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1.  The Relative Performance of Targeted Maximum Likelihood Estimators 
There is an active debate in the literature on censored data about the relative performance of model based maximum likelihood estimators, IPCW-estimators, and a variety of double robust semiparametric efficient estimators. Kang and Schafer (2007) demonstrate the fragility of double robust and IPCW-estimators in a simulation study with positivity violations. They focus on a simple missing data problem with covariates where one desires to estimate the mean of an outcome that is subject to missingness. Responses by Robins, et al. (2007), Tsiatis and Davidian (2007), Tan (2007) and Ridgeway and McCaffrey (2007) further explore the challenges faced by double robust estimators and offer suggestions for improving their stability. In this article, we join the debate by presenting targeted maximum likelihood estimators (TMLEs). We demonstrate that TMLEs that guarantee that the parametric submodel employed by the TMLE procedure respects the global bounds on the continuous outcomes, are especially suitable for dealing with positivity violations because in addition to being double robust and semiparametric efficient, they are substitution estimators. We demonstrate the practical performance of TMLEs relative to other estimators in the simulations designed by Kang and Schafer (2007) and in modified simulations with even greater estimation challenges.
PMCID: PMC3173607  PMID: 21931570
censored data; collaborative double robustness; collaborative targeted maximum likelihood estimation; double robust; estimator selection; inverse probability of censoring weighting; locally efficient estimation; maximum likelihood estimation; semiparametric model; targeted maximum likelihood estimation; targeted minimum loss based estimation; targeted nuisance parameter estimator selection
2.  A two-stage validation study for determining sensitivity and specificity. 
Environmental Health Perspectives  1994;102(Suppl 8):11-14.
A two-stage procedure for estimating sensitivity and specificity is described. The procedure is developed in the context of a validation study for self-reported atypical nevi, a potentially useful measure in the study of risk factors for malignant melanoma. The first stage consists of a sample of N individuals classified only by the test measure. The second stage is a subsample of size m, stratified according the information collected in the first stage, in which the presence of atypical nevi is determined by clinical examination. Using missing data methods for contingency tables, maximum likelihood estimators for the joint distribution of the test measure and the "gold standard" clinical evaluation are presented, along with efficient estimators for the sensitivity and specificity. Asymptotic coefficients of variation are computed to compare alternative sampling strategies for the second stage.
PMCID: PMC1566548  PMID: 7851324
3.  An Application of Collaborative Targeted Maximum Likelihood Estimation in Causal Inference and Genomics 
A concrete example of the collaborative double-robust targeted likelihood estimator (C-TMLE) introduced in a companion article in this issue is presented, and applied to the estimation of causal effects and variable importance parameters in genomic data. The focus is on non-parametric estimation in a point treatment data structure. Simulations illustrate the performance of C-TMLE relative to current competitors such as the augmented inverse probability of treatment weighted estimator that relies on an external non-collaborative estimator of the treatment mechanism, and inefficient estimation procedures including propensity score matching and standard inverse probability of treatment weighting. C-TMLE is also applied to the estimation of the covariate-adjusted marginal effect of individual HIV mutations on resistance to the anti-retroviral drug lopinavir. The influence curve of the C-TMLE is used to establish asymptotically valid statistical inference. The list of mutations found to have a statistically significant association with resistance is in excellent agreement with mutation scores provided by the Stanford HIVdb mutation scores database.
PMCID: PMC3126668  PMID: 21731530
causal effect; cross-validation; collaborative double robust; double robust; efficient influence curve; penalized likelihood; penalization; estimator selection; locally efficient; maximum likelihood estimation; model selection; super efficiency; super learning; targeted maximum likelihood estimation; targeted nuisance parameter estimator selection; variable importance
4.  Efficient Logistic Regression Designs Under an Imperfect Population Identifier 
Biometrics  2013;70(1):175-184.
