verbal subgroups in the group of triangular matrices over field of characteristic 0 in this paper we describe the verbal subgroups of groups utn k and tn k which are the groups of respectively unitriangular and triangular matrices over field k of characteristic 0 we prove that every verbal subgroup of utn k coincides with a term of the lower central series of the group and that every verbal subgroup v tn k either is of the form v k utn k or coincides with a verbal subgroup of utn k we also describe the width of these verbal subgroups and make a few remarks regarding the groups of infinite triangular matrices

the bias due to incomplete matching observational studies comparing groups of treated and control units are often used to estimate the effects caused by treatments matching is a method for sampling a large reservoir of potential controls to produce a control group of modest size that is ostensibly similar to the treated group in practice there is a trade off between the desires to find matches for all treated units and to obtain matched treated control pairs that are extremely similar to each other we derive expressions for the bias in the average matched pair difference due to the failure to match all treated units incomplete matching and the failure to obtain exact matches inexact matching a practical example shows that the bias due to incomplete matching can be severe and moreover can be avoided entirely by using an appropriate multivariate nearest available matching algorithm which in the example leaves only a small residual bias due to inexact matching

exogeneity in structural equation models the practical relevance of several concepts of exogeneity of treatments for the estimation of causal parameters based on observational data are discussed we show that the traditional concepts such as strong ignorability and weak and super exogeneity are too restrictive if interest lies in average effects i e not on distributional effects of the treatment we suggest a new definition of exogeneity kl exogeneity it does not rely on distributional assumptions and is not based on counterfactual random variables as a consequence it can be empirically tested using a proposed test that is simple to implement and is distribution free

two stage least squares estimation of average causal effects in models with variable treatment intensity abstract two stage least squares tsls is widely used in econometrics to estimate parameters in systems of linear simultaneous equations and to solve problems of omitted variables bias in single equation estimation we show here that tsls can also be used to estimate the average causal effect of variable treatments such as drug dosage hours of exam preparation cigarette smoking and years of schooling the average causal effect in which we are interested is a conditional expectation of the difference between the outcomes of the treated and what these outcomes would have been in the absence of treatment given mild regularity assumptions the probability limit of tsls is a weighted average of per unit average causal effects along the length of an appropriately defined causal response function the weighting function is illustrated in an empirical example based on the relationship between schooling and earnings

conditional permutation tests and the propensity score in observational studies abstract in observational studies the distribution of treatment assignments is unknown and therefore randomization tests are not generally applicable however permutation tests that condition on sample information about the treatment assignment mechanism can be applicable in observational studies provided treatment assignment is strongly ignorable these tests use the conditional distribution of the treatment assignments given a sufficient statistic for the unknown parameter of the propensity score several tests that are commonly used in observational studies are particular instances of this general procedure moreover conditional permutation tests and covariance adjustment are closely related in the sense that the conditional permutation distribution of the covariance adjusted difference leads to the same inferences as the conditional permutation distribution of the unadjusted difference of sample means a backtrack algorithm is developed to permit efficient calculation of the exact conditional si

causal inference in paired two arm experimental studies under noncompliance with application to prognosis of myocardial infarction motivated by a study about prompt coronary angiography in myocardial infarction we propose a method to estimate the causal effect of a treatment in two arm experimental studies with possible noncompliance in both treatment and control arms we base the method on a causal model for repeated binary outcomes before and after the treatment which includes individual covariates and latent variables for the unobserved heterogeneity between subjects moreover given the type of noncompliance the model assumes the existence of three subpopulations of subjects compliers never takers and always takers we estimate the model using a two step estimator at the first step we estimate the probability that a subject belongs to one of the three subpopulations on the basis of the available covariates at the second step we estimate the causal effects through a conditional logistic method the implementation of which depends on the results from the first step the estimator is approximately consistent and under certain circumstances exactly consistent we provide evidence that the bias is negligible in relevant situations we compute standard errors on the basis of a sandwich formula the application shows that prompt coronary angiography in patients with myocardial infarction may significantly decrease the risk of other events within the next 2 years with a log odds of about 2 given that noncompliance is significant for patients being given the treatment because of high risk conditions classical estimators fail to detect or at least underestimate this effect copyright 2013 john wiley sons ltd

