a family of iterative methods for computing the approximate inverse of a square matrix and inner inverse of a non square matrix based on a quadratical convergence method a family of iterative methods to compute the approximate inverse of square matrix are presented the theoretical proofs and numerical experiments show that these iterative methods are very effective and more importantly these methods can be used to compute the inner inverse and their convergence proofs are given by fundamental matrix tools

bayesian analysis of multivariate probit models with surrogate outcome data a new class of parametric models that generalize the multivariate probit model and the errors in variables model is developed to model and analyze ordinal data a general model structure is assumed to accommodate the information that is obtained via surrogate variables a hybrid gibbs sampler is developed to estimate the model parameters to obtain a rapidly converged algorithm the parameter expansion technique is applied to the correlation structure of the multivariate probit models the proposed model and method of analysis are demonstrated with real data examples and simulation studies

binary regression models for contaminated data on etudie la nature des observations aberrantes dans le contexte des donnees de regression binaires on propose un modele simple qui permet qu un petit nombre de reponses binaires soit mal enregistre

double sampling for exact values in some multivariate measurement error problems abstract increasing attention is being given to measurement error models in which the dimension of the proxy or surrogate values is different from that of the missing true values even if they are of the same dimension the error model may not be the simple additive one of observed true error where the error has mean 0 the use of broader models relating true and observed values requires the use of external or internal data containing some true values to calibrate validate the measurement error model this article considers a multivariate normal framework in which measurement error is allowed in any subset of the variables with a broad class of multivariate regression models relating true and observed values the classical additive model is a special case multiple regression with random regressors the structural case is treated within this framework correcting for measurement error is made possible through double sampling in which true values are obtained for a randomly chosen subset of the main

local central limit theorems the high order correlations of rejective sampling and logistic likelihood asymptotics let i 1 i n be independent but not necessarily identically distributed bernoulli random variables and let x n n j 1 i j for v in a bounded region a local central limit theorem expansion of p x n ex n v is developed to any given degree by conditioning this expansion provides information on the high order correlation structure of dependent weighted sampling schemes of a population e a special case of which is simple random sampling where a set d e is sampled with probability proportional to a d x a where x a are positive weights associated with individuals a e these results are used to determine the asymptotic information and demonstrate the consistency and asymptotic normality of the conditional and unconditional logistic likelihood estimator for unmatched case control study designs in which sets of controls of the same size are sampled with equal probability

bayesian sample size determination for inference on two binomial populations with no gold standard classifier we consider the impact of test properties on the required sample size for the bayesian design problem for comparing two proportions with error prone data specifically we examine four cases a single diagnostic test and two independent diagnostic tests both when the test properties are identical across populations and when they differ interval based and moment based sample size determination criteria are contrasted using monte carlo simulation methods we consider an application in which strongyloides infections are compared in two populations

conditional resampling for misclassified multinomial data with applications to sampling inspection a conditional resampling methodology is proposed for improving the estimators of multinomial classification probabilities in the presence of a fallible classifier the unbiased estimator requires knowledge of the misclassification error rates which are obtained by resampling from the initial sample and applying an infallible classifier tenenbein s double sampling methodology resamples randomly from the initial sample we show that substantial gains in efficiency are obtainable if instead of simple random resampling different resampling rates are allowed for the fallibly determined classes in the initial sample in most situations this resampling methodology is neither more difficult nor more expensive than the double sampling methodology a very promising area of application of the proposed methodology is sampling inspection in quality control where products can either be classified quickly by inspectors or by a more thorough perhaps destructive inspection

determining the value of additional surrogate exposure data for improving the estimate of an odds ratio we consider the design of both cohort and case control studies in which an initial stage 1 sample of complete data on an error free disease indicator d a correct gold standard dichotomous exposure measurement x and an error prone exposure measurement z are available we calculate the amount of additional information on the odds ratio relating d to x that one can obtain from a second stage 2 sample of measurements only on d and z if one allows for differential measurement error in z there is often little advantage in having more than four times as much data in stage 2 data as in stage 1 with the assumption that a non differential measurement error model is reasonable larger amounts of stage 2 data can be useful simulations indicate that stage 1 samples of modest size 50 cases in case control studies and 50 failures in cohort studies yield sufficiently reliable estimates of needed parameters to assist in determining an appropriate size for the stage 2 sample these ideas apply in settings either where the amount of stage 1 data is limited and fixed by external constraints or where one has gathered stage 1 data in advance to avoid collecting superfluous stage 2 data