Motivated by actual study designs, this article considers efficient logistic regression designs where the population is identified with a binary test that is subject to diagnostic error. We consider the case where the imperfect test is obtained on all participants, while the gold standard test is measured on a small chosen subsample. Under maximum-likelihood estimation, we evaluate the optimal design in terms of sample selection as well as verification. We show that there may be substantial efficiency gains by choosing a small percentage of individuals who test negative on the imperfect test for inclusion in the sample (e.g., verifying 90% test-positive cases). We also show that a two-stage design may be a good practical alternative to a fixed design in some situations. Under optimal and nearly optimal designs, we compare maximum-likelihood and semi-parametric efficient estimators under correct and misspecified models with simulations. The methodology is illustrated with an analysis from a diabetes behavioral intervention trial.
PMCID: PMC3954435  PMID: 24261471
Case-control designs; Diagnostic accuracy; Epidemiologic designs; Misclassification; Measurement error
5.  Collaborative Double Robust Targeted Maximum Likelihood Estimation* 
Collaborative double robust targeted maximum likelihood estimators represent a fundamental further advance over standard targeted maximum likelihood estimators of a pathwise differentiable parameter of a data generating distribution in a semiparametric model, introduced in van der Laan, Rubin (2006). The targeted maximum likelihood approach involves fluctuating an initial estimate of a relevant factor (Q) of the density of the observed data, in order to make a bias/variance tradeoff targeted towards the parameter of interest. The fluctuation involves estimation of a nuisance parameter portion of the likelihood, g. TMLE has been shown to be consistent and asymptotically normally distributed (CAN) under regularity conditions, when either one of these two factors of the likelihood of the data is correctly specified, and it is semiparametric efficient if both are correctly specified.
In this article we provide a template for applying collaborative targeted maximum likelihood estimation (C-TMLE) to the estimation of pathwise differentiable parameters in semi-parametric models. The procedure creates a sequence of candidate targeted maximum likelihood estimators based on an initial estimate for Q coupled with a succession of increasingly non-parametric estimates for g. In a departure from current state of the art nuisance parameter estimation, C-TMLE estimates of g are constructed based on a loss function for the targeted maximum likelihood estimator of the relevant factor Q that uses the nuisance parameter to carry out the fluctuation, instead of a loss function for the nuisance parameter itself. Likelihood-based cross-validation is used to select the best estimator among all candidate TMLE estimators of Q0 in this sequence. A penalized-likelihood loss function for Q is suggested when the parameter of interest is borderline-identifiable.
We present theoretical results for “collaborative double robustness,” demonstrating that the collaborative targeted maximum likelihood estimator is CAN even when Q and g are both mis-specified, providing that g solves a specified score equation implied by the difference between the Q and the true Q0. This marks an improvement over the current definition of double robustness in the estimating equation literature.
We also establish an asymptotic linearity theorem for the C-DR-TMLE of the target parameter, showing that the C-DR-TMLE is more adaptive to the truth, and, as a consequence, can even be super efficient if the first stage density estimator does an excellent job itself with respect to the target parameter.
This research provides a template for targeted efficient and robust loss-based learning of a particular target feature of the probability distribution of the data within large (infinite dimensional) semi-parametric models, while still providing statistical inference in terms of confidence intervals and p-values. This research also breaks with a taboo (e.g., in the propensity score literature in the field of causal inference) on using the relevant part of likelihood to fine-tune the fitting of the nuisance parameter/censoring mechanism/treatment mechanism.
PMCID: PMC2898626  PMID: 20628637
asymptotic linearity; coarsening at random; causal effect; censored data; crossvalidation; collaborative double robust; double robust; efficient influence curve; estimating function; estimator selection; influence curve; G-computation; locally efficient; loss-function; marginal structural model; maximum likelihood estimation; model selection; pathwise derivative; semiparametric model; sieve; super efficiency; super-learning; targeted maximum likelihood estimation; targeted nuisance parameter estimator selection; variable importance
6.  Non-parametric Estimation of a Survival Function with Two-stage Design Studies 
The two-stage design is popular in epidemiology studies and clinical trials due to its cost effectiveness. Typically, the first stage sample contains cheaper and possibly biased information, while the second stage validation sample consists of a subset of subjects with accurate and complete information. In this paper, we study estimation of a survival function with right-censored survival data from a two-stage design. A non-parametric estimator is derived by combining data from both stages. We also study its large sample properties and derive pointwise and simultaneous confidence intervals for the survival function. The proposed estimator effectively reduces the variance and finite-sample bias of the Kaplan–Meier estimator solely based on the second stage validation sample. Finally, we apply our method to a real data set from a medical device post-marketing surveillance study.
