stable weights that balance covariates for estimation with incomplete outcome data weighting methods that adjust for observed covariates such as inverse probability weighting are widely used for causal inference and estimation with incomplete outcome data part of the appeal of such methods is that one set of weights can be used to estimate a range of treatment effects based on different outcomes or a variety of population means for several variables however this appeal can be diminished in practice by the instability of the estimated weights and by the difficulty of adequately adjusting for observed covariates in some settings to address these limitations this article presents a new weighting method that finds the weights of minimum variance that adjust or balance the empirical distribution of the observed covariates up to levels prespecified by the researcher this method allows the researcher to balance very precisely the means of the observed covariates and other features of their marginal and joint distributions such as variances and correlations and also for example the

multiple imputation when records used for imputation are not used or disseminated for analysis summary when some of the records used to estimate the imputation models in multiple imputation are not used or available for analysis the usual multiple imputation variance estimator has positive bias we present an alternative approach that enables unbiased estimation of variances and hence calibrated inferences in such contexts first using all records the imputer samples m values of the parameters of the imputation model second for each parameter draw the imputer simulates the missing values for all records n times from these mn completed datasets the imputer can analyse or disseminate the appropriate subset of records we develop methods for interval estima tion and significance testing for this approach methods are presented in the context of multiple imputation for measurement error

a note on multiple imputation for method of moments estimation multiple imputation is a popular imputation method for general purpose estimation rubin 1987 provided an easily applicable formula for the variance estimation of multiple imputation however the validity of the multiple imputation inference requires the congeniality condition of meng 1994 which is not necessarily satisfied for method of moments estimation this paper presents the asymptotic bias of rubin s variance estimator when the method of moments estimator is used as a complete sample estimator in the multiple imputation procedure a new variance estimator based on over imputation is proposed to provide asymptotically valid inference for method of moments estimation

robust restricted maximum likelihood in mixed linear models summary definitions of robust maximum likelihood robust ml and robust restricted maximum likelihood robust reml are introduced and the definitions are applied to data from biological and chemical experiments a simulation study is undertaken to investigate the asymptotic properties of robust ml and robust reml in small samples and to examine the advantages of using robust methods 1 introduction and definitions linear models with multiple sources of error are widely used in designed experiments across many scientific fields an example of such an experiment is described by patterson and nabugoomu 1992 from patterson and silvey 1980 six varieties of wheat were grown at ten centres that formed a sample of the main types of growing area for wheat in scotland and the yields in tonnes hectare were recorded the experiment is unbalanced because of 60 possible variety centre combinations only 46 were used at seven centres four varieties were grown and at the remaining three centres all six varieties were grown in this paper we fit the simplest mixed linear model proposed for this data by patterson and nabugoomu namely

incorporating baseline measurements into the analysis of crossover trials with time to event endpoints two period two treatment 2 2 crossover designs are commonly used in clinical trials for continuous endpoints it has been shown that baseline pretreatment measurements collected before the start of each treatment period can be useful in improving the power of the analysis methods to achieve a corresponding gain for censored time to event endpoints have not been adequately studied we propose a method in which censored values are treated as missing data and multiply imputed using prespecified parametric event time models the event times in each imputed data set are then log transformed and analyzed using a linear model suitable for a 2 2 crossover design with continuous endpoints with the difference in period specific baselines included as a covariate results obtained from the imputed data sets are synthesized for point and confidence interval estimation of the treatment ratio of geometric mean event times using model averaging in conjunction with rubin s combination rule we use simulations to illustrate the favorable operating characteristics of our method relative to two other methods for crossover trials with censored time to event data ie a hierarchical rank test that ignores the baselines and a stratified cox model that uses each study subject as a stratum and includes period specific baselines as a covariate application to a real data example is provided

finite element approximation of the transport of reactive solutes in porous media part 1 error estimates for nonequilibrium adsorption processes in this paper we analyze a fully practical piecewise linear finite element approximation involving numerical integration backward euler time discretization and possibly regularization of the following degenerate parabolic system arising in a model of reactive solute transport in porous media find u x t v x t such that begin eqnarray partial t u partial t v delta u f quad mbox in omega times 0 t qquad u 0 quad mbox on partial omega times 0 t partial tv k varphi u v quad mbox in omega times 0 t u cdot 0 g 1 cdot quad v cdot 0 g 2 cdot quad mbox in omega subset real d quad 1 le d le 3 end eqnarray for given data k in real f g 1 g 2 and a monotonically increasing varphi in c 0 real cap c 1 infty 0 cup 0 infty satisfying varphi 0 0 which is only locally holder continuous with exponent p in 0 1 at the origin e g varphi s equiv s p this lack of lipschitz continuity at the origin limits the regularity of the unique solution u v and leads to difficulties in the finite element error analysis

on linear and quadratic estimating functions summary in some cases maximum quasi likelihood estimation which is at the core of glim can fail to give reasonable results a more general class of estimating functions is investigated which avoids such failure and also is not restricted to the particular forms of mean and variance function of glim

multiply robust estimation in regression analysis with missing data doubly robust estimators are widely used in missing data analysis they provide double protection on estimation consistency against model misspecifications however they allow only a single model for the missingness mechanism and a single model for the data distribution and the assumption that one of these two models is correctly specified is restrictive in practice for regression analysis with possibly missing outcome we propose an estimation method that allows multiple models for both the missingness mechanism and the data distribution the resulting estimator is consistent if any one of those multiple models is correctly specified and thus provides multiple protection on consistency this estimator is also robust against extreme values of the fitted missingness probability which for most doubly robust estimators can lead to erroneously large inverse probability weights that may jeopardize the numerical performance the numerical implementation of the proposed method through a modified newton ra

finite element runge kutta discretizations of porous medium type equations in this paper we analyze the convergence properties of full discretizations of a class of generalized porous medium equations for the spatial and time discretizations we use continuous piecewise linear finite elements and algebraically stable runge kutta methods respectively we prove our convergence result without any assumption on the spatial regularity it is shown that under a certain stability assumption the temporal order of convergence is given by the stage order of the method whereas the spatial order is essentially one numerical experiments illustrate our stability assumption and the convergence result

analysis of semi parametric regression models with non ignorable non response in this article we consider the problem of making inferences about the parameter 0 indexing the conditional mean of an outcome given a vector of regressors when a subset of the variables outcome or covariates are missing for some study subjects and the probability of non response depends upon both observed and unobserved data values that is non response is non ignorable we propose a new class of inverse probability of censoring weighted estimators that are consistent and asymptotically normal can for estimating 0 when the non response probabilities can be parametrically modelled and a can estimator exists the proposed estimators do not require full specification of the likelihood and their computation does not require numerical integration we show that the asymptotic variance of the optimal estimator in our class attains the semi parametric variance bound for the model in some models no can estimator of 0 exists we provide a general algorithm for determining when can estimators of 0 exist our results follow after specializing a general representation described in the article for the efficient score and the influence function of regular asymptotically linear estimators in an arbitrary semi parametric model with non ignorable non response in which the probability of observing complete data is bounded away from zero and the non response probabilities can be parametrically modelled 1997 by john wiley sons ltd

