sensitivity analysis for certain permutation inferences in matched observational studies summary in observational studies treatments are not randomly assigned to experimental units so that randomization tests and their associated interval estimates are not generally applicable in an effort to compensate for the lack of randomization treated and control units are often matched on the basis of observed covariates however the possibility remains of bias due to residual imbalances in unobserved covariates a general though simple method is proposed for displaying the sensitivity of permutation inferences to a range of assumptions about unobserved covariates in matched observational studies the sensitivity analysis is applicable to wilcoxon s signed rank test to the mcnemar cox test for paired binary responses and to some matching problems with a variable number of controls

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

some inverse problems with spectral limitations on data we consider the recovery of the impedance a y for a one dimensional seismic problem v n av y y a where y is travel time with impulse input v y 0 delta y and readout v 0 t delta t g t g t 2 pi int 0 infty hat g lambda operatorname cos lambda t d lambda hat g being the fourier cosine transform first it follows that for delta denoting change delta a y leqq tilde c y delta g infty on 0 2y the problem of estimating delta g infty on 0 2y is ill posed if only estimates delta g 2 leqq varepsilon can be obtained using regularization and summability we construct a sequence of approximants g n which converges to g pointwise on 0 2y uniformly thus g n to g in l infty 0 2y and the corresponding a n y converse to a y pointwise this paper also serves as an introduction to the connection of regularization and summability ideas with transmutation formulas applications to

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

bayesian approach for addressing differential covariate measurement error in propensity score methods propensity score methods are an important tool to help reduce confounding in non experimental studies and produce more accurate causal effect estimates most propensity score methods assume that covariates are measured without error however covariates are often measured with error recent work has shown that ignoring such error could lead to bias in treatment effect estimates in this paper we consider an additional complication that of differential measurement error across treatment groups such as can occur if a covariate is measured differently in the treatment and control groups we propose two flexible bayesian approaches for handling differential measurement error when estimating average causal effects using propensity score methods we consider three scenarios systematic i e a location shift heteroscedastic i e different variances and mixed both systematic and heteroscedastic measurement errors we also explore various prior choices i e weakly informative or point mass on the sensitivity parameters related to the differential measurement error we present results from simulation studies evaluating the performance of the proposed methods and apply these approaches to an example estimating the effect of neighborhood disadvantage on adolescent drug use disorders

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

maximum likelihood estimation via the ecm algorithm a general framework two major reasons for the popularity of the em algorithm are that its maximum step involves only complete data maximum likelihood estimation which is often computationally simple and that its convergence is stable with each iteration increasing the likelihood when the associated complete data maximum likelihood estimation itself is complicated em is less attractive because the m step is computationally unattractive in many cases however complete data maximum likelihood estimation is relatively simple when conditional on some function of the parameters being estimated

inversion of an integral transform related to a general form of heat equation inversion algorithms have been derived by d v widder for the weierstrass transform and by d t haimo for the dual weierstrass laguerre transform for the genera1 form of heat equation l u u l a self adjoint operator an integral transform is introduced with kernel related to the fundamental solution of the equation an inversion formula for the transform is derived which includes the previous resuks as special cases cf 1988 academic press inc

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

estimating time to event from longitudinal categorical data an analysis of multiple sclerosis progression the expanded disability status scale edss is an ordinal score that measures progression in multiple sclerosis ms progression is defined as reaching edss of a certain level absolute progression or increasing edss by one point relative progression survival methods for time to progression are not adequate for such data because they do not exploit the edss level at the end of follow up instead we suggest a markov transitional model applicable for repeated categorical or ordinal data this approach enables derivation of covariate specific survival curves obtained after estimation of the regression coefficients and manipulations of the resulting transition matrix large sample theory and resampling methods are employed to derive pointwise confidence intervals which perform well in simulation methods for generating survival curves for time to edss of a certain level time to increase edss by at least one point and time to two consecutive visits with edss greater than 3 are described explicitly t

propensity score weighting with multilevel data propensity score methods are being increasingly used as a less parametric alternative to traditional regression to balance observed differences across groups in both descriptive and causal comparisons data collected in many disciplines often have analytically relevant multilevel or clustered structure the propensity score however was developed and has been used primarily with unstructured data we present and compare several propensity score weighted estimators for clustered data including marginal cluster weighted and doubly robust estimators using both analytical derivations and monte carlo simulations we illustrate bias arising when the usual assumptions of propensity score analysis do not hold for multilevel data we show that exploiting the multilevel structure either parametrically or nonparametrically in at least one stage of the propensity score analysis can greatly reduce these biases these methods are applied to a study of racial disparities in breast cancer screening among beneficiaries in medicare health plans

