nonparametric variance estimation under fine stratification an alternative to collapsed strata abstractfine stratification is commonly used to control the distribution of a sample from a finite population and to improve the precision of resulting estimators one per stratum designs represent the finest possible stratification and occur in practice but designs with very low numbers of elements per stratum say two or three are also common the classical variance estimator in this context is the collapsed stratum estimator which relies on creating larger pseudo strata and computing the sum of the squared differences between estimated stratum totals across the pseudo strata we propose here a nonparametric alternative that replaces the pseudo strata by kernel weighted stratum neighborhoods and uses deviations from a fitted mean function to estimate the variance we establish the asymptotic behavior of the kernel based estimator and show its superior practical performance relative to the collapsed stratum variance estimator in a simulation study an application to data from the u s consumer expe

sample correlation coefficients based on survey data under regression imputation regression imputation is commonly used to compensate for item nonresponse when auxiliary data are available it is common practice to compute survey estimators by treating imputed values as observed data and using the standard unbiased or nearly unbiased estimation formulas designed for the case of no nonresponse although the commonly used regression imputation method preserves unbiasedness for population marginal totals i e survey estimators computed from imputed data are still nearly unbiased it does not preserve unbiasedness for population correlation coefficients a joint regression imputation method is proposed that preserves unbiasedness for marginal totals second moments and correlation coefficients some simulation results show that the joint regression imputation method produces not only sample correlation coefficients that are nearly unbiased but also estimates that are more stable than those produced by marginal nonrandom regression imputation when correlation coefficients are in a c

algorithms for constructing combined strata variance estimators a jackknife or balanced repeated replication variance estimator in a large survey typically requires a large number of replicates and replicate weights reducing the number of replicates may have important advantages for computations and for limiting the risk of data disclosure from public use data files this article proposes algorithms adapted from scheduling theory to combine variance strata and thus reduce the number of replicates the algorithms are simple and efficient and can be adapted to easily account for vector characteristics and analytic domains an important concern with combining strata is that the resulting variance estimators may be inconsistent we establish conditions for the consistency of the combined variance estimator and show that the proposed algorithms ensure they are met we also derive bounds on the degrees of freedom that the algorithms will assure the algorithms are applied both to a real sample survey and to samples from simulated populations and the algorithms perform v

implementation and applications of bootstrap methods for the national immunization survey in complex probability sample surveys numerous adjustments are customarily made to the survey weights to reduce potential bias in survey estimates these adjustments include sampling design sd weight adjustments which account for features of the sampling plan and non sampling design nsd weight adjustments which account for non sampling errors and other effects variance estimates prepared from complex survey data customarily account for sd weight adjustments but rarely account for all nsd weight adjustments as a result variance estimates may be biased and standard confidence intervals may not achieve their nominal coverage levels we describe the implementation of the bootstrap method to account for the sd and nsd weight adjustments for complex survey data using data from the national immunization survey nis we illustrate the use of the bootstrap i for evaluating the use of standard confidence intervals that use taylor series approximations to variance estimators that do not account for nsd weight adjustments ii for obtaining confidence intervals for ranks estimated from weighted survey data and iii for evaluating the predictive power of logistic regressions using receiver operating characteristic curve analyses that account for the sd and nsd adjustments made to the survey weights copyright 2003 john wiley sons ltd

asymptotic properties of the balanced repeated replication method for sample quantiles inference including variance estimation can be made from stratified samples by selecting a balanced set of subsamples this balanced subsampling method is generically called the balanced repeated replication method in survey data analysis which includes mccarthy s balanced half samples method and its extensions for more general stratified designs we establish the asymptotic consistency of the balanced repeated replication variance estimators when the parameter of interest is the population quantile the consistency results also hold when balanced subsampling is replaced by random subsampling as a key technical prerequisite we prove a bahadurtype representation for sample quantiles in stratified random sampling