PMCID: PMC2729091  PMID: 19696901
censoring; Kaplan–Meier estimator; martingale; Nelson–Aalen estimator; truncation
7.  On semiparametric efficient inference for two-stage outcome-dependent sampling with a continuous outcome 
Biometrika  2009;96(1):221.
Outcome-dependent sampling designs have been shown to be a cost effective way to enhance study efficiency. We show that the outcome-dependent sampling design with a continuous outcome can be viewed as an extension of the two-stage case-control designs to the continuous-outcome case. We further show that the two-stage outcome-dependent sampling has a natural link with the missing-data and biased-sampling framework. Through the use of semiparametric inference and missing-data techniques, we show that a certain semiparametric maximum likelihood estimator is computationally convenient and achieves the semiparametric efficient information bound. We demonstrate this both theoretically and through simulation.
PMCID: PMC2761000  PMID: 20107493
Biased sampling; Empirical process; Maximum likelihood estimation; Missing data; Outcome-dependent; Profile likelihood; Two-stage
8.  Targeted Maximum Likelihood Estimation of Effect Modification Parameters in Survival Analysis 
The Cox proportional hazards model or its discrete time analogue, the logistic failure time model, posit highly restrictive parametric models and attempt to estimate parameters which are specific to the model proposed. These methods are typically implemented when assessing effect modification in survival analyses despite their flaws. The targeted maximum likelihood estimation (TMLE) methodology is more robust than the methods typically implemented and allows practitioners to estimate parameters that directly answer the question of interest. TMLE will be used in this paper to estimate two newly proposed parameters of interest that quantify effect modification in the time to event setting. These methods are then applied to the Tshepo study to assess if either gender or baseline CD4 level modify the effect of two cART therapies of interest, efavirenz (EFV) and nevirapine (NVP), on the progression of HIV. The results show that women tend to have more favorable outcomes using EFV while males tend to have more favorable outcomes with NVP. Furthermore, EFV tends to be favorable compared to NVP for individuals at high CD4 levels.
PMCID: PMC3083138  PMID: 21556287
causal effect; semi-parametric; censored longitudinal data; double robust; efficient influence curve; influence curve; G-computation; Targeted Maximum Likelihood Estimation; Cox-proportional hazards; survival analysis
9.  Nonparametric estimation for censored mixture data with application to the Cooperative Huntington’s Observational Research Trial 
This work presents methods for estimating genotype-specific distributions from genetic epidemiology studies where the event times are subject to right censoring, the genotypes are not directly observed, and the data arise from a mixture of scientifically meaningful subpopulations. Examples of such studies include kin-cohort studies and quantitative trait locus (QTL) studies. Current methods for analyzing censored mixture data include two types of nonparametric maximum likelihood estimators (NPMLEs) which do not make parametric assumptions on the genotype-specific density functions. Although both NPMLEs are commonly used, we show that one is inefficient and the other inconsistent. To overcome these deficiencies, we propose three classes of consistent nonparametric estimators which do not assume parametric density models and are easy to implement. They are based on the inverse probability weighting (IPW), augmented IPW (AIPW), and nonparametric imputation (IMP). The AIPW achieves the efficiency bound without additional modeling assumptions. Extensive simulation experiments demonstrate satisfactory performance of these estimators even when the data are heavily censored. We apply these estimators to the Cooperative Huntington’s Observational Research Trial (COHORT), and provide age-specific estimates of the effect of mutation in the Huntington gene on mortality using a sample of family members. The close approximation of the estimated non-carrier survival rates to that of the U.S. population indicates small ascertainment bias in the COHORT family sample. Our analyses underscore an elevated risk of death in Huntington gene mutation carriers compared to non-carriers for a wide age range, and suggest that the mutation equally affects survival rates in both genders. The estimated survival rates are useful in genetic counseling for providing guidelines on interpreting the risk of death associated with a positive genetic testing, and in facilitating future subjects at risk to make informed decisions on whether to undergo genetic mutation testings.
PMCID: PMC3905630  PMID: 24489419
Censored data; Finite mixture model; Huntington’s disease; Kin-cohort design; Quantitative trait locus
10.  Dimension reduction with gene expression data using targeted variable importance measurement 
BMC Bioinformatics  2011;12:312.