inference in semiparametric regression models under partial questionnaire design and nonmonotone missing data in epidemiologic studies partial questionnaire design pqd can reduce cost time and other practical burdens associated with lengthy questionnaires by assigning different subsets of the questionnaire to different but overlapping subsets of the study participants in this article we describe methods for semiparametric inference for regression model under pqd and other study settings that can generate nonmonotone missing data in covariates in particular motivated from methods for multiphase designs we develop three estimators namely mean score pseudo likelihood and semiparametric maximum likelihood each of which has some unique advantages we develop the asymptotic theory and a sandwich variance estimator for each of the estimators under the underlying semiparametric model that allows the distribution of the covariates to remain nonparametric we study the finite sample performances and relative efficiencies of the methods using simulation studies we illustrate the methods using data from a ca

on the extension of householder s method for weighted moore penrose inverse this note discusses how to extend and use the well known method of householder 1964 6 for finding weighted moore penrose inverse the discussions cover both theoretical and numerical aspects an approach for accelerating the initial phase of convergence will be contributed some tests will also be employed to check the validity and superiority of the results over the schulz method

an improved newton iteration for the weighted moore penrose inverse abstract in this paper we showed that newton iteration may be used to compute the weighted moore penrose inverse of an arbitrary matrix we also take advantage of several new acceleration procedures to reduce the cost of newton iteration which extends the earlier work r schreiber computing generalized inverses and eigenvalues of symmetric matrices using systolic arrays in r glowinski j l lious eds computing methods in applied science and engineering north holland amsterdam 1984 t soderstorm g w stewant on the numerical properties of an iterative method for computing the moore penrose generalized inverse siam j numer anal 11 1974 61 74

testing hypotheses with binary data subject to misclassification errors analysis and experimental design summary consider a binary random variable having outcomes subject to misclassification errors where the probabilities of misclassification errors depend on outcome the asymptotic relative efficiencies are derived for comparing two binomial populations comparing stratified binomial populations and for paired comparisons the probabilities of misclassification can vary with strata or be different for each pair the criterion for optimum experimental designs is shown to be the ratio of the asymptotic relative efficiency to expected cost per observation this criterion implies that mixtures of different experiments can never be optimal a number of experimental strategies is discussed

efficient estimation in multi phase case control studies in this paper we discuss the analysis of multi phase or multi stage case control studies and present an efficient semiparametric maximum likelihood approach that unifies and extends earlier work including the seminal case control paper by prentice pyke 1979 work by breslow cain 1988 scott wild 1991 breslow holubkov 1997 and others the theoretical derivations apply to arbitrary binary regression models but we present results for logistic regression and show that the approach can be implemented by including additional intercept terms in the logistic model and then making some simple corrections to the score and information equations used in a newton raphson or fisher scoring maximization of the prospective loglikelihood

optimal designs of two stage studies for estimation of sensitivity specificity and positive predictive value the cost efficiency of estimation of sensitivity specificity and positive predictive value from two stage sampling designs is considered assuming a relatively cheap test classifies first stage subjects into several categories and an expensive gold standard is applied at stage two simple variance formulae are derived and used to find optimal designs for a given cost ratio the utility of two stage designs is measured by the reduction in variances compared with one stage simple random designs separate second stage design is also compared with proportional allocation pa the maximum percentage reductions in variance from two stage designs for sensitivity specificity and positive predictive value estimation are p per cent 1 p per cent and w respectively where p is the population prevalence of disease and w the population percentage of test negatives the optimum allocation of stage two resources is not obvious the optimum proportion of true cases at stage two may even be less than under pa pa is near optimal for sensitivity estimation in most cases when prevalence is low but inefficient compared with the optimal scheme for specificity

a geometrical approach on generalized inverses by neumann type series the convergence of the neumann type series to 1 2 inverses has been shown by k tanabe linear algebra appl 10 1975 163 in this paper these results indicating conditions characterizing the convergence of this series to different generalized inverses are extended in addition these results for obtaining different generalized inverses from the hyperpower method are applied finally generalized involutory matrices are introduced and characterized using the obtained results