the asymptotic variance of semiparametric estimators this paper derives a general formula for the asymptotic variance of semiparametric estimators that accounts for the presence of nonparametric estimators of functions the general formula is specialized to show invariance of the asymptotic variance to the type of nonparametric estimator and to obtain correction terms for estimation of densities and mean square projections including conditional expectations regularity conditions for the validity of the formula are also given including primitive conditions for asymptotic normality when series estimators are present new examples considered include a semiparametric panel probit estimator and a series estimator of the average derivative copyright 1994 by the econometric society

optimal rates of convergence for degenerate parabolic problems in two dimensions fully discrete implicit schemes for the solution of degenerate parabolic equations in two dimensions are shown to satisfy error estimates of the form o tau ln 1 h alpha h where tau is the time step size and h is the mesh size these results apply to both the stefan problem and the porous medium equation

robust estimation of causal effects of binary treatments in unconfounded studies with dichotomous outcomes the estimation of causal effects has been the subject of extensive research in unconfounded studies with a dichotomous outcome y cangul chretien gutman and rubin 2009 demonstrated that logistic regression for a scalar continuous covariate x is generally statistically invalid for testing null treatment effects when the distributions of x in the treated and control populations differ and the logistic model for y given x is misspecified in addition they showed that an approximately valid statistical test can be generally obtained by discretizing x followed by regression adjustment within each interval defined by the discretized x this paper extends the work of cangul et al 2009 in three major directions first we consider additional estimation procedures including a new one that is based on two independent splines and multiple imputation second we consider additional distributional factors and third we examine the performance of the procedures when the treatment effect is non null of all the methods considered and in most of the experimental conditions that were examined our proposed new methodology appears to work best in terms of point and interval estimation copyright 2012 john wiley sons ltd

the multiple adaptations of multiple imputation multiple imputation was first conceived as a tool that statistical agencies could use to handle nonresponse in large sample public use surveys in the last two decades the multiple imputation framework has been adapted for other statistical contexts for example individual researchers use multiple imputation to handle missing data in small samples statistical agencies disseminate multiply imputed data sets for purposes of protecting data confidentiality and survey methodologists and epidemiologists use multiple imputation to correct for measurement errors in some of these settings rubin s original rules for combining the point and variance estimates from the multiply imputed data sets are not appropriate because what is known and thus the conditional expectations and variances used to derive inferential methods differs from that in the missing data context these applications require new combining rules and methods of inference in fact more than 10 combining rules exist in the published literatur

multiple imputation for interval estimation from simple random samples with ignorable nonresponse abstract several multiple imputation techniques are described for simple random samples with ignorable nonresponse on a scalar outcome variable the methods are compared using both analytic and monte carlo results concerning coverages of the resulting intervals for the population mean using m 2 imputations per missing value gives accurate coverages in common cases and is clearly superior to single imputation m 1 in all cases the performances of the methods for various m can be predicted well by linear interpolation in 1 m 1 between the results for m 2 and m as a rough guide to assure coverages of interval estimates within 2 of the nominal level when using the preferred methods the number of imputations per missing value should increase from 2 to 3 as the nonresponse rate increases from 10 to 60

skew circuits of small width abstract a celebrated result of barrington 1985 proved that polynomial size width 5 branching programs bp are equivalent in power to a restricted form of branching programs polynomial sized width 5 permutation branching programs pbp which in turn capture all of nc 1 on the other hand it is known that width 3 pbps require exponential size to compute the and function no such lower bound is known for width 4 pbps however it is widely conjectured that width 4 pbps will not capture all of nc 1 in this work we study the power of bounded width branching programs by comparing them with bounded width skew circuits it is well known that branching programs of bounded width have the same power as skew circuit of bounded width the naive approach converts a bp of width w to a skew circuit of width w 2 we improve this bound and show that bp of width w 5 can be converted to a skew circuit of width 7 this also implies that skew circuits of bounded width are equal in power to skew circuits of width 7 for the other way we prove that for any w 2 a skew circuit of width w can be converted into an equivalent branching program of width w we prove that width 2 skew circuits are not universal while width 3 skew circuits are universal and that any polynomial sized cnf or dnf is computable by width 3 skew circuits of polynomial size it is known that parity does not have small cnfs or dnfs it is easy to see that parity has width 4 skew circuits we prove that a width 3 skew circuit computing parity requires exponential size this gives an exponential separation between the power of width 3 skew circuits and width 4 skew circuits

finite sample properties of multiple imputation estimators finite sample properties of multiple imputation estimators under the linear regression model are studied the exact bias of the multiple imputation variance estimator is presented a method of reducing the bias is presented and simulation is used to make comparisons we also show that the suggested method can be used for a general class of linear estimators

approximation of degenerate parabolic problems using numerical integration a class of multidimensional degenerate parabolic equations is considered the two phase stefan problem and the porous medium equation are analyzed as models of singular parabolic equations nonstat

performing likelihood ratio tests with multiply imputed data sets summary existing procedures for obtaining significance levels from multiply imputed data either i require access to the completed data point estimates and variance covariance matrices which may not be available in practice when the dimensionality of the estimand is high or ii directly combine p values with less satisfactory results taking advantage of the well known relationship between the wald and log likelihood ratio test statistics we propose a complete data log likelihood ratio based procedure it is shown that for any number of multiple imputations the proposed procedure is equivalent in large samples to the existing procedure based on the point estimates and the variance covariance matrices yet it only requires the point estimates and evaluations of the complete data log likelihood ratio statistic as a function of these estimates and the completed data the proposed procedure therefore is especially attractive with highly multiparameter incomplete data problems since it does not involve the computation of any matrices

finite monoids and the fine structure of nc 1 recently a new connection was discovered between the parallel complexity class nc 1 and the theory of finite automata in the work of barrington on bounded width branching programs there nonuniform nc 1 was characterized as those languages recognized by a certain nonuniform version of a dfa here we extend this characterization to show that the internal structures of nc 1 and the class of automata are closely related in particular using therien s classification of finite monoids we give new characterizations of the classes ac 0 depth k ac 0 and acc the last being the ac 0 closure of the mod q functions for all constant q we settle some of the open questions in 3 give a new proof that the dot depth hierarchy of algebraic automata theory is infinite 8 and offer a new framework for understanding the internal structure of nc 1

joint mean covariance models with applications to longitudinal data unconstrained parameterisation summary we provide unconstrained parameterisation for and model a covariance using covariates the cholesky decomposition of the inverse of a covariance matrix is used to associate a unique unit lower triangular and a unique diagonal matrix with each covariance matrix the entries of the lower triangular and the log of the diagonal matrix are unconstrained and have meaning as regression coefficients and prediction variances when regressing a measurement on its predecessors an extended generalised linear model is introduced for joint modelling of the vectors of predictors for the mean and covariance subsuming the joint modelling strategy for mean and variance heterogeneity gabriel s antedependence models dempster s covariance selection models and the class of graphical models the likelihood function and maximum likelihood estimators of the covariance and the mean parameters are studied when the observations are normally distributed applications to modelling nonstationary dependence structures and multivariate data are discussed and illustrated using real data a graphical method similar to that based on the correlogram in time series is developed and used to identify parametric models for nonstationary covariances

