nonparametric estimation of distributional policy effects this paper proposes a fully nonparametric procedure to evaluate the effect of a counterfactual change in the distribution of some covariates on the unconditional distribution of an outcome variable of interest in contrast to other methods we do not restrict attention to the effect on the mean in particular our method can be used to conduct inference on the change of the distribution function as a whole its moments and quantiles inequality measures such as the lorenz curve or gini coefficient and to test for stochastic dominance the practical applicability of our procedure is illustrated via a simulation study and an empirical example

too many covariates and too few cases a comparative study prior research indicates that 10 15 cases or controls whichever fewer are required per parameter to reliably estimate regression coefficients in multivariable logistic regression models this condition may be difficult to meet even in a well designed study when the number of potential confounders is large the outcome is rare and or interactions are of interest various propensity score approaches have been implemented when the exposure is binary recent work on shrinkage approaches like lasso were motivated by the critical need to develop methods for the p n situation where p is the number of parameters and n is the sample size those methods however have been less frequently used when p n and in this situation there is no guidance on choosing among regular logistic regression models propensity score methods and shrinkage approaches to fill this gap we conducted extensive simulations mimicking our motivating clinical data estimating vaccine effectiveness for preventing influenza hospitalizations in the 2011 2012 influenza season ridge regression and penalized logistic regression models that penalize all but the coefficient of the exposure may be considered in these types of studies copyright 2016 john wiley sons ltd

model averaged double robust estimation researchers estimating causal effects are increasingly challenged with decisions on how to best control for a potentially high dimensional set of confounders typically a single propensity score model is chosen and used to adjust for confounding while the uncertainty surrounding which covariates to include into the propensity score model is often ignored and failure to include even one important confounder will results in bias we propose a practical and generalizable approach that overcomes the limitations described above through the use of model averaging we develop and evaluate this approach in the context of double robust estimation more specifically we introduce the model averaged double robust ma dr estimators which account for model uncertainty in both the propensity score and outcome model through the use of model averaging the ma dr estimators are defined as weighted averages of double robust estimators where each double robust estimator corresponds to a specific choice of the outcome model and the propensity score model the ma dr estimators extend the desirable double robustness property by achieving consistency under the much weaker assumption that either the true propensity score model or the true outcome model be within a specified possibly large class of models using simulation studies we also assessed small sample properties and found that ma dr estimators can reduce mean squared error substantially particularly when the set of potential confounders is large relative to the sample size we apply the methodology to estimate the average causal effect of temozolomide plus radiotherapy versus radiotherapy alone on one year survival in a cohort of 1887 medicare enrollees who were diagnosed with glioblastoma between june 2005 and december 2009

estimating treatment effects of longitudinal designs using regression models on propensity scores we derive regression estimators that can compare longitudinal treatments using only the longitudinal propensity scores as regressors these estimators which assume knowledge of the variables used in the treatment assignment are important for reducing the large dimension of covariates for two reasons first if the regression models on the longitudinal propensity scores are correct then our estimators share advantages of correctly specified model based estimators a benefit not shared by estimators based on weights alone second if the models are incorrect the misspecification can be more easily limited through model checking than with models based on the full covariates thus our estimators can also be better when used in place of the regression on the full covariates we use our methods to compare longitudinal treatments for type ii diabetes mellitus

algebraic bethe ansatz for the one dimensional bariev chain the algebraic bethe ansatz is formulated for the one dimensional bariev chain as a consequence the eigenvectors eigenvalues and bariev s bethe equations are rederived in a systematic algebraic way

semantics and pragmatics of fuzzy sets and systems abstract a philosophical formalism of a new methodological aspect of humanistic systems design and evaluation is given a requisite concept of context dependency is highlighted and some approaches to fuzzy sets and linguistics subsequently extended it is consequently shown that a mathematical theory of pragmatic fuzzy subsets is not only conceptually possible but practically implementable in man machine studies as well thus the important context dependent implication of the subjective nature of zadeh s theory of fuzzy sets can be better exploited as a pragmatic theory the approach here is a seeming connection between pragmatism and ontology concepts that are traditionally diametrically opposed to each other the attitude adopted has been the equation of pragmatism and psychophysical measurements of ontological objects noumena pragmatism is tacitly defined as a form of empiricism whereby linguistic constructs i e linguistic variable denotions that represent any aspect s of a humanistic system are nothing more than an operational procedure used to achieve psychophysical measurements of the aspect s in this fashion pragmatism would enable the contents of assertions which are made through declarative propositions about humanistic systems to be deciphered within relevant contexts for pragmatism direct sense experience provides both the meaning and the criterion of reality judgements the context dependent nature of the physical reality of ontological entities is therefore better understood pragmatically vis a vis the appropriate evaluative criteria and interpretative conventions by emphasizing the distinction between a fuzzy system naturally fuzzified and a fuzzified mathematical structure meta mathematically fuzzified the use of the standard fuzzy topologies is justified although it is conceptually possible to develop a yet more general topology or perhaps an alternative one especially in the case of a meta mathematically fuzzified structure however for the express purpose of machine implementation a naturally fuzzified system such as a complex humanistic system is more amenable to an initial test of the philosophy of pragmatic fuzzy subsets consequently the fuzzy topology and spaces employed are intended to be devoid of extensive generalities in this instance