anamorphic analysis sampling and estimation for covariate effects when both exposure and disease are known when study subjects are obtained through sampling procedures based on the level of known characteristics covariate data can be used to adjust effect estimates this principle which is well understood for both exposure based sampling cohort studies and disease based sampling case control studies can also be applied to cross classifications of the source population an opportunity for exposure disease based sampling arises whenever data on exposure and disease are both available on a group of subjects who are incompletely characterized in terms of other covariates this is a common intermediate stage in epidemiologic studies of many kinds

a characterization and representation of the generalized inverse a 2 t s and its applications abstract this paper presents an explicit expression for the generalized inverse a 2 t s based on this we established the characterization the representation theorem and the limiting process for a 2 t s as an application we estimate the error bound of the iterative method for approximating a 2 t s

the representation and approximation for the generalized inverse at s 2 we present a unified representation theorem for the generalized inverses a t s 2 which are 2 or outer inverses having the prescribed range t and null space s specific expressions and computational procedures for the generalized inverses a t s 2 can be uniformly derived

semiparametric methods for response selective and missing data problems in regression summary suppose that data are generated according to the model f ylx 0 g x where y is a response and x are covariates we derive and compare semiparametric likelihood and pseudolikelihood methods for estimating 0 for situations in which units generated are not fully observed and in which it is impossible or undesirable to model the covariate distribution the probability that a unit is fully observed may depend on y and there may be a subset of covariates which is observed only for a subsample of individuals our key assumptions are that the probability that a unit has missing data depends only on which of a finite number of strata that y x belongs to and that the stratum membership is observed for every unit applications include case control studies in epidemiology field reliability studies and broad classes of missing data and measurement error problems our results make fully efficient estimation of 0 feasible and they generalize and provide insight into a variety of methods that have been proposed for specific problems

efficient analysis of case control studies with sample weights analysis of population based case control studies with complex sampling designs is challenging because the sample selection probabilities and therefore the sample weights depend on the response variable and covariates commonly the design consistent weighted estimators of the parameters of the population regression model are obtained by solving sample weighted estimating equations weighted estimators however are known to be inefficient when the weights are highly variable as is typical for case control designs in this paper we propose two alternative estimators that have higher efficiency and smaller finite sample bias compared with the weighted estimator both methods incorporate the information included in the sample weights by modeling the sample expectation of the weights conditional on design variables we discuss benefits and limitations of each of the two proposed estimators emphasizing efficiency and robustness we compare the finite sample properties of the two new estimators and traditionally used weighted estimators with the use of simulated data under various sampling scenarios we apply the methods to the u s kidney cancer case control study to identify risk factors published 2012 this article is a us government work and is in the public domain in the usa

from zhang neural network to scaled hyperpower iterations abstract a class of scaled hyperpower iterative methods for computing outer inverses is considered this class appears during the construction of the discrete time zhang neural network for computing the usual matrix inverse the usual hyperpower iterative methods belong to this class additionally a more general class of scaled iterative methods which includes the scaled hyperpower method is defined and studied different values of the real scaling parameter are investigated both theoretically and numerically

biometrika centenary sample surveys summary despite karl pearson s stated aim that the evolutionist has to become in the widest sense of the words a registrar general for all forms of life the papers published in biometrika on sample surveys have been mainly theoretical and do not start to appear until the 1940s there has been little work on sampling in the life sciences with the exception of capture recapture methods the major areas have been in variable probability designs the foundations of inference for finite populations analytic surveys and the role of selection mechanisms in inference in recent years the work on survey analysis has been the outcome of an interaction between sample selection mechanisms and mainstream theory with interesting consequences for both areas

estimation and inference for logistic regression with covariate misclassification and measurement error in main study validation study designs abstract in epidemiological studies continuous covariates often are measured with error and categorical covariates often are misclassified using the logistic regression model to represent the relationship between the binary outcome and the perfectly measured and classified covariates the model for the observed main study data is derived this derivation relies on the assumption that the error in the continuous covariates is multivariate normally distributed and uses a chain of logistic regression models to describe the misclassification processes these model assumptions are empirically verified in the validation study where the misclassified and mismeasured covariates are validated using perfectly measured and classified data the full data likelihood including contributions from both the main study and the validation study is maximized to obtain the maximum likelihood estimates for the parameters of the underlying logistic regression model and of the measurement error model and reclassification mo