inference for imputation estimators we derive an estimator of the asymptotic variance of both single and multiple imputation estimators we assume a parametric imputation model but allow for non and semiparametric analysis models our variance estimator in contrast to the estimator proposed by rubin 1987 is consistent even when the imputation and analysis models are misspecified and incompatible with one another

misspecifying the covariance structure in a linear mixed model under mar drop out misspecification of the covariance structure in a linear mixed model lmm can lead to biased population parameters estimates under mar drop out in our motivating example of modeling cd4 cell counts during untreated hiv infection random intercept and slope lmms are frequently used in this article we evaluate the performance of lmms with specific covariance structures in terms of bias in the fixed effects estimates under specific mar drop out mechanisms and adopt a bayesian model comparison criterion to discriminate between the examined approaches in real data applications we analytically show that using a random intercept and slope structure when the true one is more complex can lead to seriously biased estimates with the degree of bias depending on the magnitude of the mar drop out under misspecified covariance structure we compare in terms of induced bias the approach of adding a fractional brownian motion bm process on top of random intercepts and slopes with the approach of using splines for the random effects in general the performance of both approaches was satisfactory with the bm model leading to smaller bias in most cases a simulation study is carried out to evaluate the performance of the proposed bayesian criterion in identifying the model with the correct covariance structure overall the proposed method performs better than the aic and bic criteria under our specific simulation setting the models under consideration are applied to real data from the cascade study the most plausible model is identified by all examined criteria

censored correlated cytokine concentrations multivariate tobit regression using clustered variance estimation interest in cytokines as markers for the function of the immune system is increasing methods quantifying cytokine concentrations are often subject to detection limits which lead to non detectable observations and censored distributions when distributions are skewed geometric mean ratios gmrs can be used to describe the relative concentration between two cytokines and the gmr ratio gmrr can be used to compare two groups the problem is how to estimate gmrrs from censored distributions we evaluated methods including simple deletion and substitution in simulated and real data one method applies tobit directly to the censored difference between the two cytokine log concentrations diff however censoring is correlated to the outcome and is therefore not independent the correlation increases as the correlation between the two log concentrations decreases we propose a tobit stacking method that uses clustered variance covariance estimation allowing homogeneous stackc or inhomogeneous stackh variances we compare it with direct estimation of the bivariate tobit likelihood function bitobit and multiple imputation we assess sensitivity to inhomogeneity and non normality simulations show that deletion and substitution are empirically biased and that diff has an empirical bias which increases as the correlation between the log concentrations decreases estimates from multiple imputation stackh and bitobit are almost identical the estimates exhibit small empirical bias for both homogeneous and inhomogeneous normal distributions for skewed mixture and heavy tailed distributions they perform reasonably well if censoring is less than 30 we recommend these methods to estimate gmrrs at least one of the methods is available in stata r or sas copyright 2012 john wiley sons ltd

on the multiple imputation variance estimator for control based and delta adjusted pattern mixture models control based pattern mixture models pmm and delta adjusted pmms are commonly used as sensitivity analyses in clinical trials with non ignorable dropout these pmms assume that the statistical behavior of outcomes varies by pattern in the experimental arm in the imputation procedure but the imputed data are typically analyzed by a standard method such as the primary analysis model in the multiple imputation mi inference rubin s variance estimator is generally biased when the imputation and analysis models are uncongenial one objective of the article is to quantify the bias of rubin s variance estimator in the control based and delta adjusted pmms for longitudinal continuous outcomes these pmms assume the same observed data distribution as the mixed effects model for repeated measures mmrm we derive analytic expressions for the mi treatment effect estimator and the associated rubin s variance in these pmms and mmrm as functions of the maximum likelihood estimator from the mmrm analysis and the observed proportion of subjects in each dropout pattern when the number of imputations is infinite the asymptotic bias is generally small or negligible in the delta adjusted pmm but can be sizable in the control based pmm this indicates that the inference based on rubin s rule is approximately valid in the delta adjusted pmm a simple variance estimator is proposed to ensure asymptotically valid mi inferences in these pmms and compared with the bootstrap variance the proposed method is illustrated by the analysis of an antidepressant trial and its performance is further evaluated via a simulation study

analysis of longitudinal clinical trials with missing data using multiple imputation in conjunction with robust regression summary in a typical randomized clinical trial a continuous variable of interest e g bone density is measured at baseline and fixed postbaseline time points the resulting longitudinal data often incomplete due to dropouts and other reasons are commonly analyzed using parametric likelihood based methods that assume multivariate normality of the response vector if the normality assumption is deemed untenable then semiparametric methods such as weighted generalized estimating equations are considered we propose an alternate approach in which the missing data problem is tackled using multiple imputation and each imputed dataset is analyzed using robust regression m estimation huber 1973 annals of statistics 1 799 821 to protect against potential non normality outliers in the original or imputed dataset the robust analysis results from each imputed dataset are combined for overall estimation and inference using either the simple rubin 1987 multiple imputation for nonresponse in surveys new york wiley method or the more complex but potentially more accurate robins and wang 2000 biometrika 87 113 124 method we use simulations to show that our proposed approach performs at least as well as the standard methods under normality but is notably better under both elliptically symmetric and asymmetric non normal distributions a clinical trial example is used for illustration

discussion of the frangakis and rubin article in clinical survival studies conducted in the united states rich data are frequently available on variables recording time to drop out and the evolution over time of a patient s clinical signs symptoms and laboratory measurements motivated by a greek study frangakis and rubin fr consider estimation of a marginal survival curve under a double sampling design from severely impoverished reduced data that included none of the aforementioned variables due to confidentiality restrictions to make our discussion relevant to settings with and without confidentiality restrictions we shall consider survival curve estimation both from rich data that includes the aforementioned variables and from reduced data that does not let t l and c be continuous failure dropout and administrative censoring times respectively with time measured from date of enrollment the goal is to estimate the cumulative net marginal hazard of failure a t fj at u du and the survival function s t e a t under a double sampling design in which a subset of the dropouts is followed up in a secondphase sample in their analysis fr assumed a all dropouts had the same chance of being selected into the second stage sample b all dropouts selected into the second phase had their censoring indicator a 6 t c and their minimum x min t c of censoring and failure successfully ascertained and c c was independent of underlying variables such as t and l it was necessary for fr to impose these assumptions because fr s estimator of a t is inconsistent unless a c hold in practice one or more of assumptions a c may often fail to hold for instance in the first paragraph of their section 5 fr noted it will often happen that a subset of the dropouts pursued in the second phase will fail to have x i ascertained violating assumption b fr recommend that members of this subset be treated in the analysis as having been administratively censored however were this recommendation to be followed fr s survival estimator would be inconsistent and could be severely biased if the number of secondstage subjects who do not have x i ascertained is large assumption a will be false if a potentially more efficient design has been employed in which subjects who dropped out early are oversampled in the second phase assumption c will be false when there are secular trends in the distribution of t as was the case during the 1980s and 1990s for the survival time t of aids patients if only assumption a were false then as fr note identification of a t could be restored by addition of the known second phase sampling probabilities to fr s reduced data however when assumptions b and or c are false both additional assumptions and rich data are required to restore identification in this discussion we make the following points first we show in section 4 that if assumptions a c hold then i when only the reduced data are available for analysis fr s estimator is algebraically identical to the efficient inverse prow ability weighted ipw estimator but ii when rich data are available fr s estimator is inefficient for this case we provide in section 8 a locally semiparametric efficient lse estimator of s t that exploits the information in the rich data second we consider the more realistic setting in which i fr s estimator is inconsistent because one or more of assumptions a c fail to hold and ii rich data are available in sections 6 and 7 we derive doubly robust lse survival curve estimators under the assumption that the missingness process is ignorable in summary we describe aspects of a powerful methodology for the analysis of doubly sampled censored survival data that can resolve the three principal problems left open by fr i how to construct locally lse estimators of s t when fr s assumptions a c hold and as would typically be the case in the united states rich data are available for analysis ii how to construct lse doubly robust estimators of s t when fr s assumptions b and or c fail but missingness remains ignorable and iii how to conduct a sensitivity analysis when missingness may be nonignorable this methodology is in effect a subset of the general theory developed in robins and rotnitzky 1992 robins 1993a b robins rotnitzky and scharfstein 1999 and scharfstein rotnitzky and robins 1999a for the analysis of semiand nonparametric rightcensored data models specialized to the case of doubly sampled censored survival data because of space limitations our resolution of problem iii will be described elsewhere in order to successfully exploit our general theory in the context of doubly sampled data an additional problem must be faced specifically in this context the distribution of the censoring variable has both discrete and continuous components as a result none of the estimators of s t previously proposed in the aforementioned papers are directly applicable in section 6 and the appendix we provide a survival estimator that allows for a mixed discrete and continuous censor