When a large number of candidate variables are present, a dimension reduction procedure is usually conducted to reduce the variable space before the subsequent analysis is carried out. The goal of dimension reduction is to find a list of candidate genes with a more operable length ideally including all the relevant genes. Leaving many uninformative genes in the analysis can lead to biased estimates and reduced power. Therefore, dimension reduction is often considered a necessary predecessor of the analysis because it can not only reduce the cost of handling numerous variables, but also has the potential to improve the performance of the downstream analysis algorithms.
We propose a TMLE-VIM dimension reduction procedure based on the variable importance measurement (VIM) in the frame work of targeted maximum likelihood estimation (TMLE). TMLE is an extension of maximum likelihood estimation targeting the parameter of interest. TMLE-VIM is a two-stage procedure. The first stage resorts to a machine learning algorithm, and the second step improves the first stage estimation with respect to the parameter of interest.
We demonstrate with simulations and data analyses that our approach not only enjoys the prediction power of machine learning algorithms, but also accounts for the correlation structures among variables and therefore produces better variable rankings. When utilized in dimension reduction, TMLE-VIM can help to obtain the shortest possible list with the most truly associated variables.
PMCID: PMC3166941  PMID: 21849016
11.  Marginal analysis of panel counts through estimating functions 
Biometrika  2009;96(2):445-456.
We develop nonparametric estimation procedures for the marginal mean function of a counting process based on periodic observations, using two types of self-consistent estimating equations. The first is derived from the likelihood studied in Wellner & Zhang (2000), assuming a Poisson counting process, and gives a nondecreasing estimator, which is the same as the nonparametric maximum likelihood estimator of Wellner & Zhang and thus is consistent without the Poisson assumption. Motivated by the construction of parametric generalized estimating equations, the second type is a set of data-adaptive quasi-score functions, which are likelihood estimating functions under a mixed-Poisson assumption. We evaluate the procedures via simulation, and illustrate them with the data from a bladder cancer study.
PMCID: PMC2698463  PMID: 19543426
Counting process; Interval censoring; Marginal mean function; Nonparametric estimation; Quasi-score function
12.  A two-level structural equation model approach for analyzing multivariate longitudinal responses 
Statistics in medicine  2008;27(16):3017-3041.
The analysis of longitudinal data to study changes in variables measured repeatedly over time has received considerable attention in many fields. This paper proposes a two-level structural equation model for analyzing multivariate longitudinal responses that are mixed continuous and ordered categorical variables. The first-level model is defined for measures taken at each time point nested within individuals for investigating their characteristics that are changed with time. The second level is defined for individuals to assess their characteristics that are invariant with time. The proposed model accommodates fixed covariates, nonlinear terms of the latent variables, and missing data. A maximum likelihood (ML) approach is developed for the estimation of parameters and model comparison. Results of a simulation study indicate that the performance of the ML estimation is satisfactory. The proposed methodology is applied to a longitudinal study concerning cocaine use.
PMCID: PMC2836235  PMID: 18416447
latent variables; longitudinal study on cocaine use; maximum likelihood; MCEM algorithm; model comparison; ordered categorical variables
13.  Semiparametric inference for a 2-stage outcome-auxiliary-dependent sampling design with continuous outcome 
Biostatistics (Oxford, England)  2011;12(3):521-534.
Two-stage design has long been recognized to be a cost-effective way for conducting biomedical studies. In many trials, auxiliary covariate information may also be available, and it is of interest to exploit these auxiliary data to improve the efficiency of inferences. In this paper, we propose a 2-stage design with continuous outcome where the second-stage data is sampled with an “outcome-auxiliary-dependent sampling” (OADS) scheme. We propose an estimator which is the maximizer for an estimated likelihood function. We show that the proposed estimator is consistent and asymptotically normally distributed. The simulation study indicates that greater study efficiency gains can be achieved under the proposed 2-stage OADS design by utilizing the auxiliary covariate information when compared with other alternative sampling schemes. We illustrate the proposed method by analyzing a data set from an environmental epidemiologic study.
PMCID: PMC3114654  PMID: 21252082
Auxiliary covariate; Kernel smoothing; Outcome-auxiliary-dependent sampling; 2-stage sampling design
14.  Cox Regression in Nested Case-Control Studies with Auxiliary Covariates 
Biometrics  2009;66(2):374-381.
Nested case-control (NCC) design is a popular sampling method in large epidemiologic studies for its cost effectiveness to investigate the temporal relationship of diseases with environmental exposures or biological precursors. Thomas’ maximum partial likelihood estimator is commonly used to estimate the regression parameters in Cox’s model for NCC data. In this paper, we consider a situation that failure/censoring information and some crude covariates are available for the entire cohort in addition to NCC data and propose an improved estimator that is asymptotically more efficient than Thomas’ estimator. We adopt a projection approach that, heretofore, has only been employed in situations of random validation sampling and show that it can be well adapted to NCC designs where the sampling scheme is a dynamic process and is not independent for controls. Under certain conditions, consistency and asymptotic normality of the proposed estimator are established and a consistent variance estimator is also developed. Furthermore, a simplified approximate estimator is proposed when the disease is rare. Extensive simulations are conducted to evaluate the finite sample performance of our proposed estimators and to compare the efficiency with Thomas’ estimator and other competing estimators. Moreover, sensitivity analyses are conducted to demonstrate the behavior of the proposed estimator when model assumptions are violated, and we find that the biases are reasonably small in realistic situations. We further demonstrate the proposed method with data from studies on Wilms’ tumor.
PMCID: PMC2889133  PMID: 19508242
Counting process; Cox proportional hazards model; Martingale; Risk set sampling; Survival analysis
15.  A Method for Subsampling Terrestrial Invertebrate Samples in the Laboratory: Estimating Abundance and Taxa Richness 
Significant progress has been made in developing subsampling techniques to process large samples of aquatic invertebrates. However, limited information is available regarding subsampling techniques for terrestrial invertebrate samples. Therefore a novel subsampling procedure was evaluated for processing samples of terrestrial invertebrates collected using two common field techniques: pitfall and pan traps. A three-phase sorting protocol was developed for estimating abundance and taxa richness of invertebrates. First, large invertebrates and plant material were removed from the sample using a sieve with a 4 mm mesh size. Second, the sample was poured into a specially designed, gridded sampling tray, and 16 cells, comprising 25% of the sampling tray, were randomly subsampled and processed. Third, the remainder of the sample was scanned for 4–7 min to record rare taxa missed in the second phase. To compare estimated abundance and taxa richness with the true values of these variables for the samples, the remainder of each sample was processed completely. The results were analyzed relative to three sample size categories: samples with less than 250 invertebrates (low abundance samples), samples with 250–500 invertebrates (moderate abundance samples), and samples with more than 500 invertebrates (high abundance samples). The number of invertebrates estimated after subsampling eight or more cells was highly precise for all sizes and types of samples. High accuracy for moderate and high abundance samples was achieved after even as few as six subsamples. However, estimates of the number of invertebrates for low abundance samples were less reliable. The subsampling technique also adequately estimated taxa richness; on average, subsampling detected 89% of taxa found in samples. Thus, the subsampling technique provided accurate data on both the abundance and taxa richness of terrestrial invertebrate samples. Importantly, subsampling greatly decreased the time required to process samples, cutting the time per sample by up to 80%. Based on these data, this subsampling technique is recommended to minimize the time and cost of processing moderate to large samples without compromising the integrity of the data and to maximize the information extracted from large terrestrial invertebrate samples. For samples with a relatively low number of invertebrates, complete counting is preferred.
PMCID: PMC3014723  PMID: 20578889
pitfall traps; laboratory sampling techniques
16.  Two-stage hierarchical modeling for analysis of subpopulations in conditional distributions 
Journal of applied statistics  2011;39(2):445-460.