conditioning on the propensity score can result in biased estimation of common measures of treatment effect a monte carlo study propensity score methods are increasingly being used to estimate causal treatment effects in the medical literature conditioning on the propensity score results in unbiased estimation of the expected difference in observed responses to two treatments the degree to which conditioning on the propensity score introduces bias into the estimation of the conditional odds ratio or conditional hazard ratio which are frequently used as measures of treatment effect in observational studies has not been extensively studied we conducted monte carlo simulations to determine the degree to which propensity score matching stratification on the quintiles of the propensity score and covariate adjustment using the propensity score result in biased estimation of conditional odds ratios hazard ratios and rate ratios we found that conditioning on the propensity score resulted in biased estimation of the true conditional odds ratio and the true conditional hazard ratio in all scenarios examined treatment effects were biased towards the null treatment effect however conditioning on the propensity score did not result in biased estimation of the true conditional rate ratio in contrast conventional regression methods allowed unbiased estimation of the true conditional treatment effect when all variables associated with the outcome were included in the regression model the observed bias in propensity score methods is due to the fact that regression models allow one to estimate conditional treatment effects whereas propensity score methods allow one to estimate marginal treatment effects in several settings with non linear treatment effects marginal and conditional treatment effects do not coincide

bias reduction in effectiveness analyses of longitudinal ordinal doses with a mixed effects propensity adjustment a mixed effects propensity adjustment is described that can reduce bias in longitudinal studies involving non equivalent comparison groups there are two stages in this data analytic strategy first a model of propensity for treatment intensity examines variables that distinguish among subjects who receive various ordered doses of treatment across time using mixed effects ordinal logistic regression second the effectiveness model examines multiple times until recurrence to compare the ordered doses using a mixed effects grouped time survival model effectiveness analyses are initially stratified by propensity quintile then the quintile specific results are pooled assuming that there is not a propensity x treatment interaction a monte carlo simulation study compares bias reduction in fully specified propensity model relative to misspecified models in addition type i error rate and statistical power are examined the approach is illustrated by applying it to a longitudinal observational study of maintenance treatment of major depression

the performance of different propensity score methods for estimating differences in proportions risk differences or absolute risk reductions in observational studies propensity score methods are increasingly being used to estimate the effects of treatments on health outcomes using observational data there are four methods for using the propensity score to estimate treatment effects covariate adjustment using the propensity score stratification on the propensity score propensity score matching and inverse probability of treatment weighting iptw using the propensity score when outcomes are binary the effect of treatment on the outcome can be described using odds ratios relative risks risk differences or the number needed to treat several clinical commentators suggested that risk differences and numbers needed to treat are more meaningful for clinical decision making than are odds ratios or relative risks however there is a paucity of information about the relative performance of the different propensity score methods for estimating risk differences we conducted a series of monte carlo simulations to examine this issue we examined bias variance estimation coverage of confidence intervals mean squared error mse and type i error rates a doubly robust version of iptw had superior performance compared with the other propensity score methods it resulted in unbiased estimation of risk differences treatment effects with the lowest standard errors confidence intervals with the correct coverage rates and correct type i error rates stratification matching on the propensity score and covariate adjustment using the propensity score resulted in minor to modest bias in estimating risk differences estimators based on iptw had lower mse compared with other propensity score methods differences between iptw and propensity score matching may reflect that these two methods estimate the average treatment effect and the average treatment effect for the treated respectively copyright 2010 john wiley sons ltd

quantile regression in the secondary analysis of case control data case control design is widely used in epidemiology and other fields to identify factors associated with a disease data collected from existing case control studies can also provide a cost effective way to investigate the association of risk factors with secondary outcomes when the secondary outcome is a continuous random variable most of the existing methods focus on the statistical inference on the mean of the secondary outcome in this article we propose a quantile based approach to facilitating a comprehensive investigation of covariates effects on multiple quantiles of the secondary outcome we construct a new family of estimating equations combining observed and pseudo outcomes which lead to consistent estimation of conditional quantiles using case control data simulations are conducted to evaluate the performance of our proposed approach and a case control study on genetic association with asthma is used to demonstrate the method supplementary materials for this article are available online

infinite conservation laws in the one dimensional hubbard model we show that the hamiltonian of the 1d hubbard model commutes with a one parameter family of transfer matrices of a new 2d classical model corresponding to two coupled six vertex models central to this result is a new local algebraic relation a generalization of the infinitesimal startriangle relation

the graded yang baxter relation for the one dimensional bariev chain of fermions the graded version of quantum yang baxter relation for the one dimensional bariev chain of fermions is formulated from this relation the commutativity of the transfer matrices for different values of the spectral parameter follows our result is consistent with the applicability of the traditional bethe ansatz method further the first non trivial conserved current next to the hamiltonian is constructed