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

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

repeated measures in clinical trials analysis using mean summary statistics and its implications for design this paper explores the use of simple summary statistics for analysing repeated measurements in randomized clinical trials with two treatments quite often the data for each patient may be effectively summarized by a pre treatment mean and a post treatment mean analysis of covariance is the method of choice and its superiority over analysis of post treatment means or analysis of mean changes is quantified as regards both reduced variance and avoidance of bias using a simple model for the covariance structure between time points quantitative consideration is also given to practical issues in the design of repeated measures studies the merits of having more than one pre treatment measurement are demonstrated and methods for determining sample sizes in repeated measures designs are provided several examples from clinical trials are presented and broad practical recommendations are made the examples support the value of the compound symmetry assumption as a realistic simplification in quantitative planning of repeated measures trials the analysis using summary statistics makes no such assumption however allowance in design for alternative non equal correlation structures can and should be made when necessary

combining multiple imputation and inverse probability weighting two approaches commonly used to deal with missing data are multiple imputation mi and inverse probability weighting ipw ipw is also used to adjust for unequal sampling fractions mi is generally more efficient than ipw but more complex whereas ipw requires only a model for the probability that an individual has complete data a univariate outcome mi needs a model for the joint distribution of the missing data a multivariate outcome given the observed data inadequacies in either model may lead to important bias if large amounts of data are missing a third approach combines mi and ipw to give a doubly robust estimator a fourth approach ipw mi combines mi and ipw but unlike doubly robust methods imputes only isolated missing values and uses weights to account for remaining larger blocks of unimputed missing data such as would arise e g in a cohort study subject to sample attrition and or unequal sampling fractions in this article we examine the performance in terms of bias and efficiency of ipw mi relative to mi and ipw alone and investigate whether the rubin s rules variance estimator is valid for ipw mi we prove that the rubin s rules variance estimator is valid for ipw mi for linear regression with an imputed outcome we present simulations supporting the use of this variance estimator in more general settings and we demonstrate that ipw mi can have advantages over alternatives ipw mi is applied to data from the national child development study

survival analysis using auxiliary variables via non parametric multiple imputation we develop an approach based on multiple imputation that estimates the marginal survival distribution in survival analysis using auxiliary variables to recover information for censored observations to conduct the imputation we use two working survival models to define a nearest neighbour imputing risk set one model is for the event times and the other for the censoring times based on the imputing risk set two non parametric multiple imputation methods are considered risk set imputation and kaplan meier imputation for both methods a future event or censoring time is imputed for each censored observation with a categorical auxiliary variable we show that with a large number of imputes the estimates from the kaplan meier imputation method correspond to the weighted kaplan meier estimator we also show that the kaplan meier imputation method is robust to mis specification of either one of the two working models in a simulation study with time independent and time dependent auxiliary variables we compare the multiple imputation approaches with an inverse probability of censoring weighted method we show that all approaches can reduce bias due to dependent censoring and improve the efficiency we apply the approaches to aids clinical trial data comparing zdv and placebo in which cd4 count is the time dependent auxiliary variable

robust estimating functions and bias correction for longitudinal data analysis robust methods are useful in making reliable statistical inferences when there are small deviations from the model assumptions the widely used method of the generalized estimating equations can be robustified by replacing the standardized residuals with the m residuals if the pearson residuals are assumed to be unbiased from zero parameter estimators from the robust approach are asymptotically biased when error distributions are not symmetric we propose a distribution free method for correcting this bias our extensive numerical studies show that the proposed method can reduce the bias substantially examples are given for illustration

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

random effects models for longitudinal data models for the analysis of longitudinal data must recognize the relationship between serial observations on the same unit multivariate models with general covariance structure are often difficult to apply to highly unbalanced data whereas two stage random effects models can be used easily in two stage models the probability distributions for the response vectors of different individuals belong to a single family but some random effects parameters vary across individuals with a distribution specified at the second stage a general family of models is discussed which includes both growth models and repeated measures models as special cases a unified approach to fitting these models based on a combination of empirical bayes and maximum likelihood estimation of model parameters and using the em algorithm is discussed two examples are taken from a current epidemiological study of the health effects of air pollution

small sample degrees of freedom for multi component significance tests with multiple imputation for missing data summary when performing multi component significance tests with multiply imputed datasets analysts can use a wald like test statistic and a reference f distribution the currently employed degrees of freedom in the denominator of this f distribution are derived assuming an infinite sample size for modest complete data sample sizes this degrees of freedom can be unrealistic for example it may exceed the complete data degrees of freedom this paper presents an alternative denominator degrees of freedom that is always less than or equal to the complete data denominator degrees of freedom and equals the currently employed denominator degrees of freedom for infinite sample sizes its advantages over the currently employed degrees of freedom are illustrated with a simulation

nonuniform acc circuit lower bounds the class acc consists of circuit families with constant depth over unbounded fan in and or not and modm gates where m 1 is an arbitrary constant we prove the following nexp the class of languages accepted in nondeterministic exponential time does not have nonuniform acc circuits of polynomial size the size lower bound can be slightly strengthened to quasipolynomials and other less natural functions enp the class of languages recognized in 2o n time with an np oracle doesn t have nonuniform acc circuits of 2no 1 size the lower bound gives an exponential size depth tradeoff for every d m there is a 0 such that enp doesn t have depth d acc circuits of size 2n with modm gates previously it was not known whether expnp had depth 3 polynomial size circuits made out of only mod6 gates the high level strategy is to design faster algorithms for the circuit satisfiability problem over acc circuits then prove that such algorithms entail these lower bounds the algorithms combine known properties of acc with fast rectangular matrix multiplication and dynamic programming while the second step requires a strengthening of the author s prior work

