an approach to complete convergence theorems for dependent random fields via application of fuk nagaev inequality abstract let x n n n d be a random field i e a family of random variables indexed by n d d 2 complete convergence convergence rates for non identically distributed negatively dependent and martingale random fields are studied by application of fuk nagaev inequality the results are proved in asymmetric convergence case i e for the norming sequence equal n 1 1 n 2 2 n d d where n 1 n 2 n d n n d and min 1 i d i 1 2

eigenvalue assignment by minimal state feedback gain in lti multivariable systems in this article an improved method for eigenvalue assignment via state feedback in the linear time invariant multivariable systems is proposed this method is based on elementary similarity operations and involves mainly utilisation of vector companion forms and thus is very simple and easy to implement on a digital computer in addition to the controllable systems the proposed method can be applied for the stabilisable ones and also systems with linearly dependent inputs moreover two types of state feedback gain matrices can be achieved by this method 1 the numerical one which is unique and 2 the parametric one in which its parameters are determined in order to achieve a gain matrix with minimum frobenius norm the numerical examples are presented to demonstrate the advantages of the proposed method

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

propensity scores used for analysis of cluster randomized trials with selection bias a simulation study cluster randomized trials crts are often prone to selection bias despite randomization using a simulation study we investigated the use of propensity score ps based methods in estimating treatment effects in crts with selection bias when the outcome is quantitative of four ps based methods adjustment on ps inverse weighting stratification and optimal full matching method three successfully corrected the bias as did an approach using classical multivariable regression however they showed poorer statistical efficiency than classical methods with higher standard error for the treatment effect and type i error much smaller than the 5 nominal level

improving efficiency and robustness of the doubly robust estimator for a population mean with incomplete data considerable recent interest has focused on doubly robust estimators for a population mean response in the presence of incomplete data which involve models for both the propensity score and the regression of outcome on covariates the usual doubly robust estimator may yield severely biased inferences if neither of these models is correctly specified and can exhibit nonnegligible bias if the estimated propensity score is close to zero for some observations we propose alternative doubly robust estimators that achieve comparable or improved performance relative to existing methods even with some estimated propensity scores close to zero

parametric time optimal control of linear discrete time systems by state feedback part 2 irregular kronecker invariants in an earlier paper part 1 a parametric state feedback time optimal control law was devised for the regular kronecker invariants on the basis of parametrization of the vector companion form the present paper describes a straight forward technique for obtaining the parametric feedback matrix by exploiting the parametric vector companion forms in the case of irregular kronecker invariants hidden properties of vector companion forms are also given which result in entirely different and interesting techniques for parametrization of the control law an illustrative example is presented

parameterization of the class of deadbeat controllers via the theory of decoupling a novel approach is presented for parameterizing the class of minimum time deadbeat controllers mtdc through the minimum number of parameters the approach is based on the theory of decoupling and the properties of square decouplable systems the main result is a compact parametric form for the class of mtdc of a discrete time system contrary to many existing techniques no special assumption on the invertibility of the transition matrix is required

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

accounting for the black white wealth gap a nonparametric approach many applications involve a decomposition of the mean intergroup difference in a given variable into the portion attributable to differences in the distribution of one or more explanatory variables and that due to differences in the conditional expectation function this article notes two interrelated reasons why the blinder oaxaca b o method the approach most commonly used in the literature may yield misleading results we suggest a natural solution that both provides a more reliable answer to the original problem and affords a richer examination of the sources of intergroup differences in the variable of interest the conventional application of the b o method requires a parametric assumption about the form of the conditional expectation function furthermore it often uses estimates based on that functional form to extrapolate outside the range of the observed explanatory variables we show that misspecification of the conditional expectation function is likely to result in nontrivial errors in infer

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

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

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

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

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

on the hsu robbins erdős spitzer baum katz theorem for random fields abstract the by now classical results on convergence rates in the law of large numbers involving the sums n 1 n r 2 p s n n e where r 0 1 2 such that r 1 has been extended to the case 1 2 by adding additional logarithms all of this has been generalized to random fields by the first named author in a gut marcinkiewicz laws and convergence rates in the law of large numbers for random variables with multidimensional indices ann probab 6 1978 469 482 a gut convergence rates for probabilities of moderate deviations for sums of random variables with multidimensional indices ann probab 8 1980 298 313 the purpose of the present paper is to treat the case when the s differ in the different directions of the field as well as mixed cases with some s equal to 1 2 with added logarithms and or iterated ones

convergence rates in the slln for some classes of dependent random fields abstract let x n n n r be a random field i e a family of random variables indexed by n r r 2 we discuss complete convergence and convergence rates under assumption on dependence structure of random fields in the case of nonidentical distributions results are obtained for negatively associated random fields mixing random fields having maximal coefficient of correlation strictly smaller then 1 and martingale random fields