jackknife variance estimation with survey data under hot deck imputation summary hot deck imputation is commonly employed for item nonresponse in sample surveys it is also a common practice to treat the imputed values as if they are true values and then compute the variance estimates using standard formulae this procedure however could lead to serious underestimation of the true variance when the proportion of missing values for an item is appreciable we propose a jackknife variance estimator for stratified multistage surveys which is obtained by first adjusting the imputed values for each pseudo replicate and then applying the standard jackknife formula the proposed jackknife variance estimator is shown to be consistent as the sample size increases assuming equal response probabilities within imputation classes and using a particular hot deck imputation

inference from stratified samples properties of the linearization jackknife and balanced repeated replication methods the asymptotic normality of both linear and nonlinear statistics and the consistency of the variance estimators obtained using the linearization jackknife and balanced repeated replication brr methods in stratified samples are established the results are obtained as l oa within the context of a sequence of finite populations jil with l strata in ji l and are valid for any stratified multistage design in which the primary sampling units psu s are selected with replacement and in which independent subsamples are taken within those psu s selected more than once in addition some exact analytical results on the bias and stability of these alternative variance estimators in the case of ratio estimation are obtained for small l under a general linear regression model

temporal order discrimination of tonal sequences by younger and older adults the role of duration and rate the effect of tone duration and presentation rate on the discrimination of the temporal order of the middle two tones of a four tone sequence was investigated in young normal hearing ynh and older hearing impaired ohi listeners the frequencies and presentation level of the tone sequences were selected to minimize the effect of hearing loss on the performance of the ohi listeners tone durations varied from 20to400ms and presentation rates from 2 5to25tones s two experiments were conducted with anisochronous nonuniform duration and rate across entire sequence and isochronous uniform rate and duration sequences respectively for the ynh listeners performance for both isochronous and anisochronous sequences was determined primarily by presentation rate such that performance decreased at rates faster than 5tones s for anisochronous tone sequences alone the effects of rate were more pronounced at short tone durations for the ohi listeners both presentation rate and tone duration had an impact o

the analysis of categorical data from complex sample surveys chi squared tests for goodness of fit and independence in two way tables abstract the effect of stratification and clustering on the asymptotic distributions of standard pearson chi squared test statistics for goodness of fit simple hypothesis and independence in a two way contingency table denoted as x 2 and xi 2 respectively is investigated it is shown that both x 2 and xi 2 are asymptotically distributed as weighted sums of independent 1 2 random variables the weights are then related to the familiar design effects deffs used by survey samplers a simple correction to x 2 which requires only the knowledge of variance estimates or deffs for individual cells in the goodness of fit problem is proposed and empirical results on the performance of corrected x 2 provided empirical work on xi 2 indicated that the distortion of nominal significance level is substantially smaller with xi 2 than with x 2 some results under simple models for clustering are also given

testing hardy weinberg equilibrium and homogeneity of hardy weinberg disequilibrium using complex survey data summary for studies on population genetics the use of representative random samples of the target population can avoid ascertainment bias genetic variation data from over a hundred genes were collected in a u s nationally representative sample in the third national health and nutrition examination survey nhanes iii surveys such as the nhanes have complex stratified multistage cluster sample designs with sample weighting that can inflate variances and alter the expectations of test statistics thus classical statistical tests of hardy weinberg equilibrium hwe and homogeneity of hw disequilibrium hhwd for simple random samples are not suitable for data from complex samples we propose using wald tests for hwe and generalized score tests for hhwd that have been modified for complex samples monte carlo simulation studies are used to investigate the finite sample properties of the proposed tests rao scott corrections applied to the tests were found to improve their type i error properties our methods are applied to the nhanes iii genetic data for three loci involved in metabolizing lead in the body

jackknife variance estimation under imputation for estimators using poststratification information abstract poststratified estimators are commonly used in sample surveys to improve the efficiency of estimators and to ensure calibration to known poststrata counts similarly generalized regression estimators are used to handle two or more poststratifiers with known marginal counts in addition weighting adjustment within weighting classes is used to handle unit nonresponse and imputation within imputation classes is used to handle item nonresponse for the full response case asymptotic consistency of the jackknife variance estimator under stratified multistage sampling is established using mild regularity conditions on residuals similar to those of scott and wu for ratio and regression estimation under simple random sampling a jackknife linearization variance estimator obtained by linearizing the jackknife variance estimator is also given for unit nonresponse the general case of poststrata cutting across weighting classes is considered and a jackknife variance estimator and the corresponding