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

using mixed integer programming for matching in an observational study of kidney failure after surgery this article presents a new method for optimal matching in observational studies based on mixed integer programming unlike widely used matching methods based on network algorithms which attempt to achieve covariate balance by minimizing the total sum of distances between treated units and matched controls this new method achieves covariate balance directly either by minimizing both the total sum of distances and a weighted sum of specific measures of covariate imbalance or by minimizing the total sum of distances while constraining the measures of imbalance to be less than or equal to certain tolerances the inclusion of these extra terms in the objective function or the use of these additional constraints explicitly optimizes or constrains the criteria that will be used to evaluate the quality of the match for example the method minimizes or constrains differences in univariate moments such as means variances and skewness differences in multivariate moments such as correlations between covari

bias associated with using the estimated propensity score as a regression covariate the use of propensity score methods to adjust for selection bias in observational studies has become increasingly popular in public health and medical research a substantial portion of studies using propensity score adjustment treat the propensity score as a conventional regression predictor through a monte carlo simulation study austin and colleagues investigated the bias associated with treatment effect estimation when the propensity score is used as a covariate in nonlinear regression models such as logistic regression and cox proportional hazards models we show that the bias exists even in a linear regression model when the estimated propensity score is used and derive the explicit form of the bias we also conduct an extensive simulation study to compare the performance of such covariate adjustment with propensity score stratification propensity score matching inverse probability of treatment weighted method and nonparametric functional estimation using splines the simulation scenarios are designed to reflect real data analysis practice instead of specifying a known parametric propensity score model we generate the data by considering various degrees of overlap of the covariate distributions between treated and control groups propensity score matching excels when the treated group is contained within a larger control pool while the model based adjustment may have an edge when treated and control groups do not have too much overlap overall adjusting for the propensity score through stratification or matching followed by regression or using splines appears to be a good practical strategy

on a preference based instrumental variable approach in reducing unmeasured confounding by indication treatment preferences of groups e g clinical centers have often been proposed as instruments to control for unmeasured confounding by indication in instrumental variable iv analyses however formal evaluations of these group preference based instruments are lacking unique challenges include the following i correlations between outcomes within groups ii the multi value nature of the instruments iii unmeasured confounding occurring between and within groups we introduce the framework of between group and within group confounding to assess assumptions required for the group preference based iv analyses our work illustrates that when unmeasured confounding effects exist only within groups but not between groups preference based ivs can satisfy assumptions required for valid instruments we then derive a closed form expression of asymptotic bias of the two stage generalized ordinary least squares estimator when the ivs are valid simulations demonstrate that the asymptotic bias formula approximates bias in finite samples quite well particularly when the number of groups is moderate to large the bias formula shows that when the cluster size is finite the iv estimator is asymptotically biased only when both the number of groups and cluster size go to infinity the bias disappears however the iv estimator remains advantageous in reducing bias from confounding by indication the bias assessment provides practical guidance for preference based iv analyses to increase their performance one should adjust for as many measured confounders as possible consider groups that have the most random variation in treatment assignment and increase cluster size to minimize the likelihood for these ivs to be invalid one should minimize unmeasured between group confounding copyright 2014 john wiley sons ltd

inversion of integral transforms associated with a class of perturbed heat equations abstract let lx be the sturm liouville differential operator l x d 2 dx 2 q x x 0 we assume that lx has either a purely discrete spectrum that is bounded from below by zero or a continuous spectrum that fills up the interval 0 with possibly a finite number of negative eigenvalues the w transform of y l2 0 is defined by x 0 y g x y 1 dy where g x y t is a function associated with the fundamental solution of the perturbed equation l x u x t u x t t the main purpose of this paper is to derive an inversion formula for the w transform this inversion formula generalizes the known inversion formulae for the weierstrass weierstrass hankel convolution and weierstrass laguerre transforms the results of this paper are easily extended to the case where lx is considered over the entire line

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

model averaging in semiparametric estimation of treatment effects choosing the covariates and functional form of the propensity score is an important choice for estimating treatment effects this paper proposes a data driven way of averaging the estimators over candidate specifications to resolve the specification uncertainty in the propensity score weighting estimation of the att the proposed procedures minimize the estimated mse of the att estimator in a local asymptotic framework we formulate model averaging as a statistical decision problem in a limit experiment and derive an averaging scheme that is bayes optimal with respect to a given prior the averaging estimator outperforms selection estimators and the estimators in any of the candidate models in terms of bayes asymptotic mse our monte carlo studies illustrate the size of the mse gains we apply the averaging procedure to evaluate the effect of a labor market program

a case cohort design for epidemiologic cohort studies and disease prevention trials summary suppose that a cohort of individuals is to be followed in order to relate failure rates to preceding covariate histories a design is proposed which involves covariate data only for cases experiencing failure and for members of a randomly selected subcohort odds ratio and relative risk estimation procedures are presented for such a case cohort design a small simulation study compares case cohort relative risk estimation procedures to full cohort and synthetic case control analyses relevance to epidemiologic cohort studies and disease prevention trials is discussed

