Biometrika - current issue

Optimal sampling and estimation strategies under the linear model

Nedyalkova, D., Tille, Y. — 2008-08-29

In some cases model-based and model-assisted inferences can lead to very different estimators. These two paradigms are not so different if we search for an optimal strategy rather than just an optimal estimator, a strategy being a pair composed of a sampling design and an estimator. We show that, under a linear model, the optimal model-assisted strategy consists of a balanced sampling design with inclusion probabilities that are proportional to the standard deviations of the errors of the model and the Horvitz–Thompson estimator. If the heteroscedasticity of the model is 'fully explainable’ by the auxiliary variables, then this strategy is also optimal in a model-based sense. Moreover, under balanced sampling and with inclusion probabilities that are proportional to the standard deviation of the model, the best linear unbiased estimator and the Horvitz–Thompson estimator are equal. Finally, it is possible to construct a single estimator for both the design and model variance. The inference can thus be valid under the sampling design and under the model.

A new approach to weighting and inference in sample surveys

Beaumont, J.-F. — 2008-08-29

The validity of design-based inference is not dependent on any model assumption. However, it is well known that estimators derived through design-based theory may be inefficient for the estimation of population totals when the design weights are weakly related to the variables of interest and have widely dispersed values. We propose estimators that have the potential to improve the efficiency of any estimator derived under the design-based theory. Our main focus is limited to the improvement of the Horvitz–Thompson estimator, but we also discuss the extension to calibration estimators. The new estimators are obtained by smoothing design or calibration weights using an appropriate model. Our approach to inference requires the modelling of only one variable, the weight, and it leads to a single set of smoothed weights in multipurpose surveys. This is to be contrasted with other model-based approaches, such as the prediction approach, in which it is necessary to postulate and validate a model for each variable of interest leading potentially to variable-specific sets of weights. Our proposed approach is first justified theoretically and then evaluated through a simulation study.

Using calibration weighting to adjust for nonresponse under a plausible model

Chang, T., Kott, P. S. — 2008-08-29

When we estimate the population total for a survey variable or variables, calibration forces the weighted estimates of certain covariates to match known or alternatively estimated population totals called benchmarks. Calibration can be used to correct for sample-survey nonresponse, or for coverage error resulting from frame undercoverage or unit duplication. The quasi-randomization theory supporting its use in nonresponse adjustment treats response as an additional phase of random sampling. The functional form of a quasi-random response model is assumed to be known, its parameter values estimated implicitly through the creation of calibration weights. Unfortunately, calibration depends upon known benchmark totals while the covariates in a plausible model for survey response may not be the benchmark covariates. Moreover, it may be prudent to keep the number of covariates in a response model small. We use calibration to adjust for nonresponse when the benchmark model and covariates may differ, provided the number of the former is at least as great as that of the latter. We discuss the estimation of a total for a vector of survey variables that do not include the benchmark covariates, but that may include some of the model covariates. We show how to measure both the additional asymptotic variance due to the nonresponse in a calibration-weighted estimator and the full asymptotic variance of the estimator itself. All variances are determined with respect to the randomization mechanism used to select the sample, the response model generating the subset of sample respondents, or both. Data from the U.S. National Agricultural Statistical Service's 2002 Census of Agriculture and simulations are used to illustrate alternative adjustments for nonresponse. The paper concludes with some remarks about adjustment for coverage error.

Influence functions and robust Bayes and empirical Bayes small area estimation

Ghosh, M., Maiti, T., Roy, A. — 2008-08-29

We introduce new robust small area estimation procedures based on area-level models. We first find influence functions corresponding to each individual area-level observation by measuring the divergence between the posterior density functions of regression coefficients with and without that observation. Next, based on these influence functions, properly standardized, we propose some new robust Bayes and empirical Bayes small area estimators. The mean squared errors and estimated mean squared errors of these estimators are also found. A small simulation study compares the performance of the robust and the regular empirical Bayes estimators. When the model variance is larger than the sample variance, the proposed robust empirical Bayes estimators are superior.

Robust functional estimation using the median and spherical principal components

Gervini, D. — 2008-08-29

We present robust estimators for the mean and the principal components of a stochastic process in . Robustness and asymptotic properties of the estimators are studied theoretically, by simulation and by example. It is shown that the proposed estimators are generally more robust to outliers than the commonly used sample mean and principal components, although their properties depend on the spacings of the eigenvalues of the covariance function.

Joint modelling of paired sparse functional data using principal components

Zhou, L., Huang, J. Z., Carroll, R. J. — 2008-08-29

We propose a modelling framework to study the relationship between two paired longitudinally observed variables. The data for each variable are viewed as smooth curves measured at discrete time-points plus random errors. While the curves for each variable are summarized using a few important principal components, the association of the two longitudinal variables is modelled through the association of the principal component scores. We use penalized splines to model the mean curves and the principal component curves, and cast the proposed model into a mixed-effects model framework for model fitting, prediction and inference. The proposed method can be applied in the difficult case in which the measurement times are irregular and sparse and may differ widely across individuals. Use of functional principal components enhances model interpretation and improves statistical and numerical stability of the parameter estimates.

