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1.  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.
doi:10.2202/1557-4679.1181
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
2.  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.
doi:10.2202/1557-4679.1182
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
3.  A Targeted Maximum Likelihood Estimator for Two-Stage Designs 
We consider two-stage sampling designs, including so-called nested case control studies, where one takes a random sample from a target population and completes measurements on each subject in the first stage. The second stage involves drawing a subsample from the original sample, collecting additional data on the subsample. This data structure can be viewed as a missing data structure on the full-data structure collected in the second-stage of the study. Methods for analyzing two-stage designs include parametric maximum likelihood estimation and estimating equation methodology. We propose an inverse probability of censoring weighted targeted maximum likelihood estimator (IPCW-TMLE) in two-stage sampling designs and present simulation studies featuring this estimator.
doi:10.2202/1557-4679.1217
PMCID: PMC3083136  PMID: 21556285
two-stage designs; targeted maximum likelihood estimators; nested case control studies; double robust estimation
4.  Dimension reduction with gene expression data using targeted variable importance measurement 
BMC Bioinformatics  2011;12:312.
Background
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.
Results
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.
Conclusions
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.
doi:10.1186/1471-2105-12-312
PMCID: PMC3166941  PMID: 21849016
5.  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.
doi:10.2202/1557-4679.1307
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
6.  Covariate adjustment in randomized trials with binary outcomes: Targeted maximum likelihood estimation 
Statistics in medicine  2009;28(1):39-64.
SUMMARY
Covariate adjustment using linear models for continuous outcomes in randomized trials has been shown to increase efficiency and power over the unadjusted method in estimating the marginal effect of treatment. However, for binary outcomes, investigators generally rely on the unadjusted estimate as the literature indicates that covariate-adjusted estimates based on the logistic regression models are less efficient. The crucial step that has been missing when adjusting for covariates is that one must integrate/average the adjusted estimate over those covariates in order to obtain the marginal effect. We apply the method of targeted maximum likelihood estimation (tMLE) to obtain estimators for the marginal effect using covariate adjustment for binary outcomes. We show that the covariate adjustment in randomized trials using the logistic regression models can be mapped, by averaging over the covariate(s), to obtain a fully robust and efficient estimator of the marginal effect, which equals a targeted maximum likelihood estimator. This tMLE is obtained by simply adding a clever covariate to a fixed initial regression. We present simulation studies that demonstrate that this tMLE increases efficiency and power over the unadjusted method, particularly for smaller sample sizes, even when the regression model is mis-specified.
doi:10.1002/sim.3445
PMCID: PMC2857590  PMID: 18985634
clinical trails; efficiency; covariate adjustment; variable selection
7.  Efficient and robust method for comparing the immunogenicity of candidate vaccines in randomized clinical trials 
Vaccine  2008;27(3):396-401.
In randomized clinical trials designed to compare the magnitude of vaccine-induced immune responses between vaccination regimens, the statistical method used for the analysis typically does not account for baseline participant characteristics. This article shows that incorporating baseline variables predictive of the immunogenicity study endpoint can provide large gains in precision and power for estimation and testing of the group mean difference (requiring fewer subjects for the same scientific output) compared to conventional methods, and recommends the “semiparametric efficient” method described in Tsiatis et al. [Tsiatis AA, Davidian M, Zhang M, Lu X. Covariate adjustment for two-sample treatment comparisons in randomized clinical trials: a principled yet flexible approach. Stat Med 2007. doi:10.1002/sim.3113] for practical use. As such, vaccine clinical trial programs can be improved (1) by investigating baseline predictors (e.g., readouts from laboratory assays) of vaccine-induced immune responses, and (2) by implementing the proposed semiparametric efficient method in trials where baseline predictors are available.
