the existence of subspace wavelet sets let h be a reducing subspace of l2 rd that is a closed subspace of l2 rd with the property that f amt l h for any f h m z and l zd where a is a d d expansive matrix it is known that h is a reducing subspace if and only if there exists a measurable subset m of rd such that atm m and f h l2 rd m under some given conditions of m it is known that there exist a dilation subspace wavelet sets iwith respect to h in this paper we prove that this holds in general

estimating and comparing the distribution of onset of disease from prevalence case control data age at ascertainment from prevalence case control data identifies the age specific odds of disease when age at onset is available from the cases the conditional distribution of age at onset given that disease occurs is identifiable combining both kinds of information by introducing a multiplicative intercept allows identification of the marginal distribution of age at onset here the approach is extended to the two sample setting through a generalization of the multiplicative intercept model the efficiency of the approach is explored and a test statistic based on the integrated difference between distribution function estimates is proposed an approach to regularization of the likelihood is discussed the methods are illustrated through an application to data on colorectal polyps obtained from a case control study of individuals undergoing colonoscopy

on case control sampling of clustered data summary we derive the relationship between the population model and the sample model in the case of case control sampling of clustered responses the derived formulae provide insight into why methods that ignore the sampling scheme fail to provide consistent estimators in this case the differences between clustered and independent responses in the context of the case control design are discussed the case control design is known for its efficiency at a fixed sample size when the disease under study is rare it is also potentially useful in clustered data set ups where the interest is in covariate effects or in dependence within clusters neuhaus jewell 1990 showed that sampling clusters instead of individuals can bias the estimation of population parameters if standard logistic regression techniques are used even if the individuals within a cluster are independent they investigated the estimation problem under marginal conditional and random effect models for marginal models they presented expressions for the distribution of disease in the sampled observations in the case of within cluster independence we restrict attention to marginal also called population averaged models zeger liang albert 1988 for these models we extend the results of neuhaus jewell 1990 by presenting expressions for the sample mean and dependence structure in the general case allowing dependence among members of a cluster we envisage a population consisting of a large number of clusters observations within a cluster are dependent but clusters are independent of each other let yij be a binary random variable denoting the jth response in the ith cluster xij a p x 1 covariate vector i 1 2 k j 1 2 ni furthermore let y and xi be the response vector and covariate matrix for the ith cluster respectively suppose that the response is coded so that yj 1 indicates the presence of disease and yij 0 its absence we consider designs in which clusters rather than individual observations are the sampling units specifically let si 1 and si 0 indicate whether the ith cluster was sampled or not respectively the sampling probabilities are defined as oi pr si 1i1 y xi we consider only the case in which oi depends on yi only through t n i 1 and xi neuhaus jewell 1990 refer to this case as sampling with probability dependent on the number of responses in that case oi g t7 xi ni for a known function g we shall write gi ti for brevity a special case is one in which clusters with one or more affected members are sampled with probability rc and clusters with no affected member are sampled with probability co

wavelet frames for not necessarily reducing affine subspaces ii the structure of affine subspaces abstract this is a continuation of the investigation into the theory of wavelet frames for general affine subspaces the main focus of this paper is on the structural properties of affine subspaces we show that every affine subspace is the orthogonal direct sum of at most three purely non reducing subspaces while every reducing subspace with respect to the dilation and translation operators is the orthogonal direct sum of two purely non reducing ones this result is obtained through considering the basic question as to when the orthogonal complement of an affine subspace in another one is still affine motivated by the fundamental question as to whether every affine subspace is singly generated and by a recent result that every singly generated purely non reducing subspace admits a singly generated wavelet frame we prove that every affine subspace can be decomposed into the direct sum of a singly generated affine subspace and some space of small size as a consequence we establish a connection between the above mentioned two questions

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

the structure of shift invariant subspaces of l2 rn using the range function approach to shift invariant spaces in l2 rn we give a simple characterization of frames and riesz families generated by shifts of a countable set of generators in terms of their behavior on subspaces of l2 zn this in turn gives a simplified approach to the analysis of frames and riesz families done by gramians and dual gramians we prove a decomposition of a shift invariant space into the orthogonal sum of spaces each of which is generated by a quasi orthogonal generator as an application of this fact we characterize shift preserving operators in terms of range operators and prove some facts about the dimension function

on chi squared tests for multiway contingency tables with cell proportions estimated from survey data the impact of survey design on standard multinomial based methods for a multiway contingency table is studied under nested loglinear models the asymptotic null distribution of the pearson chi squared test statistic x 2 or the likelihood ratio test statistic g 2 is obtained as a weighted sum of independent chi 2 1 random variables and the weights are then related to the familiar design effects deffs used by survey samplers a simple correction to x 2 or g 2 is also obtained which requires the knowledge of only the cell deffs and the deffs for collapsed tables marginals whenever the model admits a direct solution of likelihood equations under multinomial sampling finally an example on the relative performance of x 2 and some corrected x 2 statistics in a three way table is given using some data from the canada health survey 1978 1979

weighted likelihood estimation under two phase sampling we develop asymptotic theory for weighted likelihood esti mators wle under two phase stratied sampling without replacement we also consider several variants of wle s involving estimated weights and calibration a set of empirical process tools are developed including a glivenko cantelli theorem a theorem for rates of convergence of z estimators and a donsker theorem for the inverse probability weighted em pirical processes under two phase sampling and sampling without replace ment at the second phase using these general results we derive asymptotic distributions of the wle of a nite dimensional parameter in a general semiparametric model where an estimator of a nuisance parameter is es timable either at regular or non regular rates we illustrate these results and methods in the cox model with right censoring and interval censor ing we compare the methods via their asymptotic variances under both sampling without replacement and the more usual and easier to analyze assumption of bernoulli sampling at the second phase ams 2000 subject classications primary 62e20 secondary 62g20 62d99 62n01