a numerical approach to degenerate parabolic equations in this work we propose a numerical approach to solve some kind of degenerate parabolic equations the underlying idea is based on the maximum principle more precisely we locally perturb the initial and boundary data instead of the nonlinear diffusion coefficients so that the resulting problem is not degenerate the efficiency of this method is shown analytically as well as numerically the numerical experiments show that this new approach is comparable with the existing ones

estimation of regression coefficients when some regressors are not always observed abstract in applied problems it is common to specify a model for the conditional mean of a response given a set of regressors a subset of the regressors may be missing for some study subjects either by design or happenstance in this article we propose a new class of semiparametric estimators based on inverse probability weighted estimating equations that are consistent for parameter vector 0 of the conditional mean model when the data are missing at random in the sense of rubin and the missingness probabilities are either known or can be parametrically modeled we show that the asymptotic variance of the optimal estimator in our class attains the semiparametric variance bound for the model by first showing that our estimation problem is a special case of the general problem of parameter estimation in an arbitrary semiparametric model in which the data are missing at random and the probability of observing complete data is bounded away from 0 and then deriving a representation for the efficient score

improved double robust estimation in missing data and causal inference models recently proposed double robust estimators for a population mean from incomplete data and for a finite number of counterfactual means can have much higher efficiency than the usual double robust estimators under misspecification of the outcome model in this paper we derive a new class of double robust estimators for the parameters of regression models with incomplete cross sectional or longitudinal data and of marginal structural mean models for cross sectional data with similar efficiency properties unlike the recent proposals our estimators solve outcome regression estimating equations in a simulation study the new estimator shows improvements in variance relative to the standard double robust estimator that are in agreement with those suggested by asymptotic theory

on the stationary distribution of iterative imputations iterative imputation in which variables are imputed one at a time each given a model predicting from all the others is a popular technique that can be convenient and exible as it replaces a potentially dicult multivariate modeling problem with relatively simple univariate regressions in this paper we begin to characterize the stationary distributions of iterative imputations and their statistical properties more precisely when the conditional models are compatible dened in the text we give a set of sucient conditions under which the imputation distribution converges in total variation to the posterior distribution of a bayesian model when the conditional models are incompatible but are valid we show that the combined imputation estimator is consistent

bounds for width two branching programs branching programs have been studied as a fundamental model for space bounded computations and in particular as a model in which to try to establish nontrivial space lower bounds and time space trade offs at present there still do not exist any results for single output functions we consider a class of severely constrained programs those having width 2 and establish characterizations as well as lower bounds for some boolean functions computable within this model

small sample inference for fixed effects from restricted maximum likelihood restricted maximum likelihood reml is now well established as a method for estimating the parameters of the general gaussian linear model with a structured covariance matrix in particular for mixed linear models conventionally estimates of precision and inference for fixed effects are based on their asymptotic distribution which is known to be inadequate for some small sample problems in this paper we present a scaled wald statistic together with an f approximation to its sampling distribution that is shown to perform well in a range of small sample settings the statistic uses an adjusted estimator of the covariance matrix that has reduced small sample bias this approach has the advantage that it reproduces both the statistics and f distributions in those settings where the latter is exact namely for hotelling t2 type statistics and for analysis of variance f ratios the performance of the modified statistics is assessed through simulation studies of four different reml analyses and the methods are illustrated using three examples

partial f tests with multiply imputed data in the linear regression framework via coefficient of determination tests for regression coefficients such as global local and partial f tests are common in applied research in the framework of multiple imputation there are several papers addressing tests for regression coefficients however for simultaneous hypothesis testing the existing methods are computationally intensive because they involve calculation with vectors and inversion of matrices in this paper we propose a simple method based on the scalar entity coefficient of determination to perform global local and partial f tests with multiply imputed data the proposed method is evaluated using simulated data and applied to suicide prevention data

on the bias of the multiple imputation variance estimator in survey sampling summary multiple imputation is a method of estimating the variances of estimators that are constructed with some imputed data we give an expression for the bias of the multiple imputation variance estimator for data that are collected with a complex sample design the bias may be sizable for certain estimators such as domain means when a large fraction of the values are imputed a bias adjusted variance estimator is suggested

combining fractional polynomial model building with multiple imputation multivariable fractional polynomial mfp models are commonly used in medical research the datasets in which mfp models are applied often contain covariates with missing values to handle the missing values we describe methods for combining multiple imputation with mfp modelling considering in turn three issues first how to impute so that the imputation model does not favour certain fractional polynomial fp models over others second how to estimate the fp exponents in multiply imputed data and third how to choose between models of differing complexity two imputation methods are outlined for different settings for model selection methods based on wald type statistics and weighted likelihood ratio tests are proposed and evaluated in simulation studies the wald based method is very slightly better at estimating fp exponents type i error rates are very similar for both methods although slightly less well controlled than analysis of complete records however there is potential for substantial gains in power over the analysis of complete records we illustrate the two methods in a dataset from five trauma registries for which a prognostic model has previously been published contrasting the selected models with that obtained by analysing the complete records only 2015 the authors statistics in medicine published by john wiley sons ltd

a combined finite volume nonconforming mixed hybrid finite element scheme for degenerate parabolic problems we propose and analyze a numerical scheme for nonlinear degenerate parabolic convection diffusion reaction equations in two or three space dimensions we discretize the diffusion term which generally involves an inhomogeneous and anisotropic diffusion tensor over an unstructured simplicial mesh of the space domain by means of the piecewise linear nonconforming crouzeix raviart finite element method or using the stiffness matrix of the hybridization of the lowest order raviart thomas mixed finite element method the other terms are discretized by means of a cell centered finite volume scheme on a dual mesh where the dual volumes are constructed around the sides of the original mesh checking the local peclet number we set up the exact necessary amount of upstream weighting to avoid spurious oscillations in the convection dominated case this technique also ensures the validity of the discrete maximum principle under some conditions on the mesh and the diffusion tensor we prove the convergence of the scheme only supposing the shape regularity condition for the original mesh we use a priori estimates and the kolmogorov relative compactness theorem for this purpose the proposed scheme is robust only 5 point 7 point in space dimension three locally conservative efficient and stable which is confirmed by numerical experiments

non commutative arithmetic circuits depth reduction and size lower bounds abstract we investigate the phenomenon of depth reduction in commutative and non commutative arithmetic circuits we prove that in the commutative setting uniform semi unbounded arithmetic circuits of logarithmic depth are as powerful as uniform arithmetic circuits of polynomial degree and unrestricted depth earlier proofs did not work in the uniform setting this also provides a unified proof of the circuit characterizations of the class logcfl and its counting variant logcfl we show that ac1 has no more power than arithmetic circuits of polynomial size and degree no log log n improving the trivial bound of no log n connections are drawn between tc1 and arithmetic circuits of polynomial size and degree then we consider non commutative computation we show that over the algebra max concat arithmetic circuits of polynomial size and polynomial degree can be reduced to o log2 n depth and even to o log n depth if unbounded fanin gates are allowed this establishes that optlogcfl is in ac1 this is the first depth reduction result for arithmetic circuits over a non commutative semiring and it complements the lower bounds of kosaraju and nisan showing that depth reduction cannot be done in the general non commutative setting we define new notions called short left paths and short right paths and we show that these notions provide a characterization of the classes of arithmetic circuits for which optimal depth reduction is possible this class also can be characterized using the auxpda model finally we characterize the languages generated by efficient circuits over the semiring 2 union concat in terms of simple one way machines and we investigate and extend earlier lower bounds on non commutative circuits

