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The commonly used two-sample tests of equal area-under-the-curve (AUC), where AUC is based on the linear trapezoidal rule, may have poor properties when observations are missing, even if they are missing completely at random (MCAR). We propose two tests: one that has good properties when data are MCAR and another that has good properties when the data are missing at random (MAR), provided that the pattern of missingness is monotonic. In addition, we discuss other non-parametric tests of hypotheses that are similar, but not identical, to the hypothesis of equal AUCs, but that often have better statistical properties than do AUC tests and may be more scientifically appropriate for many settings.

We propose an interaction tree (IT) procedure to optimize the subgroup analysis in comparative studies that involve censored survival times. The proposed method recursively partitions the data into two subsets that show the greatest interaction with the treatment, which results in a number of objectively defined subgroups: in some of them the treatment effect is prominent while in others the treatment may have a negligible or even negative effect. The resultant tree structure can be used to explore the overall interaction between treatment and other covariates and help identify and describe possible target populations on which an experimental treatment demonstrates desired efficacy. We follow the standard CART (Breiman, et al., 1984) methodology to develop the interaction tree structure. Variable importance information is extracted via random forests of interaction trees. Both simulated experiments and an analysis of the primary billiary cirrhosis (PBC) data are provided for evaluation and illustration of the proposed procedure.

A challenge in microarray data analysis concerns discovering local structures composed by sets of genes that show homogeneous expression patterns across subsets of conditions. We present an extension of the mixture of factor analyzers model (MFA) allowing for simultaneous clustering of genes and conditions. The proposed model is rather flexible since it models the density of high-dimensional data assuming a mixture of Gaussian distributions with a particular omponent-specific covariance structure. Specifically, a binary and row stochastic matrix representing tissue membership is used to cluster tissues (experimental conditions), whereas the traditional mixture approach is used to define the gene clustering. An alternating expectation conditional maximization (AECM) algorithm is proposed for parameter estimation; experiments on simulated and real data show the efficiency of our method as a general approach to biclustering. The Matlab code of the algorithm is available upon request from authors.

We consider a method for extending instrumental variables methods in order to estimate the overall effect of a treatment or exposure. The approach is designed for settings in which the instrument influences both the treatment of interest and a secondary treatment also influenced by the primary treatment. We demonstrate that, while instrumental variables methods may be used to estimate the joint effects of the primary and secondary treatments, they cannot by themselves be used to estimate the overall effect of the primary treatment. However, instrumental variables methods may be used in conjunction with approaches for estimating the effect of the primary on the secondary treatment to estimate the overall effect of the primary treatment. We consider extending the proposed methods to deal with confounding of the effect of the instrument, mediation of the effect of the instrument by other variables, failure-time outcomes, and time-varying secondary treatments. We motivate our discussion by considering estimation of the overall effect of the type of vascular access among hemodialysis patients.

It has long been recognized that covariate adjustment can increase precision in randomized experiments, even when it is not strictly necessary. Adjustment is often straightforward when a discrete covariate partitions the sample into a handful of strata, but becomes more involved with even a single continuous covariate such as age. As randomized experiments remain a gold standard for scientific inquiry, and the information age facilitates a massive collection of baseline information, the longstanding problem of if and how to adjust for covariates is likely to engage investigators for the foreseeable future.In the locally efficient estimation approach introduced for general coarsened data structures by James Robins and collaborators, one first fits a relatively small working model, often with maximum likelihood, giving a nuisance parameter fit in an estimating equation for the parameter of interest. The usual advertisement is that the estimator will be asymptotically efficient if the working model is correct, but otherwise will still be consistent and asymptotically Gaussian.However, by applying standard likelihood-based fits to misspecified working models in covariate adjustment problems, one can poorly estimate the parameter of interest. We propose a new method, empirical efficiency maximization, to optimize the working model fit for the resulting parameter estimate.In addition to the randomized experiment setting, we show how our covariate adjustment procedure can be used in survival analysis applications. Numerical asymptotic efficiency calculations demonstrate gains relative to standard locally efficient estimators.

