differences in means variance ratios kolmogorov smirnov distances and cross match test statistics is better with cardinality matching because by construction it satisfies balance requirements ii for given levels of covariate balance the matched samples are larger with cardinality matching than with the other methods iii in terms of covariate distances optimal subset matching performs best iv treatment effect estimates from cardinality matching have lower root mean square errors provided strong requirements for balance specifically fine balance or strength k balance plus close mean balance in standard practice a matched sample is considered to be balanced if the absolute differences in means of the covariates across treatment groups are smaller than 0 1 standard deviations however the simulation results suggest that stronger forms of balance should be pursued in order to remove systematic biases due to observed covariates when a difference in means treatment effect estimator is used in particular if the true outcome model is additive then marginal distributions should be balanced and if the true outcome model is additive with interactions then low dimensional joints should be balanced copyright 2016 john wiley sons ltd

exact solutions of the generalized dugdale model of two dimensional decagonal quasicrystals abstract the basic properties of the generalized dugdale model of quasicrystals have significance in the theory of fracture mechanics of this solids this study suggests a conformal mapping which transforms the complicated region in physical plane onto the exterior of the unit circle in the plane then in terms of cauchy integral theory the complicated boundary value problem is solved and the solutions are expressed in close form exactly the crack tip opening displacement ctod cohesive force zone and the most important physical quantity stress intensity factor are obtained with exact analytic form which are useful in materials science and engineering

estimation of the effect of interventions that modify the received treatment motivated by a study of surgical operating time and post operative outcomes for lung cancer we consider the estimation of causal effects of continuous point exposure treatments to investigate causality the standard paradigm postulates a series of treatment specific counterfactual outcomes and establishes conditions under which we may learn about them from observational study data while many choices are possible causal effects are typically defined in terms of variation of the mean of counterfactual outcomes in hypothetical worlds in which specific treatment strategies are applied to all individuals for example one might compare two worlds one where each individual receives some specific dose and a second where each individual receives some other dose for our motivating study defining causal effects in this way corresponds to hypothetical interventions that could not conceivably be implemented in the real world in this work we consider an alternative complimentary framework that investigates variation in the mean of counterfactual outcomes under hypothetical treatment strategies where each individual receives a treatment dose corresponding to that actually received but modified in some pre specified way quantification of this variation is defined in terms of contrasts for specific interventions as well as in terms of the parameters of a new class of marginal structural mean models within this framework we propose three estimators an outcome regression estimator an inverse probability of treatment weighted estimator and a doubly robust estimator we illustrate the methods with an analysis of the motivating data copyright 2013 john wiley sons ltd

asymptotic separability in sensitivity analysis summary in an observational study in which each treated subject is matched to several untreated controls by using observed pretreatment covariates a sensitivity analysis asks how hidden biases due to unobserved covariates might alter the conclusions the bounds required for a sensitivity analysis are the solution to an optimization problem in general this optimization problem is not separable in the sense that one cannot find the needed optimum by performing a separate optimization in each matched set and combining the results we show however that this optimization problem is asymptotically separable so that when there are many matched sets a separate optimization may be performed in each matched set and the results combined to yield the correct optimum with negligible error this is true when the wilcoxon rank sum test or the hodgeslehmann aligned rank test is applied in matching with multiple controls numerical calculations show that the asymptotic approximation performs well with as few as 10 matched sets in the case of the rank sum test a table is given containing the separable solution with this table only simple arithmetic is required to conduct the sensitivity analysis the method also supplies estimates such as the hodges lehmann estimate and confidence intervals associated with rank tests the method is illustrated in a study of dropping out of us high schools and the effects on cognitive test scores

reducing bias in observational studies using subclassification on the propensity score abstract the propensity score is the conditional probability of assignment to a particular treatment given a vector of observed covariates previous theoretical arguments have shown that subclassification on the propensity score will balance all observed covariates subclassification on an estimated propensity score is illustrated using observational data on treatments for coronary artery disease five subclasses defined by the estimated propensity score are constructed that balance 74 covariates and thereby provide estimates of treatment effects using direct adjustment these subclasses are applied within sub populations and model based adjustments are then used to provide estimates of treatment effects within these sub populations two appendixes address theoretical issues related to the application the effectiveness of subclassification on the propensity score in removing bias and balancing properties of propensity scores with incomplete data

studentized sensitivity analysis for the sample average treatment effect in paired observational studies abstracta fundamental limitation of causal inference in observational studies is that perceived evidence for an effect might instead be explained by factors not accounted for in the primary analysi

dealing with limited overlap in estimation of average treatment effects summary estimation of average treatment effects under unconfounded or ignorable treatment assignment is often hampered by lack of overlap in the covariate distributions between treatment groups this lack of overlap can lead to imprecise estimates and can make commonly used estimators sensitive to the choice of specification in such cases researchers have often used ad hoc methods for trimming the sample we develop a systematic approach to addressing lack of overlap we characterize optimal subsamples for which the average treatment effect can be estimated most precisely under some conditions the optimal selection rules depend solely on the propensity score for a wide range of distributions a good approximation to the optimal rule is provided by the simple rule of thumb to discard all units with estimated propensity scores outside the range 0 1 0 9

on the interpretation and observation of laws a law with factors x and concomitants z specifies a distribution given z of a potential value yx that is defined for each x whether or not it is observed an observed distribution of y given x and z agrees with the law if and only if given z the observed x is independent of yx or equivalently of the joint effect of ux of excluded variables on yx to establish such independence in non experimental data requires exhaustive exploration of the effects of concomitants causal and non causal r2 and f are irrelevant we show how the model free theory applies to linear models time series and simultaneous equations and point out its bayesian implications

a convergence result for nonautonomous subgradient evolution equations and its application to the steepest descent exponential penalty trajectory in linear programming abstract we present a new result on the asymptotic behavior of nonautonomous subgradient evolution equations of the form u t t u t where t t 0 is a family of closed proper convex functions the result is used to study the flow generated by the family t x f x r t where f x r ctx r exp aix bi r is the exponential penalty approximation of the linear program min ctx ax b and r t is a positive function tending to 0 when t we prove that the trajectory u t converges to an optimal solution u of the linear program and we give conditions for the convergence of an associated dual trajectory t toward an optimal solution of the dual program

bias in estimating the causal hazard ratio when using two stage instrumental variable methods two stage instrumental variable methods are commonly used to estimate the causal effects of treatments on survival in the presence of measured and unmeasured confounding two stage residual inclusion 2sri has been the method of choice over two stage predictor substitution 2sps in clinical studies we directly compare the bias in the causal hazard ratio estimated by these two methods under a principal stratification framework we derive a closed form solution for asymptotic bias of the causal hazard ratio among compliers for both the 2sps and 2sri methods when survival time follows the weibull distribution with random censoring when there is no unmeasured confounding and no always takers our analytic results show that 2sri is generally asymptotically unbiased but 2sps is not however when there is substantial unmeasured confounding 2sps performs better than 2sri with respect to bias under certain scenarios we use extensive simulation studies to confirm the analytic results from our closed form solutions we apply these two methods to prostate cancer treatment data from surveillance epidemiology and end results medicare and compare these 2sri and 2sps estimates with results from two published randomized trials copyright 2015 john wiley sons ltd

asymptotic behavior of gradient like dynamical systems involving inertia and multiscale aspects abstract in a hilbert space h we study the asymptotic behavior as time variable t goes to of nonautonomous gradient like dynamical systems involving inertia and multiscale features given h r and h r two convex differentiable functions a positive damping parameter and t a function of t which tends to zero as t goes to we consider the second order differential equation x t x t x t t x t 0 this system models the emergence of various collective behaviors in game theory as well as the asymptotic control of coupled nonlinear oscillators assuming that t tends to zero moderately slowly as t goes to infinity we show that the trajectories converge weakly in h the limiting equilibria are solutions of the hierarchical minimization problem which consists in minimizing over the set c of minimizers of as key assumptions we suppose that 0 t d t and that for every p belonging to a convex cone c depending on the data and 0 t p c t p d t where is the fenchel conjugate of and c is the support function of c an application is given to coupled oscillators