multiple imputation after 18 years abstract multiple imputation was designed to handle the problem of missing data in public use data bases where the data base constructor and the ultimate user are distinct entities the objective is valid frequency inference for ultimate users who in general have access only to complete data software and possess limited knowledge of specific reasons and models for nonresponse for this situation and objective i believe that multiple imputation by the data base constructor is the method of choice this article first provides a description of the assumed context and objectives and second reviews the multiple imputation framework and its standard results these preliminary discussions are especially important because some recent commentaries on multiple imputation have reflected either misunderstandings of the practical objectives of multiple imputation or misunderstandings of fundamental theoretical results then criticisms of multiple imputation are considered and finally comparisons are made to alt

correlated binary regression with covariates specific to each binary observation regression methods are considered for the analysis of correlated binary data when each binary observation may have its own covariates it is argued that binary response models that condition on some or all binary responses in a given block are useful for studying certain types of dependencies but not for the estimation of marginal response probabilities or pairwise correlations fully parametric approaches to these latter problems appear to be unduly complicated except in such special cases as the analysis of paired binary data hence a generalized estimating equation approach is advocated for inference on response probabilities and correlations illustrations involving both small and large block sizes are provided

orthogonality of estimating functions and nuisance parameters summary cox reid 1987 proposed the technique of orthogonalizing parameters to deal with the general problem of nuisance parameters within fully parametric models they obtained a large sample approximation to the conditional likelihood along the same lines davison 1988 studied generalized linear models in the present paper we deal with the problem of nuisance parameters within a semiparametric setup which includes the class of distributions associated with generalized linear models the technique used is that of optimum orthogonal estimating functions godambe thompson 1989 the results are related to those of cox reid 1987

log depth circuits for division and related problems we present optimal depth boolean circuits depth o log n for integer division powering and multiple products we also show that these three problems are of equivalent uniform depth and space complexity in addition we describe an algorithm for testing divisibility that is optimal for both depth and space

alternative paradigms for the analysis of imputed survey data abstract rubin has offered multiple imputation as a general approach to inference from survey data sets with missing values filled in through imputation in many situations the multiple imputation variance estimator is consistent in turn this observation has lent support to a number of complex applications in fact however the multiple imputation variance estimator is inconsistent under some simple conditions this article extends previous work of rao and shao and of fay directed toward consistent variance estimation under wider conditions extensions of rao and shao s results to fractionally weighted imputation combines the estimation efficiency of multiple imputation and the consistency of the rao shao variance estimator

analysis of longitudinal binary data from multiphase sampling the efficient use of surrogate or auxiliary information has been investigated within both model based and design based approaches to data analysis particularly in the context of missing data here we consider the use of such data in epidemiological studies of disease incidence in which surrogate measures of disease status are available for all subjects at two time points but definitive diagnoses are available only in stratified subsamples we briefly review methods for the analysis of two phase studies of disease prevalence at a single time point and we discuss the extension of four of these methods to the analysis of incidence studies their performance is compared with special reference to a study of the incidence of senile dementia

fractional hot deck imputation to compensate for item nonresponse hot deck imputation procedures replace missing values with values that occur in the sample fractional hot deck imputation replaces each missing observation with a set of imputed values and assigns a weight to each imputed value under the model in which observations in an imputation cell are independently and identically distributed fractional hot deck imputation is shown to be an effective imputation procedure a consistent replication variance estimation procedure for estimators computed with fractional imputation is suggested simulations show that fractional imputation and the suggested variance estimator are superior to multiple imputation estimators in general and much superior to multiple imputation for estimating the variance of a domain mean

missing data dial m for the question mark is common notation for the missing data that occur in most applied statistical analyses over the past century statisticians and other scientists not only have invented numerous methods for handling missing incomplete data but also have invented many forms of missing data including data augmentation hidden states latent variables potential outcome and auxiliary variables purposely constructing unobserved unobservable variables offers an extraordinarily flexible and powerful framework for both scientific modeling and computation and is one of the central statistical contributions to natural engineering and social sciences in parallel much research has been devoted to better understanding and modeling of real life missing data mechanisms that is the unintended data selection process that prevents us from observing our intended data this article is a very brief and personal tour of these developments and thus necessarily has much missing history and citations the tour consists of a number of m s starting with a historic story of the mysterious method of mckendrick for analyzing an epidemic study and its link to the em algorithm the most popular and powerful method of the twentieth century for fitting models involving missing data and latent variables the remaining m s touch on theoretical methodological and practical aspects of missingdata problems highlighted with some common applications in social computational biological medical and physical sciences

the generalized estimating equation approach when data are not missing completely at random abstract we propose two methods for handling missing data in generalized estimating equation gee analyses mean imputation and multiple imputation each provides valid gee estimates when data are missing at random missing outcomes are imputed sequentially starting from the outcome nearest in time to the observed outcome the estimators from the two kinds of imputation are compared with the weighting method of robins et al we show that multiple imputation with an infinite number of replications is asymptotically equivalent to mean imputation the methods are applied to a stroke study in which neurological outcomes are measured over time after stroke but some outcomes are missing due to death or loss to follow up

a modification of the redistribution to the right algorithm using disease markers summary disease markers are time dependent covariates which describe progression towards development of disease traditional methods in survival analysis do not make use of available data on these markers to recover additional information from censored individuals using a heuristic modification of the redistribution to the right algorithm efron 1967 a new approach for recovering information for censored individuals using disease markers is proposed additionally the statistical properties of the proposed method are examined there are two possible advantages to this modification i bias reduction when censoring is informative and ii an increase in efficiency in the case of truly noninformative censoring

application of robust estimating equations to the analysis of quantitative longitudinal data a model fit by general estimating equations gee has been used extensively for the analysis of longitudinal data in medical studies to some extent gee tries to minimize a quadratic form of the residuals and therefore is not robust in the sense that it like least squares estimates is sensitive to heavy tailed distributions contaminated distributions and extreme values this paper describes the family of truncated robust estimating equations and its properties for the analysis of quantitative longitudinal data like gee the robust estimating equations aim to assess the covariate effects in the generalized linear model in the complete population of observations but in a manner that is more robust to the influence of aberrant observations a simulation study has been conducted to compare the finite sample performance of gee and the robust estimating equations under a variety of error distributions and data structures it shows that the parameter estimates based on gee and the robust estimating equations are approximately unbiased and the type i errors of wald tests do not tend to be inflated gee is slightly more efficient with pure normal data but the efficiency of gee declines much more quickly than the robust estimating equations when the data become contaminated or have heavy tails which makes the robust estimating equations advantageous with non normal data both gee and the robust estimating equations are applied to a longitudinal analysis of renal function in the diabetes control and complications trial dcct for this application gee seems to be sensitive to the working correlation specification in that different working correlation structures may lead to different conclusions about the effect of intensive diabetes treatment on the other hand the robust estimating equations consistently conclude that the treatment effect is highly significant no matter which working correlation structure is used the dcct research group also demonstrated a significant effect using a mixed effects longitudinal model