doi:10.1016/j.vaccine.2008.10.083
PMCID: PMC2653280  PMID: 19022314
Immune responses; Statistical analysis; Vaccine trial
8.  Increasing the Efficiency of Prevention Trials by Incorporating Baseline Covariates 
Summary
Most randomized efficacy trials of interventions to prevent HIV or other infectious diseases have assessed intervention efficacy by a method that either does not incorporate baseline covariates, or that incorporates them in a non-robust or inefficient way. Yet, it has long been known that randomized treatment effects can be assessed with greater efficiency by incorporating baseline covariates that predict the response variable. Tsiatis et al. (2007) and Zhang et al. (2008) advocated a semiparametric efficient approach, based on the theory of Robins et al. (1994), for consistently estimating randomized treatment effects that optimally incorporates predictive baseline covariates, without any parametric assumptions. They stressed the objectivity of the approach, which is achieved by separating the modeling of baseline predictors from the estimation of the treatment effect. While their work adequately justifies implementation of the method for large Phase 3 trials (because its optimality is in terms of asymptotic properties), its performance for intermediate-sized screening Phase 2b efficacy trials, which are increasing in frequency, is unknown. Furthermore, the past work did not consider a right-censored time-to-event endpoint, which is the usual primary endpoint for a prevention trial. For Phase 2b HIV vaccine efficacy trials, we study finite-sample performance of Zhang et al.'s (2008) method for a dichotomous endpoint, and develop and study an adaptation of this method to a discrete right-censored time-to-event endpoint. We show that, given the predictive capacity of baseline covariates collected in real HIV prevention trials, the methods achieve 5-15% gains in efficiency compared to methods in current use. We apply the methods to the first HIV vaccine efficacy trial. This work supports implementation of the discrete failure time method for prevention trials.
doi:10.2202/1948-4690.1002
PMCID: PMC2997740  PMID: 21152074
Auxiliary; Covariate Adjustment; Intermediate-sized Phase 2b Efficacy Trial; Semiparametric Efficiency
9.  Repeated Measures Semiparametric Regression Using Targeted Maximum Likelihood Methodology with Application to Transcription Factor Activity Discovery 
In longitudinal and repeated measures data analysis, often the goal is to determine the effect of a treatment or aspect on a particular outcome (e.g., disease progression). We consider a semiparametric repeated measures regression model, where the parametric component models effect of the variable of interest and any modification by other covariates. The expectation of this parametric component over the other covariates is a measure of variable importance. Here, we present a targeted maximum likelihood estimator of the finite dimensional regression parameter, which is easily estimated using standard software for generalized estimating equations.
The targeted maximum likelihood method provides double robust and locally efficient estimates of the variable importance parameters and inference based on the influence curve. We demonstrate these properties through simulation under correct and incorrect model specification, and apply our method in practice to estimating the activity of transcription factor (TF) over cell cycle in yeast. We specifically target the importance of SWI4, SWI6, MBP1, MCM1, ACE2, FKH2, NDD1, and SWI5.
The semiparametric model allows us to determine the importance of a TF at specific time points by specifying time indicators as potential effect modifiers of the TF. Our results are promising, showing significant importance trends during the expected time periods. This methodology can also be used as a variable importance analysis tool to assess the effect of a large number of variables such as gene expressions or single nucleotide polymorphisms.
doi:10.2202/1544-6115.1553
PMCID: PMC3122882  PMID: 21291412
targeted maximum likelihood; semiparametric; repeated measures; longitudinal; transcription factors
10.  Effect Modification by Sex and Baseline CD4+ Cell Count Among Adults Receiving Combination Antiretroviral Therapy in Botswana: Results from a Clinical Trial 
Abstract
The Tshepo study was the first clinical trial to evaluate outcomes of adults receiving nevirapine (NVP)-based versus efavirenz (EFV)-based combination antiretroviral therapy (cART) in Botswana. This was a 3 year study (n=650) comparing the efficacy and tolerability of various first-line cART regimens, stratified by baseline CD4+: <200 (low) vs. 201-350 (high). Using targeted maximum likelihood estimation (TMLE), we retrospectively evaluated the causal effect of assigned NNRTI on time to virologic failure or death [intent-to-treat (ITT)] and time to minimum of virologic failure, death, or treatment modifying toxicity [time to loss of virological response (TLOVR)] by sex and baseline CD4+. Sex did significantly modify the effect of EFV versus NVP for both the ITT and TLOVR outcomes with risk differences in the probability of survival of males versus the females of approximately 6% (p=0.015) and 12% (p=0.001), respectively. Baseline CD4+ also modified the effect of EFV versus NVP for the TLOVR outcome, with a mean difference in survival probability of approximately 12% (p=0.023) in the high versus low CD4+ cell count group. TMLE appears to be an efficient technique that allows for the clinically meaningful delineation and interpretation of the causal effect of NNRTI treatment and effect modification by sex and baseline CD4+ cell count strata in this study. EFV-treated women and NVP-treated men had more favorable cART outcomes. In addition, adults initiating EFV-based cART at higher baseline CD4+ cell count values had more favorable outcomes compared to those initiating NVP-based cART.