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

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

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

an alternative empirical likelihood method in missing response problems and causal inference missing responses are common problems in medical social and economic studies when responses are missing at random a complete case data analysis may result in biases a popular debias method is inverse probability weighting proposed by horvitz and thompson to improve efficiency robins et al proposed an augmented inverse probability weighting method the augmented inverse probability weighting estimator has a double robustness property and achieves the semiparametric efficiency lower bound when the regression model and propensity score model are both correctly specified in this paper we introduce an empirical likelihood based estimator as an alternative to qin and zhang 2007 our proposed estimator is also doubly robust and locally efficient simulation results show that the proposed estimator has better performance when the propensity score is correctly modeled moreover the proposed method can be applied in the estimation of average treatment effect in observational causal inferences finally we apply our method to an observational study of smoking using data from the cardiovascular outcomes in renal atherosclerotic lesions clinical trial copyright 2016 john wiley sons ltd

estimating propensity scores with missing covariate data using general location mixture models in many observational studies analysts estimate causal effects using propensity scores e g by matching sub classifying or inverse probability weighting based on the scores estimation of propensity scores is complicated when some values of the covariates are missing analysts can use multiple imputation to create completed data sets from which propensity scores can be estimated we propose a general location mixture model for imputations that assumes that the control units are a latent mixture of i units whose covariates are drawn from the same distributions as the treated units covariates and ii units whose covariates are drawn from different distributions this formulation reduces the influence of control units outside the treated units region of the covariate space on the estimation of parameters in the imputation model which can result in more plausible imputations in turn this can result in more reliable estimates of propensity scores and better balance in the true covariate distributions when matching or sub classifying we illustrate the benefits of the latent class modeling approach with simulations and with an observational study of the effect of breast feeding on children s cognitive abilities

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

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

extensions of the penalized spline of propensity prediction method of imputation little and an 2004 statistica sinica 14 949 968 proposed a penalized spline of propensity prediction pspp method of imputation of missing values that yields robust model based inference under the missing at random assumption the propensity score for a missing variable is estimated and a regression model is fitted that includes the spline of the estimated logit propensity score as a covariate the predicted unconditional mean of the missing variable has a double robustness dr property under misspecification of the imputation model we show that a simplified version of pspp which does not center other regressors prior to including them in the prediction model also has the dr property we also propose two extensions of pspp namely stratified pspp and bivariate pspp that extend the dr property to inferences about conditional means these extended pspp methods are compared with the pspp method and simple alternatives in a simulation study and applied to an online weight loss study conducted by kaiser permanente

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

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

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

time optimal control of disturbance rejection tracking systems for discrete time time delayed systems by state feedback in this paper we solve the disturbance rejection and tracking problem for linear multivariable discrete time systems with time delayed controlled inputs a set of necessary and sufficient conditions under which the proposed problem is controllable is defined also the nilpotency properties of such systems is established and used as the basis of a comprehensive design procedure this general procedure is illustrated by designing a time optimal disturbance rejection tracking system for a stirred tank with time delayed control inputs

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

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

nonparametric identification of a binary random factor in cross section data suppose v and u are two independent mean zero random variables where v has an asymmetric distribution with two mass points and u has a symmetric distribution we show that the distributions of v and u are nonparametrically identifled just from observing the sum v u and provide a rate root n estimator we apply these results to the world income distribution to measure the extent of convergence over time where the values v can take on correspond to country types i e wealthy versus poor countries we also extend our results to include covariates x showing that we can nonparametrically identify and estimate cross section regression models of the form y g x d u where d is an unobserved