Typically locus specific genotype data do not contain information regarding the gametic phase of haplotypes, especially when an individual is heterozygous at more than one locus among a large number of linked polymorphic loci. Thus, studying disease-haplotype association using unphased genotype data is essentially a problem of handling a missing covariate in a case-control design. There are several methods for estimating a disease-haplotype association parameter in a matched case-control study. Here we propose a conditional likelihood approach for inference regarding the disease-haplotype association using unphased genotype data arising from a matched case-control study design. The proposed method relies on a logistic disease risk model and a Hardy-Weinberg equilibrium (HWE) among the control population only. We develop an expectation and conditional maximization (ECM) algorithm for jointly estimating the haplotype frequency and the disease-haplotype association parameter(s). We apply the proposed method to analyze the data from the Alpha-Tocopherol, Beta-Carotene Cancer prevention study, and a matched case-control study of breast cancer patients conducted in Israel. The performance of the proposed method is evaluated via simulation studies.

In the analysis of dependence of bivariate correlated failure time data, a popular model is a gamma frailty model proposed by Clayton and Oakes. An alternative approach is using a Plackett distribution, whose dependence parameter has a very appealing odds ratio interpretation for dependence between the two failure times. In this article, we develop novel semiparametric estimation and inference procedures for the model. The asymptotic results of the estimator are developed; in addition, a goodness of fit test is also developed. We also discuss a regression extension to adjust for covariates using the linear regression model as well as applications to semi-competing risks data. The performance of the proposed techniques in finite samples is examined using simulation studies. Several real-data examples are used to illustrate the methodology.

Attenuation of the exposure-response relationship due to exposure measurement error is often encountered in epidemiology. Given that error cannot be totally eliminated, bias correction methods of analysis are needed. Many methods require more than one exposure measurement per person to be made, but the `group mean OLS method,' in which subjects are grouped into several a priori defined groups followed by ordinary least squares (OLS) regression on the group means, can be applied with one measurement. An alternative approach is to use an instrumental variable (IV) method in which both the single error-prone measure and an IV are used in IV analysis. In this paper we show that the `group mean OLS' estimator is equal to an IV estimator with the group mean used as IV, but that the variance estimators for the two methods are different. We derive a simple expression for the bias in the common estimator which is a simple function of group size, reliability and contrast of exposure between groups, and show that the bias can be very small when group size is large. We compare this method with a new proposal (group mean ranking method), also applicable with a single exposure measurement, in which the IV is the rank of the group means. When there are two independent exposure measurements per subject, we propose a new IV method (EVROS IV) and compare it with Carroll and Stefanski's (CS IV) proposal in which the second measure is used as an IV; the new IV estimator combines aspects of the `group mean' and `CS' strategies. All methods are evaluated in terms of bias, precision and root mean square error via simulations and a dataset from occupational epidemiology. The `group mean ranking method' does not offer much improvement over the `group mean method.' Compared with the `CS' method, the `EVROS' method is less affected by low reliability of exposure. We conclude that the group IV methods we propose may provide a useful way to handle mismeasured exposures in epidemiology with or without replicate measurements. Our finding may also have implications for the use of aggregate variables in epidemiology to control for unmeasured confounding.

In designing longitudinal studies, researchers must determine the number of subjects to randomize based on the power to detect a clinically meaningful treatment difference and a proposed analysis plan. In this paper, we present formulas for sample size estimation and an assessment of statistical power for a two-treatment repeated measures design allowing for subject attrition. These formulas can be used for comparing two treatment groups across time in terms of linear contrasts. Subjects are assumed to drop out of the study at random so that the missing data do not alter the parameters of interest.

Exact analytic expressions are developed for the average power of the Benjamini and Hochberg false discovery control procedure. The result is based on explicit computation of the joint probability distribution of the total number of rejections and the number of false rejections, and expressed in terms of the cumulative distribution functions of the p-values of the hypotheses. An example of analytic evaluation of the average power is given. The result is confirmed by numerical experiments and applied to a meta-analysis of three clinical studies in mammography.

Standard statistical analyses of observational data often exclude valuable information from individuals with incomplete measurements. This may lead to biased estimates of the treatment effect and loss of precision. The issue of missing data for inverse probability of treatment weighted estimation of marginal structural models (MSMs) has often been addressed, though little has been done to compare different missing data techniques in this relatively new method of analysis. We propose a method for systematically dealing with missingness in MSMs by treating missingness as a cause for censoring and weighting subjects by the inverse probability of missingness. We developed a series of simulations to systematically compare the effect of using case deletion, our inverse weighting approach, and multiple imputation in a MSM when there is missing information on an important confounder. We found that multiple imputation was slightly less biased and considerably less variable than the inverse probability approach. Thus, the lower variability achieved through multiple imputation makes it desirable in most practical cases where the missing data are strongly predicted by the available data. Inverse probability weighting is, however, a superior alternative to naive approaches such as complete-case analysis.