a mean score method for missing and auxiliary covariate data in regression models summary we consider regression analysis when incomplete or auxiliary covariate data are available for all study subjects and in addition for a subset called the validation sample true covariate data of interest have been ascertained the term auxiliary data refers to data not in the regression model but thought to be informative about the true missing covariate data of interest we discuss a method which is nonparametric with respect to the association between available and missing data allows missingness to depend on available response and covariate values and is applicable to both cohort and case control study designs the method previously proposed by flanders greenland 1991 and by zhao lipsitz 1992 is generalised and asymptotic theory is derived our expression for the asymptotic variance of the estimator provides intuition regarding performance of the method optimal sampling strategies for the validation set are also suggested by the asymptotic results

the fisher information matrix for log linear models arguing conditionally on observed explanatory variable summary for poisson or multinomial contingency table data the conditional distribution is product multinomial when conditioning on observed values of explanatory variables birch 1963 showed that under the restriction formed by keeping the marginal totals of one margin fixed at their observed values the poisson multinomial and product multinomial likelihoods are proportional and give the same estimates for common parameters in the log linear model here the inverses of the fisher information matrices are shown to be identical over common parameters so that the asymptotic covariance matrices of the estimates correspond

prospective analysis of logistic case control studies abstract in a classical case control study prentice and pyke proposed to ignore the study design and instead base estimation and inference on a random sampling i e prospective formulation we generalize this prospective formulation of case control studies to include multiplicative models stratification missing data measurement error robustness and other examples the resulting estimators which ignore the case control study aspect and instead are based on a random sampling formulation are typically consistent for nonintercept parameters and are asymptotically normally distributed we derive the resulting asymptotic covariance matrix of the parameter estimates the covariance matrix obtained by ignoring the case control sampling scheme and using prospective formulas instead is shown to be at worst asymptotically conservative and asymptotically correct in a variety of problems a simple sufficient condition guaranteeing the latter is obtained

two improvements of the iterative method for computing moore penrose inverse based on penrose equations abstract two improvements of the iterative method for computing the moore penrose inverse introduced in petkovic and stanimirovic 2011 are introduced the first improvement defines new choice for the initial approximation and ensures better stability of the method the second one defines its extension to the set of outer inverses with prescribed range and null space numerical examples for both of these improvements are presented

cost efficient study designs for binary response data with gaussian covariate measurement error summary when mismeasurement of the exposure variable is anticipated epidemiologic cohort studies may be augmented to include a validation study where a small sample of data relating the imperfect exposure measurement method to the better method is collected optimal study designs i e least expensive subject to specified power constraints are developed that give the overall sample size and proportion of the overall sample size allocated to the validation study if better exposure measurements can be collected on a sample of subjects an optimal design can be suggested that conforms to realistic budgetary constraints the properties of three designs those that include an internal validation study those where the validated subsample is derived from subjects external to the primary investigation and those that use the better method of exposure assessment on all subjects are compared the proportion of overall study resources allocated to the validation substudy increases with increasing sample disease frequency decreasing unit cost of the superior exposure measurement relative to the imperfect one increasing unit cost of outcome ascertainment increasing distance between two alternative values of the relative risk between which the study is designed to discriminate and increasing magnitude of hypothesized values this proportion also depends in a nonlinear fashion on the severity of measurement error and when the validation study is internal measurement error reaches a point after which the optimal design is the smaller fully validated one

semiparametric inference in matched case control studies with missing covariate data summary we consider the problem of matched studies with a binary outcome that are analysed using conditional logistic regression and for which data on some covariates are missing for some study participants methods for this problem involve either modelling the distribution of missing covariates or modelling the probability of data being missing for this second approach the previously proposed method did not make use of data for those persons with missing covariate data except in the model for the missingness we propose a new class of estimators that use outcome and available covariate data for all study participants and show that a particular member of this class always has better efficiency than the previously proposed estimator we illustrate the efficiency gains that are possible with our approach using simulated data

full rank representation of generalized inverse at s 2 and its application in this paper we introduce a full rank representation of the generalized inverse a t s 2 of a given complex matrix a which is based on an arbitrary full rank decomposition of g where g is a matrix such that r g t and n g s using this representation we introduce the minor of the generalized inverse a t s 2 as a special case of the minor a determinantal representation of the generalized inverse a t s 2 is obtained as an application we use an example to demonstrate that this representation is correct