confidence intervals associated with tests for bioequivalence summary previously the connection between bioequivalence tests and their associated confidence intervals was not well understood in fact the oc level two one sided tests approach was thought to be associated with a 1 20c confidence interval in this paper we build up connections which allow us to construct 1 oc confidence intervals from cx level tests and vice versa when applied to the cc level two one sided tests the resultant 1 oc confidence intervals are properly contained in westlake s 1 oc symmetric interval 1976 our approach is readily generalized to different settings including the nonparametric setting the ratio parameter setting and the repeated confidence interval setting

balanced risk set matching a new form of matching optimal balanced risk set matching is applied in an observational study of a treatment cystoscopy and hydrodistention given in response to the symptoms of the chronic nonlethal disease interstitial cystitis when a patient receives the treatment at time t that patient is matched to another patient with a similar history of symptoms up to time t who has not received the treatment up to time t this is risk set matching by using a penalty function in integer programming in a new way we force the marginal distributions of symptoms to be balanced in the matched treated and control groups among all balanced matchings we pick the one that is optimal in the sense of minimizing the multivariate pretreatment covariate distance within matched pairs under a simple model for the treatment assignment mechanism we study the sensitivity of the findings to hidden biases in particular we show that a simple conventional sensitivity analysis is appropriate with risk set matching when the

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

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

on regression adjustment for the propensity score propensity scores are widely adopted in observational research because they enable adjustment for high dimensional confounders without requiring models for their association with the outcome of interest the results of statistical analyses based on stratification matching or inverse weighting by the propensity score are therefore less susceptible to model extrapolation than those based solely on outcome regression models this is attractive because extrapolation in outcome regression models may be alarming yet difficult to diagnose when the exposed and unexposed individuals have very different covariate distributions standard regression adjustment for the propensity score forms an alternative to the aforementioned propensity score methods but the benefits of this are less clear because it still involves modelling the outcome in addition to the propensity score in this article we develop novel insights into the properties of this adjustment method we demonstrate that standard tests of the null hypothesis of no exposure effect based on robust variance estimators as well as particular standardised effects obtained from such adjusted regression models are robust against misspecification of the outcome model when a propensity score model is correctly specified they are thus not vulnerable to the aforementioned problem of extrapolation we moreover propose efficient estimators for these standardised effects which retain a useful causal interpretation even when the propensity score model is misspecified provided the outcome regression model is correctly specified

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

the use of propensity scores and observational data to estimate randomized controlled trial generalizability bias although randomized controlled trials are considered the gold standard for clinical studies the use of exclusion criteria may impact the external validity of the results it is unknown whether estimators of effect size are biased by excluding a portion of the target population from enrollment we propose to use observational data to estimate the bias due to enrollment restrictions which we term generalizability bias in this paper we introduce a class of estimators for the generalizability bias and use simulation to study its properties in the presence of non constant treatment effects we find the surprising result that our estimators can be unbiased for the true generalizability bias even when all potentially confounding variables are not measured in addition our proposed doubly robust estimator performs well even for mis specified models

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

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

high order difference schemes for the system of two space second order nonlinear hyperbolic equations with variable coefficients abstract in this paper we develop implicit difference schemes of o k4 k2h2 h4 where k 0 h 0 are grid sizes in time and space coordinates respectively for solving the system of two space dimensional second order nonlinear hyperbolic partial differential equations with variable coefficients having mixed derivatives subject to appropriate initial and boundary conditions the proposed difference method for the scalar equation is applied for the solution of wave equation in polar coordinates to obtain three level conditionally stable adi method of o k4 k2h2 h4 some physical nonlinear problems are provided to demonstrate the accuracy of the implementation

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

paley wiener type theorems for a class of integral transforms abstract a characterization of weighted l 2 i spaces in terms of their images under various integral transformations is derived where i is an interval finite or infinite this characterization is then used to derive paley wiener type theorems for these spaces unlike the classical paley wiener theorem our theorems use real variable techniques and do not require analytic continuation to the complex plane the class of integral transformations considered is related to singular sturm liouville boundary value problems on a half line and on the whole line