empirical likelihood confidence intervals for complex sampling designs we define an empirical likelihood approach which gives consistent design based confidence intervals which can be calculated without the need of variance estimates design effects resampling joint inclusion probabilities and linearization even when the point estimator is not linear it can be used to construct confidence intervals for a large class of sampling designs and estimators which are solutions of estimating equations it can be used for means regressions coefficients quantiles totals or counts even when the population size is unknown it can be used with large sampling fractions and naturally includes calibration constraints it can be viewed as an extension of the empirical likelihood approach to complex survey data this approach is computationally simpler than the pseudoempirical likelihood and the bootstrap approaches the simulation study shows that the confidence interval proposed may give better coverages than the confidence intervals based on linearization bootstrap and pseudoempirical likelihood our simulation study shows that under complex sampling designs standard confidence intervals based on normality may have poor coverages because point estimators may not follow a normal sampling distribution and their variance estimators may be biased

replication variance estimation for two phase stratified sampling in two phase sampling the second phase sample is often a stratified sample based on the information observed in the first phase sample for the total of a population characteristic either the double expansion estimator or the reweighted expansion estimator can be used given a consistent first phase replication variance estimator we propose a consistent variance estimator that is applicable to both the double expansion estimator and the reweighted expansion estimator the proposed method can be extended to multiphase sampling

hypothesis testing with complex survey data the use of classical quadratic test statistics with particular reference to regression problems abstract sample surveys often have complex sample designs with multistage cluster sampling stratification and differential selection probabilities this article is concerned with testing the null hypothesis h 0 where the p dimensional parameter g and is a q dimensional vector of means the asymptotic framework that consists of a sequence of increasing finite populations is used to define as the limit of finite population means as part of the inference we use replicated estimates of variances that take into account the complex sample design the wald statistic can be used to test h 0 but inference for based on the wald statistic can have low power thus an alternative to using a wald test is pursued in this article first define a classical quadratic test statistic that would be used if one had a simple random sample of the population second treating this quadratic form as a population parameter use design based methods to estimate it from the observed survey data last use

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

inference from stratified samples second order analysis of three methods for nonlinear statistics abstract for stratified samples and nonlinear statistics that can be expressed as functions of estimated totals second order asymptotic expansions of the linearization jackknife and balanced repeated replication variance estimators are obtained based on these comparisons are made in terms of their biases some higher order asymptotic equivalence results are also established the special case of a combined ratio estimator is investigated in detail some results on bias reduction achieved by the jack knife and balanced repeated replication estimators of a nonlinear function of totals are also given

a resampling procedure for complex survey data abstract in complex survey data often the sampling design induces a non iid structure to the data e g without replacement sampling stratification multistage or unequal probability of selection though techniques for variance estimation and confidence intervals do exist they often are cumbersome to implement or do not extend to complex designs it would be desirable to have resampling methods that reuse the existing estimation system repeatedly using computing power to avoid theoretical work and that can be applied to such data in recognition of this need various resampling procedures for variance estimation and confidence intervals in sample survey data where the sampling is without replacement have been proposed in the literature these include the jackknife the with replacement bootstrap bwr the without replacement bootstrap bwo and the rescaling bootstrap the bwr and bwo are applicable only to simple sampling designs others have shown the asymptotic consistency of jackknife vari