generalized meta analysis for multiple regression models across studies with disparate covariate information meta analysis is widely popular for synthesizing information on common parameters of interest across multiple studies because of its logistical convenience and statistical efficiency we develop a generalized meta analysis approach to combining information on multivariate regression parameters across multiple studies that have varying levels of covariate information using algebraic relationships among regression parameters in different dimensions we specify a set of moment equations for estimating parameters of a maximal model through information available from sets of parameter estimates for a series of reduced models from the different studies the specification of the equations requires a reference dataset for estimating the joint distribution of the covariates we propose to solve these equations using the generalized method of moments approach with the optimal weighting of the equations taking into account uncertainty associated with estimates of the parameters of the reduced models we describe extensions of the iterated reweighted least squares algorithm for fitting generalized linear regression models using the proposed framework based on the same moment equations we also develop a diagnostic test for detecting violations of underlying model assumptions such as those arising from heterogeneity in the underlying study populations the proposed methods are illustrated with extensive simulation studies and a real data example involving the development of a breast cancer risk prediction model using disparate risk factor information from multiple studies

successive matrix squaring algorithm for computing outer inverses abstract in this paper we derive a successive matrix squaring sms algorithm to approximate an outer generalized inverse with prescribed range and null space of a given matrix a c r m n we generalize the results from the papers l chen e v krishnamurthy i macleod generalized matrix inversion and rank computation by successive matrix powering parallel computing 20 1994 297 311 y wei successive matrix squaring algorithm for computing drazin inverse appl math comput 108 2000 67 75 y wei h wu j wei successive matrix squaring algorithm for parallel computing the weighted generalized inverse a mn appl math comput 116 2000 289 296 and obtain an algorithm for computing various classes of outer generalized inverses of a instead of particular matrices used in these articles we use an appropriate matrix r c s n m s r numerical examples are presented

survival curve estimation with partial non random exposure information the objective of this paper is to estimate survival curves for two different exposure groups when the exposure group is not known for all observations and the data is subject to left truncation and right censoring the situation we consider is when the probability that the exposure group is missing may depend on whether the observation is censored or uncensored in which case the exposure is not missing at random the problem was motivated by a study of alzheimer s disease to estimate the distribution of ages at diagnosis for individuals with and without an apolipoprotein e4 allele the exposure group genotyping for this risk factor was incomplete and performed more frequently on the cases of alzheimer s disease the uncensored observations than the censored observations the survival curves are estimated in discrete time using an em algorithm a bootstrapping procedure is proposed that guarantees each bootstrap sample has the same proportion of observations with missing exposure a simulation is performed to evaluate the bias of the estimators and to investigate design and efficiency issues the methods are applied to the alzheimer s disease study

designing a logistic regression study using surrogate measures for exposure and outcome summary suppose that a binary or polytomous outcome variable is related to a possibly continuous exposure variable through a logistic regression model and that prior to sample selection subjects can be characterized according to surrogates for outcome and exposure the surrogate variables are assumed to satisfy certain conditional independence properties commonly used in errors in variables models this paper considers the problem of using the surrogate variables to select subjects for measurement of the true values with the goal of minimizing the variance of an estimate of the log odds ratio we describe three possible sampling plans and show that the log odds ratio parameter can be estimated in each plan using ordinary logistic regression

locally optimal design for comparing two probabilities from binomial data subject to misclassification optimal experimental designs are found for estimating the difference 1 2 of two population probabilities using measuring instruments possibly subject to known error rates the available experiments considered are to measure for i 1 or 2 the outcome of a trial on the population i without error or with error or with error with the possibility of checking either positive or negative responses given that the cost per observation depends on which experiment is carried out an asymptotically locally optimal allocation is derived in terms of results of elfving which provide a simple geometrical solution

a family of iterative methods for computing moore penrose inverse of a matrix abstract this paper improves on generalized properties of a family of iterative methods to compute the approximate inverses of square matrices originally proposed in 1 and while the methods of 1 can be used to compute the inner inverses of any matrix it has not been proved that these sequences converge in norm to a fixed inner inverse of the matrix in this paper it is proved that the sequences indeed are convergent to a fixed inner inverse of the matrix which is the moore penrose inverse of the matrix the convergence proof of these sequences is given by fundamental matrix calculus and numerical experiments show that the third order iterations are as good as the second order iterations