comparison of adaptive treatment strategies based on longitudinal outcomes in sequential multiple assignment randomized trials in sequential multiple assignment randomized trials longitudinal outcomes may be the most important outcomes of interest because this type of trials is usually conducted in areas of chronic diseases or conditions we propose to use a weighted generalized estimating equation gee approach to analyzing data from such type of trials for comparing two adaptive treatment strategies based on generalized linear models although the randomization probabilities are known we consider estimated weights in which the randomization probabilities are replaced by their empirical estimates and prove that the resulting weighted gee estimator is more efficient than the estimators with true weights the variance of the weighted gee estimator is estimated by an empirical sandwich estimator the time variable in the model can be linear piecewise linear or more complicated forms this provides more flexibility that is important because in the adaptive treatment setting the treatment changes over time and hence a single linear trend over the whole period of study may not be practical simulation results show that the weighted gee estimators of regression coefficients are consistent regardless of the specification of the correlation structure of the longitudinal outcomes the weighted gee method is then applied in analyzing data from the clinical antipsychotic trials of intervention effectiveness copyright 2016 john wiley sons ltd

sample size formulae for two stage randomized trials with survival outcomes two stage randomized trials are growing in importance in developing adaptive treatment strategies i e treatment policies or dynamic treatment regimes usually the first stage involves randomization to one of the several initial treatments the second stage of treatment begins when an early nonresponse criterion or response criterion is met in the second stage nonresponding subjects are re randomized among second stage treatments sample size calculations for planning these two stage randomized trials with failure time outcomes are challenging because the variances of common test statistics depend in a complex manner on the joint distribution of time to the early nonresponse criterion or response criterion and the primary failure time outcome we produce simple albeit conservative sample size formulae by using upper bounds on the variances the resulting formulae only require the working assumptions needed to size a standard single stage randomized trial and in common settings are only mildly conservative these sample size formulae are based on either a weighted kaplan meier estimator of survival probabilities at a fixed time point or a weighted version of the log rank test

structural nested mean models for assessing time varying effect moderation this article considers the problem of assessing causal effect moderation in longitudinal settings in which treatment or exposure is time varying and so are the covariates said to moderate its effect intermediate causal effects that describe time varying causal effects of treatment conditional on past covariate history are introduced and considered as part of robins structural nested mean model two estimators of the intermediate causal effects and their standard errors are presented and discussed the first is a proposed two stage regression estimator the second is robins g estimator the results of a small simulation study that begins to shed light on the small versus large sample performance of the estimators and on the bias variance trade off between the two estimators are presented the methodology is illustrated using longitudinal data from a depression study

on the matrix equation f x a abstract this paper deals with the matrix equation f x a where a c n n is a given matrix and is a complex holomorphic function defined on an open subset d of c it furnishes information concerning the set s of all solutions x c n n of x a the subset sp s of all solutions that are polynomials in a and the subset sd s of all diagonalizable solutions a necessary and sufficient condition for s to be nonempty is established relations among all solutions of x a lying in the same similarity class are established in the particular case where a r n n d is symmetric with respect to the real line r c and z f z for every z d a necessary and sufficient condition for s r n n to be nonempty is established necessary and sufficient conditions for sp and sd to be empty finite infinite and commutative are established explicit computations of all the elements of sp and sd are furnished and the numbers of elements of these two sets are determined necessary and sufficient conditions for the relations s sp s sd or sp sd to be satisfied are established necessary and sufficient conditions for the existence of normal hermitian skew hermitian positive definite and unitary solutions are established it is shown that the set p a of all polynomials p c x such that x p a is a solution of x a depends only on the minimal polynomial of a and not on the size of a an example is furnished of a matrix equation x a that has no solution in f n n for any field f but has a solution in r n n when r is a certain ring these results are applied to the particular equations xm a ex a and the fixed point equation x x

volatility and correlation based systemic risk measures in the us market this paper deals with the problem of how to use simple systemic risk measures to assess portfolio risk characteristics using three simple examples taken from previous literature one based on raw and partial correlations another based on the eigenvalue decomposition of the covariance matrix and the last one based on an eigenvalue entropy a granger causation analysis revealed some of them are not always a good measure of risk in the sp therefore in the sense of risk as volatility the indicators are not always suitable nevertheless their results towards returns are similar to previous works that accept them a deeper analysis has shown that any symmetric measure based on eigenvalue decomposition of correlation matrices however is not useful as a measure of correlation risk the empirical counterpart analysis of this proposition stated that negative correlations are usually small and therefore do not heavily distort the behavior of the indicator

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

on lattices of closed subgroups in the group of infinite triangular matrices over a field abstract we investigate a special type of closed subgroups of the topological group ut k of infinite dimensional unitriangular matrices over a field k k 2 considered with the natural inverse limit topology namely we generalize the concept of partition subgroups introduced in 23 and define partition subgroups in ut k we show that they are all closed and discuss the problem of their invariance to various group homomorphisms we prove that a characteristic subgroup of ut k is necessarily a partition subgroup and characterize the lattices of characteristic and fully characteristic subgroups in ut k we conclude with some implications of the given characterization on verbal structure of ut k and t k and use some topological properties to discuss the problem of the width of verbal subgroups in groups defined over a finite field k

estimating marginal causal effects in a secondary analysis of case control data when an initial case control study is performed data can be used in a secondary analysis to evaluate the effect of the case defining event on later outcomes in this paper we study the example in

identifiability stratification and minimum variance estimation of causal effects the weakest sufficient condition for the identifiability of causal effects is the weakly ignorable treatment assignment which implies that potential responses are independent of treatment assignment in each fine subpopulation stratified by a covariate in this paper we expand the independence that holds in fine subpopulations to the case that the independence may also hold in several coarse subpopulations each of which consists of several fine subpopulations and may have overlaps with other coarse subpopulations we first show that the identifiability of causal effects occurs if and only if the coarse subpopulations partition the whole population we then propose a principle called minimum variance principle which says that the estimator possessing the minimum variance is preferred in dealing with the stratification and the estimation of the causal effects the simulation results with the detail programming and a practical example demonstrate that it is a feasible and reasonable way to achieve our goals

efficient nonparametric estimation of causal effects in randomized trials with noncompliance causal approaches based on the potential outcome framework provide a useful tool for addressing noncompliance problems in randomized trials we propose a new estimator of causal treatment effects in randomized clinical trials with noncompliance we use the empirical likelihood approach to construct a profile random sieve likelihood and take into account the mixture structure in outcome distributions so that our estimator is robust to parametric distribution assumptions and provides substantial finite sample efficiency gains over the standard instrumental variable estimator our estimator is asymptotically equivalent to the standard instrumental variable estimator and it can be applied to outcome variables with a continuous ordinal or binary scale we apply our method to data from a randomized trial of an intervention to improve the treatment of depression among depressed elderly patients in primary care practices

from association to causation in observational studies the role of tests of strongly ignorable treatment assignment abstract if treatment assignment is strongly ignorable then adjustment for observed covariates is sufficient to produce consistent estimates of treatment effects in observational studies a general approach to testing this critical assumption is developed and applied to a study of the effects of nuclear fallout on the risk of childhood leukemia r a fisher s advice on the interpretation of observational studies was make your theories elaborate formally make causal theories sufficiently detailed that under the theory strongly ignorable assignment has testable consequences

sampling bias and logistic models summary in a regression model the joint distribution for each finite sample of units is determined by a function px y depending only on the list of covariate values x x u1 x un on the sampled units no random sampling of units is involved in biological work random sampling is frequently unavoidable in which case the joint distribution p y x depends on the sampling scheme regression models can be used for the study of dependence provided that the conditional distribution p y x for random samples agrees with px y as determined by the regression model for a fixed sample having a non random configuration x the paper develops a model that avoids the concept of a fixed population of units thereby forcing the sampling plan to be incorporated into the sampling distribution for a quota sample having a predetermined covariate configuration x the sampling distribution agrees with the standard logistic regression model with correlated components for most natural sampling plans such as sequential or simple random sampling the conditional distribution p y x is not the same as the regression distribution unless px y has independent components in this sense most natural sampling schemes involving binary random effects models are biased the implications of this formulation for subject specific and population averaged procedures are explored

accounting for covariates observed post randomization for discrete and continuous repeated measures data adjusting for covariates observed post randomization is a difficult issue in randomized clinical trials one approach is to enter these covariates as time dependent covariates in a general linear model however this approach fails to account for interactions between the primary end point and confounding variable as they evolve over time in this paper we adopt the view that since the confounder is observed following randomization it should be treated as an outcome and analysed accordingly we consider the repeated measures design where both the primary end point and the confounding measure are observed repeatedly over patient follow up a generalized estimating equation model is applied to allow each set of repeated measures to be modelled in terms of important explanatory variables seemingly unrelated regression combines the two models into an overall framework for analysis inference is then performed by imposing restrictions on the confounder regression parameters to reflect the behaviour profile of interest estimation and identification procedures are described and the methodology is illustrated with an example

sharp sensitivity bounds for mediation under unmeasured mediator outcome confounding it is often of interest to decompose the total effect of an exposure into a component that acts on the outcome through some mediator and a component that acts independently through other pathways said another way we are interested in the direct and indirect effects of the exposure on the outcome even if the exposure is randomly assigned it is often infeasible to randomize the mediator leaving the mediator outcome confounding not fully controlled we develop a sensitivity analysis technique that can bound the direct and indirect effects without parametric assumptions about the unmeasured mediator outcome confounding