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

semiparametric double balancing score estimation for incomplete data with ignorable missingness when estimating the marginal mean response with missing observations a critical issue is robustness to model misspecification in this article we propose a semiparametric estimation method with extended double robustness that attains the optimal efficiency under less stringent requirement for model specifications than the doubly robust estimators in this semiparametric estimation covariate information is collapsed into a two dimensional score s with one dimension for i the pattern of missingness and the other for ii the pattern of response both estimated from some working parametric models the mean response e y is then estimated by the sample mean of e y s which is estimated via nonparametric regression the semiparametric estimator is consistent if either the core of i or the core of ii is captured by s and attains the optimal efficiency if both are captured by s as the cores can be obtained without correctly specifying the full parametric models for i or ii the proposed es

fourth order approximations at first time level linear stability analysis and the numerical solution of multidimensional second order nonlinear hyperbolic equations in polar coordinates in this article three level implicit difference schemes of o k4 k2h2 h4 where k 0 h 0 are grid sizes in time and space coordinates respectively are proposed for the numerical solution of one two and three space dimensional nonlinear wave equations in polar coordinates subject to appropriate initial and dirichlet boundary conditions we also discuss fourth order approximation at first time level for more general case we also obtain the stability range of the difference scheme when applied to a test equation utt urr aruu ar2u g r t a 1 and 2 numerical examples are provided to demonstrate the required order of convergence of the methods

two sample asymptotically distribution free tests for incomplete multivariate observations abstract asymptotically distribution free tests for equality of two multivariate distributions based on censored observations are proposed by choosing appropriate test scores we obtain the multivariate versions of the gehan and log rank tests an important application of the proposed tests is in the area in which multiple observations are collected on study subjects some of whom may have missing observations the patterns of missing observations are allowed to differ for the two groups to be compared these tests are particularly useful for longitudinal studies for example the analysis of repeated measurements and growth curves real examples are also provided for illustration

estimation of mean response via the effective balancing score we introduce effective balancing scores for estimation of the mean response under a missing at random mechanism unlike conventional balancing scores the effective balancing scores are constructed via dimension reduction free of model specification three types of effective balancing scores are introduced those that carry the covariate information about the missingness the response or both they lead to consistent estimation with little or no loss in efficiency compared to existing estimators the effective balancing score based estimator relieves the burden of model specification and is the most robust it is a near automatic procedure which is most appealing when high dimensional covariates are involved we investigate both the asymptotic and the numerical properties and demonstrate the proposed method in a study on human immunodeficiency virus disease

signed rank statistics for coherent predictions a generalization of wilcoxon s signed rank test is proposed for testing a dose response relationship with one or more outcomes the test is useful in matched observational studies or in nonrandomized experiments that use dose response relationships and predictions about multiple outcomes in an effort to distinguish actual treatment effects from hidden biases a sensitivity analysis indicating whether a dose response relationship or multiple predictions are confirmed with sufficient strength to reduce sensitivity to hidden bias is performed together the test and the sensitivity analysis help to quantify the degree to which a coherent pattern of associations is present or absent and the degree to which this strengthens or fails to strengthen evidence of cause and effect formal properties of tests of this kind are examined the form of the optimal test is determined though this test is not usable because it depends upon the values of the unknown parameters under study also examined are the conditions under which the proposed test resembles the optimal test as well as the impact of various violations of those conditions on power an example involving matched pairs exposed to varying doses of cadmium is considered in detail

instrumental variables as bias amplifiers with general outcome and confounding drawing causal inference with observational studies is the central pillar of many disciplines one sufficient condition for identifying the causal effect is that the treatment outcome relationship is unconfounded conditional on the observed covariates it is often believed that the more covariates we condition on the more plausible this unconfoundedness assumption is this belief has had a huge impact on practical causal inference suggesting that we should adjust for all pretreatment covariates however when there is unmeasured confounding between the treatment and outcome estimators adjusting for some pretreatment covariate might have greater bias than estimators without adjusting for this covariate this kind of covariate is called a bias amplifier and includes instrumental variables that are independent of the confounder and affect the outcome only through the treatment previously theoretical results for this phenomenon have been established only for linear models we fill in this gap in the literature by providing a general theory showing that this phenomenon happens under a wide class of models satisfying certain monotonicity assumptions we further show that when the treatment follows an additive or multiplicative model conditional on the instrumental variable and the confounder these monotonicity assumptions can be interpreted as the signs of the arrows of the causal diagrams

estimation of propensity scores using generalized additive models propensity score matching is often used in observational studies to create treatment and control groups with similar distributions of observed covariates typically propensity scores are estimated using logistic regressions that assume linearity between the logistic link and the predictors we evaluate the use of generalized additive models gams for estimating propensity scores we compare logistic regressions and gams in terms of balancing covariates using simulation studies with artificial and genuine data we find that when the distributions of covariates in the treatment and control groups overlap sufficiently using gams can improve overall covariate balance especially for higher order moments of distributions when the distributions in the two groups overlap insufficiently gam more clearly reveals this fact than logistic regression does we also demonstrate via simulation that matching with gams can result in larger reductions in bias when estimating treatment effects than matching with logistic regression

stability interval for explicit difference schemes for multi dimensional second order hyperbolic equations with significant first order space derivative terms in this piece of work we introduce a new idea and obtain stability interval for explicit difference schemes of o k2 h2 for one two and three space dimensional second order hyperbolic equations utt a x t uxx x t ux 2 2 x t u utt a x y t uxx b x y t uyy x y t ux x y t uy 2 2 x y t u and utt a x y z t uxx b x y z t uyy c x y z t uzz x y z t ux x y z t uy x y z t uz 2 2 x y z t u 0 0 subject to appropriate initial and dirichlet boundary conditions where h 0 and k 0 are grid sizes in space and time coordinates respectively a new idea is also introduced to obtain explicit difference schemes of o k2 in order to obtain numerical solution of u at first time step in a different manner