Missing data and especially dropouts frequently arise in longitudinal data. Maximum likelihood estimates are consistent when data are missing at random (MAR) but, as this assumption is not checkable, pattern mixture models (PMM) have been developed to deal with informative dropout. More recently, latent class models (LCM) have been proposed as a way to relax PMM assumptions. The aim of this paper is to compare PMM and LCM in order to tackle informative dropout in a longitudinal study of cognitive ageing measured by several psychometric tests. Using a multivariate longitudinal model with a latent process, a sensitivity analysis was performed to compare estimates under the MAR assumption, from a PMM and from two LCM. In the PMM, dropout patterns are included as covariates in the multivariate longitudinal model. In the simple LCM, they are predictors of the class membership probabilities while, in the joint LCM, the dropout time is jointly modeled using a proportional hazard model depending on latent classes. We show that parameter interpretation is different in the two kinds of models and thus can lead to different estimated values. PMM parameters are adjusted on the dropout patterns while LCM parameters are adjusted on the latent classes. This difference is highlighted in our data set because the latent classes exhibit much more heterogeneity than dropout patterns. We suggest several complementary analyses to investigate the characteristics of latent classes in order to understand the meaning of the parameters when using LCM to deal with informative dropout.

Translational research studies often involve a central study (e.g. clinical trial, cohort of patients, etc.) and multiple investigators who are each interested in addressing different research questions using the same patient population. However, it is often impossible for the investigators to include all patients in all of the ancillary translational research substudies that are part of the main study. This arises due to time and budgetary constraints and other logistical considerations. In this paper, we propose a prospective Systematic Missing-At-Random study design (SMAR) with planned partially missing covariates collected using a nested random sampling scheme that allows an integrated statistical analysis across all domains of data. We propose an algorithm for data analysis that incorporates the features of the design. We show that the SMAR design is computationally and statistically efficient as well as cost effective using simulation studies and a published data example. An extension to a two-stage prospective-retrospective design is discussed.

Method comparison studies are usually analyzed by computing limits of agreement. It is recommended that replicate measurements be taken by each method, but the resulting data are more cumbersome to analyze. We discuss the statistical model underlying the classical limits of agreement and extend it to the case with replicate measurements. As the required code to fit the models is non-trivial, we provide example computer code to fit the models, and show how to use the output to derive measures of repeatability and limits of agreement.

Regular case-control sampling is an extremely common design used to generate data to estimate effects of exposures or treatments on a binary outcome of interest when the proportion of cases (i.e., binary outcome equal to 1) in the population of interest is low. Case-control sampling represents a biased sample of a target population of interest by sampling a disproportional number of cases. Case-control studies are also commonly employed to estimate the effects of genetic markers or biomarkers on binary phenotypes.In this article we present a general method of estimation relying on knowing the prevalence probability, conditional on the matching variable if matching is used.Our general proposed methodology, involving a simple weighting scheme of cases and controls, maps any estimation method for a parameter developed for prospective sampling from the population of interest into an estimation method based on case-control sampling from this population.We show that this case-control weighting of an efficient estimator for a prospective sample from the target population of interest maps into an efficient estimator for matched and unmatched case-control sampling. In particular, we show how application of this generic methodology provides us with double robust locally efficient targeted maximum likelihood estimators of the causal relative risk and causal odds ratio for regular case control sampling and matched case control sampling.Various extensions and generalizations of our methods are discussed.

Understanding how long-term clinical outcomes relate to short-term response to therapy is an important topic of research with a variety of applications. In HIV, early measures of viral RNA levels are known to be a strong prognostic indicator of future viral load response. However, mutations observed in the high-dimensional viral genotype at an early time point may change this prognosis. Unfortunately, some subjects may not have a viral genetic sequence measured at the early time point, and the sequence may be missing for reasons related to the outcome. Complete-case analyses of missing data are generally biased when the assumption that data are missing completely at random is not met, and methods incorporating multiple imputation may not be well-suited for the analysis of high-dimensional data. We propose a semiparametric multiple testing approach to the problem of identifying associations between potentially missing high-dimensional covariates and response. Following the recent exposition by Tsiatis, unbiased nonparametric summary statistics are constructed by inversely weighting the complete cases according to the conditional probability of being observed, given data that is observed for each subject. Resulting summary statistics will be unbiased under the assumption of missing at random. We illustrate our approach through an application to data from a recent AIDS clinical trial, and demonstrate finite sample properties with simulations.

