bayesian sensitivity analysis for unmeasured confounding in observational studies we consider bayesian sensitivity analysis for unmeasured confounding in observational studies where the association between a binary exposure binary response measured confounders and a single binary unmeasured confounder can be formulated using logistic regression models a model for unmeasured confounding is presented along with a family of prior distributions that model beliefs about a possible unknown unmeasured confounder simulation from the posterior distribution is accomplished using markov chain monte carlo because the model for unmeasured confounding is not identifiable standard large sample theory for bayesian analysis is not applicable consequently the impact of different choices of prior distributions on the coverage probability of credible intervals is unknown using simulations we investigate the coverage probability when averaged with respect to various distributions over the parameter space the results indicate that credible intervals will have approximately nominal coverage probability on average when the prior distribution used for sensitivity analysis approximates the sampling distribution of model parameters in a hypothetical sequence of observational studies we motivate the method in a study of the effectiveness of beta blocker therapy for treatment of heart failure

case control studies assessing the effect of a confounding factor summary upper and lower bounds for the strength of the effect of a dichotomous confounding factor on the exposure disease relationship are developed for case control studies of a a problem in the design and analysis of a case control study is the identification of confounding factors controversies frequently arise about whether an exposure disease relationship is spurious i e caused by extraneous or confounding factors that were not measured or controlled as part of the study the effect of confounding factors has been discussed quantitatively by bross 1966 and schlesselman 1978 in this paper we develop upper and lower bounds for the stratum specific odds ratios which are applicable for assessing the effect of a confounding factor we restrict our attention to the simple case involving a dichotomous disease variable denoted by d if present and d if absent a dichotomous exposure variable denoted by e if present and e if absent and a single dichotomous extraneous factor f with f1 and fo denoting its two levels 2 bounds we parameterize the joint distributions of e and f in d and d as follows

primal dual interior point methods for semidefinite programming convergence rates stability and numerical results primal dual interior point path following methods for semidefinite programming are considered several variants are discussed based on newton s method applied to three equations primal feasibility dual feasibility and some form of centering condition the focus is on three such algorithms called the xz xz zx and q methods for the xz zx and q algorithms the newton system is well defined and its jacobian is nonsingular at the solution under nondegeneracy assumptions the associated schur complement matrix has an unbounded condition number on the central path under the nondegeneracy assumptions and an additional rank assumption practical aspects are discussed including mehrotra predictor corrector variants and issues of numerical stability compared to the other methods considered the xz zx method is more robust with respect to its ability to step close to the boundary converges more rapidly and achieves higher accuracy

facilitating the gibbs sampler the gibbs stopper and the griddy gibbs sampler abstract the article briefly reviews the history literature and form of the gibbs sampler an importance sampling device is proposed for converting the output of the gibbs sampler to a sample from the exact posterior this gibbs stopper technique is also useful for assessing convergence of the gibbs sampler for moderate sized problems also presented is an approach for implementing the gibbs sampler in nonconjugate situations the basic idea is to approximate the true cdf of each conditional distribution by a piecewise linear function and then sample from the approximation questions relating to the number of nodes in the approximation gap size between successive nodes and the treatment of unbounded intervals for a given conditional are discussed the methodology is illustrated using a genetic linkage model a nonlinear regression model and the cox model

some counterclaims undermine themselves in observational studies claims based on observational studies that a treatment has certain effects are often met with counterclaims asserting that the treatment is without effect that associations are produced by biased treatment assignment some counterclaims undermine themselves in the following specific sense presuming the counterclaim to be true may strengthen the support that the original data provide for the original claim so that the counterclaim fails in its role as a critique of the original claim in mathematics a proof by contradiction supposes a proposition to be true en route to proving that the proposition is false analogously the supposition that a particular counterclaim is true may justify an otherwise unjustified statistical analysis and this added analysis may interpret the original data as providing even stronger support for the original claim more precisely the original study is sensitive to unmeasured biases of a particular magnitude but an analysis that supposes the counterclaim to be true may be

equivalent unconstrained minimization and global error bounds for variational inequality problems new merit functions for variational inequality problems are constructed through the moreau yosida regularization of some gap functions the proposed merit functions constitute unconstrained optimization problems equivalent to the original variational inequality problem under suitable assumptions conditions are studied for those merit functions to be differentiable and for any stationary point of those functions to be a solution of the original variational inequality problem moreover those functions are shown to provide global error bounds for general variational inequality problems under the strong monotonicity assumption only