design of validation studies for estimating the odds ratio of exposure disease relationships when exposure is misclassified summary we compared several validation study designs for estimating the odds ratio of disease with misclassified exposure we assumed that the outcome and misclassified binary covariate are available and that the error free binary covariate is measured in a subsample the validation sample we considered designs in which the total size of the validation sample is fixed and the probability of selection into the validation sample may depend on outcome and misclassified covariate values design comparisons were conducted for rare and common disease scenarios where the optimal design is the one that minimizes the variance of the maximum likelihood estimator of the true log odds ratio relating the outcome to the exposure of interest misclassification rates were assumed to be independent of the outcome we used a sensitivity analysis to assess the effect of misspecifying the misclassification rates under the scenarios considered our results suggested that a balanced design which allocates equal numbers of validation subjects into each of the four outcome mismeasured covariate categories is preferable for its simplicity and good performance a user friendly fortran program is available from the second author which calculates the optimal sampling fractions for all designs considered and the efficiencies of these designs relative to the optimal hybrid design for any scenario of interest

likelihood methods for missing covariate data in highly stratified studies summary the paper considers canonical link generalized linear models with stratum specific nuisance intercepts and missing covariate data this family includes the conditional logistic regression model existing methods for this problem each of which uses a conditioning argu ment to eliminate the nuisance intercept model either the missing covariate data or the missingness process the paper compares these methods under a common likelihood framework the semiparametric efficient estimator is identified and a new estimator which reduces dependence on the model for the missing covariate is proposed a simulation study compares the methods with respect to efficiency and robustness to model misspecification

matrix methods for estimating odds ratios with misclassified exposure data extensions and comparisons misclassification of exposure variables is a common problem in epidemiologic studies this paper compares the matrix method barron 1977 biometrics 33 414 418 greenland 1988a statistics in medicine 7 745 757 and the inverse matrix method marshall 1990 journal of clinical epidemiology 43 941 947 to the maximum likelihood estimator mle that corrects the odds ratio for bias due to a misclassified binary covariate under the assumption of differential misclassification the inverse matrix method is always more efficient than the matrix method however the efficiency depends strongly on the values of the sensitivity specificity baseline probability of exposure the odds ratio case control ratio and validation sampling fraction in a study on sudden infant death syndrome sids an estimate of the asymptotic relative efficiency are of the inverse matrix estimate was 0 99 while the matrix method s are was 0 19 under nondifferential misclassification neither the matrix nor the inverse matrix estimator is uniformly more efficient than the other the efficiencies again depend on the underlying parameters in the sids data the mle was more efficient than the matrix method are 0 39 in a study investigating the effect of vitamin a intake on the incidence of breast cancer the mle was more efficient than the matrix method are 0 75

two phase sampling for simultaneous prevalence estimation and case detection two phase designs for estimation of prevalence where the first phase classification is fallible and the second is accurate but relatively expensive are not necessarily justified on efficiency grounds however they might be advantageous for dual purpose studies for example where prevalence estimation is followed by a clinical trial or case control study if they can identify cases of disease for the second study in a cost effective way alternatively they may be justified on ethical grounds if they can identify more previously undetected but treatable cases of disease than a simple random sample design an approach to sampling is proposed which formally combines the goals of efficient prevalence estimation and case detection by setting different notional study costs for investigating cases and noncases two variants of the method are compared with an ethical two phase scheme proposed by shrout and newman 1989 biometrics 45 549 555 and with the most efficient scheme for prevalence estimation alone in terms of the standard error of the prevalence estimate the expected number of cases and the fraction of cases among second phase subjects given a fixed budget one variant yields the highest fraction and expected number of cases but also the largest standard errors the other yields a higher fraction than shrout and newman s scheme and a similar number of cases but appears to do so more efficiently

nonparametric regression with missing outcomes using weighted kernel estimating equations we consider nonparametric regression of a scalar outcome on a covariate when the outcome is missing at random mar given the covariate and other observed auxiliary variables we propose a class of augmented inverse probability weighted aipw kernel estimating equations for nonparametric regression under mar we show that aipw kernel estimators are consistent when the probability that the outcome is observed that is the selection probability is either known by design or estimated under a correctly specified model in addition we show that a specific aipw kernel estimator in our class that employs the fitted values from a model for the conditional mean of the outcome given covariates and auxiliaries is double robust that is it remains consistent if this model is correctly specified even if the selection probabilities are modeled or specified incorrectly furthermore when both models happen to be right this double robust estimator attains the smallest possible asymptotic variance of all aipw kernel