Pointwise testing with functional data using the Westfall-Young randomization method

Cox, D. D., Lee, J. S. — 2008-08-29

We consider hypothesis testing with smooth functional data by performing pointwise tests and applying a multiple comparisons procedure. Methods based on general inequalities, such as Bonferroni’s method, do not perform well because of the high correlation between observations at nearby points. We consider the multiple comparison procedure proposed by Westfall & Young (1993) and show that it approximates a multiple comparison correction for a continuum of comparisons as the grid for pointwise comparisons becomes finer. Simulations and an application verify that this result applies in practical settings.

Adjustment uncertainty in effect estimation

Crainiceanu, C. M., Dominici, F., Parmigiani, G. — 2008-08-29

Often there is substantial uncertainty in the selection of confounders when estimating the association between an exposure and health. We define this type of uncertainty as `adjustment uncertainty'. We propose a general statistical framework for handling adjustment uncertainty in exposure effect estimation for a large number of confounders, we describe a specific implementation, and we develop associated visualization tools. Theoretical results and simulation studies show that the proposed method provides consistent estimators of the exposure effect and its variance. We also show that, when the goal is to estimate an exposure effect accounting for adjustment uncertainty, Bayesian model averaging with posterior model probabilities approximated using information criteria can fail to estimate the exposure effect and can over- or underestimate its variance. We compare our approach to Bayesian model averaging using time series data on levels of fine particulate matter and mortality.

Generalized varying coefficient models for longitudinal data

Senturk, D., Muller, H.-G. — 2008-08-29

We propose a generalization of the varying coefficient model for longitudinal data to cases where not only current but also recent past values of the predictor process affect current response. More precisely, the targeted regression coefficient functions of the proposed model have sliding window supports around current time t. A variant of a recently proposed two-step estimation method for varying coefficient models is proposed for estimation in the context of these generalized varying coefficient models, and is found to lead to improvements, especially for the case of additive measurement errors in both response and predictors. The proposed methodology for estimation and inference is also applicable for the case of additive measurement error in the common versions of varying coefficient models that relate only current observations of predictor and response processes to each other. Asymptotic distributions of the proposed estimators are derived, and the model is applied to the problem of predicting protein concentrations in a longitudinal study. Simulation studies demonstrate the efficacy of the proposed estimation procedure.

Additive partial linear models with measurement errors

Liang, H., Thurston, S. W., Ruppert, D., Apanasovich, T., Hauser, R. — 2008-08-29

We consider statistical inference for additive partial linear models when the linear covariate is measured with error. We propose attenuation-to-correction and simulation-extrapolation, simex, estimators of the parameter of interest. It is shown that the first resulting estimator is asymptotically normal and requires no undersmoothing. This is an advantage of our estimator over existing backfitting-based estimators for semiparametric additive models which require undersmoothing of the nonparametric component in order for the estimator of the parametric component to be root-n consistent. This feature stems from a decrease of the bias of the resulting estimator, which is appropriately derived using a profile procedure. A similar characteristic in semiparametric partially linear models was obtained by Wang et al. (2005). We also discuss the asymptotics of the proposed simex approach. Finite-sample performance of the proposed estimators is assessed by simulation experiments. The proposed methods are applied to a dataset from a semen study.

Improving the efficiency of the log-rank test using auxiliary covariates

Lu, X., Tsiatis, A. A. — 2008-08-29

Under the assumption of proportional hazards, the log-rank test is optimal for testing the null hypothesis , where denotes the logarithm of the hazard ratio. However, if there are additional covariates that correlate with survival times, making use of their information will increase the efficiency of the log-rank test. We apply the theory of semiparametrics to characterize a class of regular and asymptotically linear estimators for when auxiliary covariates are incorporated into the model, and derive estimators that are more efficient. The Wald tests induced by these estimators are shown to be more powerful than the log-rank test. Simulation studies are used to illustrate the gains in efficiency.

Supremum weighted log-rank test and sample size for comparing two-stage adaptive treatment strategies

Feng, W., Wahed, A. S. — 2008-08-29

In two-stage adaptive treatment strategies, patients receive an induction treatment followed by a maintenance therapy, given that the patient responded to the induction treatment they received. To test for a difference in the effects of different induction and maintenance treatment combinations, a modified supremum weighted log-rank test is proposed. The test is applied to a dataset from a two-stage randomized trial and the results are compared to those obtained using a standard weighted log-rank test. A sample-size formula is proposed based on the limiting distribution of the supremum weighted log-rank statistic. The sample-size formula reduces to Eng and Kosorok's sample-size formula for a two-sample supremum log-rank test when there is no second randomization. Monte Carlo studies show that the proposed test provides sample sizes that are close to those obtained by standard weighted log-rank test under a proportional hazards alternative. However, the proposed test is more powerful than the standard weighted log-rank test under non-proportional hazards alternatives.