Researchers of uncommon diseases are often interested in assessing potential risk factors. Given the low incidence of disease, these studies are frequently case-control in design. Such a design allows a sufficient number of cases to be obtained without extensive sampling and can increase efficiency; however, these case-control samples are then biased since the proportion of cases in the sample is not the same as the population of interest. Methods for analyzing case-control studies have focused on utilizing logistic regression models that provide conditional and not causal estimates of the odds ratio. This article will demonstrate the use of the prevalence probability and case-control weighted targeted maximum likelihood estimation (MLE), as described by van der Laan (2008), in order to obtain causal estimates of the parameters of interest (risk difference, relative risk, and odds ratio). It is meant to be used as a guide for researchers, with step-by-step directions to implement this methodology. We will also present simulation studies that show the improved efficiency of the case-control weighted targeted MLE compared to other techniques.

Bayesian hierarchical models that characterize the distributions of (transformed) gene profiles have been proven very useful and flexible in selecting differentially expressed genes across different types of tissue samples (e.g. Lo and Gottardo, 2007). However, the marginal mean and variance of these models are assumed to be the same for different gene clusters and for different tissue types. Moreover, it is not easy to determine which of the many competing Bayesian hierarchical models provides the best fit for a specific microarray data set. To address these two issues, we propose a marginal mixture model that directly models the marginal distribution of transformed gene profiles. Specifically, we approximate the marginal distributions of transformed gene profiles via a mixture of three-component multivariate Normal distributions, each component of which has the same structures of marginal mean vector and covariance matrix as those for Bayesian hierarchical models, but the values can differ. Based on the proposed model, a method is derived to select genes differentially expressed across two types of tissue samples. The derived gene selection method performs well on a real microarray data set and consistently has the best performance (based on class agreement indices) compared with several other gene selection methods on simulated microarray data sets generated from three different mixture models.

This paper considers joint analysis of current status and marker data using a threshold model based on first hitting times. A failure time is defined as the time at which a subject's latent health status process first decreases to zero. We extend the bivariate Wiener process model in Whitmore et al. (1998) to the case when only current status data are available. We develop maximum likelihood estimation procedures and provide simulation studies. We apply our methods to a motivating example involving liver tumors in mice.

We describe analytic approaches for study designs that, like large simple trials, can be better characterized as longitudinal studies with baseline randomization than as either a pure randomized experiment or a purely observational study. We (i) discuss the intention-to-treat effect as an effect measure for randomized studies, (ii) provide a formal definition of causal effect for longitudinal studies, (iii) describe several methods -- based on inverse probability weighting and g-estimation -- to estimate such effect, (iv) present an application of these methods to a naturalistic trial of antipsychotics on symptom severity of schizophrenia, and (v) discuss the relative advantages and disadvantages of each method.

The causal effect of a treatment on an outcome is generally mediated by several intermediate variables. Estimation of the component of the causal effect of a treatment that is not mediated by an intermediate variable (the direct effect of the treatment) is often relevant to mechanistic understanding and to the design of clinical and public health interventions. Robins, Greenland and Pearl develop counterfactual definitions for two types of direct effects, natural and controlled, and discuss assumptions, beyond those of sequential randomization, required for the identifiability of natural direct effects. Building on their earlier work and that of others, this article provides an alternative counterfactual definition of a natural direct effect, the identifiability of which is based only on the assumption of sequential randomization. In addition, a novel approach to direct effect estimation is presented, based on assuming a model directly on the natural direct effect, possibly conditional on a subset of the baseline covariates. Inverse probability of censoring weighted estimators, double robust inverse probability of censoring weighted estimators, likelihood-based estimators, and targeted maximum likelihood-based estimators are proposed for the unknown parameters of this novel causal model.

For missing data and causal inference problems, Rubin and van der Laan (2008) proposed estimators to achieve so-called improved local efficiency. We show that their estimators agree with existing estimators in the case of linear models, point out that one particular version of their estimators is also doubly robust, and suggest an extension for where the propensity score is estimated.

We respond to several interesting points raised by Tan regarding our article.