sensitivity analysis for m estimates tests and confidence intervals in matched observational studies huber s m estimates use an estimating equation in which observations are permitted a controlled level of influence the family of m estimates includes least squares and maximum likelihood but typical applications give extreme observations limited weight maritz proposed methods of exact and approximate permutation inference for m tests confidence intervals and estimators which can be derived from random assignment of paired subjects to treatment or control in contrast in observational studies where treatments are not randomly assigned subjects matched for observed covariates may differ in terms of unobserved covariates so differing outcomes may not be treatment effects in observational studies a method of sensitivity analysis is developed for m tests m intervals and m estimates it shows the extent to which inferences would be altered by biases of various magnitudes due to nonrandom treatment assignment the method is developed for both matched pairs with one treated subject matched to one control and for matched sets with one treated subject matched to one or more controls the method is illustrated using two studies i a paired study of damage to dna from exposure to chromium and nickel and ii a study with one or two matched controls comparing side effects of two drug regimes to treat tuberculosis the approach yields sensitivity analyses for i m tests with huber s weight function and other robust weight functions ii the permutational t test which uses the observations directly and iii various other procedures such as the sign test noether s test and the permutation distribution of the efficient score test for a location family of distributions permutation inference with covariance adjustment is briefly discussed

merit functions for semi definite complementarity problems merit functions such as the gap function the regularized gap function the implicit lagrangian and the norm squared of the fischer burmeister function have played an important role in the solution of complementarity problems defined over the cone of nonnegative real vectors we study the extension of these merit functions to complementarity problems defined over the cone of block diagonal symmetric positive semi definite real matrices the extension suggests new solution methods for the latter problems

getting it right joint distribution tests of posterior simulators analytical or coding errors in posterior simulators can produce reasonable but incorrect approximations of posterior moments this article develops simple tests of posterior simulators that detect both kinds of errors and uses them to detect and correct errors in two previously published articles the tests exploit the fact that a bayesian model specifies the joint distribution of observables data and unobservables parameters there are two joint distribution simulators the marginal conditional simulator draws unobservables from the prior and then observables conditional on unobservables the successive conditional simulator alternates between the posterior simulator and an observables simulator formal comparison of moment approximations of the two simulators reveals existing analytical or coding errors in the posterior simulator

error bounds for the linear complementarity problem with a p matrix abstract we introduce a fundamental quantity associated with a p matrix and show how this quantity is useful in deriving error bounds for the linear complementarity problem of the p type we also obtain upper and lower bounds for the quantity introduced

a superlinearly convergent primal dual infeasible interior point algorithm for semidefinite programming a primal dual infeasible interior point path following algorithm is proposed for solving semidefinite programming sdp problems if the problem has a solution then the algorithm is globally convergent if the starting point is feasible or close to being feasible the algorithm finds an optimal solution in at most o sqrt n l iterations where n is the size of the problem and l is the logarithm of the ratio of the initial error and the tolerance if the starting point is large enough then the algorithm terminates in at most o nl steps either by finding a solution or by determining that the primal dual problem has no solution of norm less than a given number moreover we propose a sufficient condition for the superlinear convergence of the algorithm in addition we give two special cases of sdp for which the algorithm is quadratically convergent

simulated latent variable estimation of models with ordered categorical data this paper addresses the problem that occurs when estimating a linear regression model in which some of the explanatory variables are represented by categorical indicators we replace the categorical variable with a simulated latent variable slv drawn from a truncated multivariate normal distribution using the gibbs sampler consistent and efficient parameter estimates are derived using generalized two stage least squares where the instrument set contains a second slv drawn from the same distribution consistent estimates of the parameter covariance matrix are derived using a two stage procedure we illustrate this methodology using a monte carlo simulation and also investigate the consequences of incorrect distributional assumptions the simulated latent variable procedure is shown to be superior to conventional methods involving dummy variables or conditional means

weighted m statistics with superior design sensitivity in matched observational studies with multiple controls in a nonrandomized or observational study a weak association between receipt of the treatment and an outcome may be explained not as effects caused by the treatment but rather by a small bias in the assignment of individuals to treatment or control however a strong association may be explained as noncausal only by a large bias the strength of the association between treatment and outcome is not uniform across the data from a study and this motivates giving greater weight where the association is stronger in an observational study with treated control matched pairs it is known that results are less sensitive to unmeasured biases if pairs with small absolute differences in outcomes are given little weight in the analysis more precisely such a test statistic has superior design sensitivity how should outcomes be weighted if an observational study is matched in sets with one treated subject and several controls an m statistic is the quantity equated to zero in defining huber s m estimates includin