a kernel method for incorporating information on disease progression in the analysis of survival summary this paper considers incorporating information on disease progression in the analysis of survival a three state model is assumed with the distribution of each transition estimated separately the distribution of survival following progression can depend on the time of progression kernel methods are used to give consistent estimators under general forms of dependence the estimators for the individual transitions are then combined into an overall estimator of the survival distribution a test statistic for equality of survival between treatment groups is proposed based on the tests of pepe fleming 1989 1991 in simulations the kernel method successfully incorporated dependence on the time of progression in some reasonable settings but under extreme forms of dependence the tests had substantial bias if survival beyond progression can be predicted fairly accurately then gains in power over standard methods that ignore progression can be substantial but the gains are smaller when survival beyond progression is more variable the methodology is illustrated with an application to a breast cancer clinical trial

an approach to the analysis of repeated measurements summary a linear model for repeated measurements is proposed in which the correlation structure within each time sequence of measurements includes parameters for measurement error variation between experimental units and serial correlation within units an approach to data analysis is presented which involves preliminary analysis by ordinary least squares use of the empirical semi variogram of residuals to suggest a suitable correlation structure and formal inference using likelihood based methods applications to two biological data sets are described

circuit definitions of nondeterministic complexity classes we consider restictions on boolean circuits and use them to obtain new uniform circuit characterizations of nondeterministic space and time classes we also obtain characterizations of counting classes based on nondeterministic time bounded computations on the arithmetic circuit model it is shown how the notion of semiunboundedness unifies the definitions of many natural complexity classes

multiple imputation approaches for the analysis of dichotomized responses in longitudinal studies with missing data often a binary variable is generated by dichotomizing an underlying continuous variable measured at a specific time point according to a prespecified threshold value in the event that the underlying continuous measurements are from a longitudinal study one can use the repeated measures model to impute missing data on responder status as a result of subject dropout and apply the logistic regression model on the observed or otherwise imputed responder status standard bayesian multiple imputation techniques rubin 1987 in multiple imputation for nonresponse in surveys that draw the parameters for the imputation model from the posterior distribution and construct the variance of parameter estimates for the analysis model as a combination of within and between imputation variances are found to be conservative the frequentist multiple imputation approach that fixes the parameters for the imputation model at the maximum likelihood estimates and construct the variance of parameter estimates for the analysis model using the results of robins and wang 2000 biometrika 87 113 124 is shown to be more efficient we propose to apply kenward and roger 1997 biometrics 53 983 997 degrees of freedom to account for the uncertainty associated with variance covariance parameter estimates for the repeated measures model

toward a curse of dimensionality appropriate coda asymptotic theory for semi parametric models we argue that due to the curse of dimensionality there are major difficulties with any pure or smoothed likelihood based method of inference in designed studies with randomly missing data when missingness depends on a high dimensional vector of variables we study in detail a semi parametric superpopulation version of continuously stratified random sampling we show that all estimators of the population mean that are uniformly consistent or that achieve an algebraic rate of convergence no matter how slow require the use of the selection randomization probabilities we argue that in contrast to likelihood methods which ignore these probabilities inverse selection probability weighted estimators continue to perform well achieving uniform n 1 2 rates of convergence we propose a curse of dimensionality appropriate coda asymptotic theory for inference in non and semi parametric models in an attempt to formalize our arguments we discuss whether our results constitute a fatal blow to the likelihood principle and study the attitude toward these that a committed subjective bayesian would adopt finally we apply our coda theory to analyse the effect of the curse of dimensionality in several interesting semi parametric models including a model for a two armed randomized trial with randomization probabilities depending on a vector of continuous pretreatment covariates x we provide substantive settings under which a subjective bayesian would ignore the randomization probabilities in analysing the trial data we then show that any statistician who ignores the randomization probabilities is unable to construct nominal 95 per cent confidence intervals for the true treatment effect that have both i an expected length which goes to zero with increasing sample size and ii a guaranteed expected actual coverage rate of at least 95 per cent over the ensemble of trials analysed by the statistician during his or her lifetime however we derive a new interval estimator depending on the randomization probabilities that satisfies i and ii

small sample degrees of freedom with multiple imputation an appealing feature of multiple imputation is the simplicity of the rules for combining the multiple complete data inferences into a final inference the repeated imputation inference rubin 1987 this inference is based on a t distribution and is derived from a bayesian paradigm under the assumption that the complete data degrees of freedom com are infinite but the number of imputations m is finite when com is small and there is only a modest proportion of missing data the calculated repeated imputation degrees of freedom m for the t reference distribution can be much larger than com which is clearly inappropriate following the bayesian paradigm we derive an adjusted degrees of freedom m with the following three properties for fixed m and estimated fraction of missing information m monotonically increases in com m is always less than or equal to com and m equals m when com is infinite a small simulation study demonstrates the superior frequentist performance when using m rather than m

robust estimation through estimating equations summary utilizing the theory of estimating equations godambe 1960 godambe thompson 1978 this paper develops concepts of parameter defining function and effective parameter the paper provides theory and techniques for choosing from a given set of robust parameters one that is most effective or one that can most efficiently be estimated the word parameter is used in statistics in two rather distinct contexts in the data analytic context a parameter characterizes some interesting aspect of the distribution such as its mean its median or some percentile in the context of modern statistical theory founded by k pearson r a fisher and others the parameter is related to and derives its meaning from a specific probabilistic model in this second context the model is of central importance and one seeks to complete it or perfect it by estimating its unknown parameter in the first context there is considerable flexibility and a parameter can typically be defined in many ways the choice of a parameter definition involves the following considerations a how much does it tell about the interesting or practically important part of the distribution b how insensitive or robust it is with respect to the uninteresting part of the distribution c how best can it be estimated the two contexts described above may of course overlap in the present paper we develop a theory which emphasizes the data analytic meaning of the parameter the approach on the other hand is more in line with the modern statistical theory mentioned above than that of most other works on robust estimation in particular asymptotic minimaxity plays no role the theory is designed to incorporate questions a b and c raised above it is based on the theory of estimating equations godambe 1960 godambe thompson 1978 the new concepts employed are those of parameter defining function and effectiveness of a parameter these are defined in 2 1 within

parametric fractional imputation for missing data analysis under a parametric model for missing data the em algorithm is a popular tool for flnding the maximum likelihood estimates mle of the parameters of the model imputation when carefully done can be used to facilitate the parameter estimation by applying the complete sample estimators to the imputed dataset the basic idea is to generate the imputed values from the conditional distribution of the missing data given the observed data multiple imputation is a bayesian approach to generate the imputed values from the conditional distribution in this article parametric fractional imputation is proposed as a parametric approach for generating imputed values using fractional weights the e step of the em algorithm can be approximated by the weighted mean of the imputed data likelihood where the fractional weights are computed from the current value of the parameter estimates some computational e ciency can be achieved using the idea of importance sampling in the monte carlo approximation of the conditional expectation the resulting estimator of the specifled parameters will be identical to the mle under missing data if the fractional weights are adjusted using a calibration step the proposed imputation method provides e cient parameter estimates for the model parameters specifled and also provides reasonable estimates for parameters that are not part of the imputation model for example domain means thus the proposed imputation method is a useful tool for general purpose data analysis variance estimation is covered and results from a limited simulation study are presented

