Multilevel modelling of complex survey data

Summary.  Multilevel modelling is sometimes used for data from complex surveys involving multistage sampling, unequal sampling probabilities and stratification. We consider generalized linear mixed models and particularly the case of dichotomous responses. A pseudolikelihood approach for accommodating inverse probability weights in multilevel models with an arbitrary number of levels is implemented by using adaptive quadrature. A sandwich estimator is used to obtain standard errors that account for stratification and clustering. When level 1 weights are used that vary between elementary units in clusters, the scaling of the weights becomes important. We point out that not only variance components but also regression coefficients can be severely biased when the response is dichotomous. The pseudolikelihood methodology is applied to complex survey data on reading proficiency from the American sample of the 'Program for international student assessment' 2000 study, using the Stata program gllamm which can estimate a wide range of multilevel and latent variable models. Performance of pseudo-maximum-likelihood with different methods for handling level 1 weights is investigated in a Monte Carlo experiment. Pseudo-maximum-likelihood estimators of (conditional) regression coefficients perform well for large cluster sizes but are biased for small cluster sizes. In contrast, estimators of marginal effects perform well in both situations. We conclude that caution must be exercised in pseudo-maximum-likelihood estimation for small cluster sizes when level 1 weights are used.     

Conditionally Efficient Estimation of Long-Run Relationships Using Mixed-Frequency Time Series

I analyze efficient estimation of a cointegrating vector when the regressand and regressor are observed at different frequencies. Previous authors have examined the effects of specific temporal aggregation or sampling schemes, finding conventionally efficient techniques to be efficient only when both the regressand and the regressors are average sampled. Using an alternative method for analyzing aggregation under more general weighting schemes, I derive an efficiency bound that is conditional on the type of aggregation used on the low-frequency series and differs from the unconditional bound defined by the full-information high-frequency data-generating process, which is infeasible due to aggregation of at least one series. I modify a conventional estimator, canonical cointegrating regression (CCR), to accommodate cases in which the aggregation weights are known. The correlation structure may be utilized to offset the potential information loss from aggregation, resulting in a conditionally efficient estimator. In the case of unknown weights, the correlation structure of the error term generally confounds identification of conditionally efficient weights. Efficiency is illustrated using a simulation study and an application to estimating a gasoline demand equation.     

Extending permutation conditional inference to unconditional ones

In this presentation we discuss the extension of permutation conditional inferences to unconditional or population ones. Within the parametric approach this extension is possible when the data set is randomly selected by well-designed sampling procedures on well-defined population distributions, provided that their nuisance parameters have boundely complete statistics in the null hypothesis or are provided with invariant statistics. When these conditions fail, especially if selection-bias procedures are used for data collection processes, in general most of the parametric inferential extensions are wrong or misleading. We will see that, since they are provided with similarity and conditional unbiasedness properties and if correctly applicable, permutation tests may extend, at least in a weak sense, conditional to unconditional inferences.     

CONDITIONING VS. STANDARDIZATION FOR CONTRASTS ON ERRORS ATTRACTED TO STABLE LAWS

In multiple regression and other settings one encounters the problem of estimating sampling distributions for contrast operations applied to i.i.d. errors. Permutation bootstrap applied to least squares residuals has been proven to consistently estimate conditionalsampling distributions of contrasts, conditional upon order statistics of errors, even for long-tailed error distributions. How does this compare with the unconditional sampling distribution of the contrast when standardizing by the sample s.d. of the errors (or the residuals)? For errors belonging to the domain of attraction of a normal we present a limit theorem proving that these distributions are far closer to one another than they are to the limiting standard normal distribution. For errors attracted to α-stable laws with α ≤ 2 we construct random variables possessing these conditional and unconditional sampling distributions and develop a Poisson representation for their a.s. limit correlation ρ_α. We prove that ρ₂= 1, ρ_α→ 1 for α → 0 + or 2 ?, and ρ_α< 1 a.s. for α < 2.     

A conditional perspective of weighted variance estimation of the optimal regression estimator

The estimation of the variance for the GREG (general regression) estimator by weighted residuals is widely accepted as a method which yields estimators with good conditional properties. Since the optimal (regression) estimator shares the properties of GREG estimators which are used in the construction of weighted variance estimators, we introduce the weighting procedure also for estimating the variance of the optimal estimator. This method of variance estimation was originally presented in a seemingly ad hoc manner, and we shall discuss it from a conditional point of view and also look at an alternative way of utilizing the weights. Examples that stress conditional behaviour of estimators are then given for elementary sampling designs such as simple random sampling, stratified simple random sampling and Poisson sampling, where for the latter design we have conducted a small simulation study.     

Notes on Kolassa’s Method for Estimating the Power of Wilcoxon–Mann–Whitney Test

The Kolassa method implemented in the nQuery Advisor software has been widely used for approximating the power of the Wilcoxon–Mann–Whitney (WMW) test for ordered categorical data, in which Edgeworth approximation is used to estimate the power of an unconditional test based on the WMW U statistic. When the sample size is small or when the sizes in the two groups are unequal, Kolassa’s method may yield quite poor approximation to the power of the conditional WMW test that is commonly implemented in statistical packages. Two modifications of Kolassa’s formula are proposed and assessed by simulation studies.     

Variance reduction of estimators arising from Metropolis–Hastings algorithms

The Metropolis–Hastings algorithm is one of the most basic and well-studied Markov chain Monte Carlo methods. It generates a Markov chain which has as limit distribution the target distribution by simulating observations from a different proposal distribution. A proposed value is accepted with some particular probability otherwise the previous value is repeated. As a consequence, the accepted values are repeated a positive number of times and thus any resulting ergodic mean is, in fact, a weighted average. It turns out that this weighted average is an importance sampling-type estimator with random weights. By the standard theory of importance sampling, replacement of these random weights by their (conditional) expectations leads to more efficient estimators. In this paper we study the estimator arising by replacing the random weights with certain estimators of their conditional expectations. We illustrate by simulations that it is often more efficient than the original estimator while in the case of the independence Metropolis–Hastings and for distributions with finite support we formally prove that it is even better than the “optimal” importance sampling estimator.     

Highly Efficient Designs to Handle the Incorrect Specification of Linear Mixed Models

The asymptotic chi-square test for testing the Hardy–Weinberg law is unreliable in either small or unbalanced samples. As an alternative, either the unconditional or conditional exact test might be used. It is known that the unconditional exact test has greater power than the conditional exact test in small samples. In this article, we show that the conditional exact test is more powerful than the unconditional exact test in large samples. This result is useful in extremely unbalanced cases with large sample sizes which are often obtained when a rare allele exists.     

动态资产定价理论中风险估计及其统计性质分析

一、引言古典资产定价理论,如Ｓｈａｒｐｅ的资产定价模型（ＣＡＰＭ）和Ｒｏｓ的套利定价理论（ＡＰＴ）,有助于大家对风险定价及其概念的理解。这些理论解释了资产回报的横截面行为,允许根据不同的风险水平对资产进行评价和计算。这些理论的经验证明必须对风险进行估...     

Bootstrapping the conditional copula

This paper is concerned with inference about the dependence or association between two random variables conditionally upon the given value of a covariate. A way to describe such a conditional dependence is via a conditional copula function. Nonparametric estimators for a conditional copula then lead to nonparametric estimates of conditional association measures such as a conditional Kendall's tau. The limiting distributions of nonparametric conditional copula estimators are rather involved. In this paper we propose a bootstrap procedure for approximating these distributions and their characteristics, and establish its consistency. We apply the proposed bootstrap procedure for constructing confidence intervals for conditional association measures, such as a conditional Blomqvist beta and a conditional Kendall's tau. The performances of the proposed methods are investigated via a simulation study involving a variety of models, ranging from models in which the dependence (weak or strong) on the covariate is only through the copula and not through the marginals, to models in which this dependence appears in both the copula and the marginal distributions. As a conclusion we provide practical recommendations for constructing bootstrap-based confidence intervals for the discussed conditional association measures.     

Sampling of pairs in pairwise likelihood estimation for latent variable models with categorical observed variables

Pairwise likelihood is a limited information estimation method that has also been used for estimating the parameters of latent variable and structural equation models. Pairwise likelihood is a special case of composite likelihood methods that uses lower-order conditional or marginal log-likelihoods instead of the full log-likelihood. The composite likelihood to be maximized is a weighted sum of marginal or conditional log-likelihoods. Weighting has been proposed for increasing efficiency, but the choice of weights is not straightforward in most applications. Furthermore, the importance of leaving out higher-order scores to avoid duplicating lower-order marginal information has been pointed out. In this paper, we approach the problem of weighting from a sampling perspective. More specifically, we propose a sampling method for selecting pairs based on their contribution to the total variance from all pairs. The sampling approach does not aim to increase efficiency but to decrease the estimation time, especially in models with a large number of observed categorical variables. We demonstrate the performance of the proposed methodology using simulated examples and a real application.

     

Multilevel modelling of survey data: impact of the two-level weights used in the pseudolikelihood

Approaches that use the pseudolikelihood to perform multilevel modelling on survey data have been presented in the literature. To avoid biased estimates due to unequal selection probabilities, conditional weights can be introduced at each level. Less-biased estimators can also be obtained in a two-level linear model if the level-1 weights are scaled. In this paper, we studied several level-2 weights that can be introduced into the pseudolikelihood when the sampling design and the hierarchical structure of the multilevel model do not match. Two-level and three-level models were studied. The present work was motivated by a study that aims to estimate the contributions of lead sources to polluting the interior floor dust of the rooms within dwellings. We performed a simulation study using the real data collected from a French survey to achieve our objective. We conclude that it is preferable to use unweighted analyses or, at the most, to use conditional level-2 weights in a two-level or a three-level model. We state some warnings and make some recommendations.     

Unconditional empirical likelihood approach for analytic use of public survey data

Modeling survey data often requires having the knowledge of design and weighting variables. With public-use survey data, some of these variables may not be available for confidentiality reasons. The proposed approach can be used in this situation, as long as calibrated weights and variables specifying the strata and primary sampling units are available. It gives consistent point estimation and a pivotal statistics for testing and confidence intervals. The proposed approach does not rely on with-replacement sampling, single-stage, negligible sampling fractions, or noninformative sampling. Adjustments based on design effects, eigenvalues, joint-inclusion probabilities or bootstrap, are not needed. The inclusion probabilities and auxiliary variables do not have to be known. Multistage designs with unequal selection of primary sampling units are considered. Nonresponse can be easily accommodated if the calibrated weights include reweighting adjustment for nonresponse. We use an unconditional approach, where the variables and sample are random variables. The design can be informative.     

The Use of Sample Weights in Hot Deck Imputation

A common strategy for handling item nonresponse in survey sampling is hot deck imputation, where each missing value is replaced with an observed response from a "similar" unit. We discuss here the use of sampling weights in the hot deck. The naive approach is to ignore sample weights in creation of adjustment cells, which effectively imputes the unweighted sample distribution of respondents in an adjustment cell, potentially causing bias. Alternative approaches have been proposed that use weights in the imputation by incorporating them into the probabilities of selection for each donor. We show by simulation that these weighted hot decks do not correct for bias when the outcome is related to the sampling weight and the response propensity. The correct approach is to use the sampling weight as a stratifying variable alongside additional adjustment variables when forming adjustment cells.     

The distribution of a linear predictor after model selection: conditional finite-sample distributions and asymptotic approximations

This paper studies the distribution of a linear predictor that is constructed after a data-driven model selection step in a linear regression model. The finite-sample cumulative distribution function (cdf) of the linear predictor is derived and a detailed analysis of the effects of the model selection step is given. Moreover, a simple approximation to the (complicated) finite-sample cdf is proposed. This approximation facilitates the study of the large-sample limit behavior of the linear predictor and its cdf, in the fixed-parameter case and under local alternatives. The focus of this paper is on the conditional distribution of a linear predictor, conditional on the event that a fixed (possibly incorrect) model has been selected. The unconditional distribution of a linear predictor is studied in the companion paper Leeb (The distribution of a linear predictor after model selection: unconditional finite-sample distributions and asymptotic approximations, Technical Report, Department of Statistics, University of Vienna, 2002).     

Some remarks on conditional and unconditional inference for location-scale models

Conditional and unconditional confidence intervals have been compared by Grice, Bain, and Engelhardt (Commun. Statist. B7 (1978), 515–524) in terms of the location-scale model with double-exponential distribution form. Preference was found for the conditional intervals based on mean length and coverage probability for untrue parameters values. These two criteria for a location-scale system are shown to be inappropriate criteria for assessing the conditional versus unconditional approaches to inference. The usual ancillarity concept is also noted to be inappropriate. Support for many conditional analyses, however, is found in a more careful formulation of the statistical model.     