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

some approximate evidence factors in observational studies an observational study or nonrandomized experiment has exact evidence factors if it permits several statistically independent tests of the same null hypothesis h0 of no treatment effect where these several tests depend upon different assumptions about bias from nonrandom treatment assignment in an observational study we are typically uncertain about what assumptions truly describe treatment assignment if independent tests each reject h0 and if each of these tests is valid under assumptions about treatment assignment that would invalidate the others then rejection of h0 does not depend upon the truth of any one of these assumptions although exact evidence factors do exist the requirement of exact independence may limit the class of test statistics in ways that reduce pitman efficiency and design sensitivity so one might wish to be free of these limitations a much larger class of statistics permits approximate evidence factors which are not independent but which preserve some of the important con

strong control of the familywise error rate in observational studies that discover effect modification by exploratory methods an effect modifier is a pretreatment covariate that affects the magnitude of the treatment effect or its stability when there is effect modification an overall test that ignores an effect modifier may be more sensitive to unmeasured bias than a test that combines results from subgroups defined by the effect modifier if there is effect modification one would like to identify specific subgroups for which there is evidence of effect that is insensitive to small or moderate biases in this paper we propose an exploratory method for discovering effect modification and combine it with a confirmatory method of simultaneous inference that strongly controls the familywise error rate in a sensitivity analysis despite the fact that the groups being compared are defined empirically a new form of matching strength k matching permits a search through more than k covariates for effect modifiers in such a way that no pairs are lost provided that at most k covariates are selected to group the pairs in a strength k match each set of k covariates is exactly balanced although a set of more than k covariates may exhibit imbalance we apply the proposed method to study the effects of the earthquake that struck chile in 2010

sensitivity analysis for multiple comparisons in matched observational studies through quadratically constrained linear programming abstracta sensitivity analysis in an observational study assesses the robustness of significant findings to unmeasured confounding while sensitivity analyses in matched observational studies have been well addressed when there is a single outcome variable accounting for multiple comparisons through the existing methods yields overly conservative results when there are multiple outcome variables of interest this stems from the fact that unmeasured confounding cannot affect the probability of assignment to treatment differently depending on the outcome being analyzed existing methods allow this to occur by combining the results of individual sensitivity analyses to assess whether at least one hypothesis is significant which in turn results in an overly pessimistic assessment of a study s sensitivity to unobserved biases by solving a quadratically constrained linear program we are able to perform a sensitivity analysis while enforcing that unmeasured confounding must have the same impact on the treatm

a posteriori error bounds for the linearly constrained varitional inequality problem in this paper we derive a posteriori error bound of an approximate solution to a linearly constrained variational inequality problem three measures of error are used with each depending on a particular characterization of an exact solution and related to a certain class of iterative methods

equivalence of variational inequality problems to unconstrained minimization in this paper we propose a class of merit functions for variational inequality problems vi through these merit functions the variational inequality problem is cast as unconstrained minimization problem we estimate the growth rate of these merit functions and give conditions under which the stationary points of these functions are the solutions of vi

explaining individual response using aggregated data abstract empirical analysis of individual response behavior is sometimes limited due to the lack of explanatory variables at the individual level in this paper we put forward a new approach to estimate the effects of covariates on individual response where the covariates are unknown at the individual level but observed at some aggregated level this situation may for example occur when the response variable is available at the household level but covariates only at the zip code level we describe the missing individual covariates by a latent variable model which matches the sample information at the aggregate level parameter estimates can be obtained using maximum likelihood or a bayesian analysis we illustrate the approach estimating the effects of household characteristics on donating behavior to a dutch charity donating behavior is observed at the household level while the covariates are only observed at the zip code level