estimators and confidence intervals for the marginal odds ratio using logistic regression and propensity score stratification propensity score methods are widely used to estimate treatment or exposure effects in observational studies in studies with binary response the effect can be described as an odds ratio and the mantel haenszel estimator is traditionally used for stratified data although propensity score methods are designed for marginal treatment effects it has been shown that the mantel haenszel estimator stratified for propensity score is a questionable estimator for the marginal odds ratio which describes the change in odds of response if everybody versus nobody were treated we studied recently proposed alternative estimators for the marginal odds ratio one stratified for propensity score the other derived from logistic regression additionally we adapted the methodology of the logistic regression based estimator for the derivation of a marginal odds ratio estimator to covariate adjustment by the propensity score we also derived corresponding variance estimators using the delta method the estimators were illustrated and compared to the inverse probability weighted estimator and the stratified mantel haenszel estimator in a study dealing with respiratory tract infections in children in germany furthermore simulation studies that were carried out to investigate relative bias variance and coverage probability showed reasonable performance of marginal odds ratio estimators if response rates or regression based approaches were used their variances were accurately estimated in contrast the stratified mantel haenszel estimator was substantially biased in some situations due to problems of non collapsibility and thus it is generally inappropriate for a reliable estimation of the marginal odds ratio

stratification and weighting via the propensity score in estimation of causal treatment effects a comparative study estimation of treatment effects with causal interpretation from observational data is complicated because exposure to treatment may be confounded with subject characteristics the propensity score the probability of treatment exposure conditional on covariates is the basis for two approaches to adjusting for confounding methods based on stratification of observations by quantiles of estimated propensity scores and methods based on weighting observations by the inverse of estimated propensity scores we review popular versions of these approaches and related methods offering improved precision describe theoretical properties and highlight their implications for practice and present extensive comparisons of performance that provide guidance for practical use

bayesian estimation of the average treatment effect on the treated using inverse weighting we develop a bayesian approach to estimate the average treatment effect on the treated in the presence of confounding the approach builds on developments proposed by saarela et al in the context of marginal structural models using importance sampling weights to adjust for confounding and estimate a causal effect the bayesian bootstrap is adopted to approximate posterior distributions of interest and avoid the issue of feedback that arises in bayesian causal estimation relying on a joint likelihood we present results from simulation studies to estimate the average treatment effect on the treated evaluating the impact of sample size and the strength of confounding on estimation we illustrate our approach using the classic right heart catheterization data set and find a negative causal effect of the exposure on 30 day survival in accordance with previous analyses of these data we also apply our approach to the data set of the national center for health statistics birth data and obtain a negative effect of maternal smoking during pregnancy on birth weight

estimation and inference for distribution functions and quantile functions in treatment effect models abstract we propose inverse probability weighted estimators for the distribution functions of the potential outcomes under the unconfoundedness assumption and apply the inverse mapping to obtain the quantile functions we show that these estimators converge weakly to zero mean gaussian processes a simulation method is proposed to approximate these limiting processes based on these results we construct tests for stochastic dominance relations between the potential outcomes monte carlo simulations are conducted to examine the finite sample properties of our tests we apply our test in an empirical example and find that a job training program had a positive effect on incomes

metrics for covariate balance in cohort studies of causal effects inferring causation from non randomized studies of exposure requires that exposure groups can be balanced with respect to prognostic factors for the outcome although there is broad agreement in the literature that balance should be checked there is confusion regarding the appropriate metric we present a simulation study that compares several balance metrics with respect to the strength of their association with bias in estimation of the effect of a binary exposure on a binary count or continuous outcome the simulations utilize matching on the propensity score with successively decreasing calipers to produce datasets with varying covariate balance we propose the post matching c statistic as a balance metric and found that it had consistently strong associations with estimation bias even when the propensity score model was misspecified as long as the propensity score was estimated with sufficient study size this metric along with the average standardized difference and the general weighted difference outperformed all other metrics considered in association with bias including the unstandardized absolute difference kolmogorov smirnov and levy distances overlapping coefficient mahalanobis balance and l1 metrics of the best performing metrics the c statistic and general weighted difference also have the advantage that they automatically evaluate balance on all covariates simultaneously and can easily incorporate balance on interactions among covariates therefore when combined with the usual practice of comparing individual covariate means and standard deviations across exposure groups these metrics may provide useful summaries of the observed covariate imbalance

estimation of causal effects of binary treatments in unconfounded studies estimation of causal effects in non randomized studies comprises two distinct phases design without outcome data and analysis of the outcome data according to a specified protocol recently gutman and rubin 2013 proposed a new analysis phase method for estimating treatment effects when the outcome is binary and there is only one covariate which viewed causal effect estimation explicitly as a missing data problem here we extend this method to situations with continuous outcomes and multiple covariates and compare it with other commonly used methods such as matching subclassification weighting and covariance adjustment we show using an extensive simulation that of all methods considered and in many of the experimental conditions examined our new multiple imputation using two subclassification splines method appears to be the most efficient and has coverage levels that are closest to nominal in addition it can estimate finite population average causal effects as well as non linear causal estimands this type of analysis also allows the identification of subgroups of units for which the effect appears to be especially beneficial or harmful copyright 2015 john wiley sons ltd