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

utilizing propensity scores to estimate causal treatment effects with censored time lagged data observational studies frequently are conducted to compare long term effects of treatments without randomization patients receiving one treatment are not guaranteed to be prognostically comparable to those receiving another treatment furthermore the response of interest may be right censored because of incomplete follow up statistical methods that do not account for censoring and confounding may lead to biased estimates this article presents a method for estimating treatment effects in nonrandomized studies with right censored responses we review the assumptions required to estimate average causal effects and derive an estimator for comparing two treatments by applying inverse weights to the complete cases the weights are determined according to the estimated probability of receiving treatment conditional on covariates and the estimated treatment specific censoring distribution by utilizing martingale representations the estimator is shown to be asymptotically normal and an estimator for the asymptotic variance is derived simulation results are presented to evaluate the properties of the estimator these methods are applied to an observational data set of acute coronary syndrome patients from duke university medical center to estimate the effect of a treatment strategy on the mean 5 year medical cost

moving towards best practice when using inverse probability of treatment weighting iptw using the propensity score to estimate causal treatment effects in observational studies the propensity score is defined as a subject s probability of treatment selection conditional on observed baseline covariates weighting subjects by the inverse probability of treatment received creates a synthetic sample in which treatment assignment is independent of measured baseline covariates inverse probability of treatment weighting iptw using the propensity score allows one to obtain unbiased estimates of average treatment effects however these estimates are only valid if there are no residual systematic differences in observed baseline characteristics between treated and control subjects in the sample weighted by the estimated inverse probability of treatment we report on a systematic literature review in which we found that the use of iptw has increased rapidly in recent years but that in the most recent year a majority of studies did not formally examine whether weighting balanced measured covariates between treatment groups we then proceed to describe a suite of quantitative and qualitative methods that allow one to assess whether measured baseline covariates are balanced between treatment groups in the weighted sample the quantitative methods use the weighted standardized difference to compare means prevalences higher order moments and interactions the qualitative methods employ graphical methods to compare the distribution of continuous baseline covariates between treated and control subjects in the weighted sample finally we illustrate the application of these methods in an empirical case study we propose a formal set of balance diagnostics that contribute towards an evolving concept of best practice when using iptw to estimate causal treatment effects using observational data

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

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

the performance of different propensity score methods for estimating marginal hazard ratios propensity score methods are increasingly being used to reduce or minimize the effects of confounding when estimating the effects of treatments exposures or interventions when using observational or non randomized data under the assumption of no unmeasured confounders previous research has shown that propensity score methods allow for unbiased estimation of linear treatment effects e g differences in means or proportions however in biomedical research time to event outcomes occur frequently there is a paucity of research into the performance of different propensity score methods for estimating the effect of treatment on time to event outcomes furthermore propensity score methods allow for the estimation of marginal or population average treatment effects we conducted an extensive series of monte carlo simulations to examine the performance of propensity score matching 1 1 greedy nearest neighbor matching within propensity score calipers stratification on the propensity score inverse probability of treatment weighting iptw using the propensity score and covariate adjustment using the propensity score to estimate marginal hazard ratios we found that both propensity score matching and iptw using the propensity score allow for the estimation of marginal hazard ratios with minimal bias of these two approaches iptw using the propensity score resulted in estimates with lower mean squared error when estimating the effect of treatment in the treated stratification on the propensity score and covariate adjustment using the propensity score result in biased estimation of both marginal and conditional hazard ratios applied researchers are encouraged to use propensity score matching and iptw using the propensity score when estimating the relative effect of treatment on time to event outcomes copyright 2012 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

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

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

using inverse weighting and predictive inference to estimate the effects of time varying treatments on the discrete time hazard we estimate the effects of non randomized time varying treatments on the discrete time hazard using inverse weighting we consider the special monotone pattern of treatment that develops over time as subjects permanently discontinue an initial treatment and assume that treatment selection is sequentially ignorable we use a propensity score in the hazard model to reduce the potential for finite sample bias due to inverse weighting when the number of subjects who discontinue treatment at any given time is small we impose scientific restrictions on the potentially observable discontinuation hazards to improve efficiency we use predictive inference to account for the correlation of the potential hazards when comparing outcomes under different durations of initial treatment

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

simultaneous control of linear systems by state feedback in this paper a new method for finding a state feedback matrix in order to control simultaneously a collection of linear systems of the same size by using similarity operations is presented for stabilization of all the systems it is necessary that the eigenvalues of the closed loop systems lie inside a specified region in the left hand side of the complex plane this aim is achieved by solving linear and nonlinear parametric systems of equations using nonlinear programming the presented method is implemented in two examples and the results are verified in view of the norm of the state feedback matrix and stabilizability

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

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

robust pole assignment in linear state feedback numerical methods are described for determining robust or well conditioned solutions to the problem of pole assignment by state feedback the solutions obtained are such that the sensitivity of the assigned poles to perturbations in the system and gain matrices is minimized it is shown that for these solutions upper bounds on the norm of the feedback matrix and on the transient response are also minimized and a lower bound on the stability margin is maximized a measure is derived which indicates the optimal conditioning that may be expected for a particular system with a given set of closed loop poles and hence the suitability of the given poles for assignment

