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A general framework is introduced to estimate how much external information has been infused into a search algorithm, the so-called active information. This is rephrased as a test of fine-tuning, where tuning corresponds to the amount of pre-specified knowledge that the algorithm makes use of in order to reach a certain target. A function f quantifies specificity for each possible outcome x of a search, so that the target of the algorithm is a set of highly specified states, whereas fine-tuning occurs if it is much more likely for the algorithm to reach the target as intended than by chance. The distribution of a random outcome X of the algorithm involves a parameter θ that quantifies how much background information has been infused. A simple choice of this parameter is to use θf in order to exponentially tilt the distribution of the outcome of the search algorithm under the null distribution of no tuning, so that an exponential family of distributions is obtained. Such algorithms are obtained by iterating a Metropolis-Hastings type of Markov chain, which makes it possible to compute their active information under the equilibrium and non-equilibrium of the Markov chain, with or without stopping when the targeted set of fine-tuned states has been reached. Other choices of tuning parameters θ are discussed as well. Nonparametric and parametric estimators of active information and tests of fine-tuning are developed when repeated and independent outcomes of the algorithm are available. The theory is illustrated with examples from cosmology, student learning, reinforcement learning, a Moran type model of population genetics, and evolutionary programming.

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In this paper, we present a method for conducting global sensitivity analysis of randomized trials in which binary outcomes are scheduled to be collected on participants at prespecified points in time after randomization and these outcomes may be missing in a nonmonotone fashion. We introduce a class of missing data assumptions, indexed by sensitivity parameters, which are anchored around the missing not at random assumption introduced by Robins (Statistics in Medicine, 1997). For each assumption in the class, we establish that the joint distribution of the outcomes is identifiable from the distribution of the observed data. Our estimation procedure uses the plug-in principle, where the distribution of the observed data is estimated using random forests. We establish n$\sqrt {n}$ asymptotic properties for our estimation procedure. We illustrate our methodology in the context of a randomized trial designed to evaluate a new approach to reducing substance use, assessed by testing urine samples twice weekly, among patients entering outpatient addiction treatment. We evaluate the finite sample properties of our method in a realistic simulation study. Our methods have been implemented in an R package entitled slabm.

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Typically, case-control studies to estimate odds-ratios associating risk factors with disease incidence only include newly diagnosed cases. Recently proposed methods allow incorporating information on prevalent cases, individuals who survived from disease diagnosis to sampling, into cross-sectionally sampled case-control studies under parametric assumptions for the survival time after diagnosis. Here we propose and study methods to additionally use prospectively observed survival times from prevalent and incident cases to adjust logistic models for the time between diagnosis and sampling, the backward time, for prevalent cases. This adjustment yields unbiased odds-ratio estimates from case-control studies that include prevalent cases. We propose a computationally simple two-step generalized method-of-moments estimation procedure. First, we estimate the survival distribution assuming a semiparametric Cox model using an expectation-maximization algorithm that yields fully efficient estimates and accommodates left truncation for prevalent cases and right censoring. Then, we use the estimated survival distribution in an extension of the logistic model to three groups (controls, incident, and prevalent cases), to adjust for the survival bias in prevalent cases. In simulations, under modest amounts of censoring, odds-ratios from the two-step procedure were equally efficient as those estimated from a joint logistic and survival data likelihood under parametric assumptions. This indicates that utilizing the cases' prospective survival data lessens model dependencies and improves precision of association estimates for case-control studies with prevalent cases. We illustrate the methods by estimating associations between single nucleotide polymorphisms and breast cancer risk using controls, and incident and prevalent cases sampled from the US Radiologic Technologists Study cohort.

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Data analyses typically rely upon assumptions about the missingness mechanisms that lead to observed versus missing data, assumptions that are typically unassessable. We explore an approach where the joint distribution of observed data and missing data are specified in a nonstandard way. In this formulation, which traces back to a representation of the joint distribution of the data and missingness mechanism, apparently first proposed by J. W. Tukey, the modeling assumptions about the distributions are either assessable or are designed to allow relatively easy incorporation of substantive knowledge about the problem at hand, thereby offering a possibly realistic portrayal of the data, both observed and missing. We develop Tukey's representation for exponential-family models, propose a computationally tractable approach to inference in this class of models, and offer some general theoretical comments. We then illustrate the utility of this approach with an example in systems biology.

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We propose and study a fully efficient method to estimate associations of an exposure with disease incidence when both, incident cases and prevalent cases, i.e., individuals who were diagnosed with the disease at some prior time point and are alive at the time of sampling, are included in a case-control study. We extend the exponential tilting model for the relationship between exposure and case status to accommodate two case groups, and correct for the survival bias in the prevalent cases through a tilting term that depends on the parametric distribution of the backward time, i.e., the time from disease diagnosis to study enrollment. We construct an empirical likelihood that also incorporates the observed backward times for prevalent cases, obtain efficient estimates of odds ratio parameters that relate exposure to disease incidence and propose a likelihood ratio test for model parameters that has a standard chi-squared distribution. We quantify the changes in efficiency of association parameters when incident cases are supplemented with, or replaced by, prevalent cases in simulations. We illustrate our methods by estimating associations of single nucleotide polymorphisms (SNPs) with breast cancer incidence in a sample of controls, incident and prevalent cases from the U.S. Radiologic Technologists Health Study.