estimating exposure effects by modelling the expectation of exposure conditional on confounders in order to estimate the causal effects of one or more exposures or treatments on an outcome of interest one has to account for the effect of confounding factors which both covary with the exposures or treatments and are independent predictors of the outcome in this paper we present regression methods which in contrast to standard methods adjust for the confounding effect of multiple continuous or discrete covariates by modelling the conditional expectation of the exposures or treatments given the confounders in the special case of a univariate dichotomous exposure or treatment this conditional expectation is identical to what rosenbaum and rubin have called the propensity score they have also proposed methods to estimate causal effects by modelling the propensity score our methods generalize those of rosenbaum and rubin in several ways first our approach straightforwardly allows for multivariate exposures or treatments each of which may be continuous ordinal or discrete second even in the case of a single dichotomous exposure our approach does not require subclassification or matching on the propensity score so that the potential for residual confounding i e bias due to incomplete matching is avoided third our approach allows a rather general formalization of the idea that it is better to use the estimated propensity score than the true propensity score even when the true score is known the additional power of our approach derives from the fact that we assume the causal effects of the exposures or treatments can be described by the parametric component of a semiparametric regression model to illustrate our methods we reanalyze the effect of current cigarette smoking on the level of forced expiratory volume in one second in a cohort of 2 713 adult white males we compare the results with those obtained using standard methods

propensity score methods for bias reduction in the comparison of a treatment to a non randomized control group in observational studies investigators have no control over the treatment assignment the treated and non treated that is control groups may have large differences on their observed covariates and these differences can lead to biased estimates of treatment effects even traditional covariance analysis adjustments may be inadequate to eliminate this bias the propensity score defined as the conditional probability of being treated given the covariates can be used to balance the covariates in the two groups and therefore reduce this bias in order to estimate the propensity score one must model the distribution of the treatment indicator variable given the observed covariates once estimated the propensity score can be used to reduce bias through matching stratification subclassification regression adjustment or some combination of all three in this tutorial we discuss the uses of propensity score methods for bias reduction give references to the literature and illustrate the uses through applied examples

efficient estimation of average treatment effects using the estimated propensity score we are interested in estimating the average effect of a binary treatment on a scalar outcome if assignment to the treatment is exogenous or unconfounded that is independent of the potential outcomes given covariates biases associated with simple treatmentcontrol average comparisons can be removed by adjusting for differences in the covariates rosenbaum and rubin 1983 show that adjusting solely for differences between treated and control units in the propensity score removes all biases associated with differences in covariates although adjusting for differences in the propensity score removes all the bias this can come at the expense of efficiency as shown by hahn 1998 heckman ichimura and todd 1998 and robins mark and newey 1992 we show that weighting by the inverse of a nonparametric estimate of the propensity score rather than the true propensity score leads to an efficient estimate of the average treatment effect we provide intuition for this result by showing that this estimator can be interpreted as an empirical likelihood estimator that efficiently incorporates the information about the propensity score

non additive fusion hubbard models and non locality within the framework of quantum groups and additive r matrices the fusion procedure allows one to construct higher dimensional solutions of the yang baxter equation these solutions lead to integrable one dimensional spin chain hamiltonians here fusion is shown to generalize naturally to non additive r matrices which therefore do not have a quantum group symmetry this method is then applied to the generalized hubbard models although the resulting integrable models are not as simple as the starting ones the general structure is that of two spin s s sl 2 models coupled at the free fermion point an important issue is the probable lack of regular points which give local hamiltonians this problem is related to the existence of second order zeros in the unitarity equation and arises for the xx models of higher spins the building blocks of the hubbard models a possible connection between some lax operators l and r matrices is noted

estimating and using propensity scores with partially missing data abstract investigators in observational studies have no control over treatment assignment as a result large differences can exist between the treatment and control groups on observed covariates which can lead to badly biased estimates of treatment effects propensity score methods are an increasingly popular method for balancing the distribution of the covariates in the two groups to reduce this bias for example using matching or subclassification sometimes in combination with model based adjustment to estimate propensity scores which are the conditional probabilities of being treated given a vector of observed covariates we must model the distribution of the treatment indicator given these observed covariates much work has been done in the case where covariates are fully observed we address the problem of calculating propensity scores when covariates can have missing values in such cases which commonly arise in practice the pattern of missing covariates can be prognostically important and