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

on parametric state feedback design an explicit parametric expression for the controller gain matrix in linear state variable feedback is derived it involves both the case where the set of closed loop eigenvalues does not include any open loop eigenvalue and the case where the two eigenvalue sets have elements in common some advantages of this new explicit parameterization of the controller gain matrix are illustrated by two numerical examples

estimation of relative risk and prevalence ratio relative risks rrs and prevalence ratios prs are measures of association that are more intuitively interpretable than odds ratios ors many health science studies report or estimates however even when their designs permit and study questions target rrs and or prs this is partially attributable to the popularity and technical advantage i e no restriction on the parameter space of logistic regression for estimating ors to improve this practice several biostatistical approaches for estimating rr pr adjusting for potential confounders have been proposed in this paper we consider two rr pr estimating methods 1 the modification of log binomial regression with the copy method and 2 an inverse probability of treatment weighted iptw log binomial regression we newly propose for the copy method we rigorously establish the existence and uniqueness of the maximum likelihood estimator provided certain degeneracies in the data do not occur moreover the global maximum of the copy modified likelihood is shown to occur at an interior point of the restricted parameter space this result explains why the copy method avoids convergence problems of log binomial models frequently for the iptw estimator we show that its simple procedure results in standardized estimates of rr pr and discuss its potential challenges extensions and an improvement through propensity score based grouping of observations furthermore we compare the performances of four rr pr estimation methods including the copy method and iptw log binomial regression on simulated data we demonstrate a lack of robustness of the copy method against misspecification of the true relationship between binary outcome and explanatory variables and show robustness of the iptw approach in this regard

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

partial distributional policy effects in this paper we propose a method to evaluate the effect of a counterfactual change in the unconditional distribution of a single covariate on the unconditional distribution of an outcome variable of interest both fixed and infinitesimal changes are considered we show that such effects are point identified under general conditions if the covariate affected by the counterfactual change is continuously distributed but are typically only partially identified if its distribution is discrete for the latter case we derive informa tive bounds making use of the available information we also discuss estimation and inference

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

causal inference on quantiles with an obstetric application the current statistical literature on causal inference is primarily concerned with population means of potential outcomes while the current statistical practice also involves other meaningful quantities such as quantiles motivated by the consortium on safe labor csl a large observational study of obstetric labor progression we propose and compare methods for estimating marginal quantiles of potential outcomes as well as quantiles among the treated by adapting existing methods and techniques we derive estimators based on outcome regression or inverse probability weighting and stratification as well as a doubly robust dr estimator by incorporating stratification into the dr estimator we further develop a hybrid estimator with enhanced numerical stability at the expense of a slight bias under misspecification of the or model the proposed methods are illustrated with the csl data and evaluated in simulation experiments mimicking the csl

pole assignment by state transition graph the pole assignment problem is considered using the graph representation of a matrix the parametrization of controllers which have a specified characteristic polynomial is given a simple algorithm based on graphs is presented and two examples are given

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

extension of the peters belson method to estimate health disparities among multiple groups using logistic regression with survey data determining the extent of a disparity if any between groups of people for example race or gender is of interest in many fields including public health for medical treatment and prevention of disease an observed difference in the mean outcome between an advantaged group ag and disadvantaged group dg can be due to differences in the distribution of relevant covariates the peters belson pb method fits a regression model with covariates to the ag to predict for each dg member their outcome measure as if they had been from the ag the difference between the mean predicted and the mean observed outcomes of dg members is the unexplained disparity of interest we focus on applying the pb method to estimate the disparity based on binary multinomial proportional odds logistic regression models using data collected from complex surveys with more than one dg estimators of the unexplained disparity an analytic variance covariance estimator that is based on the taylor linearization variance covariance estimation method as well as a wald test for testing a joint null hypothesis of zero for unexplained disparities between two or more minority groups and a majority group are provided simulation studies with data selected from simple random sampling and cluster sampling as well as the analyses of disparity in body mass index in the national health and nutrition examination survey 1999 2004 are conducted empirical results indicate that the taylor linearization variance covariance estimation is accurate and that the proposed wald test maintains the nominal level