In this work, we develop modeling and estimation approach for the analysis of cross-sectional clustered data with multimodal conditional distributions where the main interest is in analysis of subpopulations. It is proposed to model such data in a hierarchical model with conditional distributions viewed as finite mixtures of normal components. With a large number of observations in the lowest level clusters, a two-stage estimation approach is used. In the first stage, the normal mixture parameters in each lowest level cluster are estimated using robust methods. Robust alternatives to the maximum likelihood estimation are used to provide stable results even for data with conditional distributions such that their components may not quite meet normality assumptions. Then the lowest level cluster-specific means and standard deviations are modeled in a mixed effects model in the second stage. A small simulation study was conducted to compare performance of finite normal mixture population parameter estimates based on robust and maximum likelihood estimation in stage 1. The proposed modeling approach is illustrated through the analysis of mice tendon fibril diameters data. Analyses results address genotype differences between corresponding components in the mixtures and demonstrate advantages of robust estimation in stage 1.
PMCID: PMC3329128  PMID: 22523443
Robust finite normal mixture; Weighted likelihood estimator; Hierarchical models; Mixed effects models; Two-stage estimation
17.  Likelihood-based analysis of longitudinal data from outcome-related sampling designs 
Biometrics  2013;70(1):44-52.
Investigators commonly gather longitudinal data to assess changes in responses over time and to relate these changes to within-subject changes in predictors. With rare or expensive outcomes such as uncommon diseases and costly radiologic measurements, outcome-dependent, and more generally outcome-related, sampling plans can improve estimation efficiency and reduce cost. Longitudinal follow up of subjects gathered in an initial outcome-related sample can then be used to study the trajectories of responses over time and to assess the association of changes in predictors within subjects with change in response. In this paper we develop two likelihood-based approaches for fitting generalized linear mixed models (GLMMs) to longitudinal data from a wide variety of outcome-related sampling designs. The first is an extension of the semi-parametric maximum likelihood approach developed in and applies quite generally. The second approach is an adaptation of standard conditional likelihood methods and is limited to random intercept models with a canonical link. Data from a study of Attention Deficit Hyperactivity Disorder in children motivates the work and illustrates the findings.
PMCID: PMC3954410  PMID: 24571396
Conditional likelihood; Retrospective sampling; Subject-specific models
18.  Choosing Models for Health Care Cost Analyses: Issues of Nonlinearity and Endogeneity 
Health services research  2012;47(6):10.1111/j.1475-6773.2012.01414.x.
To compare methods of analyzing endogenous treatment effect models for nonlinear outcomes and illustrate the impact of model specification on estimates of treatment effects such as healthcare costs.
Data Sources
Secondary data on cost and utilization for inpatients hospitalized in five Veterans Affairs acute care facilities in 2005–2006.
Study Design
We compare results from analyses with full information maximum simulated likelihood (FIMSL); control function (CF) approaches employing different types and functional forms for the residuals, including the special case of two-stage residual inclusion; and two-stage least squares (2SLS). As an example, we examine the effect of an inpatient palliative care (PC) consultation on direct costs of care per day.
Data Collection/Extraction Methods
We analyzed data for 3,389 inpatients with one or more life-limiting diseases.
Principal Findings
The distribution of average treatment effects on the treated and local average treatment effects of a PC consultation depended on model specification. CF and FIMSL estimates were more similar to each other than to 2SLS estimates. CF estimates were sensitive to choice and functional form of residual.
When modeling cost or other nonlinear data with endogeneity, one should be aware of the impact of model specification and treatment effect choice on results.
PMCID: PMC3523380  PMID: 22524165
Costs; Endogeneity; Non-Linear Models; Treatment Effects; Palliative Care
19.  A weighted combination of pseudo-likelihood estimators for longitudinal binary data subject to nonignorable non-monotone missingness 
Statistics in medicine  2010;29(14):1511-1521.
For longitudinal binary data with non-monotone non-ignorably missing outcomes over time, a full likelihood approach is complicated algebraically, and with many follow-up times, maximum likelihood estimation can be computationally prohibitive. As alternatives, two pseudo-likelihood approaches have been proposed that use minimal parametric assumptions. One formulation requires specification of the marginal distributions of the outcome and missing data mechanism at each time point, but uses an “independence working assumption,” i.e., an assumption that observations are independent over time. Another method avoids having to estimate the missing data mechanism by formulating a “protective estimator.” In simulations, these two estimators can be very inefficient, both for estimating time trends in the first case and for estimating both time-varying and time-stationary effects in the second. In this paper, we propose use of the optimal weighted combination of these two estimators, and in simulations we show that the optimal weighted combination can be much more efficient than either estimator alone. Finally, the proposed method is used to analyze data from two longitudinal clinical trials of HIV-infected patients.