extraction of mth roots in matrix rings over fields abstract a matrix a m n f f an arbitrary field with characteristic p not necessarily positive is an m th power in m n f if and only if each of its p x primary components is where p x runs through the irreducible factors of the minimal or characteristic polynomial of a this paper establishes necessary and sufficient criteria for determining when such a p x primary matrix is an m th power the criteria fall into three cases 1 p x x if e 1 e 2 is the sequence of exponents of p x which form the elementary divisors of a extended by adding infinitely many 0 terms the criterion states that for all i 1 e i 1 m 1 e im 0 or 1 2 p x x m not divisible by p if is a root of the separable core q x of p x and e f then the multiplicity of each elementary divisor of a must be representable as a sum of integers not necessarily distinct each of which is the degree of some irreducible factor over e of the polynomial x m 3 p x x m a power of p in this case the criterion combines the criteria in the first two cases if e 1 e 2 is the sequence of exponents of p x which form the elementary divisors of a extended by adding infinitely many 0 terms then for all i 1 e i 1 m 1 e im 0 or 1 and the multiplicity of each e i must be representable as a sum of integers not necessarily distinct each of which is the degree of some irreducible factor over e of the polynomial x m

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

a general dynamical statistical model with causal interpretation summary we develop a general dynamical model as a framework for causal interpretation we first state a criterion of local independence in terms of measurability of processes that are involved in the doob meyer decomposition of stochastic processes then we define direct and indirect influence we propose a definition of causal influence using the concepts of a physical system this framework makes it possible to link descriptive and explicative statistical models and encompasses quantitative processes and events one of the features of the paper is the clear distinction between the model for the system and the model for the observation we give a dynamical representation of a conventional joint model for human immunodeficiency virus load and cd4 cell counts we show its inadequacy to capture causal influences whereas in contrast known mechanisms of infection by the human immunodeficiency virus can be expressed directly through a system of differential equations

improved estimation of controlled direct effects in the presence of unmeasured confounding of intermediate variables adjusting for a causal intermediate is a common analytic strategy for estimating an average causal direct effect acde the acde is the component of the total exposure effect that is not relayed through the specified intermediate even if the total effect is unconfounded the usual acde estimate may be biased when an unmeasured variable affects the intermediate and outcome variables using linear programming optimization to compute non parametric bounds we develop new acde estimators for binary measured variables in this causal structure and use root mean square confounding bias rmsb to compare their performance with the usual stratified estimator in simulated distributions of target populations comprised of the 64 possible potential response types as well as distributions of target populations restricted to subsets of 18 or 12 potential response types defined by monotonicity or no interactions assumptions of unit level causal effects we also consider target population distributions conditioned on fixed outcome risk among the unexposed or fixed true acde in one stratum of the intermediate results show that a midpoint estimator constructed from the optimization bounds has consistently lower rmsb than the usual stratified estimator both unconditionally and conditioned on any risk in the unexposed when conditioning on true acde this midpoint estimator performs more poorly only when conditioned on an extreme true acde in one stratum of the intermediate yet outperforms the stratified estimator in the other stratum when interaction is permitted an alternate limit modified crude estimator can never perform less favourably than the stratified estimator and often has lower rmsb

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

augmented designs to assess principal strata direct effects summary many studies involving causal questions are often concerned with understanding the causal pathways by which a treatment affects an outcome thus the concept of direct versus indirect effects comes into play we tackle the problem of disentangling direct and indirect effects by investigating new augmented experimental designs where the treatment is randomized and the mediating variable is not forced but only randomly encouraged there are two key features of our framework we adopt a principal stratification approach and we mainly focus on principal strata effects avoiding involving a priori counterfactual outcomes using non parametric identification strategies we provide a set of assumptions which allow us to identify partially the causal estimands of interest the principal strata direct effects some examples are shown to illustrate our design and causal estimands of interest large sample bounds for the principal strata average direct effects are provided and a simple hypothetical example is used to show how our augmented design can be implemented and how the bounds can be calculated finally our augmented design is compared and contrasted with a standard randomized design

stratification for the propensity score compared with linear regression techniques to assess the effect of treatment or exposure stratifying and matching by the propensity score are increasingly popular approaches to deal with confounding in medical studies investigating effects of a treatment or exposure a more traditional alternative technique is the direct adjustment for confounding in regression models this paper discusses fundamental differences between the two approaches with a focus on linear regression and propensity score stratification and identifies points to be considered for an adequate comparison the treatment estimators are examined for unbiasedness and efficiency this is illustrated in an application to real data and supplemented by an investigation on properties of the estimators for a range of underlying linear models we demonstrate that in specific circumstances the propensity score estimator is identical to the effect estimated from a full linear model even if it is built on coarser covariate strata than the linear model as a consequence the coarsening property of the propensity score adjustment for a one dimensional confounder instead of a high dimensional covariate may be viewed as a way to implement a pre specified richly parametrized linear model we conclude that the propensity score estimator inherits the potential for overfitting and that care should be taken to restrict covariates to those relevant for outcome