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

functional central limit theorems for single stage sampling designs for a joint model based and design based inference we establish functional central limit theorems for the horvitz thompson empirical process and the h ajek empirical process centered by their finite population mean as well as by their super population mean in a survey sampling framework the results apply to single stage unequal probability sampling designs and essentially only require conditions on higher order correlations we apply our main results to a hadamard differentiable statistical functional and illustrate its limit behavior by means of a computer simulation

nonparametric model calibration estimation in survey sampling calibration is commonly used in survey sampling to include auxiliary information at the estimation stage of a population parameter calibrating the observation weights on population means totals of a set of auxiliary variables implies building weights that when applied to the auxiliaries give exactly their population mean total implicitly calibration techniques rely on a linear relation between the survey variable and the auxiliary variables however when auxiliary information is available for all units in the population more complex modeling can be handled by means of model calibration auxiliary variables are used to obtain fitted values of the survey variable for all units in the population and estimation weights are sought to satisfy calibration constraints on the fitted values population mean rather than on the auxiliary variables one in this work we extend model calibration considering more general superpopulation models and use nonparametric methods to obtain the fitted values on which t

a jackknife variance estimator for unistage stratified samples with unequal probabilities summary existing jackknife variance estimators used with sample surveys can seriously overestimate the true variance under unistage stratified sampling without replacement with unequal probabilities a novel jackknife variance estimator is proposed which is as numerically simple as existing jackknife variance estimators under certain regularity conditions the proposed variance estimator is consistent under stratified sampling without replacement with unequal probabilities the high entropy regularity condition necessary for consistency is shown to hold for the rao sampford design an empirical study of three unequal probability sampling designs supports our findings

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

jackknifing in the presence of inhomogeneity under the classical assumption that data are a random sample from a distribution with cumulative distribution function f the jackknife generally yields bias reduction an asymptotically pivotal statistic and a variance estimator for an estimator of an unknown parameter f in this article the classical assumption is relaxed to allow for inhomogeneous subpopulations the jackknife is been to account for these inhomogeneities automatically and so is valid in a class of problems much larger than that for which it was originally intended data from experiments to determine the acceleration of gravity at washington d c are analyzed a family of weighted mean estimators is considered and recommendations are made regarding which estimators yield both valid and efficient jackknife based inferences

resampling inference with complex survey data abstract methods for standard errors and confidence intervals for nonlinear statistics such as ratios regression and correlation coefficients have been extensively studied for stratified multistage designs in which the clusters are sampled with replacement in particular the important special case of two sampled clusters per stratum these methods include the customary linearization or taylor method and resampling methods based on the jackknife and balanced repeated replication brr unlike the jackknife or the brr the linearization method is applicable to general sampling designs but it involves a separate variance formula for each nonlinear statistic thereby requiring additional programming efforts both the jackknife and the brr use a single variance formula for all nonlinear statistics but they are more computing intensive the resampling methods developed here retain these features of the jackknife and the brr yet permit extension to more complex designs involving sampling without replace

design effects of two stage sampling abstract in sampe surveys the design effect of a statistic is usually defined as the ratio of its true variance under the given sample design to its variance had the sample been obtained by simple random sampling empirical work suggests certain patterns for design effects of different types of statistics under different designs but theoretical work explaining these patterns is limited this paper obtains general theoretical results on the sructure of design effects for a broad class of statistical inference under a two stage sampling design in particular it discusses the relation between design effects of multivariate and of univariate statistics this relation is of practical interest because it is of relevance to the imputation of standard errors for multivariate statistics such as correlation coefficients or regression coefficients using design effects of univariate statistics the latter quantities are often routinely derived on completion of the survey the former may be difficult to compute by standard procedures either because of the absence of the necessary design information or because of software or degrees of freedom limitations

balanced repeated replication for stratified multistage survey data under imputation abstract balanced repeated replication brr is a popular method for variance estimation in surveys the standard brr method works by first creating a set of balanced pseudoreplicated datasets from the original dataset for a survey estimator the brr variance estimator is the average of squared deviations r where r is the same as but based on the data in the rth pseudoreplicated dataset only but when there are a large number of imputed missing values nonrespondents treating the imputed values as observed data and applying the standard brr variance estimation formula does not produce valid variance estimators intuitively the variation due to imputation can be captured by the brr method if every pseudoreplicated dataset is imputed in exactly the same way as the original dataset is imputed assuming that the dataset contains flags for nonrespondents but when a random imputation method such as random hot deck imputation random ratio imputation or random regression imputation is u