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

the central role of the propensity score in observational studies for causal effects abstract the results of observational studies are often disputed because of nonrandom treatment assignment for example patients at greater risk may be overrepresented in some treatment group this paper discusses the central role of propensity scores and balancing scores in the analysis of observational studies the propensity score is the estimated conditional probability of assignment to a particular treatment given a vector of observed covariates both large and small sample theory show that adjustment for the scalar propensity score is sufficient to remove bias due to all observed covariates applications include matched sampling on the univariate propensity score which is equal percent bias reducing under more general conditions than required for discriminant matching multivariate adjustment by subclassification on balancing scores where the same subclasses are used to estimate treatment effects for all outcome variables and in all subpopulations and visual representation of multivariate adjustment by a two dimensional plot author

propensity score estimation with missing values using a multiple imputation missingness pattern mimp approach propensity scores have been used widely as a bias reduction method to estimate the treatment effect in nonrandomized studies since many covariates are generally included in the model for estimating the propensity scores the proportion of subjects with at least one missing covariate could be large while many methods have been proposed for propensity score based estimation in the presence of missing covariates little has been published comparing the performance of these methods in this article we propose a novel method called multiple imputation missingness pattern mimp and compare it with the naive estimator ignoring propensity score and three commonly used methods of handling missing covariates in propensity score based estimation separate estimation of propensity scores within each pattern of missing data multiple imputation and discarding missing data under different mechanisms of missing data and degree of correlation among covariates simulation shows that all adjusted estimators are much less biased than the naive estimator under certain conditions mimp provides benefits smaller bias and mean squared error compared with existing alternatives

variance reduction in randomised trials by inverse probability weighting using the propensity score in individually randomised controlled trials adjustment for baseline characteristics is often undertaken to increase precision of the treatment effect estimate this is usually performed using covariate adjustment in outcome regression models an alternative method of adjustment is to use inverse probability of treatment weighting iptw on the basis of estimated propensity scores we calculate the large sample marginal variance of iptw estimators of the mean difference for continuous outcomes and risk difference risk ratio or odds ratio for binary outcomes we show that iptw adjustment always increases the precision of the treatment effect estimate for continuous outcomes we demonstrate that the iptw estimator has the same large sample marginal variance as the standard analysis of covariance estimator however ignoring the estimation of the propensity score in the calculation of the variance leads to the erroneous conclusion that the iptw treatment effect estimator has the same variance as an unadjusted estimator thus it is important to use a variance estimator that correctly takes into account the estimation of the propensity score the iptw approach has particular advantages when estimating risk differences or risk ratios in this case non convergence of covariate adjusted outcome regression models frequently occurs such problems can be circumvented by using the iptw adjustment approach 2013 the authors statistics in medicine published by john wiley sons ltd

sensitivity of the hazard ratio to nonignorable treatment assignment in an observational study in non randomized studies estimation of treatment effects generally requires adjustment for imbalances in observed covariates one such method based on the propensity score is useful in many applications but may be biased when the assumption of strongly ignorable treatment assignment is violated because it is not possible to evaluate this assumption from the data it is advisable to assess the sensitivity of conclusions to violations of strong ignorability lin et al biomet 1998 54 948 963 have implemented this idea by investigating how an unmeasured covariate may affect the conclusions of an observational study we extend their method to assess sensitivity of the treatment hazard ratio to hidden bias under a range of covariate distributions we derive simple formulas for approximating the true from the apparent treatment hazard ratio estimated under a specific survival model and assess the validity of these formulas in simulation studies we demonstrate the method in an analysis of seer medicare data on the effects of chemotherapy in elderly colon cancer patients