doi:10.1089/aid.2011.0349
PMCID: PMC3423643  PMID: 22309114
11.  On doubly robust estimation in a semiparametric odds ratio model 
Biometrika  2009;97(1):171-180.
We consider the doubly robust estimation of the parameters in a semiparametric conditional odds ratio model. Our estimators are consistent and asymptotically normal in a union model that assumes either of two variation independent baseline functions is correctly modelled but not necessarily both. Furthermore, when either outcome has finite support, our estimators are semiparametric efficient in the union model at the intersection submodel where both nuisance functions models are correct. For general outcomes, we obtain doubly robust estimators that are nearly efficient at the intersection submodel. Our methods are easy to implement as they do not require the use of the alternating conditional expectations algorithm of Chen (2007).
doi:10.1093/biomet/asp062
PMCID: PMC3412601  PMID: 23049119
Doubly robust; Generalized odds ratio; Locally efficient; Semiparametric logistic regression
12.  Estimation and inference based on Neumann series approximation to locally efficient score in missing data problems 
Theory on semiparametric efficient estimation in missing data problems has been systematically developed by Robins and his coauthors. Except in relatively simple problems, semiparametric efficient scores cannot be expressed in closed forms. Instead, the efficient scores are often expressed as solutions to integral equations. Neumann series was proposed in the form of successive approximation to the efficient scores in those situations. Statistical properties of the estimator based on the Neumann series approximation are difficult to obtain and as a result, have not been clearly studied. In this paper, we reformulate the successive approximation in a simple iterative form and study the statistical properties of the estimator based on the reformulation. We show that a doubly-robust locally-efficient estimator can be obtained following the algorithm in robustifying the likelihood score. The results can be applied to, among others, the parametric regression, the marginal regression, and the Cox regression when data are subject to missing values and the missing data are missing at random. A simulation study is conducted to evaluate the performance of the approach and a real data example is analyzed to demonstrate the use of the approach.
doi:10.1111/j.1467-9469.2009.00646.x
PMCID: PMC2811346  PMID: 20161609
auxiliary covariates; information operator; non-monotone missing pattern; weighted estimating equations
13.  Semiparametric Maximum Likelihood Estimation in Normal Transformation Models for Bivariate Survival Data 
Biometrika  2008;95(4):947-960.
SUMMARY
We consider a class of semiparametric normal transformation models for right censored bivariate failure times. Nonparametric hazard rate models are transformed to a standard normal model and a joint normal distribution is assumed for the bivariate vector of transformed variates. A semiparametric maximum likelihood estimation procedure is developed for estimating the marginal survival distribution and the pairwise correlation parameters. This produces an efficient estimator of the correlation parameter of the semiparametric normal transformation model, which characterizes the bivariate dependence of bivariate survival outcomes. In addition, a simple positive-mass-redistribution algorithm can be used to implement the estimation procedures. Since the likelihood function involves infinite-dimensional parameters, the empirical process theory is utilized to study the asymptotic properties of the proposed estimators, which are shown to be consistent, asymptotically normal and semiparametric efficient. A simple estimator for the variance of the estimates is also derived. The finite sample performance is evaluated via extensive simulations.
doi:10.1093/biomet/asn049
PMCID: PMC2600666  PMID: 19079778
Asymptotic normality; Bivariate failure time; Consistency; Semiparametric efficiency; Semiparametric maximum likelihood estimate; Semiparametric normal transformation
14.  A Semiparametric Regression Cure Model for Interval-Censored Data 
Motivated by medical studies in which patients could be cured of disease but the disease event time may be subject to interval censoring, we presents a semiparametric non-mixture cure model for the regression analysis of interval-censored time-to-event datxa. We develop semiparametric maximum likelihood estimation for the model using the expectation-maximization method for interval-censored data. The maximization step for the baseline function is nonparametric and numerically challenging. We develop an efficient and numerically stable algorithm via modern convex optimization techniques, yielding a self-consistency algorithm for the maximization step. We prove the strong consistency of the maximum likelihood estimators under the Hellinger distance, which is an appropriate metric for the asymptotic property of the estimators for interval-censored data. We assess the performance of the estimators in a simulation study with small to moderate sample sizes. To illustrate the method, we also analyze a real data set from a medical study for the biochemical recurrence of prostate cancer among patients who have undergone radical prostatectomy. Supplemental materials for the computational algorithm are available online.