the role of the propensity score in estimating dose response functions summary estimation of average treatment effects in observational studies often requires adjustment for differences in pre treatment variables if the number of pre treatment variables is large standard covariance adjustment methods are often inadequate rosenbaum rubin 1983 propose an alternative method for adjusting for pre treatment variables for the binary treatment case based on the so called propensity score here an extension of the propensity score methodology is proposed that allows for estimation of average casual effects with multi valued treatments estimation of average treatment effects in observational studies often requires adjustment for differences in pre treatment variables if the number of pre treatment variables is large and their distribution varies substantially with treatment status standard adjustment methods such as covari ance adjustment are often inadequate rosenbaum rubin 1983 1984 propose an alternative method for adjusting for pre treatment variables based on the propensity score the conditional probability of receiving the treatment given pre treatment variables they demonstrate that adjusting solely for the propensity score removes all bias associated with differences in the pre treatment variables the rosenbaum rubin proposals deal exclusively with binary valued treat ments in many cases of interest however treatments take on more than two values here an extension of the propensity score methodology is proposed that allows for estimation of average causal effects with multi valued treatments the key insight is that for estimation of average causal effects it is not necessary to divide the population into subpopulations where causal comparisons are valid as the propensity score does it is sufficient to divide the population into subpopulations where average potential outcomes can be estimated

regression and weighting methods for causal inference using instrumental variables please see the supplementary material for a correction and a new result recent researches in econometrics and statistics have gained considerable insights into the use of instrumental variables ivs for causal inference a basic idea is that ivs serve as an experimental handle the turning of which may change each individual s treatment status and through and only through this effect also change observed outcome the average difference in observed outcome relative to that in treatment status gives the average treatment effect for those whose treatment status is changed in this hypothetical experiment we build on the modern iv framework and develop two estimation methods in parallel to regression adjustment and propensity score weighting in the case of treatment selection based on covariates the iv assumptions are made explicitly conditional on covariates to allow for the fact that instruments can be related to these background variables the regression method focuses on the relationship between respo

weighted semiparametric estimation in regression analysis with missing covariate data abstract this article investigates estimation of the regression coefficients in an assumed mean function when covariates on some subjects are missing we examine the performance of a horvitz and thompson 1952 type weighted estimator by using different estimates of the selection probabilities which may be treated as nuisance parameters or a nuisance function in particular we investigate the properties of the estimate of the regression parameters when the selection probabilities are estimated by kernel smoothers we present large sample theory for the new estimator and conduct simulation studies comparing the proposed estimator to the maximum likelihood estimator and multiple imputation under various model assumptions and different missingness mechanisms in addition we provide two real examples that motivate this investigation

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

estimation of conditional and marginal odds ratios using the prognostic score introduced by hansen in 2008 the prognostic score pgs has been presented as the prognostic analogue of the propensity score pps pps based methods are intended to estimate marginal effects most previous studies evaluated the performance of existing pgs based methods adjustment stratification and matching using the pgs in situations in which the theoretical conditional and marginal effects are equal i e collapsible situations to support the use of pgs framework as an alternative to the pps framework applied researchers must have reliable information about the type of treatment effect estimated by each method we propose four new pgs based methods each developed to estimate a specific type of treatment effect we evaluated the ability of existing and new pgs based methods to estimate the conditional treatment effect cte the marginal average treatment effect on the whole population ate and the marginal average treatment effect on the treated population att when the odds ratio a non collapsible estimator is the measure of interest the performance of pgs based methods was assessed by monte carlo simulations and compared with pps based methods and multivariate regression analysis existing pgs based methods did not allow for estimating the ate and showed unacceptable performance when the proportion of exposed subjects was large when estimating marginal effects pps based methods were too conservative whereas the new pgs based methods performed better with low prevalence of exposure and had coverages closer to the nominal value when estimating cte the new pgs based methods performed as well as traditional multivariate regression copyright 2016 john wiley sons ltd

decorated star triangle relations and exact integrability of the one dimensional hubbard model the exact integrability of the one dimensional hubbard model is demonstrated with the help of a novel set of triangle relations the decorated star triangle relations the covering two dimensional statistical mechanical model obeys the star triangle or yang baxter relation a conjecture is presented for the eigenvalues of the transfer matrix

estimation of average treatment effects with misclassification this paper considers identification and estimation of the effect of a mismeasured binary regressor in a nonparametric or semiparametric regression or the conditional average effect of a binary treatment or policy on some outcome where treatment may be misclassified failure to account for misclassification is shown to result in attenua tion bias in the estimated treatment effect an identifying assumption that overcomes this bias is the existence of an instrument for the binary regressor that is conditionally independent of the treatment effect a discrete instrument suffices for nonparametric identification this paper provides conditions for identification and estimation of the av erage effect conditioned on covariates of a binary treatment program or pol icy on a scalar outcome when treatment may be misclassified more generally what is provided is an estimator of the effect of a binary regressor in a con ditional mean regression which may also include other regressors when the binary regressor is observed with error this equals the conditional average treatment effect given a weak unconfoundedness assumption misclassification occurs when a binary variable the treatment indicator is measured with error that is some units are reported to have received treat ment when they actually have not and vice versa for example in a returns to schooling analysis the outcome could be wages the binary variable could be attaining a college degree and misclassification could arise from school tran script reporting errors see e g kane and rouse 1995 and kane rouse and staiger 1999 bollinger 1996 considers misclassification of union status in wage equations if treatment is a job training program misclassification may arise if individuals assigned to the program fail to attend or if those not as signed obtain training elsewhere similarly for medical treatment individuals assigned a drug might fail to take it and those assigned a placebo might obtain treatment from another source more generally misclassification describes any binary variable that is sometimes mismeasured this paper first shows that misclassification causes attenuation bias in esti mated treatment effects analogous to the attenuation bias of classically mis measured variables in linear regression models additional assumptions on