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

armitage lecture 2010 understanding treatment effects the value of integrating longitudinal data and survival analysis there is a single minded focus on events in survival analysis and we often ignore longitudinal data that are collected together with the event data this is due to a lack of methodology but also a result of the artificial distinction between survival and longitudinal data analyses understanding the dynamics of such processes is important but has been hampered by a lack of appreciation of the difference between confirmatory and exploratory causal inferences the latter represents an attempt at elucidating mechanisms by applying mediation analysis to statistical data and will usually be of a more tentative character than a confirmatory analysis the concept of local independence and the associated graphs are useful this is related to granger causality an important method from econometrics that is generally undervalued by statisticians this causality concept is different from the counterfactual one since it lacks lacks the intervention aspect the notion that one can intervene at will in naturally occurring processes which seems to underly much of modern causal inference is problematic when studying mediation and mechanisms it is natural to assume a stochastic process point of view when analyzing dynamic relationships we present some examples to illustrate this it is not clear how survival analysis must be developed to handle the complex life history data that are increasingly being collected today we give some suggestions copyright 2012 john wiley sons ltd

randomization inference with natural experiments an analysis of ballot effects in the 2003 california recall election since the 2000 u s presidential election social scientists have rediscovered a long tradition of research examining the effects of ballot format on voting using a new dataset collected by the new york times we investigate the causal effect of being listed on the first ballot page in the 2003 california gubernatorial recall election california law mandates a unique randomization procedure of ballot order that when appropriately modeled can be used to approximate a classical randomized experiment in a real world setting we apply randomization inference based on fisher s exact test which directly incorporates the exact randomization procedure and yields accurate nonparametric confidence intervals our results suggest that being listed on the first ballot page causes a statistically significant increase in vote shares for more than 40 of the minor candidates whereas there is no significant effect for the top two candidates we also investigate how randomization inference differs from conventional e

regression analysis with clustered data clustered data are found in many different types of studies for example studies involving repeated measures inter rater agreement studies household surveys crossover designs and community randomized trials analyses based on population average and cluster specific models are commonly used for estimating treatment exposure effects with clustered data this paper discusses conditions when one or both types of models are appropriate for estimating causal effects and when there is agreement between population average and cluster specific analyses applications of survey sampling methods to the robust estimation of standard errors of estimated treatment parameters are discussed

causality conditional independence and graphical separation in settable systems we study the connections between causal relations and conditional independence within the settable systems extension of the pearl causal model pcm our analysis clearly distinguishes between causal notions and probabilistic notions and it does not formally rely on graphical representations as a foundation we provide definitions in terms of suitable functional dependence for direct causality and for indirect and total causality via and exclusive of a set of variables based on these foundations we provide causal and stochastic conditions formally characterizing conditional dependence among random vectors of interest in structural systems by stating and proving the conditional reichenbach principle of common cause obtaining the classical reichenbach principle as a corollary we apply the conditional reichenbach principle to show that the useful tools of d separation and d separation can be employed to establish conditional independence within suitably restricted settable systems analogous to markovian pcms

lower bounds on communication complexity abstract we prove the following four results on communication complexity 1 for every k 2 the language of encodings of directed graphs of out degree one that contain a path of length k 1 from the first vertex to the last vertex can be recognized by exchanging o k log n 1 bits using a simple k round protocol and requires the exchange of n 1 2 k 4 log 3 n bits by any k 1 round protocol 2 for every k 1 and for infinitely many n 1 there exists a collection of sets lkn 0 1 2n that can be recognized by exchanging o k log n bits using a k round protocol and any k 1 round protocol recognizing lkn requires the exchange of n k bits 3 given a set l 0 1 2n there is a set l 0 1 8n such that any k round protocol recognizing l can be transformed to a k round fixed partition protocol recognizing l with the same communication complexity and vice versa 4 for every integer function f 1 f n n there are languages recognizable by a one round deterministic protocol exchanging f n bits but not by any nondeterministic protocol exchanging f n 1 bits the first two results show in an incomparable way an exponential gap between k 1 round and k round protocols settling a conjecture by papadimitriou and sipser the third result shows that as long as we are interested in existence proofs a fixed partition of the input is not a restriction the fourth result extends a result by papadimitriou and sipser who showed that for every integer function f 1 f n n there is a language accepted by a deterministic protocol exchanging f n bits but not by any deterministic protocol exchanging f n 1 bits