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

survival analysis without survival data connecting length biased and case control data we show that relative mean survival parameters of a semiparametric log linear model can be estimated using covariate data from an incident sample and a prevalent sample even when there is no prospective follow up to collect any survival data estimation is based on an induced semiparametric density ratio model for covariates from the two samples and it shares the same structure as for a logistic regression model for case control data likelihood inference coincides with well established methods for case control data we show two further related results first estimation of interaction parameters in a survival model can be performed using covariate information only from a prevalent sample analogous to a case only analysis furthermore propensity score and conditional exposure effect parameters on survival can be estimated using only covariate data collected from incident and prevalent samples

propensity score methods for estimating relative risks in cluster randomized trials with low incidence binary outcomes and selection bias despite randomization selection bias may occur in cluster randomized trials classical multivariable regression usually allows for adjusting treatment effect estimates with unbalanced covariates however for binary outcomes with low incidence such a method may fail because of separation problems this simulation study focused on the performance of propensity score ps based methods to estimate relative risks from cluster randomized trials with binary outcomes with low incidence the results suggested that among the different approaches used multivariable regression direct adjustment on ps inverse weighting on ps and stratification on ps only direct adjustment on the ps fully corrected the bias and moreover had the best statistical properties

endogenous selection or treatment model estimation in a sample selection or treatment effects model common unobservables may affect both the outcome and the probability of selection in unknown ways this paper shows that the distribution function of potential outcomes conditional on covariates can be identified given an observed variable v that affects the treatment or selection probability in certain ways and is conditionally independent of the error terms in a model of potential outcomes selection model estimators based on this identification are provided which take the form of simple weighted averages gmm or two stage least squares these estimators permit endogenous and mismeasured regressors empirical applications are provided to estimation of a firm investment model and a schooling effects on wages model

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

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

variance estimation for stratified propensity score estimators propensity score methods are increasingly used to estimate the effect of a treatment or exposure on an outcome in non randomised studies we focus on one such method stratification on the propensity score comparing it with the method of inverse probability weighting by the propensity score the propensity score the conditional probability of receiving the treatment given observed covariates is usually an unknown probability estimated from the data estimators for the variance of treatment effect estimates typically used in practice however do not take into account that the propensity score itself has been estimated from the data by deriving the asymptotic marginal variance of the stratified estimate of treatment effect correctly taking into account the estimation of the propensity score we show that routinely used variance estimators are likely to produce confidence intervals that are too conservative when the propensity score model includes variables that predict cause the outcome but only weakly predict the treatment in contrast a comparison with the analogous marginal variance for the inverse probability weighted ipw estimator shows that routinely used variance estimators for the ipw estimator are likely to produce confidence intervals that are almost always too conservative because exact calculation of the asymptotic marginal variance is likely to be complex particularly for the stratified estimator we suggest that bootstrap estimates of variance should be used in practice

matching and weighting with functions of error prone covariates for causal inference abstractmatching estimators are commonly used to estimate causal effects in nonexperimental settings covariate measurement error can be problematic for matching estimators when observational treatment groups differ on latent quantities observed only through error prone surrogates we establish necessary and sufficient conditions for matching and weighting with functions of observed covariates to yield unconfounded causal effect estimators generalizing results from the standard i e no measurement error case we establish that in common covariate measurement error settings including continuous variables with continuous measurement error discrete variables with misclassification and factor and item response theory models no single function of the observed covariates computed for all units in a study is appropriate for matching however we demonstrate that in some circumstances it is possible to create different functions of the observed covariates for treatment and control units to construct a va

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

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

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

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

parameterizations of the state feedback controllers for linear multivariable systems abstract a general framework is constructed upon which an explicit parametric formula can be derived for state feedback controllers containing all the possible combination of parameters the relation between the parameters is nonlinear in general and therefore many different constraint may be imposed by the designer to obtain desired performance criteria a couple of illustrative examples are presented

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

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

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

the prognostic analogue of the propensity score summary the propensity score collapses the covariates of an observational study into a single measure summarizing their joint association with treatment conditions prognostic scores summarize covariates association with potential responses as with propensity scores stratification on prognostic scores brings to uncontrolled studies a concrete and desirable form of balance a balance that is more familiar as an objective of experimental control like propensity scores prognostic scores can reduce the dimension of the covariate yet causal inferences conditional on them are as valid as are inferences conditional only on the unreduced covariate as a method of adjustment unto itself prognostic scoring has limitations not shared with propensity scoring but it holds promise as a complement to the propensity score particularly in certain designs for which unassisted propensity adjustment is difficult or infeasible