large sample theory for parametric multiple imputation procedures we consider the asymptotic behaviour of various parametric multiple imputation procedures which include but are not restricted to the proper imputation procedures proposed by rubin 1978 the asymptotic variance structure of the resulting estimators is provided this result is used to compare the relative efficiencies of different imputation procedures it also provides a basis to understand the behaviour of two monte carlo iterative estimators stochastic em celeux diebolt 1985 wei tanner 1990 and simulated em ruud 1991 we further develop properties of these estimators when they stop at iteration k with imputation size m an application to a mcasurement error problem is used to illustrate the results

unbalanced repeated measures models with structured covariance matrices the question of how to analyze unbalanced or incomplete repeated measures data is a common problem facing analysts we address this problem through maximum likelihood analysis using a general linear model for expected responses and arbitrary structural models for the within subject covariances models that can be fit include standard univariate and multivariate models with incomplete data random effects models and models with time series and factor analytic error structures we describe newton raphson and fisher scoring algorithms for computing maximum likelihood estimates and generalized em algorithms for computing restricted and unrestricted maximum likelihood estimates an example fitting several models to a set of growth data is included

specification of covariance structure in longitudinal data analysis for randomized clinical trials misspecification of the covariance structure for repeated measurements in longitudinal analysis may lead to biased estimates of the regression parameters and under or overestimation of the corresponding standard errors in the presence of missing data the so called sandwich estimator can correct the variance but it does not reduce the bias in point estimates removing all assumptions from the covariance structure i e using an unstructured un covariance will remove such biases however an excessive amount of missing data may cause convergence problems for iterative algorithms such as the default newton raphson algorithm in the popular sas proc mixed this article examines both through theory and simulations the existence and the magnitude of these biases we recommend the use of un covariance as the default strategy for analyzing longitudinal data from randomized clinical trials with moderate to large number of subjects and small to moderate number of time points we also present an algorithm to assist in the convergence when the un covariance is used

modelling covariance structure in the analysis of repeated measures data the term repeated measures refers to data with multiple observations on the same sampling unit in most cases the multiple observations are taken over time but they could be over space it is usually plausible to assume that observations on the same unit are correlated hence statistical analysis of repeated measures data must address the issue of covariation between measures on the same unit until recently analysis techniques available in computer software only offered the user limited and inadequate choices one choice was to ignore covariance structure and make invalid assumptions another was to avoid the covariance structure issue by analysing transformed data or making adjustments to otherwise inadequate analyses ignoring covariance structure may result in erroneous inference and avoiding it may result in inefficient inference recently available mixed model methodology permits the covariance structure to be incorporated into the statistical model the mixed procedure of the sas r system provides a rich selection of covariance structures through the random and repeated statements modelling the covariance structure is a major hurdle in the use of proc mixed however once the covariance structure is modelled inference about fixed effects proceeds essentially as when using proc glm an example from the pharmaceutical industry is used to illustrate how to choose a covariance structure the example also illustrates the effects of choice of covariance structure on tests and estimates of fixed effects in many situations estimates of linear combinations are invariant with respect to covariance structure yet standard errors of the estimates may still depend on the covariance structure

analysis of semiparametric regression models for repeated outcomes in the presence of missing data abstract we propose a class of inverse probability of censoring weighted estimators for the parameters of models for the dependence of the mean of a vector of correlated response variables on a vector of explanatory variables in the presence of missing response data the proposed estimators do not require full specification of the likelihood they can be viewed as an extension of generalized estimating equations estimators that allow for the data to be missing at random but not missing completely at random these estimators can be used to correct for dependent censoring and nonrandom noncompliance in randomized clinical trials studying the effect of a treatment on the evolution over time of the mean of a response variable the likelihood based parametric g computation algorithm estimator may also be used to attempt to correct for dependent censoring and nonrandom noncompliance but because of possible model misspecification the parametric g computation algorithm estimator in contrast with the proposed w

runge kutta time discretizations of nonlinear dissipative evolution equations global error bounds are derived for runge kutta time discretizations of fully nonlinear evolution equations governed by m dissipative vector fields on hilbert spaces in contrast to earlier studies the analysis presented here is not based on linearization procedures but on the fully nonlinear framework of logarithmic lipschitz constants in order to extend the classical b convergence theory to infinite dimensional spaces an algebraically stable runge kutta method with stage order q is derived to have a global error which is at least of order q 1 or q depending on the monotonicity properties of the method

standardized regression coefficients and newly proposed estimators for r 2 r2 in multiply imputed data whenever statistical analyses are applied to multiply imputed datasets specific formulas are needed to combine the results into one overall analysis also called combination rules in the context of regression analysis combination rules for the unstandardized regression coefficients the t tests of the regression coefficients and the f tests for testing r 2 for significance have long been established however there is still no general agreement on how to combine the point estimators of r 2 in multiple regression applied to multiply imputed datasets additionally no combination rules for standardized regression coefficients and their confidence intervals seem to have been developed at all in the current article two sets of combination rules for the standardized regression coefficients and their confidence intervals are proposed and their statistical properties are discussed additionally two improved point estimators of r 2 in multiply imputed data are proposed which in their computation use the pooled standardized regression coefficients simulations show that the proposed pooled standardized coefficients produce only small bias and that their 95 confidence intervals produce coverage close to the theoretical 95 furthermore the simulations show that the newly proposed pooled estimates for r 2 are less biased than two earlier proposed pooled estimates

a fast improvement to the em algorithm on its own terms the em algorithm is a numerical technique for the evaluation of maximum likelihood estimates for parameters describing incomplete data models it is easy to apply in many problems and is stable but slow the algorithm fails to provide a consistent estimator of the standard errors of the maximum likelihood estimates unless the additional analysis required by the louis method is performed newton type or other gradient methods are faster and provide error estimates but tend to be unstable and require the analytical evaluation of likelihoods to derive expressions for the score function and at least approximations to the fisher information matrix the purpose of this paper is to expand on a result by fisher that permits a unification of em methodology and newton methods the evaluation of the individual observation by observation score functions of the incomplete data is a byproduct of the application of the e step of the em algorithm once these become available the fisher information matrix may be consistently estimated and the m step may be replaced by a fast newton type step

error estimates for a class of degenerate parabolic equations the degenerate parabolic equation u t delta beta u is considered in the cylinder omega times 0 t where omega subset r n is bounded beta 0 beta 0 0 and beta geq0 e g beta u u u m 1 m 1 a fully discrete scheme is given using c0 piecewise linear finite elements in space and a backward difference scheme in time l2 error estimates for the approximation of u and of beta u are proven for that purpose we employ the fact that the free boundary has a finite speed of propagation

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures