Statistical Models Commons^™

Targeted Maximum Likelihood Estimation For Dynamic And Static Longitudinal Marginal Structural Working Models, Maya L. Petersen, Joshua Schwab, Susan Gruber, Nello Blaser, Michael Schomaker, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

This paper describes a targeted maximum likelihood estimator (TMLE) for the parameters of longitudinal static and dynamic marginal structural models. We consider a longitudinal data structure consisting of baseline covariates, time-dependent intervention nodes, intermediate time-dependent covariates, and a possibly time dependent outcome. The intervention nodes at each time point can include a binary treatment as well as a right-censoring indicator. Given a class of dynamic or static interventions, a marginal structural model is used to model the mean of the intervention specific counterfactual outcome as a function of the intervention, time point, and possibly a subset of baseline covariates. Because …

Confidence Intervals For Negative Binomial Random Variables Of High Dispersion, David Shilane, Alan E. Hubbard, S N. Evans

U.C. Berkeley Division of Biostatistics Working Paper Series

This paper considers the problem of constructing confidence intervals for the mean of a Negative Binomial random variable based upon sampled data. When the sample size is large, we traditionally rely upon a Normal distribution approximation to construct these intervals. However, we demonstrate that the sample mean of highly dispersed Negative Binomials exhibits a slow convergence to the Normal in distribution as a function of the sample size. As a result, standard techniques (such as the Normal approximation and bootstrap) that construct confidence intervals for the mean will typically be too narrow and significantly undercover in the case of high …

Supervised Distance Matrices: Theory And Applications To Genomics, Katherine S. Pollard, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

We propose a new approach to studying the relationship between a very high dimensional random variable and an outcome. Our method is based on a novel concept, the supervised distance matrix, which quantifies pairwise similarity between variables based on their association with the outcome. A supervised distance matrix is derived in two stages. The first stage involves a transformation based on a particular model for association. In particular, one might regress the outcome on each variable and then use the residuals or the influence curve from each regression as a data transformation. In the second stage, a choice of distance …

Confidence Intervals For The Population Mean Tailored To Small Sample Sizes, With Applications To Survey Sampling, Michael Rosenblum, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

The validity of standard confidence intervals constructed in survey sampling is based on the central limit theorem. For small sample sizes, the central limit theorem may give a poor approximation, resulting in confidence intervals that are misleading. We discuss this issue and propose methods for constructing confidence intervals for the population mean tailored to small sample sizes.

We present a simple approach for constructing confidence intervals for the population mean based on tail bounds for the sample mean that are correct for all sample sizes. Bernstein's inequality provides one such tail bound. The resulting confidence intervals have guaranteed coverage probability …

Using Regression Models To Analyze Randomized Trials: Asymptotically Valid Hypothesis Tests Despite Incorrectly Specified Models, Michael Rosenblum, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

Regression models are often used to test for cause-effect relationships from data collected in randomized trials or experiments. This practice has deservedly come under heavy scrutiny, since commonly used models such as linear and logistic regression will often not capture the actual relationships between variables, and incorrectly specified models potentially lead to incorrect conclusions. In this paper, we focus on hypothesis test of whether the treatment given in a randomized trial has any effect on the mean of the primary outcome, within strata of baseline variables such as age, sex, and health status. Our primary concern is ensuring that such …

Super Learner, Mark J. Van Der Laan, Eric C. Polley, Alan E. Hubbard

U.C. Berkeley Division of Biostatistics Working Paper Series

Previous articles (van der Laan and Dudoit (2003); van der Laan et al. (2006); Sinisi et al. (2007)) advertised and theoretically validated the use of cross-validation to select among many candidate estimators to compute a so called super learner which outperforms any of the given candidate estimators. The theoretical basis was provided for this super learner based on oracle results for the cross-validation selector (e.g., van der Laan and Dudoit (2003); van der Laan et al. (2006)) and in Sinisi et al. (2007). In addition, these papers contained a practical demonstration of the adaptivity of this so called super learner …

Multiple Tests Of Association With Biological Annotation Metadata, Sandrine Dudoit, Sunduz Keles, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

We propose a general and formal statistical framework for the multiple tests of associations between known fixed features of a genome and unknown parameters of the distribution of variable features of this genome in a population of interest. The known fixed gene-annotation profiles, corresponding to the fixed features of the genome, may concern Gene Ontology (GO) annotation, pathway membership, regulation by particular transcription factors, nucleotide sequences, or protein sequences. The unknown gene-parameter profiles, corresponding to the variable features of the genome, may be, for example, regression coefficients relating genome-wide transcript levels or DNA copy numbers to possibly censored biological and …

A Fine-Scale Linkage Disequilibrium Measure Based On Length Of Haplotype Sharing, Yan Wang, Lue Ping Zhao, Sandrine Dudoit

U.C. Berkeley Division of Biostatistics Working Paper Series

High-throughput genotyping technologies for single nucleotide polymorphisms (SNP) have enabled the recent completion of the International HapMap Project (Phase I), which has stimulated much interest in studying genome-wide linkage disequilibrium (LD) patterns. Conventional LD measures, such as D' and r-square, are two-point measurements, and their relationship with physical distance is highly noisy. We propose a new LD measure, defined in terms of the correlation coefficient for shared haplotype lengths around two loci, thereby borrowing information from multiple loci. A U-statistic-based estimator of the new LD measure, which takes into consideration the dependence structure of the observed data, is developed and …

Population Intervention Models In Causal Inference, Alan E. Hubbard, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

Marginal structural models (MSM) provide a powerful tool for estimating the causal effect of a] treatment variable or risk variable on the distribution of a disease in a population. These models, as originally introduced by Robins (e.g., Robins (2000a), Robins (2000b), van der Laan and Robins (2002)), model the marginal distributions of treatment-specific counterfactual outcomes, possibly conditional on a subset of the baseline covariates, and its dependence on treatment. Marginal structural models are particularly useful in the context of longitudinal data structures, in which each subject's treatment and covariate history are measured over time, and an outcome is recorded at …

Direct Effect Models, Mark J. Van Der Laan, Maya L. Petersen

U.C. Berkeley Division of Biostatistics Working Paper Series

The causal effect of a treatment on an outcome is generally mediated by several intermediate variables. Estimation of the component of the causal effect of a treatment that is mediated by a given intermediate variable (the indirect effect of the treatment), and the component that is not mediated by that intermediate variable (the direct effect of the treatment) is often relevant to mechanistic understanding and to the design of clinical and public health interventions. Under the assumption of no-unmeasured confounders for treatment and the intermediate variable, Robins & Greenland (1992) define an individual direct effect as the counterfactual effect of …

Application Of A Multiple Testing Procedure Controlling The Proportion Of False Positives To Protein And Bacterial Data, Merrill D. Birkner, Alan E. Hubbard, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

Simultaneously testing multiple hypotheses is important in high-dimensional biological studies. In these situations, one is often interested in controlling the Type-I error rate, such as the proportion of false positives to total rejections (TPPFP) at a specific level, alpha. This article will present an application of the E-Bayes/Bootstrap TPPFP procedure, presented in van der Laan et al. (2005), which controls the tail probability of the proportion of false positives (TPPFP), on two biological datasets. The two data applications include firstly, the application to a mass-spectrometry dataset of two leukemia subtypes, AML and ALL. The protein data measurements include intensity and …

Cross-Validating And Bagging Partitioning Algorithms With Variable Importance, Annette M. Molinaro, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

We present a cross-validated bagging scheme in the context of partitioning algorithms. To explore the benefits of the various bagging scheme, we compare via simulations the predictive ability of single Classification and Regression (CART) Tree with several previously suggested bagging schemes and with our proposed approach. Additionally, a variable importance measure is explained and illustrated.

Test Statistics Null Distributions In Multiple Testing: Simulation Studies And Applications To Genomics, Katherine S. Pollard, Merrill D. Birkner, Mark J. Van Der Laan, Sandrine Dudoit

U.C. Berkeley Division of Biostatistics Working Paper Series

Multiple hypothesis testing problems arise frequently in biomedical and genomic research, for instance, when identifying differentially expressed or co-expressed genes in microarray experiments. We have developed generally applicable resampling-based single-step and stepwise multiple testing procedures (MTP) for control of a broad class of Type I error rates, defined as tail probabilities and expected values for arbitrary functions of the numbers of false positives and rejected hypotheses (Dudoit and van der Laan, 2005; Dudoit et al., 2004a,b; Pollard and van der Laan, 2004; van der Laan et al., 2005, 2004a,b). As argued in the early article of Pollard and van der …

Cross-Validated Bagged Learning, Mark J. Van Der Laan, Sandra E. Sinisi, Maya L. Petersen

U.C. Berkeley Division of Biostatistics Working Paper Series

Many applications aim to learn a high dimensional parameter of a data generating distribution based on a sample of independent and identically distributed observations. For example, the goal might be to estimate the conditional mean of an outcome given a list of input variables. In this prediction context, Breiman (1996a) introduced bootstrap aggregating (bagging) as a method to reduce the variance of a given estimator at little cost to bias. Bagging involves applying the estimator to multiple bootstrap samples, and averaging the result across bootstrap samples. In order to deal with the curse of dimensionality, typical practice has been to …

Causal Inference In Longitudinal Studies With History-Restricted Marginal Structural Models, Romain Neugebauer, Mark J. Van Der Laan, Ira B. Tager

U.C. Berkeley Division of Biostatistics Working Paper Series

Causal Inference based on Marginal Structural Models (MSMs) is particularly attractive to subject-matter investigators because MSM parameters provide explicit representations of causal effects. We introduce History-Restricted Marginal Structural Models (HRMSMs) for longitudinal data for the purpose of defining causal parameters which may often be better suited for Public Health research. This new class of MSMs allows investigators to analyze the causal effect of a treatment on an outcome based on a fixed, shorter and user-specified history of exposure compared to MSMs. By default, the latter represents the treatment causal effect of interest based on a treatment history defined by the …

Data Adaptive Estimation Of The Treatment Specific Mean, Yue Wang, Oliver Bembom, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

An important problem in epidemiology and medical research is the estimation of the causal effect of a treatment action at a single point in time on the mean of an outcome, possibly within strata of the target population defined by a subset of the baseline covariates. Current approaches to this problem are based on marginal structural models, i.e., parametric models for the marginal distribution of counterfactural outcomes as a function of treatment and effect modifiers. The various estimators developed in this context furthermore each depend on a high-dimensional nuisance parameter whose estimation currently also relies on parametric models. Since misspecification …

History-Adjusted Marginal Structural Models And Statically-Optimal Dynamic Treatment Regimes, Mark J. Van Der Laan, Maya L. Petersen

U.C. Berkeley Division of Biostatistics Working Paper Series

Marginal structural models (MSM) provide a powerful tool for estimating the causal effect of a treatment. These models, introduced by Robins, model the marginal distributions of treatment-specific counterfactual outcomes, possibly conditional on a subset of the baseline covariates. Marginal structural models are particularly useful in the context of longitudinal data structures, in which each subject's treatment and covariate history are measured over time, and an outcome is recorded at a final time point. However, the utility of these models for some applications has been limited by their inability to incorporate modification of the causal effect of treatment by time-varying covariates. …

Estimating A Survival Distribution With Current Status Data And High-Dimensional Covariates, Mark J. Van Der Laan, Aad Van Der Vaart

U.C. Berkeley Division of Biostatistics Working Paper Series

We consider the inverse problem of estimating a survival distribution when the survival times are only observed to be in one of the intervals of a random bisection of the time axis. We are particularly interested in the case that high-dimensional and/or time-dependent covariates are available, and/or the survival events and censoring times are only conditionally independent given the covariate process. The method of estimation consists of regularizing the survival distribution by taking the primitive function or smoothing, estimating the regularized parameter by using estimating equations, and finally recovering an estimator for the parameter of interest.

Linear Life Expectancy Regression With Censored Data, Ying Qing Chen, Su-Chun Cheng

U.C. Berkeley Division of Biostatistics Working Paper Series

Life expectancy, i.e., mean residual life function, has been of important practical and scientific interests to characterise the distribution of residual life. Regression models are often needed to model the association between life expectancy and its covariates. In this article, we consider a linear mean residual life model and further developed some inference procedures in presence of censoring. The new model and proposed inference procedure will be demonstrated by numerical examples and application to the well-known Stanford heart transplant data. Additional semiparametric efficiency calculation and information bound are also considered.