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

hypothesis testing in case control studies summary prentice pyke 1979 have shown that valid estimators of the odds ratio parameters in a logistic regression model may be obtained from case control data by fitting the model as if the data had been obtained in a prospective study the resulting estimates of standard errors are also valid asymptotically we extend the prentice pyke result to likelihood ratio tests obtained from fitting a prospective model these tests are asymptotically exact if the hypothesis involves only the odds ratio parameters and are asymptotically conservative if stratum effects are involved

the structure of finitely generated shift invariant spaces in l2 rd a simple characterization is given of finitely generated subspaces of l2 rd which are invariant under translation by any multi integer and is used to give conditions under which such a space has a particularly nice generating set namely a basis and more than that a basis with desirable properties such as stability orthogonality or linear independence the last property makes sense only for local spaces i e shift invariant spaces generated by finitely many compactly supported functions and special attention is paid to such spaces as an application we prove that the approximation order provided by a given local space is already provided by the shift invariant space generated by just one function with this function constructible as a finite linear combination of the finite generating set for the whole space hence compactly supported this settles a question of some 20 years standing

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

fitting logistic models under case control or choice based sampling on compare deux approches dans le cas important de l ajustement des modeles de regression logistique

model checking techniques for stratified case control studies we present graphical and numerical methods for assessing the adequacy of the logistic regression model for stratified case control data the proposed methods are derived from the cumulative sum of residuals over the covariate or linear predictor under the assumed model the cumulative residual process converges weakly to a zero mean gaussian process whose distribution can be approximated via monte carlo simulation the observed cumulative residual pattern can then be compared both visually and analytically to a number of simulated realizations from the approximate null distribution these comparisons enable one to examine the functional form of each covariate the logistic link function as well as the overall model adequacy simulation studies demonstrate that the proposed methods perform well in practical settings illustration with an oesophageal cancer study is provided

standard errors for attributable risk for simple and complex sample designs adjusted attributable risk ar is the proportion of diseased individuals in a population that is due to an exposure we consider estimates of adjusted ar based on odds ratios from logistic regression to adjust for confounding influence function methods used in survey sampling are applied to obtain simple and easily programmable expressions for estimating the variance of ar these variance estimators can be applied to data from case control cross sectional and cohort studies with or without frequency or individual matching and for sample designs with subject samples that range from simple random samples to sample weighted multistage stratified cluster samples like those used in national household surveys the variance estimation of ar is illustrated with i a weighted stratified multistage clustered cross sectional study of childhood asthma from the third national health and examination survey nhanes iii and ii a frequency matched case control study of melanoma skin cancer

on the robustness of weighted methods for fitting models to case control data summary we compare the robustness under model misspecification of two approaches to fitting logistic regression models with unmatched case control data one is the standard survey approach based on weighted versions of population estimating equations the other is the likelihood based approach that is standard in medical applications the conventional view is that the less efficient survey weighted approach leads to greater robustness we conclude that this view is not always justified

analysis of clustered binary outcomes using within cluster paired resampling summary conditional logistic regression clr is useful for analyzing clustered binary outcome data when interest lies in estimating a cluster specific exposure parameter while treating the dependency arising from random cluster effects as a nuisance clr aggregates unmeasured cluster specific factors into a cluster specific baseline risk and is invalid in the presence of unmodeled heterogeneous covariate effects or within cluster dependency we propose an alternative resampling based method for analyzing clustered binary outcome data within cluster paired resampling wcpr which allows for within cluster dependency not solely due to baseline heterogeneity for example dependency may be in part caused by heterogeneity in response to an exposure across clusters due to unmeasured cofactors when both clr and wcpr are valid our simulations suggest that the two methods perform comparably when clr is invalid wcpr continues to have good operating characteristics for illustration we apply both wcpr and clr to a periodontal data set where there is heterogeneity in response to exposure across clusters

penalized maximum likelihood estimation in logistic regression and discrimination summary maximum likelihood estimation of the parameters of the binary logistic regression model for pr h i x is discussed with separate discussion of sampling from i the conditional distribution of h given x ii the joint distribution of h and x and iii the conditional distribution of x given h difficulties associated with continuous x in the latter sampling scheme are discussed to avoid these penalized maximum likelihood estimates are introduced which give estimates of the logistic parameters and a nonparametric spline estimate of the marginal distribution of x extensions to multinomial logistic regression are outlined

local uncertainty sampling for large scale multiclass logistic regression a major challenge for building statistical models in the big data era is that the available data volume far exceeds the computational capability a common approach for solving this problem is to employ a subsampled dataset that can be handled by available computational resources we propose a general subsampling scheme for large scale multiclass logistic regression and examine the variance of the resulting estimator we show that asymptotically the proposed method always achieves a smaller variance than that of the uniform random sampling moreover when the classes are conditionally imbalanced significant improvement over uniform sampling can be achieved empirical performance of the proposed method is evaluated and compared to other methods via both simulated and real world datasets and these results match and confirm our theoretical analysis

a note on the prospective analysis of outcome dependent samples summary two likelihood representations corresponding to the prospective and retrospective analyses of the case control design are derived for general outcome dependent samples with arbitrary discrete or continuous outcomes and possibly non multiplicative models parameter identification in the general outcome dependent design is reduced to the simple problem of parameter identification in the general odds ratio function both likelihoods are shown to generate the same profile likelihood for the common parameter of interest maximum like lihood estimators based on either likelihood are semiparametric efficient for the identifiable parameters

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