PMCID: PMC2996053  PMID: 20205269
20.  Maximum Likelihood Estimations and EM Algorithms with Length-biased Data 
Length-biased sampling has been well recognized in economics, industrial reliability, etiology applications, epidemiological, genetic and cancer screening studies. Length-biased right-censored data have a unique data structure different from traditional survival data. The nonparametric and semiparametric estimations and inference methods for traditional survival data are not directly applicable for length-biased right-censored data. We propose new expectation-maximization algorithms for estimations based on full likelihoods involving infinite dimensional parameters under three settings for length-biased data: estimating nonparametric distribution function, estimating nonparametric hazard function under an increasing failure rate constraint, and jointly estimating baseline hazards function and the covariate coefficients under the Cox proportional hazards model. Extensive empirical simulation studies show that the maximum likelihood estimators perform well with moderate sample sizes and lead to more efficient estimators compared to the estimating equation approaches. The proposed estimates are also more robust to various right-censoring mechanisms. We prove the strong consistency properties of the estimators, and establish the asymptotic normality of the semi-parametric maximum likelihood estimators under the Cox model using modern empirical processes theory. We apply the proposed methods to a prevalent cohort medical study. Supplemental materials are available online.
PMCID: PMC3273908  PMID: 22323840
Cox regression model; EM algorithm; Increasing failure rate; Non-parametric likelihood; Profile likelihood; Right-censored data
21.  Correlation and studies of habitat selection: problem, red herring or opportunity? 
With the advent of new technologies, animal locations are being collected at ever finer spatio-temporal scales. We review analytical methods for dealing with correlated data in the context of resource selection, including post hoc variance inflation techniques, ‘two-stage’ approaches based on models fit to each individual, generalized estimating equations and hierarchical mixed-effects models. These methods are applicable to a wide range of correlated data problems, but can be difficult to apply and remain especially challenging for use–availability sampling designs because the correlation structure for combinations of used and available points are not likely to follow common parametric forms. We also review emerging approaches to studying habitat selection that use fine-scale temporal data to arrive at biologically based definitions of available habitat, while naturally accounting for autocorrelation by modelling animal movement between telemetry locations. Sophisticated analyses that explicitly model correlation rather than consider it a nuisance, like mixed effects and state-space models, offer potentially novel insights into the process of resource selection, but additional work is needed to make them more generally applicable to large datasets based on the use–availability designs. Until then, variance inflation techniques and two-stage approaches should offer pragmatic and flexible approaches to modelling correlated data.
PMCID: PMC2894958  PMID: 20566500
generalized estimating equation; generalized linear mixed model; hierarchical model; resource-selection function; telemetry; use–availability
22.  Analysis of Two-sample Censored Data Using a Semiparametric Mixture Model 
In this article we study a semiparametric mixture model for the two-sample problem with right censored data. The model implies that the densities for the continuous outcomes are related by a parametric tilt but otherwise unspecified. It provides a useful alternative to the Cox (1972) proportional hazards model for the comparison of treatments based on right censored survival data. We propose an iterative algorithm for the semiparametric maximum likelihood estimates of the parametric and nonparametric components of the model. The performance of the proposed method is studied using simulation. We illustrate our method in an application to melanoma.
PMCID: PMC2901133  PMID: 20622987
Biased sampling; EM algorithm; maximum likelihood estimation; mixture model; semiparametric model
23.  Improved Horvitz-Thompson Estimation of Model Parameters from Two-phase Stratified Samples: Applications in Epidemiology 
The case-cohort study involves two-phase sampling: simple random sampling from an infinite super-population at phase one and stratified random sampling from a finite cohort at phase two. Standard analyses of case-cohort data involve solution of inverse probability weighted (IPW) estimating equations, with weights determined by the known phase two sampling fractions. The variance of parameter estimates in (semi)parametric models, including the Cox model, is the sum of two terms: (i) the model based variance of the usual estimates that would be calculated if full data were available for the entire cohort; and (ii) the design based variance from IPW estimation of the unknown cohort total of the efficient influence function (IF) contributions. This second variance component may be reduced by adjusting the sampling weights, either by calibration to known cohort totals of auxiliary variables correlated with the IF contributions or by their estimation using these same auxiliary variables. Both adjustment methods are implemented in the R survey package. We derive the limit laws of coefficients estimated using adjusted weights. The asymptotic results suggest practical methods for construction of auxiliary variables that are evaluated by simulation of case-cohort samples from the National Wilms Tumor Study and by log-linear modeling of case-cohort data from the Atherosclerosis Risk in Communities Study. Although not semiparametric efficient, estimators based on adjusted weights may come close to achieving full efficiency within the class of augmented IPW estimators.