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Randomized trials with patient-reported outcomes are commonly plagued by missing data. The analysis of such trials relies on untestable assumptions about the missing data mechanism. To address this issue, it has been recommended that the sensitivity of the trial results to assumptions should be a mandatory reporting requirement. In this paper, we discuss a recently developed methodology (Scharfstein et al., Biometrics, 2018) for conducting sensitivity analysis of randomized trials in which outcomes are scheduled to be measured at fixed points in time after randomization and some subjects prematurely withdraw from study participation. The methodology is explicated in the context of a placebo-controlled randomized trial designed to evaluate a treatment for bipolar disorder. We present a comprehensive data analysis and a simulation study to evaluate the performance of the method. A software package entitled SAMON (R and SAS versions) that implements our methods is available at www.missingdatamatters.org .

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As an important part of modern health care, medical imaging data, which can be regarded as densely sampled functional data, have been widely used for diagnosis, screening, treatment, and prognosis, such as finding breast cancer through mammograms. The aim of this paper is to propose a functional linear regression model for using functional (or imaging) predictors to predict clinical outcomes (e.g., disease status), while addressing missing clinical outcomes. We introduce an exponential tilting semiparametric model to account for the nonignorable missing data mechanism. We develop a set of estimating equations and its associated computational methods for both parameter estimation and the selection of the tuning parameters. We also propose a bootstrap resampling procedure for carrying out statistical inference. Under some regularity conditions, we systematically establish the asymptotic properties (e.g., consistency and convergence rate) of the estimates calculated from the proposed estimating equations. Simulation studies and a real data analysis are used to illustrate the finite sample performance of the proposed methods.

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In practice, both testable and untestable assumptions are generally required to draw inference about the mean outcome measured at the final scheduled visit in a repeated measures study with drop-out. Scharfstein et al. (2014) proposed a sensitivity analysis methodology to determine the robustness of conclusions within a class of untestable assumptions. In their approach, the untestable and testable assumptions were guaranteed to be compatible; their testable assumptions were based on a fully parametric model for the distribution of the observable data. While convenient, these parametric assumptions have proven especially restrictive in empirical research. Here, we relax their distributional assumptions and provide a more flexible, semi-parametric approach. We illustrate our proposal in the context of a randomized trial for evaluating a treatment of schizoaffective disorder.

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Recently, the scaling limit of cluster sizes for critical inhomogeneous random graphs of rank-1 type having finite variance but infinite third moment degrees was obtained in Bhamidi et al. (Ann Probab 40:2299-2361, 2012). It was proved that when the degrees obey a power law with exponent τ ∈ ( 3 , 4 ) , the sequence of clusters ordered in decreasing size and multiplied through by n - ( τ - 2 ) / ( τ - 1 ) converges as n → ∞ to a sequence of decreasing non-degenerate random variables. Here, we study the tails of the limit of the rescaled largest cluster, i.e., the probability that the scaling limit of the largest cluster takes a large value u, as a function of u. This extends a related result of Pittel (J Combin Theory Ser B 82(2):237-269, 2001) for the Erdos-Rényi random graph to the setting of rank-1 inhomogeneous random graphs with infinite third moment degrees. We make use of delicate large deviations and weak convergence arguments.

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We use exponential tilting to obtain versions of asymptotic formulae for Bayesian computation that do not involve conditional maxima of the likelihood function, yielding a more stable computational procedure and significantly reducing computational time. In particular we present an alternative version of the Laplace approximation for a marginal posterior density. Implementation of the asymptotic formulae and a modified signed root based importance sampler are illustrated with an example.

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Covariate adjustment methods are frequently used when baseline covariate information is available for randomized controlled trials. Using a simulation study, we compared the analysis of covariance (ANCOVA) with three nonparametric covariate adjustment methods with respect to point and interval estimation for the difference between means. The three alternative methods were based on important members of the generalized empirical likelihood (GEL) family, specifically on the empirical likelihood (EL) method, the exponential tilting (ET) method, and the continuous updated estimator (CUE) method. Two criteria were considered for the comparison of the four statistical methods: the root mean squared error and the empirical coverage of the nominal 95% confidence intervals for the difference between means. Based on the results of the simulation study, for sensitivity analysis purposes, we recommend the use of ANCOVA (with robust standard errors when heteroscedasticity is present) together with the CUE-based covariate adjustment method.

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We develop an empirical likelihood (EL) inference on parameters in generalized estimating equations with nonignorably missing response data. We consider an exponential tilting model for the nonignorably missing mechanism, and propose modified estimating equations by imputing missing data through a kernel regression method. We establish some asymptotic properties of the EL estimators of the unknown parameters under different scenarios. With the use of auxiliary information, the EL estimators are statistically more efficient. Simulation studies are used to assess the finite sample performance of our proposed EL estimators. We apply our EL estimators to investigate a data set on earnings obtained from the New York Social Indicators Survey.

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Large- and finite-sample efficiency and resistance to outliers are the key goals of robust statistics. Although often not simultaneously attainable, we develop and study a linear regression estimator that comes close. Efficiency obtains from the estimator's close connection to generalized empirical likelihood, and its favorable robustness properties are obtained by constraining the associated sum of (weighted) squared residuals. We prove maximum attainable finite-sample replacement breakdown point, and full asymptotic efficiency for normal errors. Simulation evidence shows that compared to existing robust regression estimators, the new estimator has relatively high efficiency for small sample sizes, and comparable outlier resistance. The estimator is further illustrated and compared to existing methods via application to a real data set with purported outliers.

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The proportional likelihood ratio model introduced in Luo & Tsai (2011) is adapted to explicitly model the means of observations. This is useful for the estimation of and inference on treatment effects, particularly in designed experiments, and allows the data analyst greater control over model specification and parameter interpretation.