comparison of drug dissolution profiles a proposal based on tolerance limits meaningful comparison of the dissolution profiles between the reference and test formulations of a drug is critical for assessing similarity between the two formulations and for quality control purposes such a dissolution profile comparison is required by regulatory authorities and the criteria used for this include the widely used difference factor f1 and a similarity factor f2 recommended by the food and drug administration in spite of their extensive use in practice the two factors have been heavily criticized on various grounds the criticisms include ignoring sampling variability and ignoring the correlations across time points while using the criteria in practice the goal of this article is to put f1 and f2 on a firm statistical footing by developing tolerance limits for the distributions of f1 and f2 so that both the sampling variability and the correlations over time points are taken into account because f1 and f2 are defined in terms of sample mean dissolution profiles they are not appropriate for comparing individual dissolution profiles for the latter we have considered similar criteria and have derived tolerance limits both parametric and nonparametric approaches are explored and a bootstrap calibration is used to improve accuracy of the tolerance limits simulated coverage probabilities show that the method leads to accurate tolerance limits two examples are used to illustrate the methodology copyright 2016 john wiley sons ltd

characterizing the effect of matching using linear propensity score methods with normal distributions summary matched sampling is a standard technique for controlling bias in observational studies due to specific covariates since rosenbaum rubin 1983 multivariate matching methods based on estimated propensity scores have been used with increasing frequency in medical educational and sociological applications we obtain analytic expressions for the effect of matching using linear propensity score methods with normal distributions these expressions cover cases where the propensity score is either known or estimated using either discriminant analysis or logistic regression as is typically done in current practice the results show that matching using estimated propensity scores not only reduces bias along the population propensity score but also controls variation of components orthogonal to it matching on estimated rather than population propensity scores can therefore lead to relatively large variance reduction as much as a factor of two in common matching settings where close matches are possible approximations are given for the magnitude of this variance reduction which can be computed using estimates obtained from the matching pools related expressions for bias reduction are also presented which suggest that in difficult matching situations the use of population scores leads to greater bias reduction than the use of estimated scores

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

logit choice models for interactive attributes abstract in multi attribute decision making madm the choice of a decision maker dm is often the result of an interplay between the dm s unique attitude the utilities specific to the dm for the attributes and the degree of positive or negative interaction among the attributes based on this conglomeration of information we introduce a few choice models to give the probabilities for different possible choices of a dm more specifically the conventional multinomial logit mnl model is extended to consider the interaction among the attributes and also the dm s unique attitudinal character the usefulness of the proposed models is illustrated in a real world application

the dual weierstrass laguerre transform an inversion algorithm is derived for the dual weierstrass laguerre transform 1o g x y 1 9 v yae a l dy where the function g x x y t is associated with the source solution of the laguerre differential heat equation xuv y x t a 1 x u x t u x t correspondingly sufficient conditions are established for a function to be represented by a weierstrass laguerre stieltjes transform 1o ga x y 1 d 8 y of a nondecreasing function b

effects attributable to treatment inference in experiments and observational studies with a discrete pivot summary in randomisation and permutation inference pivotal arguments remove the hypothesised treatment effect thereby basing inferences on the null distribution in which the treatment has no effect this is common for instance with additive treatment effects the current paper uses attributable effects to expand substantially the scope of pivotal arguments attributable effects are defined for three cases namely the 2 x 2 contingency table displacement effects and the mann whitney wilcoxon statistic and in each case removing an appropriate attributable effect restores the familiar null randomisation distribution of the associated statistic yielding exact inferences the procedure extends immediately for use in sensitivity analysis in nonrandomised observational studies

using a monotone single index model to stabilize the propensity score in missing data problems and causal inference the augmented inverse weighting method is one of the most popular methods for estimating the mean of the response in causal inference and missing data problems an important component of this method is the propensity score popular parametric models for the propensity score include the logistic probit and complementary log log models a common feature of these models is that the propensity score is a monotonic function of a linear combination of the explanatory variables to avoid the need to choose a model we model the propensity score via a semiparametric single index model in which the score is an unknown monotonic nondecreasing function of the given single index under this new model the augmented inverse weighting estimator aiwe of the mean of the response is asymptotically linear semiparametrically efficient and more robust than existing estimators moreover we have made a surprising observation the inverse probability weighting and aiwes based on a correctly specified parametric model may have worse performance than their counterparts based on a nonparametric model a heuristic explanation of this phenomenon is provided a real data example is used to illustrate the proposed methods

testing causal effects in observational survival data using propensity score matching design time to event data are very common in observational studies unlike randomized experiments observational studies suffer from both observed and unobserved confounding biases to adjust for observed confounding in survival analysis the commonly used methods are the cox proportional hazards ph model the weighted logrank test and the inverse probability of treatment weighted cox ph model these methods do not rely on fully parametric models but their practical performances are highly influenced by the validity of the ph assumption also there are few methods addressing the hidden bias in causal survival analysis we propose a strategy to test for survival function differences based on the matching design and explore sensitivity of the p values to assumptions about unmeasured confounding specifically we apply the paired prentice wilcoxon ppw test or the modified ppw test to the propensity score matched data simulation studies show that the ppw type test has higher power in situations when the ph assumption fails for potential hidden bias we develop a sensitivity analysis based on the matched pairs to assess the robustness of our finding following rosenbaum s idea for nonsurvival data for a real data illustration we apply our method to an observational cohort of chronic liver disease patients from a mayo clinic study the ppw test based on observed data initially shows evidence of a significant treatment effect but this finding is not robust as the sensitivity analysis reveals that the p value becomes nonsignificant if there exists an unmeasured confounder with a small impact