evaluation of architectural design profiles using fuzzy sets abstract a design is a statement about values and design is intricately bound up with evaluation so that any design decision in an architectural context is either explicitly or implicitly based on evaluation on the hypotheses that a multi objective system of many attributes such as architecture is describable linguistically and that all natural or artificial languages of interest can be given transformational grammars this paper sets out to give conceptual general answers to the following questions 1 a what is the meaning of evaluation 2 b what are the forms of the operational rules whereby the meaning of architectural evaluation is built up from the space elements and contexts of a complex and contradictory architecture

estimating causal effects for multivalued treatments a comparison of approaches interventions with multivalued treatments are common in medical and health research such as when comparing the efficacy of competing drugs or interventions or comparing between various doses of a particular drug in recent years there has been a growing interest in the development of multivalued treatment effect estimators using observational data in this paper we compare the performance of commonly used regression based methods that estimate multivalued treatment effects based on the unconfoundedness assumption these estimation methods fall into three general categories i estimators based on a model for the outcome variable using conventional regression adjustment ii weighted estimators based on a model for the treatment assignment and iii doubly robust estimators that model both the treatment assignment and outcome variable within the same framework we assess the performance of these models using monte carlo simulation and demonstrate their application with empirical data our results show that i when models estimating both the treatment and outcome are correctly specified all adjustment methods provide similar unbiased estimates ii when the outcome model is misspecified regression adjustment performs poorly while all the weighting methods provide unbiased estimates iii when the treatment model is misspecified methods based solely on modeling the treatment perform poorly while regression adjustment and the doubly robust models provide unbiased estimates and iv when both the treatment and outcome models are misspecified all methods perform poorly given that researchers will rarely know which of the two models is misspecified our results support the use of doubly robust estimation

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

ophthalmological prognosis via fuzzy subsets abstract this paper examines the possibility of handling treatment effects as a fuzzy relational problem with particular reference to ophthalmology an ophthalmological evaluation is considered to consist of two major stages firstly an explanation of a patient s eye health state in terms of known disease labels is formulated through diagnosis secondly the effectiveness of a therapeutic strategy designed to restore a patient s normal eye health is then predicted via prognosis the paper considers the latter aspect in the sense of assessing the effects of medical intervention by comparing pre and post therapeutic fuzzy subsets

empirical comparison of approaches to forming strata using classification trees to adjust for covariates abstract in medical studies the computation of a meaningful incidence rate frequently demands adjustment for covariates one means for achieving such adjustment is standardization the directly standardized rate is calculated by using the appropriate weighted average of rates over strata determined by the covariates thus strata formation is crucial in obtaining good results this article compares three competing methods for stratification using data from a drug study in which the incidence rate of a certain side effect is of interest the techniques examined are a contingency table cell aggregation b classification trees and c logistic regression the relative merits of the three methods are discussed

bethe ansatz solution of the t j model the t j model for strongly correlated electron systems is mapped onto a model with a generalised permutation operator for a specific value of the ratio t j this model in one space dimension is shown to be equivalent to that of lai 1974 and can be solved by the bethe ansatz this special point corresponds to supersymmetric invariance of the hamiltonian

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

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

balance diagnostics for comparing the distribution of baseline covariates between treatment groups in propensity score matched samples the propensity score is a subject s probability of treatment conditional on observed baseline covariates conditional on the true propensity score treated and untreated subjects have similar distributions of observed baseline covariates propensity score matching is a popular method of using the propensity score in the medical literature using this approach matched sets of treated and untreated subjects with similar values of the propensity score are formed inferences about treatment effect made using propensity score matching are valid only if in the matched sample treated and untreated subjects have similar distributions of measured baseline covariates in this paper we discuss the following methods for assessing whether the propensity score model has been correctly specified comparing means and prevalences of baseline characteristics using standardized differences ratios comparing the variance of continuous covariates between treated and untreated subjects comparison of higher order moments and interactions five number summaries and graphical methods such as quantile quantile plots side by side boxplots and non parametric density plots for comparing the distribution of baseline covariates between treatment groups we describe methods to determine the sampling distribution of the standardized difference when the true standardized difference is equal to zero thereby allowing one to determine the range of standardized differences that are plausible with the propensity score model having been correctly specified we highlight the limitations of some previously used methods for assessing the adequacy of the specification of the propensity score model in particular methods based on comparing the distribution of the estimated propensity score between treated and untreated subjects are uninformative