smooth unbiased multivariate probability simulators for maximum likelihood estimation of limited dependent variable models abstract we apply a new simulation method that solves the multidimensional probability integrals that arise in maximum likelihood estimation of a broad class of limited dependent variable models the simulation method has four key features the simulated choice probabilities are unbiased they are a continuous and differentiable function of the parameters of the model they are bounded between 0 and 1 and their computation takes an effort that is nearly linear in the dimension of the probability integral independent of the magnitudes of the true probabilities we also show that the new simulation method produces probability estimates with substantially smaller variance than those generated by acceptance rejection methods or by stern s 1992 method the simulated probabilities can therefore be used to revive the lerman and manski 1981 procedure of approximating the likelihood function using simulated choice probabilities by overcoming its computational disadvantages

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

analysis of multivariate probit models summary this paper provides a practical simulation based bayesian and non bayesian analysis of correlated binary data using the multivariate probit model the posterior distribution is simulated by markov chain monte carlo methods and maximum likelihood estimates are obtained by a monte carlo version of the em algorithm a practical approach for the computation of bayes factors from the simulation output is also developed the methods are applied to a dataset with a bivariate binary response to a four year longitudinal dataset from the six cities study of the health effects of air pollution and to a sevenvariate binary response dataset on the labour supply of married women from the panel survey of income dynamics

global convergence properties of some iterative methods for linear complementarity problems the subject of this work is a class of iterative methods for solving the linear complementarily problem lcp these methods are based on a reformulation of the lcp consisting of a usually differentiable system of nonlinear equations to which newton s method is applied thus the algorithms are locally q quadratically convergent furthermore global convergence results for these methods are proved for lcps associated with r 0 nondegenerate and p matrices finally some promising numerical results are reported for both constructed and randomly generated problems

a one parametric class of merit functions for the symmetric cone complementarity problem abstract in this paper we extend the one parametric class of merit functions proposed by kanzow and kleinmichel c kanzow h kleinmichel a new class of semismooth newton type methods for nonlinear complementarity problems comput optim appl 11 1998 227 251 for the nonnegative orthant complementarity problem to the general symmetric cone complementarity problem sccp we show that the class of merit functions is continuously differentiable everywhere and has a globally lipschitz continuous gradient mapping from this we particularly obtain the smoothness of the fischer burmeister merit function associated with symmetric cones and the lipschitz continuity of its gradient in addition we also consider a regularized formulation for the class of merit functions which is actually an extension of one of the ncp function classes studied by c kanzow y yamashita m fukushima new ncp functions and their properties j optim theory appl 97 1997 115 135 to the sccp by exploiting the cartesian p properties for a nonlinear transformation we show that the class of regularized merit functions provides a global error bound for the solution of the sccp and moreover has bounded level sets under a rather weak condition which can be satisfied by the monotone sccp with a strictly feasible point or the sccp with the joint cartesian r 02 property all of these results generalize some recent important works in j s chen p tseng an unconstrained smooth minimization reformulation of the second order cone complementarity problem math program 104 2005 293 327 c k sim j sun d ralph a note on the lipschitz continuity of the gradient of the squared norm of the matrix valued fischer burmeister function math program 107 2006 547 553 p tseng merit function for semidefinite complementarity problems math program 83 1998 159 185 under a unified framework

bayesian correlation estimation we propose prior probability models for variance covariance matrices in order to address two important issues first the models allow a researcher to represent substantive prior information about the strength of correlations among a set of variables secondly even in the absence of such information the increased flexibility of the models mitigates dependence on strict parametric assumptions in standard prior models for example the model allows a posteriori different levels of uncertainty about correlations among different subsets of variables we achieve this by including a clustering mechanism in the prior probability model clustering is with respect to variables and pairs of variables our approach leads to shrinkage towards a mixture structure implied by the clustering we discuss appropriate posterior simulation schemes to implement posterior inference in the proposed models including the evaluation of normalising constants that are functions of parameters of interest the normalising constants result from the restriction that the correlation matrix be positive definite we discuss examples based on simulated data a stock return dataset and a population genetics dataset