on integral transforms whose kernels are solutions of singular sturm liouville problems on etudie des transformees integrales du type f x 0 f s x s d ou x s est la solution du probleme de sturm liouville y s 2 q x y 0 0 x avec y 0 cos y 0 sin 0 y x est bornee a l infini et d est la mesure spectrale

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

interval estimation for treatment effects using propensity score matching in causal studies without random assignment of treatment causal effects can be estimated using matched treated and control samples where matches are obtained using estimated propensity scores propensity score matching can reduce bias in treatment effect estimators in cases where the matched samples have overlapping covariate distributions despite its application in many applied problems there is no universally employed approach to interval estimation when using propensity score matching in this article we present and evaluate approaches to interval estimation when using propensity score matching

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

biometrika centenary theory and general methodology note reprinted in biometrika one hundred years edited by d m titterington and d r cox oxford university press 11 50 reference stat article 2001 002doi 10 1093 biomet 88 1 13view record in web of science record created on 2006 04 21 modified on 2017 05 12

analytical linear stability criteria for the leap frog dufort frankel method abstract the criterion of linear numerical stability of the combined leap frog dufort frankel scheme for advective diffusive problems in two dimensions is x x 2 y y 2 u 2 x v 2 y t 2 1 where x and y are the grid spacings in the x and y directions u and v the velocity components x and y the diffusion coefficients and t the time step although this stability requirement does not depend explicitly on the magnitude of the diffusivity only on the ratio of the diffusivity coefficients the presence of the diffusive terms renders the criterion more severe than the one obtained for purely advective problems u x v y t 1 only in one dimension are both criteria identical therefore at more than one dimension the unconditionally stable scheme dufort frankel combined with the conditionally stable scheme leap frog leads to the more restrictive stability condition the maximum allowable time step occurs when the grid spacings in both directions have the same ratio as the square roots of the diffusivity coefficients a new method is presented for discussing the stability of a finite difference scheme without actually solving for the modulus of the amplification factor this method is also extended to more general cases

asymptotic expansions of some integral transforms by using generalized functions the technique devised by wong to derive the asymptotic expansions of multiple fourier transforms by using the theory of schwartz distributions is extended to a large class of integral transforms the extension requires establishing a general procedure to extend these integral transforms to generalized functions wong s technique is then applied to some of these integral transforms to obtain their asymptotic expansions this class of integral transforms encompasses among others the laplace the airy the k and the hankel transforms

sensitivity analysis for equivalence and difference in an observational study of neonatal intensive care units in randomized experiments it is sometimes important to demonstrate that two treatments do not differ greatly in their effects or to demonstrate that the treatments have very different effects on one outcome but similar effects on another outcome these demonstrations may take the form of rejecting a null hypothesis of inequivalence in an equivalence test or rejecting a null hypothesis of equal and inequivalent effects on two outcomes the procedures often express a complex hypothesis in terms of component hypotheses and combine the component significance levels to test the complex hypothesis if used in a randomized trial randomization provides valid significance levels for each component test and hence also for the combined procedure in an observational study that is in a study of treatments that were not randomly assigned there is typically concern that significance levels for testing hypotheses about treatment effects may be distorted by failure to control for some unobserved pretreatment cov

two step estimation of semiparametric censored regression models root n consistent estimators of the regression coecients in the linear censored regression model under conditional quantile restrictions on the error terms were pro posed by powell journal of econometrics 25 1984 303 325 32 1986a 143 155 while those estimators have desirable asymptotic properties under weak regularity conditions simulation studies have shown these estimators to exhibit a small sam ple bias in the opposite direction of the least squares bias for censored data this paper introduces two step estimators for these models which minimize convex objec tive functions and are designed to overcome thisnite sample bias the paper gives regularity conditions under which the proposed two step estimators are consistent and asymptotically normal a monte carlo study compares thenite sample behavior of the proposed methods with their one step counterparts 2001 elsevier science s a all rights reserved

on the use of high order difference methods for the system of one space second order nonlinear hyperbolic equations with variable coefficients abstract implicit difference schemes of o k 4 k 2 h 2 h 4 where k 0 h 0 are grid sizes in time and space coordinates respectively are developed for the efficient numerical integration of the system of one space second order nonlinear hyperbolic equations with variable coefficients subject to appropriate initial and dirichlet boundary conditions the proposed difference method for a scalar equation is applied for the wave equation in cylindrical and spherical symmetry the numerical examples are given to illustrate the fourth order convergence of the methods