PMCID: PMC2846840  PMID: 20354594
Convex optimization; Hellinger consistency; maximum likelihood estimation; primal-dual interior-point method; prostate cancer
15.  Adjustment for Missingness Using Auxiliary Information in Semiparametric Regression 
Biometrics  2009;66(1):115-122.
Summary
In this article, we study the estimation of mean response and regression coefficient in semiparametric regression problems when response variable is subject to nonrandom missingness. When the missingness is independent of the response conditional on high-dimensional auxiliary information, the parametric approach may misspecify the relationship between covariates and response while the nonparametric approach is infeasible because of the curse of dimensionality. To overcome this, we study a model-based approach to condense the auxiliary information and estimate the parameters of interest nonparametrically on the condensed covariate space. Our estimators possess the double robustness property, i.e., they are consistent whenever the model for the response given auxiliary covariates or the model for the missingness given auxiliary covariate is correct. We conduct a number of simulations to compare the numerical performance between our estimators and other existing estimators in the current missing data literature, including the propensity score approach and the inverse probability weighted estimating equation. A set of real data is used to illustrate our approach.
doi:10.1111/j.1541-0420.2009.01231.x
PMCID: PMC3148802  PMID: 19432773
Auxiliary covariate; High-dimensional data; Kernel estimation; Missing at random; Semiparametric regression
16.  Likelihood approaches for the invariant density ratio model with biased-sampling data 
Biometrika  2012;99(2):363-378.
The full likelihood approach in statistical analysis is regarded as the most efficient means for estimation and inference. For complex length-biased failure time data, computational algorithms and theoretical properties are not readily available, especially when a likelihood function involves infinite-dimensional parameters. Relying on the invariance property of length-biased failure time data under the semiparametric density ratio model, we present two likelihood approaches for the estimation and assessment of the difference between two survival distributions. The most efficient maximum likelihood estimators are obtained by the em algorithm and profile likelihood. We also provide a simple numerical method for estimation and inference based on conditional likelihood, which can be generalized to k-arm settings. Unlike conventional survival data, the mean of the population failure times can be consistently estimated given right-censored length-biased data under mild regularity conditions. To check the semiparametric density ratio model assumption, we use a test statistic based on the area between two survival distributions. Simulation studies confirm that the full likelihood estimators are more efficient than the conditional likelihood estimators. We analyse an epidemiological study to illustrate the proposed methods.
doi:10.1093/biomet/ass008
PMCID: PMC3635710  PMID: 23843663
Conditional likelihood; Density ratio model; em algorithm; Length-biased sampling; Maximum likelihood approach
17.  Estimating the age of retrotransposon subfamilies using maximum likelihood 
Genomics  2009;94(1):78-82.
We present a maximum likelihood model to estimate the age of retrotransposon subfamilies. This method is designed around a master-gene model which assumes a constant retrotransposition rate. The statistical properties of this model and an ad hoc estimation procedure are compared using two simulated data sets. We also test whether each estimation procedure is robust to violation of the master gene model. According to our results, both estimation procedures are accurate under the master gene model. While both methods tend to overestimate ages under the intermediate model, the maximum likelihood estimate is significantly less inflated than the ad hoc estimate. We estimate the ages of two subfamilies of human-specific LINE-I insertions using both estimation procedures. By calculating confidence intervals around the maximum likelihood estimate, our model can both provide an estimate of retrotransposon subfamily age and describe the range of subfamily ages consistent with the data.
doi:10.1016/j.ygeno.2009.04.002
PMCID: PMC2703446  PMID: 19379804
LINE-1; maximum likelihood; subfamily age; retrotransposon
18.  A Joint Model for Longitudinal Measurements and Survival Data in the Presence of Multiple Failure Types 
Biometrics  2007;64(3):762-771.