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

poststratification and conditional variance estimation poststratification estimation is a technique used in sample surveys to improve efficiency of estimators survey weights are adjusted to force the estimated numbers of units in each of a set of estimation cells to be equal to known population totals the resulting weights are then used in forming estimates of means or totals of variables collected in the survey for example in a household survey the estimation cells may be based on age race sex categories of individuals and the known totals may come from the most recent population census although the variance of a poststratified estimator can be computed over all possible sample configurations inferences made conditionally on the achieved sample configuration are desirable theory and a simulation study using data from the u s current population survey are presented to study both the conditional bias and variance of the poststratified estimator of a total the linearization balanced repeated replication and jackknife variance estimators are also exa

adjusted jackknife for imputation under unequal probability sampling without replacement imputation is commonly used to compensate for item non response in sample surveys if we treat the imputed values as if they are true values and then compute the variance estimates by using standard methods such as the jackknife we can seriously underestimate the true variances we propose a modified jackknife variance estimator which is defined for any without replacement unequal probability sampling design in the presence of imputation and non negligible sampling fraction mean ratio and random imputation methods will be considered the practical advantage of the method proposed is its breadth of applicability

on the complexity of mod 2l sum pla s consideration is given to the realization of logic functions by using plas with an exclusive or exor array where a function is represented by mod 2 exor sum of products esops and both true and complemented variables are used the authors propose a new pla structure using an exor array they derive upper bounds on the number of products of this type of pla that are useful for estimating the size of a pla as well as for assessing the minimality of the solutions obtained by heuristic esop minimization algorithms computer simulation using randomly generated functions shows that plas with the exor array require on the average fewer products than conventional plas for symmetric functions the authors conjecture that the plas with an exor array require at most as many products as the conventional plas the proposed plas can be made easily testable by adding a small amount of hardware

modified balanced repeated replication for complex survey data summary balanced repeated replication commonly used to estimate the variances of nonlinear estimators from stratified sampling designs can run into problems in estimating a ratio of small domain totals or in the case of poststratification or unit nonresponse adjustment involving ratio weighting within cells if some cell sizes are small this problem is mainly caused by a sharp perturbation of the weights to construct replicate estimates to avoid extreme replicate estimates robert fay proposed a modified balanced repeated replication method using gentler perturbation of weights theoretical properties of the modified method are studied here the method is extended to the case of imputation for item nonresponse both smooth and nonsmooth estimators are studied

bootstrap methods for finite populations abstract we show that the familiar bootstrap plug in rule of efron has a natural analog in finite population settings in our method a characteristic of the population is estimated by the average value of the characteristic over a class of empirical populations constructed from the sample our method extends that of gross to situations in which the stratum sizes are not integer multiples of their respective sample sizes moreover we show that our method can be used to generate second order correct confidence intervals for smooth functions of population means a property that has not been established for other resampling methods suggested in the literature a second resampling method is proposed that also leads to second order correct confidence intervals and is less computationally intensive than our bootstrap but a simulation study reveals that the second method can be quite unstable in some situations whereas our bootstrap performs very well

gap detection and the precedence effect in young and old adults thresholds for detecting a gap between two gaussian enveloped standard deviation 0 5 ms 2 khz tones were determined in young and old listeners the gap detection thresholds of old adults were more variable and about twice as large as those obtained from young adults moreover gap detection thresholds were not correlated with audiometric thresholds in either group estimates of the width of the temporal window of young subjects based on the detection of a gap between two tone pips were smaller than those typically obtained when a relatively long duration pure tone is interrupted moore et al j acoust soc am 85 1266 1275 1989 because the amount of time it takes to recover from an adapting stimulus is likely to affect gap detection thresholds glasberg et al j acoust soc am 81 1546 1556 1987 smaller estimates of temporal window size would be expected in this paradigm if the amount of adaptation produced by the first tone pip was negligible the larger gap detection thresholds of o