maximum likelihood estimation for mixed continuous and categorical data with missing values summary maximum likelihood procedures for analysing mixed continuous and categorical data with missing values are presented the general location model of olkin tate 1961 and extensions introduced by krzanowski 1980 1982 form the basis for our methods maximum likelihood estimation with incomplete data is achieved by an application of the em algorithm dempster laird rubin 1977 special cases of the algorithm include orchard woodbury s 1972 algorithm for incomplete normal samples fuchs s 1982 algorithms for log linear modelling of partially classified contingency tables and day s 1969 algorithm for multivariate normal mixtures applications include a imputation of missing values b logistic regression and discriminant analysis with missing predictors and unclassified observations c linear regression with missing continuous and categorical predictors and d parametric cluster analysis with incomplete data methods are illustrated using data from the st louis risk research project some key word8 cluster analysis discriminant analysis em algorithm incomplete data linear regression logistic regression log linear model mixture model

effects of misspecification of the propensity score on estimators of treatment effect a comparison was carried out between the propensity score and prognostic models in estimating treatment effects from observational studies one issue investigated was the effect of estimating the propensity score on estimators of treatment effect a second question addressed comparisons of the propensity score and prognostic approach when a confounder is omitted third misspecifications of the propensity score were compared to messpecified response models in all cases there were two types of models one involving a continuous and one a binary response

a two step bayesian approach for propensity score analysis simulations and case study a two step bayesian propensity score approach is introduced that incorporates prior information in the propensity score equation and outcome equation without the problems associated with simultaneous bayesian propensity score approaches the corresponding variance estimators are also provided the two step bayesian propensity score is provided for three methods of implementation propensity score stratification weighting and optimal full matching three simulation studies and one case study are presented to elaborate the proposed two step bayesian propensity score approach results of the simulation studies reveal that greater precision in the propensity score equation yields better recovery of the frequentist based treatment effect a slight advantage is shown for the bayesian approach in small samples results also reveal that greater precision around the wrong treatment effect can lead to seriously distorted results however greater precision around the correct treatment effect parameter yields quite good results with slight improvement seen with greater precision in the propensity score equation a comparison of coverage rates for the conventional frequentist approach and proposed bayesian approach is also provided the case study reveals that credible intervals are wider than frequentist confidence intervals when priors are non informative

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 joint use of propensity and prognostic scores in estimation of the average treatment effect on the treated a simulation study propensity and prognostic score methods seek to improve the quality of causal inference in non randomized or observational studies by replicating the conditions found in a controlled experiment at least with respect to observed characteristics propensity scores model receipt of the treatment of interest prognostic scores model the potential outcome under a single treatment condition while the popularity of propensity score methods continues to grow prognostic score methods and methods combining propensity and prognostic scores have thus far received little attention to this end we performed a simulation study that compared subclassification and full matching on a single estimated propensity or prognostic score with three approaches combining estimated propensity and prognostic scores full matching on a mahalanobis distance combining the estimated propensity and prognostic scores full mahal full matching on the estimated prognostic propensity score within propensity score calipers full pgppty and subclassification on an estimated propensity and prognostic score grid with 5 5 subclasses subclass 5 5 we considered settings in which one both or neither score model was misspecified the data generating mechanisms varied in the degree of linearity and additivity in the true treatment assignment and outcome models full mahal and full pgppty exhibited strong to superior performance in root mean square error terms across all simulation settings and scenarios methods combining propensity and prognostic scores were no less robust to model misspecification than single score methods even when both score models were incorrectly specified our findings support the joint use of propensity and prognostic scores in estimation of the average treatment effect on the treated

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

assessing sensitivity to an unobserved binary covariate in an observational study with binary outcome this paper proposes a simple technique for assessing the range of plausible causal con clusions from observational studies with a binary outcome and an observed categorical covariate the technique assesses the sensitivity of conclusions to assumptions about an unobserved binary covariate relevant to both treatment assignment and response a medical study of coronary artery disease is used to illustrate the technique inevitably the results of clinical studies are subject to dispute in observational studies one basis for dispute is obvious since patients were not assigned to treatments at random patients at greater risk may be over represented in some treatment groups this paper proposes a method for assess ing the sensitivity of causal conclusions to an unmeasured patient characteristic relevant to both treatment assignment and response despite their limitations observational studies will continue to be a valuable source of information and therefore it is prudent to develop appropriate methods of analysis for them our sensitivity analysis consists of the estimation of the average effect of a treatment on a binary outcome variable after adjustment for observed categorical covariates and an unobserved binary covariate u under several sets of assumptions about u both cornfield et al 1959 and bross 1966 have proposed guidelines for determining whether an unmeasured binary covariate having specified properties could explain all of the apparent effect of a treatment that is whether the treatment effect after adjustment for u could be zero our method has two advantages first cornfield et al 1959 and bross 1966 adjust only for the unmeasured binary covariate u whereas we adjust for measured covariates in addition to the unmeasured covariate u second cornfield et al 1959 and bross 1966 1967 only judge whether the effect of the treatment could be zero having adjusted for u where cornfield et al 1959 employ an implicit yet extreme assumption about u in contrast we provide actual estimates of the treatment effect adjusted for both u and the observed categorical covariates under any assumption about u in principle the ith of the n patients under study has both a binary response r1i that would have resulted if he had received the new treatment and a binary response ro0 that would have resulted if he had received the control treatment in this formulation treatment effects are comparisons of r1i and roi such as r1i roi since each patient receives only one treatment either rli or ro0 is observed but not both and therefore comparisons of rli and roi imply some degree of speculation treatment effects defined as comparisons of the two potential responses r1i and roi of individual patients are implicit in fisher s 1953 randomization test of the sharp null