harish chandra s method of descent 1 one of the main tools in the study of representation theory of reductive groups is harish chandra s method of descent let g be a real reductive group given a semisimple element s of g let h be the centralizer of s in g then the behavior of an invariant eigendistribution in a neighborhood of s may be described in terms of an invariant distribution of h in a neighborhood of the identity this leads to the notion of descent taking an invariant eigendistribution of g defined on a neighborhood of s to an invariant distribution of h defined in a neighborhood of the identity in h the descent of any standard character of

logistic regression for stratified case control studies fears and brown 1986 developed a procedure for logistic regression analysis of stratified case control data where the sampling fractions for cases and controls and thus their population frequencies were assumed known they fitted the usual prospective model to the case control data treating case control status as a binary outcome variable in order to adjust for the biased sampling they included the logarithm of the odds ratio relating the actual sample sizes and the population frequencies in each stratum as an offset in the regression equation however no adjustments were made to the estimated variances of the regression coefficients of variables associated with the strata to account for the information about them available in the population frequencies furthermore fears and brown incorrectly claimed that their procedure gave restricted maximum likelihood rml estimates aitchison and silvey 1958 based on the likelihood of the retrospectively sampled data breslow and cain 1988 show that the fears and brown procedure does yield consistent and asymptotically normal estimates of the regression parameters in a logistic regression model for the probability of disease development in fact it is equivalent to the conditional maximum likelihood cml estimate developed by manski and mcfadden 1981 for estimation of quantal response functions from stratified data see also hsieh manski and mcfadden 1985 breslow and cain extended the work of manski and mcfadden for use in the more realistic situation where the distribution of cases and controls in each stratum is estimated from a first stage sample rather than being assumed known they developed variance estimators for the regression coefficients that accurately reflect the additional information available in the first stage sample and that are easily modified to accommodate an infinite population at that stage we first present a reanalysis of the fears and brown data that contrasts the correct variances computed under the assumptions that the first stage sample is finite and infinite respectively with the incorrect variances obtained from the standard logistic analysis then using a subset of the data with only three strata we demonstrate the differences between the cml estimates of manski and mcfadden and rml estimates calculated from the retrospective probabilities a small scale simulation study investigates the properties of cml and rml estimators in samples of moderate size

precision of double sampling estimators for comparing two probabilities summary it is assumed that both error prone and error free observations can be made on a binary response the second type being more expensive in double sampling error prone observations are made on all sample units and for a random subsample error free observations are made as well tenenbein 1970 studied allocation of units in double sampling for estimating a single probability we consider double sampling for comparing two probabilities for different cost ratios for error prone versus error free observations and for fixed total cost we assess how the large sample variance of the log odds ratio estimator varies with the proportion of units subject to both error prone and error free observation we determine the optimal double sampling proportion and the possible gain from using double sampling versus single sampling in which only error free observations are made

higher order convergent iterative method for computing the generalized inverse and its application to toeplitz matrices abstract the main aim of this paper is to provide a higher order convergent iterative method in order to calculate the generalized inverse of a given matrix we extend the iterative method proposed in li et al w g li z li a family of iterative methods for computing the approximate inverse of a square matrix and inner inverse of a non square matrix applied mathematics and computation 215 2010 3433 3442 to compute the 2 inverse generalized inverse a t s 2 2 3 inverse and 2 4 inverse moreover we modify the iterative method to compute the generalized inverse a t s 2 of toeplitz matrices by using the displacement theory

iterative method for computing the moore penrose inverse based on penrose equations an iterative algorithm for estimating the moore penrose generalized inverse is developed the main motive for the construction of the algorithm is simultaneous usage of penrose equations 2 and 4 convergence properties of the introduced method as well as their first order and second order error terms are considered numerical experiment is also presented

log linear models for doubly sampled categorical data fitted by the em algorithm abstract double sampling experiments can be expressed as incomplete multiway contingency tables and analyzed by using techniques appropriate for fitting log linear models the framework described can be applied to experiments with any number of factors and measurement devices maximum likelihood estimation via the em algorithm leads to straightforward expressions for covariances of estimates of the parameters and functions of the parameters