impact of multiple matched controls on design sensitivity in observational studies summary in an observational study one treated subject may be matched for observed covariates to either one or several untreated controls the common motivation for using several controls rather than one is to increase the power of a test of no effect under the doubtful assumption that matching for observed covariates suffices to remove bias from nonrandom treatment assignment does the choice between one or several matched controls affect the sensitivity of conclusions to violations of this doubtful assumption with continuous responses it is known that reducing the heterogeneity of matched pair differences reduces sensitivity to unmeasured biases but increasing the sample size has a highly circumscribed effect on sensitivity to bias is the use of several controls rather than one analogous to a reduction in heterogeneity or to an increase in sample size the issue is examined for huber s m statistics including the t test the examination having three components an example asymptotic calculations using design sensitivity and a simulation use of multiple controls with continuous responses yields a nontrivial reduction in sensitivity to unmeasured biases an example looks at lead and cadmium in the blood of smokers from the 2008 national health and nutrition examination survey a by product of the discussion is a new result giving the design sensitivity for the permutation distribution of m statistics

a squared smoothing newton method for nonsmooth matrix equations and its applications in semidefinite optimization problems we study a smoothing newton method for solving a nonsmooth matrix equation that includes semidefinite programming and the semidefinite complementarity problem as special cases this method if specialized for solving semidefinite programs needs to solve only one linear system per iteration and achieves quadratic convergence under strict complementarity and nondegeneracy we also establish quadratic convergence of this method applied to the semidefinite complementarity problem under the assumption that the jacobian of the problem is positive definite on the affine hull of the critical cone at the solution these results are based on the strong semismoothness and complete characterization of the b subdifferential of a corresponding squared smoothing matrix function which are of general theoretical interest

a method for smoothing simulated moments of discrete probabilities in multinomial probit models there has been much interest over the years in estimating discrete choice models however with the exception of multinomial logit models agents have been restricted to few choices so that multivariate integrals could be feasibly integrated numerically 2 pakes and pollard 1989 and mcfadden 1989 have independently developed the method of simulated moments msm to deal with estimating a wide class of models of which high order discrete choice models are a subset one of the problems a researcher must handle when using msm for a discrete choice model is how to smooth the discrete simulated random variables this is necessary to keep small the number of draws required to simulate derivatives and to simulate variation in the data geweke 1989 and mcfadden 1989 have suggested importance sampling methods and other methods to smooth the simulated variables for general error structures in this paper i present a factor analytic smoothing method that can be applied to probit problems although it is not as general as the methods geweke and mcfadden suggest it is easy to use and has clear intuition furthermore simulated probabilities will be unbiased will be bounded between zero and unity and will have smaller variances than unsmoothed probabilities always and smaller variances than importance sampling estimates for a large class of probabilities the second section of this paper develops notation defines the problem and presents the smoothing method and an algorithm to employ it the last section presents monte carlo comparisons of the smoothing method to the importance sampling method

design sensitivity in observational studies summary outside the field of statistics the literature on observational studies offers advice about research designs or strategies for judging whether or not an association is causal such as multiple operationalism or a dose response relationship these useful suggestions are typically informal and qualitative a quantitative measure design sensitivity is proposed for measuring the contribution such strategies make in distinguishing causal effects from hidden biases several common strategies are then evaluated in terms of their contribution to design sensitivity a related method for computing the power of a sensitivity analysis is also developed

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

on a global projection type error bound for the linear complementarity problem abstract we consider a projection type error bound for the linear complementarity problem involving a matrix m and vector q first we show that the mangasarian ren sufficient condition on m for this error bound to hold globally for all q such that the problem is solvable is also necessary second we derive necessary and sufficient conditions on m and q for this error bound to hold globally thus extending a result of luo and tseng which in addition assumes m is psd plus third we derive necessary and sufficient conditions on m for this error bound to hold globally independent of q these conditions are closely related to m being lipschitzian we also discuss application of our result to the affine variational inequality problem

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

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

strong semismoothness of the fischer burmeister sdc and soc complementarity functions we show that the fischer burmeister complementarity functions associated to the semidefinite cone sdc and the second order cone soc respectively are strongly semismooth everywhere interestingly enough the proof relys on a relationship between the singular value decomposition of a nonsymmetric matrix and the spectral decomposition of a symmetric matrix

from the statistics of data to the statistics of knowledge symbolic data analysis increasingly datasets are so large they must be summarized in some fashion so that the resulting summary dataset is of a more manageable size while still retaining as much knowledge inherent to the entire dataset as possible one consequence of this situation is that the data may no longer be formatted as single values such as is the case for classical data but rather may be represented by lists intervals distributions and the like these summarized data are examples of symbolic data this article looks at the concept of symbolic data in general and then attempts to review the methods currently available to analyze such data it quickly becomes clear that the range of methodologies available draws analogies with developments before 1900 that formed a foundation for the inferential statistics of the 1900s methods largely limited to small by comparison datasets and classical data formats the scarcity of available methodologies for symbolic data also becomes clear and so draws attention to an enor

a note on the lipschitz continuity of the gradient of the squared norm of the matrix valued fischer burmeister function based on a formula of tseng we show that the squared norm of the matrix valued fischer burmeister function has a lipschitz continuous gradient

causal conclusions are most sensitive to unobserved binary covariates there is a rich literature that considers whether an observed relation between treatment and response is due to an unobserved covariate in order to quantify this unmeasured bias an assumption is made about the distribution of this unobserved covariate typically that it is either binary or at least confined to the unit interval in this paper this assumption is relaxed in the context of matched pairs with binary treatment and response one might think that a long tailed unobserved covariate could do more damage remarkably that is not the case the most harm is done by a binary covariate so the case commonly considered in the literature is most conservative this has two practical consequences i it is always safe to assume that an unobserved covariate is binary if one is content to make a conservative statement ii when another assumption seems more appropriate say normal covariate there will be less sensitivity than with a binary covariate this assumption implies that it is possible that a relation between treatment and response that is sensitive to unmeasured bias if the unobserved covariate is dichotomous ceases to be sensitive if the unobserved covariate is normally distributed these ideas are illustrated by three examples it is important to note that the claim in this paper applies to our specific setting of matched pairs with binary treatment and response whether the same conclusion holds in other settings is an open question

amplification of sensitivity analysis in matched observational studies a sensitivity analysis displays the increase in uncertainty that attends an inference when a key assumption is relaxed in matched observational studies of treatment effects a key assumption in some analyses is that subjects matched for observed covariates are comparable and this assumption is relaxed by positing a relevant covariate that was not observed and not controlled by matching what properties would such an unobserved covariate need to have to materially alter the inference about treatment effects for ease of calculation and reporting it is convenient that the sensitivity analysis be of low dimension perhaps indexed by a scalar sensitivity parameter but for interpretation in specific contexts a higher dimensional analysis may be of greater relevance an amplification of a sensitivity analysis is defined as a map from each point in a low dimensional sensitivity analysis to a set of points perhaps a curve in a higher dimensional sensitivity analysis such that the possible inferences are

a computationally practical simulation estimator for panel data in this paper i develop a practical extension of mcfadden s method of simulated moments estimator for limited dependent variable models to the panel data case the method is based on a factorization of the msm first order condition into transition probabilities along with the development of a new highly accurate method for similating these transition probabilities a series of monte carlo tests show that this msm estimator performs quite well relative to quadrature based ml estimators even when large numbers of quadrature points are employed the estimator also performs well relative to simulated ml even when a highly accurate method is used to simulate the choice probabilities in terms of computational speed complex panel data models involving random effects and arma errors may be estimated via msm in times similar to those necessary for estimation of simple random effects models via ml quadrature

the calculation of posterior distributions by data augmentation abstract the idea of data augmentation arises naturally in missing value problems as exemplified by the standard ways of filling in missing cells in balanced two way tables thus data augmentation refers to a scheme of augmenting the observed data so as to make it more easy to analyze this device is used to great advantage by the em algorithm dempster laird and rubin 1977 in solving maximum likelihood problems in situations when the likelihood cannot be approximated closely by the normal likelihood maximum likelihood estimates and the associated standard errors cannot be relied upon to make valid inferential statements from the bayesian point of view one must now calculate the posterior distribution of parameters of interest if data augmentation can be used in the calculation of the maximum likelihood estimate then in the same cases one ought to be able to use it in the computation of the posterior distribution it is the purpose of this article to explain how this can be done the basic idea