discrete optimization for interpretable study populations and randomization inference in an observational study of severe sepsis mortality abstractmotivated by an observational study of the effect of hospital ward versus intensive care unit admission on severe sepsis mortality we develop methods to address two common problems in observational studies 1 when there is a lack of covariate overlap between the treated and control groups how to define an interpretable study population wherein inference can be conducted without extrapolating with respect to important variables and 2 how to use randomization inference to form confidence intervals for the average treatment effect with binary outcomes our solution to problem 1 incorporates existing suggestions in the literature while yielding a study population that is easily understood in terms of the covariates themselves and can be solved using an efficient branch and bound algorithm we address problem 2 by solving a linear integer program to use the worst case variance of the average treatment effect among values for unobserved potential outcomes that are compatible with the null hy

randomization inference for treatment effects on a binary outcome two methods are developed for constructing randomization based confidence sets for the average effect of a treatment on a binary outcome the methods are nonparametric and require no assumptions about random sampling from a larger population both of the resulting 1 confidence sets are exact in the sense that the probability of containing the true treatment effect is at least 1 both types of confidence sets are also guaranteed to have width no greater than one in contrast a previously proposed asymptotic confidence interval is not exact and may have width greater than one the first approach combines bonferroni adjusted prediction sets for the attributable effects in the treated and untreated the second method entails inverting a permutation test simulations are presented comparing the two randomization based confidence sets with the asymptotic interval as well as the standard wald confidence interval and a commonly used exact interval for the difference in binomial proportions results show for small to moderate sample sizes that the permutation confidence set attains the narrowest width on average among the methods that maintain nominal coverage extensions that allow for stratifying on categorical baseline covariates are also discussed

sensitivity analysis for instrumental variables regression with overidentifying restrictions instrumental variables regression iv regression is a method for making causal inferences about the effect of a treatment based on an observational study in which there are unmeasured confounding variables the method requires one or more valid instrumental variables ivs a valid iv is a variable that is associated with the treatment is independent of unmeasured confounding variables and has no direct effect on the outcome often there is uncertainty about the validity of the proposed ivs when a researcher proposes more than one iv the validity of these ivs can be tested through the overidentifying restrictions test although the overidentifying restrictions test does provide some information the test has no power versus certain alternatives and can have low power versus many alternatives due to its omnibus nature to fully address uncertainty about the validity of the proposed ivs we argue that a sensitivity analysis is needed a sensitivity analysis examines the impact of plausible amounts of

evaluating markers for selecting a patient s treatment selecting the best treatment for a patient s disease may be facilitated by evaluating clinical characteristics or biomarker measurements at diagnosis we consider how to evaluate the potential impact of such measurements on treatment selection algorithms for example magnetic resonance neurographic imaging is potentially useful for deciding whether a patient should be treated surgically for carpal tunnel syndrome or should receive less invasive conservative therapy we propose a graphical display the selection impact si curve that shows the population response rate as a function of treatment selection criteria based on the marker the curve can be useful for choosing a treatment policy that incorporates information on the patient s marker value exceeding a threshold the si curve can be estimated using data from a comparative randomized trial conducted in the population as long as treatment assignment in the trial is independent of the predictive marker estimating the si curve is therefore part of a post hoc analysis to determine whether the marker identifies patients that are more likely to benefit from one treatment over another nonparametric and parametric estimates of the si curve are proposed in this article asymptotic distribution theory is used to evaluate the relative efficiencies of the estimators simulation studies show that inference is straightforward with realistic sample sizes we illustrate the si curve and statistical inference for it with data motivated by an ongoing trial of surgery versus conservative therapy for carpal tunnel syndrome

causality tests and observationally equivalent representations of econometric models abstract this article deals with the problem of observational equivalence of alternative econometric equation systems on the same data as a practical obstacle to econometricians success in applying definitions of causality by simon 1953 strotz and wold 1960 and granger 1969 to economic time series

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

full matching in an observational study of coaching for the sat among matching techniques for observational studies full matching is in principle the best in the sense that its alignment of comparable treated and control subjects is as good as that of any alternate method and potentially much better this article evaluates the practical performance of full matching for the first time modifying it in order to minimize variance as well as bias and then using it to compare coached and uncoached takers of the sat in this new version with restrictions on the ratio of treated subjects to controls within matched sets full matching makes use of many more observations than does pair matching but achieves far closer matches than does matching with k 2 controls prior to matching the coached and uncoached groups are separated on the propensity score by 1 1 sds full matching reduces this separation to 1 or 2 of an sd in older literature comparing matching and regression cochran expressed doubts that any method of adjustment could substantially reduce observed bias