latent variables causal models and overidentifying constraints abstract when is a statistical dependency between two variables best explained by the supposition that one of these variables causes the other as opposed to the supposition that there is a possibly unmeasured common cause acting on both variables in this paper we describe an approach towards model specification developed more fully in our book discovering causal structure and illustrate its application to the aforementioned question briefly the approach is to determine constraints satisfied by the variance covariance matrix of a sample and then to conduct a quasi automated search for the causal specifications that will best explain those constraints

commutators and powers of infinite unitriangular matrices abstract in the paper we consider some commutator type and power type matrix equations in the group ut k of infinite dimensional unitriangular matrices over a field k we introduce a notion of a power outer commutator k m 1 m k x 1 x k and a power engel commutator e k l m 1 m k x y as outer respectively engel commutators modified by allowing powers of letters instead of letters alone for a given infinite unitriangular matrix a we discuss the matrix equations x k a k m 1 m k x 1 x k a and e k l m 1 m k x y a in variables x x 1 x k y as a main result we provide the necessary and sufficient conditions for solvability of these equations

commutator subgroup of vershik kerov group ii abstract we describe a commutator subgroup of vershik kerov group over an infinite field and find the bound for its commutator width this gives a partial solution of the problem posed by sushchanskii in 2010 we also describe the lower central series of the group of infinite upper triangular matrices over an infinite field and find the bound for its commutator width

identifying direct and indirect effects in a non counterfactual framework summary identifying direct and indirect effects is a common problem in the social science and medical literature and can be described as follows a treatment is administered and a response is recorded however another variable mediates the effect of the treatment on the response in some way channelling a part of the treatment effect the question is how to extricate the direct and channelled indirect effects from one another when it is not possible to intervene on the mediating variable the aim of the paper is to tackle this problem by using a model for direct and indirect effects based on the decision theoretic framework for causal inference

a galois approach to mth roots of matrices with rational entries abstract let a be a given n n matrix with rational entries and irreducible characteristic polynomial f x we investigate the galois groups of f x and f x m to find necessary and sufficient conditions for the existence of a solution b to the matrix equation a b m where b is also a matrix with rational entries we do this by finding necessary and sufficient conditions for f x m to have a factor of degree n with rational coefficients

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

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

dual and simultaneous sensitivity analysis for matched pairs summary when a study shows an association between a treatment and a response before concluding that there is a causal relationship it is useful to assess whether or not an unobserved variable u might explain the observed association sensitivity analysis clarifies the properties u must have in terms of its relationship to the response and its imbalance in the groups being compared a substantial literature has investigated the imbalance or association between u and group assignment this paper develops a dual sensitivity analysis for matched pairs focusing on the strength of the relationship between u and the response required to reduce an observed association to non significance a third simultaneous form of sensitivity analysis models both relationships between u and treatment assignment and u and response the simultaneous form allows one to compare results from the sensitivity analysis to subject matter knowledge about both relationships the methods are illustrated by several examples

model misspecification and robustness in causal inference comparing matching with doubly robust estimation in this paper we compare the robustness properties of a matching estimator with a doubly robust estimator we describe the robustness properties of matching and subclassification estimators by show

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

causation entropy identifies indirect influences dominance of neighbors and anticipatory couplings abstract inference of causality is central in nonlinear time series analysis and science in general a popular approach to infer causality between two processes is to measure the information flow between them in terms of transfer entropy using dynamics of coupled oscillator networks we show that although transfer entropy can successfully detect information flow in two processes it often results in erroneous identification of network connections under the presence of indirect interactions dominance of neighbors or anticipatory couplings such effects are found to be profound for time dependent networks to overcome these limitations we develop a measure called causation entropy and show that its application can lead to reliable identification of true couplings

parabolic subgroups of groups of column finite infinite matrices abstract for a wide class of rings r that contains all local and semilocal rings we consider the group sl vk r of all matrices over r of the form a b 0 c in which a is a finite matrix with determinant 1 and c is an upper triangular infinite matrix with the main diagonal 1 1 we describe all subgroups of sl vk r that contain all its upper triangular matrices and study their properties