exact integrability of the one dimensional hubbard model the 1d hubbard model is shown to be an exactly integrable system a covering model of 2d statistical mechanics which i proposed recently was shown to provide a one parameter family of transfer matrices commuting with the hamiltonian of the hubbard model i show in this work that any two transfer matrices of a family commute mutually at the root of the commutation relation is the ubiquitous yang baxter factorization condition the form of the r operator is displayed explicitly

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

multiply robust estimation of causal quantile treatment effects in causal inference often the interest lies in the estimation of the average causal effect other quantities such as the quantile treatment effect may be of interest as well in this article we propose a multiply robust method for estimating the marginal quantiles of potential outcomes by achieving mean balance in a the propensity score and b the conditional distributions of potential outcomes an empirical likelihood or entropy measure approach can be utilized for estimation instead of inverse probability weighting which is known to be sensitive to the misspecification of the propensity score model simulation studies are conducted across different scenarios of correctness in both the propensity score models and the outcome models both simulation results and theoretical development indicate that our proposed estimator is consistent if any of the models are correctly specified in the data analysis we investigate the quantile treatment effect of mothers smoking status on infants birthweight

model based direct adjustment abstract direct adjustment or standardization applies population weights to subclass means in an effort to estimate population quantities from a sample that is not representative of the population direct adjustment has several attractive features but when there are many subclasses it can attach large weights to small quantities of data often in a fairly erratic manner in the extreme direct adjustment can attach infinite weight to nonexistent data a noticeable inconvenience in practice this article proposes a method of model based direct adjustment that preserves the attractive features of conventional direct adjustment while stabilizing the weights attached to small subclasses the sample mean and conventional direct adjustment are both special cases of model based direct adjustment under two different extreme models for the subclass specific selection probabilities the discussion of this method provides some insights into the behavior of true and estimated propensity scores the estimated scores

double robust estimator of average causal treatment effect for censored medical cost data in observational studies estimation of average causal treatment effect on a patient s response should adjust for confounders that are associated with both treatment exposure and response in addition the response such as medical cost may have incomplete follow up in this article a double robust estimator is proposed for average causal treatment effect for right censored medical cost data the estimator is double robust in the sense that it remains consistent when either the model for the treatment assignment or the regression model for the response is correctly specified double robust estimators increase the likelihood the results will represent a valid inference asymptotic normality is obtained for the proposed estimator and an estimator for the asymptotic variance is also derived simulation studies show good finite sample performance of the proposed estimator and a real data analysis using the proposed method is provided as illustration copyright 2016 john wiley sons ltd

hidden local gauge invariance in the one dimensional hubbard model and its equivalent coupled spin model hidden local gauge invariance in the one dimensional 1 d hubbard model and its equivalent coupled spin model is studied it is found that abelian u 1 u 1 gauge transformations appear in both cases furthermore it is shown that the energy spectrum is gauge invariant whereas the eigenvectors are explicitly gauge dependent however this result relies heavily on shastry s conjecture about the eigenvalue of the transfer matrix for the 1 d hubbard model lastly there is also a discrete symmetry associated to z2 z2 once this symmetry is broken one immediately obtains another nontrivial solution to the yang baxter relations ufaipxr

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

testing treatment effects in unconfounded studies under model misspecification logistic regression discretization and their combination logistic regression is commonly used to test for treatment effects in observational studies if the distribution of a continuous covariate differs between treated and control populations logistic regression yields an invalid hypothesis test even in an uncounfounded study if the link is not logistic this flaw is not corrected by the commonly used technique of discretizing the covariate into intervals a valid test can be obtained by discretization followed by regression adjustment within each interval

analysis of contingency tables by ideal point discriminant analysis cross classified data are frequently encountered in behavioral and social science research the loglinear model and dual scaling correspondence analysis are two representative methods of analyzing such data an alternative method based on ideal point discriminant analysis da is proposed for analysis of contingency tables which in a certain sense encompasses the two existing methods a variety of interesting structures can be imposed on rows and columns of the tables through manipulations of predictor variables and or as direct constraints on model parameters this along with maximum likelihood estimation of the model parameters allows interesting model comparisons this is illustrated by the analysis of several data sets

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