Summary
In this article we study a joint model for longitudinal measurements and competing risks survival data. Our joint model provides a flexible approach to handle possible nonignorable missing data in the longitudinal measurements due to dropout. It is also an extension of previous joint models with a single failure type, offering a possible way to model informatively censored events as a competing risk. Our model consists of a linear mixed effects submodel for the longitudinal outcome and a proportional cause-specific hazards frailty submodel (Prentice et al., 1978, Biometrics 34, 541-554) for the competing risks survival data, linked together by some latent random effects. We propose to obtain the maximum likelihood estimates of the parameters by an expectation maximization (EM) algorithm and estimate their standard errors using a profile likelihood method. The developed method works well in our simulation studies and is applied to a clinical trial for the scleroderma lung disease.
doi:10.1111/j.1541-0420.2007.00952.x
PMCID: PMC2751647  PMID: 18162112
Cause-specific hazard; Competing risks; EM algorithm; Joint modeling; Longitudinal data; Mixed effects model
19.  Penalized Estimating Functions and Variable Selection in Semiparametric Regression Models 
We propose a general strategy for variable selection in semiparametric regression models by penalizing appropriate estimating functions. Important applications include semiparametric linear regression with censored responses and semiparametric regression with missing predictors. Unlike the existing penalized maximum likelihood estimators, the proposed penalized estimating functions may not pertain to the derivatives of any objective functions and may be discrete in the regression coefficients. We establish a general asymptotic theory for penalized estimating functions and present suitable numerical algorithms to implement the proposed estimators. In addition, we develop a resampling technique to estimate the variances of the estimated regression coefficients when the asymptotic variances cannot be evaluated directly. Simulation studies demonstrate that the proposed methods perform well in variable selection and variance estimation. We illustrate our methods using data from the Paul Coverdell Stroke Registry.
doi:10.1198/016214508000000184
PMCID: PMC2850080  PMID: 20376193
Accelerated failure time model; Buckley-James estimator; Censoring; Least absolute shrinkage and selection operator; Least squares; Linear regression; Missing data; Smoothly clipped absolute deviation
20.  A GENERAL ASYMPTOTIC THEORY FOR MAXIMUM LIKELIHOOD ESTIMATION IN SEMIPARAMETRIC REGRESSION MODELS WITH CENSORED DATA 
Statistica Sinica  2010;20(2):871-910.
We establish a general asymptotic theory for nonparametric maximum likelihood estimation in semiparametric regression models with right censored data. We identify a set of regularity conditions under which the nonparametric maximum likelihood estimators are consistent, asymptotically normal, and asymptotically efficient with a covariance matrix that can be consistently estimated by the inverse information matrix or the profile likelihood method. The general theory allows one to obtain the desired asymptotic properties of the nonparametric maximum likelihood estimators for any specific problem by verifying a set of conditions rather than by proving technical results from first principles. We demonstrate the usefulness of this powerful theory through a variety of examples.
PMCID: PMC2888521  PMID: 20577580
Counting process; empirical process; multivariate failure times; nonparametric likelihood; profile likelihood; survival data
21.  Maximum Likelihood Estimations and EM Algorithms with Length-biased Data 
SUMMARY
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.
doi:10.1198/jasa.2011.tm10156
PMCID: PMC3273908  PMID: 22323840
Cox regression model; EM algorithm; Increasing failure rate; Non-parametric likelihood; Profile likelihood; Right-censored data
22.  Bayesian and maximum likelihood phylogenetic analyses of protein sequence data under relative branch-length differences and model violation 
Background
Bayesian phylogenetic inference holds promise as an alternative to maximum likelihood, particularly for large molecular-sequence data sets. We have investigated the performance of Bayesian inference with empirical and simulated protein-sequence data under conditions of relative branch-length differences and model violation.