logistic regression for two stage case control data summary samples of diseased cases and nondiseased controls are drawn at random from the population at risk after classification according to the exposure of interest subsamples of cases and controls are selected for purposes of covariable ascertainment a modification of the usual logistic regression analysis yields consistent estimates of covariable adjusted relative risks and their standard errors by balancing the numbers of exposed and nonexposed for whom covariable inforniation is ascertained within case and control samples some efficiency may be gained over the usual single stage design particularly when the exposure is rare and the relative risks associated with the covariables are large the procedure may be useful also when covariable information is missing for a large part of the sample

logistic analysis in case control studies under validation sampling summary this paper proposes an analysis of case control data under a double sampling scheme when covariates are missing or measured with error at the first stage of sampling and are validated at the second stage in a subsample the method combines risk information from both samples it is derived under the assumptions that i the prospective disease incidence model is of logistic form ii the proxy or partial information takes on finitely many values and iii the error is nondifferential the method of prentice pyke 1979 is extended to a two sample design to derive an estimating equation for the odds ratio parameters it provides an alternative estimator to that given by breslow cain 1988 consistency and asymptotic normality of the estimates are derived and a variance formula is presented parameters can be estimated by use of standard packages for quantalresponse data the method can easily be extended to the analysis of stratified designs with large strata

novel two phase sampling designs for studying binary outcomes in biomedical cohort studies for assessing the association between an outcome variable and a set of covariates usually some covariates can only be measured on a subgroup of study subjects an important design question is which subjects to select into the subgroup to increase statistical efficiency when the outcome is binary one may adopt a case control sampling design or a balanced case control design where cases and controls are further matched on a small number of complete discrete covariates while the latter achieves success in estimating odds ratio or parameters for the matching covariates similar two phase design options have not been explored for the remaining covariates especially the incompletely collected ones this is of great importance in studies where the covariates of interest cannot be completely collected to this end assuming that an external model is available to relate the outcome and complete covariates we propose a novel sampling scheme that oversamples cases and controls with worse goodness of fit based on the external model and further matches them on complete covariates similarly to the balanced design we develop a pseudolikelihood method for estimating or parameters through simulation studies and explorations in a real cohort study we find that our design generally leads to reduced asymptotic variances of the or estimates and the reduction for the matching covariates is comparable to that of the balanced design

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

fitting logistic regression models in stratified case control studies summary methods are developed for fitting logistic models to data in which cases and or controls are sampled from the available cases and controls within population strata particular attention is paid to models in which stratum differences are modelled as well as the effects of the different risk factors experienced by individuals within strata maximum likelihood estimation is developed for discrete explanatory variables and is compared with the method of fears and brown 1986 biometrics 42 955 960 and breslow and cain 1988 biometrika 75 11 20 in which the prospective logistic model is fitted with a fixed offset an example is explored in which maximum likelihood estimation proves to be substantially more efficient than the fears brown method and the modelling of the stratum effects leads to much more efficient estimates of regression coefficients for the remaining variables

a bayesian approach to prospective binary outcome studies with misclassification in a binary risk factor misclassification in a binary exposure variable within an unmatched prospective study may lead to a biased estimate of the disease exposure relationship it usually gives falsely small credible intervals because uncertainty in the recorded exposure is not taken into account when there are several other perfectly measured covariates interrelationships may introduce further potential for bias bayesian methods are proposed for analysing binary outcome studies in which an exposure variable is sometimes misclassified but its correct values have been validated for a random subsample of the subjects this bayesian approach can model relationships between explanatory variables and between exploratory variables and the probabilities of misclassification three logistic regressions are used to relate disease to true exposure misclassified exposure to true exposure and true exposure to other covariates credible intervals may be used to make decisions about whether certain parameters are unnecessary and hence whether the model can be reduced in complexity in the disease exposure model for parameters representing coefficients related to perfectly measured covariates the precision of posterior estimates is only slightly lower than would be found from data with no misclassification for the risk factor which has misclassification the estimates of model coefficients obtained are much less biased than those with misclassification ignored

a class of numerical algorithms for computing outer inverses the aim of this study is to present a class of numerical algorithms for finding outer inverses with prescribed range and null space general convergence theorems are proven some particular cases of the general algorithm are introduced and demonstrated theoretical convergence speed and computational efficiency will further be supported by employing some numerical tests