A modified random-effect procedure for combining risk difference in sets of 2×2 tables from clinical trials

Summary Meta-analyses of sets of clinical trials often combine risk differences from several 2×2 tables according to a random-effects model. The DerSimonian-Laird random-effects procedure, widely used for estimating the populaton mean risk difference, weights the risk difference from each primary study inversely proportional to an estimate of its variance (the sum of the between-study variance and the conditional within-study variance). Because those weights are not independent of the risk differences, however, the procedure sometimes exhibits bias and unnatural behavior. The present paper proposes a modified weighting scheme that uses the unconditional within-study variance to avoid this source of bias. The modified procedure has variance closer to that available from weighting by ideal weights when such weights are known. We studied the modified procedure in extensive simulation experiments using situations whose parameters resemble those of actual studies in medical research. For comparison we also included two unbiased procedures, the unweighted mean and a sample-size-weighted mean; their relative variability depends on the extent of heterogeneity among the primary studies. An example illustrates the application of the procedures to actual data and the differences among the results. This research was supported by Grant HS 05936 from the Agency for Health Care Policy and Research to Harvard University.     

Conditional and unconditional tests with historical controls

Score statistics utilizing historical control data have been proposed to test for increasing trend in tumour occurrence rates in laboratory carcinogenicity studies. Novel invariance arguments are used to confirm, under slightly weaker conditions, previously established asymptotic distributions (mixtures of normal distributions) of tests unconditional on the tumor response rate in the concurrent control group. Conditioning on the control response rate, an ancillary statistic, leads to a new conditional limit theorem in which the test statistic converges to an unknown random variable. Because of this, a subasymptotic approximation to the conditional limiting distribution is also considered. The adequacy of these large-sample approximations in finite samples is evaluated using computer simulation. Bootstrap methods for use in finite samples are also proposed. The application of the conditional and unconditional tests is illustrated using bioassay data taken from the literature. The results presented in this paper are used to formulate recommendations for the use of tests for trend with historical controls in practice.     

Conditional Correlation Models of Autoregressive Conditional Heteroscedasticity With Nonstationary GARCH Equations

In this article, we investigate the effects of careful modeling the long-run dynamics of the volatilities of stock market returns on the conditional correlation structure. To this end, we allow the individual unconditional variances in conditional correlation generalized autoregressive conditional heteroscedasticity (CC-GARCH) models to change smoothly over time by incorporating a nonstationary component in the variance equations such as the spline-GARCH model and the time-varying (TV)-GARCH model. The variance equations combine the long-run and the short-run dynamic behavior of the volatilities. The structure of the conditional correlation matrix is assumed to be either time independent or to vary over time. We apply our model to pairs of seven daily stock returns belonging to the S&P 500 composite index and traded at the New York Stock Exchange. The results suggest that accounting for deterministic changes in the unconditional variances improves the fit of the multivariate CC-GARCH models to the data. The effect of careful specification of the variance equations on the estimated correlations is variable: in some cases rather small, in others more discernible. We also show empirically that the CC-GARCH models with time-varying unconditional variances using the TV-GARCH model outperform the other models under study in terms of out-of-sample forecasting performance. In addition, we find that portfolio volatility-timing strategies based on time-varying unconditional variances often outperform the unmodeled long-run variances strategy out-of-sample. As a by-product, we generalize news impact surfaces to the situation in which both the GARCH equations and the conditional correlations contain a deterministic component that is a function of time.     

On the uth geometric conditional quantile

Motivated by Chaudhuri's work [1996. On a geometric notion of quantiles for multivariate data. J. Amer. Statist. Assoc. 91, 862–872] on unconditional geometric quantiles, we explore the asymptotic properties of sample geometric conditional quantiles, defined through kernel functions, in high-dimensional spaces. We establish a Bahadur-type linear representation for the geometric conditional quantile estimator and obtain the convergence rate for the corresponding remainder term. From this, asymptotic normality including bias on the estimated geometric conditional quantile is derived. Based on these results, we propose confidence ellipsoids for multivariate conditional quantiles. The methodology is illustrated via data analysis and a Monte Carlo study.