Results
With empirical protein-sequence data, Bayesian posterior probabilities provide more-generous estimates of subtree reliability than does the nonparametric bootstrap combined with maximum likelihood inference, reaching 100% posterior probability at bootstrap proportions around 80%. With simulated 7-taxon protein-sequence datasets, Bayesian posterior probabilities are somewhat more generous than bootstrap proportions, but do not saturate. Compared with likelihood, Bayesian phylogenetic inference can be as or more robust to relative branch-length differences for datasets of this size, particularly when among-sites rate variation is modeled using a gamma distribution. When the (known) correct model was used to infer trees, Bayesian inference recovered the (known) correct tree in 100% of instances in which one or two branches were up to 20-fold longer than the others. At ratios more extreme than 20-fold, topological accuracy of reconstruction degraded only slowly when only one branch was of relatively greater length, but more rapidly when there were two such branches. Under an incorrect model of sequence change, inaccurate trees were sometimes observed at less extreme branch-length ratios, and (particularly for trees with single long branches) such trees tended to be more inaccurate. The effect of model violation on accuracy of reconstruction for trees with two long branches was more variable, but gamma-corrected Bayesian inference nonetheless yielded more-accurate trees than did either maximum likelihood or uncorrected Bayesian inference across the range of conditions we examined. Assuming an exponential Bayesian prior on branch lengths did not improve, and under certain extreme conditions significantly diminished, performance. The two topology-comparison metrics we employed, edit distance and Robinson-Foulds symmetric distance, yielded different but highly complementary measures of performance.
Conclusions
Our results demonstrate that Bayesian inference can be relatively robust against biologically reasonable levels of relative branch-length differences and model violation, and thus may provide a promising alternative to maximum likelihood for inference of phylogenetic trees from protein-sequence data.
doi:10.1186/1471-2148-5-8
PMCID: PMC549035  PMID: 15676079
23.  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.
doi:10.1093/aje/kwr444
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
24.  A SIEVE M-THEOREM FOR BUNDLED PARAMETERS IN SEMIPARAMETRIC MODELS, WITH APPLICATION TO THE EFFICIENT ESTIMATION IN A LINEAR MODEL FOR CENSORED DATA* 
Annals of statistics  2011;39(6):2795-3443.
In many semiparametric models that are parameterized by two types of parameters – a Euclidean parameter of interest and an infinite-dimensional nuisance parameter, the two parameters are bundled together, i.e., the nuisance parameter is an unknown function that contains the parameter of interest as part of its argument. For example, in a linear regression model for censored survival data, the unspecified error distribution function involves the regression coefficients. Motivated by developing an efficient estimating method for the regression parameters, we propose a general sieve M-theorem for bundled parameters and apply the theorem to deriving the asymptotic theory for the sieve maximum likelihood estimation in the linear regression model for censored survival data. The numerical implementation of the proposed estimating method can be achieved through the conventional gradient-based search algorithms such as the Newton-Raphson algorithm. We show that the proposed estimator is consistent and asymptotically normal and achieves the semiparametric efficiency bound. Simulation studies demonstrate that the proposed method performs well in practical settings and yields more efficient estimates than existing estimating equation based methods. Illustration with a real data example is also provided.
PMCID: PMC3890689  PMID: 24436500
Accelerated failure time model; B-spline; bundled parameters; efficient score function; semiparametric efficiency; sieve maximum likelihood estimation
25.  Dimension reduced kernel estimation for distribution function with incomplete data 
SUMMARY
This work focuses on the estimation of distribution functions with incomplete data, where the variable of interest Y has ignorable missingness but the covariate X is always observed. When X is high dimensional, parametric approaches to incorporate X — information is encumbered by the risk of model misspecification and nonparametric approaches by the curse of dimensionality. We propose a semiparametric approach, which is developed under a nonparametric kernel regression framework, but with a parametric working index to condense the high dimensional X — information for reduced dimension. This kernel dimension reduction estimator has double robustness to model misspecification and is most efficient if the working index adequately conveys the X — information about the distribution of Y. Numerical studies indicate better performance of the semiparametric estimator over its parametric and nonparametric counterparts. We apply the kernel dimension reduction estimation to an HIV study for the effect of antiretroviral therapy on HIV virologic suppression.
doi:10.1016/j.jspi.2011.03.030
PMCID: PMC3127551  PMID: 21731174
curse of dimensionality; dimension reduction; distribution function; ignorable missingness; kernel regression; quantile

Results 1-25 (209836)