effects of age and sequence presentation rate on temporal order recognition the experiments examined the ability of younger and older listeners to identify the temporal order of sounds presented in tonal sequences the stimuli were three tone sequences that spanned two octave frequency range and listeners identified random permutations of tone order using labels of relative pitch some of the sequences featured uniform timing characteristics and the sequence duty cycle was varied across conditions to examine the relative influence of tonal durations and intertone interval on recognition performance across a range of sequence presentation rates other stimulus sequences featured nonuniform timing with unequal tone durations and intertone intervals the listeners were groups of younger and older persons with or without hearing loss results indicated that temporal order recognition was influenced primarily by sequence presentation rate independent of tonal duration tonal interval spacing or sequence timing characteristics the performance of older listeners was poorer than younger listeners but the age related recognition differences were independent of sequence presentation rate there were no consistent effects of hearing loss on temporal ordering performance

some prediction properties of balanced half sample variance estimators in single stage sampling on introduit un modele de prediction et une classe d estimateurs qui inclut plusieurs estimateurs du ratio de la regression et autres on donne une approximation a l estimateur bhs balanced half sample

l statistics in complex survey problems we study linear combinations of order statistics l statistics in survey problems with a stratified multistage sampling design two general types of l statistics are considered smooth l statistics with weights generated by a smooth function and nonsmooth l statistics sample quantiles the trimmed sample mean the decile mean and variance the sample lorenz curve and the sample gini s family parameters are examples of smooth lstatistics or functions of smooth l statistics used in survey problems it is shown that under weak conditions the smooth l statistics are asymptotically normal and their asymptotic variances can be consistently estimated by jackknifing for the sample quantiles their asymptotic normality requires more conditions on the finite population distribution functions consistent estimators for the asymptotic variances of the sample quantiles are derived asymptotic validity of the woodruff s confidence intervals for population quantiles is also proved

variance estimation for survey data with composite imputation and nonnegligible sampling fractions abstract this article considers variance estimation for horvitz thompson type estimated totals based on survey data with imputed non respondents and with nonnegligible sampling fractions a method based on a variance decomposition is proposed our method can be applied to complicated situations where a composite of some deterministic and or random imputation methods is used including using imputed data in subsequent imputations although here linearization or taylor expansion type techniques are adopted replication methods such as the jackknife balanced repeated replication and random groups can also be used in applying our method to derive variance estimators using our method variance estimators can be derived under either the customary design based approach or the model assisted approach and are asymptotically unbiased and consistent the transportation annual survey conducted at the u s census bureau in which nonrespondents are imputed using a composite of cold deck and ratio type imputation m

confidence intervals for weighted proportions we investigate methods for the construction of confidence intervals for a proportion in a stratified two stage sampling design with few events occurring in a small number of large unequal size strata the critical aspect is the incorporation of the weighting scheme into the construction of a single overall confidence interval with small numbers of events the binomial based methods may be inadequate since the normal approximation is not valid computer simulations compare coverage probability and bias for five methods of obtaining confidence intervals for proportions by combining 1 binomial variances 2 confidence intervals based on the f distribution approximation to the cumulative binomial 3 the binomial variance method with exact confidence limits when a zero prevalence occurs in any stratum 4 confidence intervals based on the f distribution using a rescaling factor and 5 the binomial variance method with exact confidence limits using a rescaling factor the method that performs best in terms of coverage probability is the combination of stratum specific confidence intervals based on the f distribution using a rescaling factor the methods involving the binomial variance tend to be negatively biased and the methods based on the f distribution tend to be positively biased application of these methods with data from a study of adolescent depression that employs a stratified two stage sampling design is consistent with these results

a jackknife variance estimator for unequal probability sampling the jackknife method is often used for variance estimation in sample surveys but has only been developed for a limited class of sampling designs we propose a jackknife variance estimator which is defined for any without replacement unequal probability sampling design we demonstrate design consistency of this estimator for a broad class of point estimators a monte carlo study shows how the proposed estimator may improve on existing estimators