PMCID: PMC2822363  PMID: 20174455
Calibration; Case-cohort; Estimation; Log-linear model; Semiparametric
24.  Modelling the initial phase of an epidemic using incidence and infection network data: 2009 H1N1 pandemic in Israel as a case study 
This paper presents new computational and modelling tools for studying the dynamics of an epidemic in its initial stages that use both available incidence time series and data describing the population's infection network structure. The work is motivated by data collected at the beginning of the H1N1 pandemic outbreak in Israel in the summer of 2009. We formulated a new discrete-time stochastic epidemic SIR (susceptible-infected-recovered) model that explicitly takes into account the disease's specific generation-time distribution and the intrinsic demographic stochasticity inherent to the infection process. Moreover, in contrast with many other modelling approaches, the model allows direct analytical derivation of estimates for the effective reproductive number (Re) and of their credible intervals, by maximum likelihood and Bayesian methods. The basic model can be extended to include age–class structure, and a maximum likelihood methodology allows us to estimate the model's next-generation matrix by combining two types of data: (i) the incidence series of each age group, and (ii) infection network data that provide partial information of ‘who-infected-who’. Unlike other approaches for estimating the next-generation matrix, the method developed here does not require making a priori assumptions about the structure of the next-generation matrix. We show, using a simulation study, that even a relatively small amount of information about the infection network greatly improves the accuracy of estimation of the next-generation matrix. The method is applied in practice to estimate the next-generation matrix from the Israeli H1N1 pandemic data. The tools developed here should be of practical importance for future investigations of epidemics during their initial stages. However, they require the availability of data which represent a random sample of the real epidemic process. We discuss the conditions under which reporting rates may or may not influence our estimated quantities and the effects of bias.
PMCID: PMC3104348  PMID: 21247949
epidemic modelling; H1N1 influenza; maximum likelihood; model fitting; next-generation matrix
25.  A Causal Framework for Understanding the Effect of Losses to Follow-up on Epidemiologic Analyses in Clinic-based Cohorts: The Case of HIV-infected Patients on Antiretroviral Therapy in Africa 
American Journal of Epidemiology  2012;175(10):1080-1087.
Although clinic-based cohorts are most representative of the “real world,” they are susceptible to loss to follow-up. Strategies for managing the impact of loss to follow-up are therefore needed to maximize the value of studies conducted in these cohorts. The authors evaluated adult patients starting antiretroviral therapy at an HIV/AIDS clinic in Uganda, where 29% of patients were lost to follow-up after 2 years (January 1, 2004–September 30, 2007). Unweighted, inverse probability of censoring weighted (IPCW), and sampling-based approaches (using supplemental data from a sample of lost patients subsequently tracked in the community) were used to identify the predictive value of sex on mortality. Directed acyclic graphs (DAGs) were used to explore the structural basis for bias in each approach. Among 3,628 patients, unweighted and IPCW analyses found men to have higher mortality than women, whereas the sampling-based approach did not. DAGs encoding knowledge about the data-generating process, including the fact that death is a cause of being classified as lost to follow-up in this setting, revealed “collider” bias in the unweighted and IPCW approaches. In a clinic-based cohort in Africa, unweighted and IPCW approaches—which rely on the “missing at random” assumption—yielded biased estimates. A sampling-based approach can in general strengthen epidemiologic analyses conducted in many clinic-based cohorts, including those examining other diseases.
PMCID: PMC3353135  PMID: 22306557
Africa; antiretroviral therapy; clinic-based cohorts; directed acyclic graphs; informative censoring; inverse probability of censoring weights; loss to follow-up; missing at random

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