on channel capacity per unit cost memoryless communication channels with arbitrary alphabets where each input symbol is assigned a cost are considered the maximum number of bits that can be transmitted reliably through the channel per unit cost is studied it is shown that if the input alphabet contains a zero cost symbol then the capacity per unit cost admits a simple expression as the maximum normalized divergence between two conditional output distributions the direct part of this coding theorem admits a constructive proof via stein s lemma on the asymptotic error probability of binary hypothesis tests single user multiple access and interference channels are studied

a tutorial on propensity score estimation for multiple treatments using generalized boosted models the use of propensity scores to control for pretreatment imbalances on observed variables in non randomized or observational studies examining the causal effects of treatments or interventions has become widespread over the past decade for settings with two conditions of interest such as a treatment and a control inverse probability of treatment weighted estimation with propensity scores estimated via boosted models has been shown in simulation studies to yield causal effect estimates with desirable properties there are tools e g the twang package in r and guidance for implementing this method with two treatments however there is not such guidance for analyses of three or more treatments the goals of this paper are twofold 1 to provide step by step guidance for researchers who want to implement propensity score weighting for multiple treatments and 2 to propose the use of generalized boosted models gbm for estimation of the necessary propensity score weights we define the causal quantities that may be of interest to studies of multiple treatments and derive weighted estimators of those quantities we present a detailed plan for using gbm to estimate propensity scores and using those scores to estimate weights and causal effects we also provide tools for assessing balance and overlap of pretreatment variables among treatment groups in the context of multiple treatments a case study examining the effects of three treatment programs for adolescent substance abuse demonstrates the methods

mediation analysis with time varying exposures and mediators in this paper we consider causal mediation analysis when exposures and mediators vary over time we give non parametric identification results discuss parametric implementation and also provide a weighting approach to direct and indirect effects based on combining the results of two marginal structural models we also discuss how our results give rise to a causal interpretation of the effect estimates produced from longitudinal structural equation models when there are time varying confounders affected by prior exposure and mediator natural direct and indirect effects are not identified however we define a randomized interventional analogue of natural direct and indirect effects that are identified in this setting the formula that identifies these effects we refer to as the mediational g formula when there is no mediation the mediational g formula reduces to robins regular g formula for longitudinal data when there are no time varying confounders affected by prior exposure and mediator values then the mediational g formula reduces to a longitudinal version of pearl s mediation formula however the mediational g formula itself can accommodate both mediation and time varying confounders and constitutes a general approach to mediation analysis with time varying exposures and mediators

addressing complications of intention to treat analysis in the combined presence of all or none treatment noncompliance and subsequent missing outcomes summary we study the combined impact that all or none compliance and subsequent missing outcomes can have on the estimation of the intention to treat effect of assignment in randomised studies in this setting a standard analysis which drops subjects with missing outcomes and ignores compliance information can be biased for the intention to treat effect to address all or none compliance that is followed by missing outcomes we construct a new estimation procedure for the intention to treat effect that maintains good randomisation based properties under more plausible nonignorable noncompliance and nonignorable missing outcome conditions the compound exclusion restriction on the effect of assignment and the latent ignorability of the missing data mechanism we present both theoretical results and a simulation study moreover we show how the two key concepts of compound exclusion and latent ignorability are relevant in more complicated settings such as right censoring of a time to event outcome

covariate selection for the nonparametric estimation of an average treatment effect observational studies in which the effect of a nonrandomized treatment on an outcome of interest is estimated are common in domains such as labour economics and epidemiology such studies often rel

on solvability of engel equations in the group of triangular matrices over a field abstract in the following paper we investigate commutator type matrix equations and discuss the existence of their solutions in particular we derive the solutions of the engel equations a e k x y where e k x y denotes the k th engel word in the groups of unitriangular and triangular matrices over field k of arbitrary characteristic the results directly apply to the discussion on the width of k engel subgroups in the considered groups we conclude with a few observations on that matter

doubly robust estimators of causal exposure effects with missing data in the outcome exposure or a confounder we consider the estimation of the causal effect of a binary exposure on a continuous outcome confounding and missing data are both likely to occur in practice when observational data are used to estimate this causal effect in dealing with each of these problems model misspecification is likely to introduce bias we present augmented inverse probability weighted aipw estimators that account for both confounding and missing data with the latter occurring in a single variable only these estimators have an element of robustness to misspecification of the models used our estimators require two models to be specified to deal with confounding and two to deal with missing data only one of each of these models needs to be correctly specified when either the outcome or the exposure of interest is missing we derive explicit expressions for the aipw estimator when a confounder is missing explicit derivation is complex so we use a simple algorithm which can be applied using standard statistical software to obtain an approximation to the aipw estimator