The Benjamini-Hochberg method with infinitely many contrasts in linear models

Westfall, P. H. — 2008-08-29

Benjamini and Hochberg's method for controlling the false discovery rate is applied to the problem of testing infinitely many contrasts in linear models. Exact, easily calculated critical values are derived, defining a new multiple comparisons method for testing contrasts in linear models. The method is adaptive, depending on the data through the F-statistic, like the Waller–Duncan Bayesian multiple comparisons method. Comparisons with Scheffé's method are given, and the method is extended to the simultaneous confidence intervals of Benjamini and Yekutieli.

Semiparametric model-based inference in the presence of missing responses

Wang, Q., Dai, P. — 2008-08-29

We consider a semiparametric model that parameterizes the conditional density of the response, given covariates, but allows the marginal distribution of the covariates to be completely arbitrary. Responses may be missing. A likelihood-based imputation estimator and a semi-empirical-likelihood-based estimator for the parameter vector describing the conditional density are defined and proved to be asymptotically normal. Semi-empirical loglikelihood functions for the parameter vector and the response mean are derived. It is shown that the two semi-empirical loglikelihood functions are distributed asymptotically as weighted ² and scaled ², respectively.

Conditionally specified continuous distributions

Wang, Y. J., Ip, E. H. — 2008-08-29

A distribution is conditionally specified when its model constraints are expressed conditionally. For example, Besag's (1974) spatial model was specified conditioned on the neighbouring states, and pseudolikelihood is intended to approximate the likelihood using conditional likelihoods. There are three issues of interest: existence, uniqueness and computation of a joint distribution. In the literature, most results and proofs are for discrete probabilities; here we exclusively study distributions with continuous state space. We examine all three issues using the dependence functions derived from decomposition of the conditional densities. We show that certain dependence functions of the joint density are shared with its conditional densities. Therefore, two conditional densities involving the same set of variables are compatible if their overlapping dependence functions are identical. We prove that the joint density is unique when the set of dependence functions is both compatible and complete. In addition, a joint density, apart from a constant, can be computed from the dependence functions in closed form. Since all of the results are expressed in terms of dependence functions, we consider our approach to be dependence-based, whereas methods in the literature are generally density-based. Applications of the dependence-based formulation are discussed.

Conditional properties of unconditional parametric bootstrap procedures for inference in exponential families

DiCiccio, T. J., Young, G. A. — 2008-08-29

Higher-order inference about a scalar parameter in the presence of nuisance parameters can be achieved by bootstrapping, in circumstances where the parameter of interest is a component of the canonical parameter in a full exponential family. The optimal test, which is approximated, is a conditional one based on conditioning on the sufficient statistic for the nuisance parameter. A bootstrap procedure that ignores the conditioning is shown to have desirable conditional properties in providing third-order relative accuracy in approximation of p-values associated with the optimal test, in both continuous and discrete models. The bootstrap approach is equivalent to third-order analytical approaches, and is demonstrated in a number of examples to give very accurate approximations even for very small sample sizes.

Extended Bayesian information criteria for model selection with large model spaces

Chen, J., Chen, Z. — 2008-08-29

The ordinary Bayesian information criterion is too liberal for model selection when the model space is large. In this paper, we re-examine the Bayesian paradigm for model selection and propose an extended family of Bayesian information criteria, which take into account both the number of unknown parameters and the complexity of the model space. Their consistency is established, in particular allowing the number of covariates to increase to infinity with the sample size. Their performance in various situations is evaluated by simulation studies. It is demonstrated that the extended Bayesian information criteria incur a small loss in the positive selection rate but tightly control the false discovery rate, a desirable property in many applications. The extended Bayesian information criteria are extremely useful for variable selection in problems with a moderate sample size but with a huge number of covariates, especially in genome-wide association studies, which are now an active area in genetics research.

A note on conditional AIC for linear mixed-effects models

Liang, H., Wu, H., Zou, G. — 2008-08-29

The conventional model selection criterion, the Akaike information criterion, aic, has been applied to choose candidate models in mixed-effects models by the consideration of marginal likelihood. Vaida & Blanchard (2005) demonstrated that such a marginal aic and its small sample correction are inappropriate when the research focus is on clusters. Correspondingly, these authors suggested the use of conditional aic. Their conditional aic is derived under the assumption that the variance-covariance matrix or scaled variance-covariance matrix of random effects is known. This note provides a general conditional aic but without these strong assumptions. Simulation studies show that the proposed method is promising.