radial bounds for perturbations of elliptic operators abstract elliptic operators a m b x d a multi index with leading term positive and constant coefficient and with lower order coefficients b x l r l with n r defined on r n or a quotient space r n r n u u r n are considered it is shown that the lp spectrum of a is contained in a parabolic region of the complex plane enclosing the positive real axis uniformly in p outside the kernel of the resolvent of a is shown to be uniformly bounded by an l1 radial convolution kernel some consequences are a can be closed in all lp 1 p and is essentially self adjoint in l2 if it is symmetric a generates an analytic semigroup e ta in the right half plane strongly lp and pointwise continuous at t 0 a priori estimates relating the leading term and remainder are obtained and summability ea e 0 0 with analytic is proved for l p with convergence in lp and on the lebesgue set of more comprehensive summability results are obtained when a has constant coefficients

new multiplier methods for summing classical eigenfunction expansions on unifie et on generalise un certain nombre de resultats en theorie de sommabilite classique pour des developpements de sturm liouville regulier et singulier

reduced sensitivity to hidden bias at upper quantiles in observational studies with dilated treatment effects summary when a treatment has a dilated effect with larger effects when responses are higher there can be much less sensitivity to bias at upper quantiles than at lower quantiles i e small plausible hidden biases might explain the ostensible effect of the treatment for many subjects and yet only quite large hidden biases could explain the effect on a few subjects having dramatically elevated responses an example concerning kidney function of cadmium workers is discussed in detail in that example the treatment effect is far from additive it is plausibly zero at the lower quartile of responses to control and it is large and fairly insensitive to bias at the upper quartile

inference in linear regression models with many covariates and heteroskedasticity the linear regression model is widely used in empirical work in economics statistics and many other disciplines researchers often include many covariates in their linear model specification in a

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

propensity score matching with time dependent covariates summary in observational studies with a time dependent treatment and time dependent covariates it is desirable to balance the distribution of the covariates at every time point a time dependent propensity score based on the cox proportional hazards model is proposed and used in risk set matching matching on this propensity score is shown to achieve a balanced distribution of the covariates in both treated and control groups optimal matching with various designs is conducted and compared in a study of a surgical treatment cystoscopy and hydrodistention given in response to a chronic bladder disease interstitial cystitis simulation studies also suggest that the statistical analysis after matching outperforms the analysis without matching in terms of both point and interval estimations

on the use of discrete choice models for causal inference methodology for causal inference based on propensity scores has been developed and popularized in the last two decades however the majority of the methodology has concentrated on binary treatments only recently have these methods been extended to settings with multi valued treatments we propose a number of discrete choice models for estimating the propensity scores the models differ in terms of flexibility with respect to potential correlation between treatments and in turn the accuracy of the estimated propensity scores we present the effects of discrete choice models used on performance of the causal estimators through a monte carlo study we also illustrate the use of discrete choice models to estimate the effect of antipsychotic drug use on the risk of diabetes in a cohort of adults with schizophrenia

affinely invariant matching methods with discriminant mixtures of proportional ellipsoidally symmetric distributions in observational studies designed to estimate the effects of interventions or exposures such as cigarette smoking it is desirable to try to control background differences between the treated group e g current smokers and the control group e g never smokers on covariates x e g age education matched sampling attempts to effect this control by selecting subsets of the treated and control groups with similar distributions of such covariates this paper examines the consequences of matching using affinely invariant methods when the covariate distributions are discriminant mixtures of proportional ellipsoidally symmetric dmpes distributions a class herein defined which generalizes the ellipsoidal symmetry class of rubin and thomas ann statist 20 1992 1079 1093 the resulting generalized results help indicate why earlier results hold quite well even when the simple assumption of ellipsoidal symmetry is not met e g biometrics 52 1996 249 264 extensions to conditionally affinely invariant matching with conditionally dmpes distributions are also discussed

control of confounding through secondary samples the control of confounding is essential in many statistical problems especially in those that attempt to estimate exposure effects in some cases in addition to the primary sample there is another secondary sample which though having no direct information about the exposure effect contains information about the confounding factors the purpose of this article is to study the influence of the secondary sample on likelihood inference for the exposure effect in particular we investigate the interplay between the efficiency improvement and the possible bias introduced by the secondary sample as a function of the degree of confounding in the primary sample and the sizes of the primary and secondary samples in the case of weak confounding the secondary sample can only little improve estimation of the exposure effect whereas with strong confounding the secondary sample can be much more useful on the other hand it will be more important to consider possible biasing effects in the latter case for illustration we use a formal example of a generalized linear model and a real example with sparse data from a case control study of the association between gastric cancer and hm cap band 120