silence based communication communication complexity the minimum amount of communication required for computing a function of data held by several parties is studied a communication model where silence is used to convey information is introduced for this model the worst case and average case complexities of symmetric functions are studied for binary input functions the average and worst case complexities are determined and the protocols achieving them are described for functions of nonbinary inputs one round communication where each party is restricted to communicate in consecutive stages is considered and the extra amount of communication required by one over multiple round communication is analyzed for the special case of ternary input functions close lower and upper bounds on the worst case one round complexity are provided and protocols achieving them are described protocols achieving the average case one round complexity for ternary input functions are also described these protocols can be generalized to inputs of arbitrary size

bayesian analysis of cross section and clustered data treatment models this paper is concerned with the problem of determining the effect of a categorical treatment variable on a response given that the treatment is non randomly assigned and the response on any given subject is observed for one setting of the treatment we consider classes of models that are designed for such problems these models are subjected to a fully bayesian analysis based on markov chain monte carlo methods the analysis of the treatment effect is then based on amongst other things the posterior distribution of the potential outcomes counter factuals at the subject level which is obtained as a by product of the mcmc simulation procedure the analysis is extended to models with categorical treatments and binary and clustered outcomes the problem of model comparisons is also considered different aspects of the methodology are illustrated through two data examples

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

statistics and causal inference abstract problems involving causal inference have dogged at the heels of statistics since its earliest days correlation does not imply causation and yet causal conclusions drawn from a carefully designed experiment are often valid what can a statistical model say about causation this question is addressed by using a particular model for causal inference holland and rubin 1983 rubin 1974 to critique the discussions of other writers on causation and causal inference these include selected philosophers medical researchers statisticians econometricians and proponents of causal modeling

mutual information based matching for causal inference with observational data this paper presents an information theory driven matching methodology for making causal inference from observational data the paper adopts a potential outcomes framework view on evaluating the strength of cause effect relationships the population wide average effects of binary treatments are estimated by comparing two groups of units the treated and untreated control to reduce the bias in such treatment effect estimation one has to compose a control group in such a way that across the compared groups of units treatment is independent of the units covariates this requirement gives rise to a subset selection matching problem this paper presents the models and algorithms that solve the matching problem by minimizing the mutual information mi between the covariates and the treatment variable such a formulation becomes tractable thanks to the derived optimality conditions that tackle the non linearity of the sample based mi function computational experiments with mixed integer programming formulations and four matching algorithms demonstrate the utility of mi based matching for causal inference studies the algorithmic developments culminate in a matching heuristic that allows for balancing the compared groups in polynomial close to linear time thus allowing for treatment effect estimation with large data sets

a bayesian perspective on the analysis of unreplicated factorial experiments using potential outcomes unreplicated factorial designs have been widely used in scientific and industrial settings when it is important to distinguish active or real factorial effects from inactive or noise factorial effects used to estimate residual or error terms we propose a new approach to screen for active factorial effects from such experiments that uses the potential outcomes framework and is based on sequential posterior predictive model checks one advantage of the proposed method is its ability to broaden the standard definition of active effects and to link their definition to the population of interest another important aspect of this approach is its conceptual connection to fisherian randomization tests extensive simulation studies are conducted which demonstrate the superiority of the proposed approach over existing ones in the situations considered

causal bounds and observable constraints for non deterministic models conditional independence relations involving latent variables do not necessarily imply observable independences they may imply inequality constraints on observable parameters and causal bounds which can be used for falsification and identification the literature on computing such constraints often involve a deterministic underlying data generating process in a counterfactual framework if an analyst is ignorant of the nature of the underlying mechanisms then they may wish to use a model which allows the underlying mechanisms to be probabilistic a method of computation for a weaker model without any determinism is given here and demonstrated for the instrumental variable model though applicable to other models the approach is based on the analysis of mappings with convex polytopes in a decision theoretic framework and can be implemented in readily available polyhedral computation software well known constraints and bounds are replicated in a probabilistic model and novel ones are computed for instrumental variable models without non deterministic versions of the randomization exclusion restriction and monotonicity assumptions respectively

a linear theory for noncausality abstract different definitions of noncausality according to granger sims haugh and pierce are analyzed in terms of orthogonality in the hilbert space of square integrable variables conditions when necessary are given for their respective equivalence some problems of testability are mentioned finally noncausality is also analyzed in terms of rational expectations extending previous results of sims author