A Note On Empirical Likelihood Inference Of Residual Life Regression, Ying Qing Chen, Yichuan Zhao

U.C. Berkeley Division of Biostatistics Working Paper Series

Mean residual life function, or life expectancy, is an important function to characterize distribution of residual life. The proportional mean residual life model by Oakes and Dasu (1990) is a regression tool to study the association between life expectancy and its associated covariates. Although semiparametric inference procedures have been proposed in the literature, the accuracy of such procedures may be low when the censoring proportion is relatively large. In this paper, the semiparametric inference procedures are studied with an empirical likelihood ratio method. An empirical likelihood confidence region is constructed for the regression parameters. The proposed method is further compared …

Semiparametric Quantitative-Trait-Locus Mapping: I. On Functional Growth Curves, Ying Qing Chen, Rongling Wu

U.C. Berkeley Division of Biostatistics Working Paper Series

The genetic study of certain quantitative traits in growth curves as a function of time has recently been of major scientific interest to explore the developmental evolution processes of biological subjects. Various parametric approaches in the statistical literature have been proposed to study the quantitative-trait-loci (QTL) mapping of the growth curves as multivariate outcomes. In this article, we view the growth curves as functional quantitative traits and propose some semiparametric models to relax the strong parametric assumptions which may not be always practical in reality. Appropriate inference procedures are developed to estimate the parameters of interest which characterise the possible …

Semiparametric Quantitative-Trait-Locus Mapping: Ii. On Censored Age-At-Onset, Ying Qing Chen, Chengcheng Hu, Rongling Wu

U.C. Berkeley Division of Biostatistics Working Paper Series

In genetic studies, the variation in genotypes may not only affect different inheritance patterns in qualitative traits, but may also affect the age-at-onset as quantitative trait. In this article, we use standard cross designs, such as backcross or F2, to propose some hazard regression models, namely, the additive hazards model in quantitative trait loci mapping for age-at-onset, although the developed method can be extended to more complex designs. With additive invariance of the additive hazards models in mixture probabilities, we develop flexible semiparametric methodologies in interval regression mapping without heavy computing burden. A recently developed multiple comparison procedures is adapted …

Semiparametric Regression Analysis Of Mean Residual Life With Censored Survival Data, Ying Qing Chen, Su-Chun Cheng

U.C. Berkeley Division of Biostatistics Working Paper Series

As a function of time t, mean residual life is the remaining life expectancy of a subject given survival up to t. The proportional mean residual life model, proposed by Oakes & Dasu (1990), provides an alternative to the Cox proportional hazards model to study the association between survival times and covariates. In the presence of censoring, we develop semiparametric inference procedures for the regression coefficients of the Oakes-Dasu model using martingale theory for counting processes. We also present simulation studies and an application to the Veterans' Administration lung cancer data.

Loss-Based Cross-Validated Deletion/Substitution/Addition Algorithms In Estimation, Sandra E. Sinisi, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

In van der Laan and Dudoit (2003) we propose and theoretically study a unified loss function based statistical methodology, which provides a road map for estimation and performance assessment. Given a parameter of interest which can be described as the minimizer of the population mean of a loss function, the road map involves as important ingredients cross-validation for estimator selection and minimizing over subsets of basis functions the empirical risk of the subset-specific estimator of the parameter of interest, where the basis functions correspond to a parameterization of a specified subspace of the complete parameter space. In this article we …

Unified Cross-Validation Methodology For Selection Among Estimators And A General Cross-Validated Adaptive Epsilon-Net Estimator: Finite Sample Oracle Inequalities And Examples, Mark J. Van Der Laan, Sandrine Dudoit

U.C. Berkeley Division of Biostatistics Working Paper Series

In Part I of this article we propose a general cross-validation criterian for selecting among a collection of estimators of a particular parameter of interest based on n i.i.d. observations. It is assumed that the parameter of interest minimizes the expectation (w.r.t. to the distribution of the observed data structure) of a particular loss function of a candidate parameter value and the observed data structure, possibly indexed by a nuisance parameter. The proposed cross-validation criterian is defined as the empirical mean over the validation sample of the loss function at the parameter estimate based on the training sample, averaged over …

Measuring Treatment Effects Using Semiparametric Models, Zhuo Yu, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

In order to estimate the causal effect of treatments on an outcome of interest, one has to account for the effect of confounding factors which covary with the treatments and also contribute to the outcome of interest. In this paper, we use the semiparametric regression model to estimate the causal parameters. We assume the causal effect of the treatments can be described by the parametric component of the semiparametric regression model. Following the general methodology which was developed in van der Laan and Robins (2002) we give the orthogonal complement of the nuisance tangent space which identifies all the estimating …

Asymptotically Optimal Model Selection Method With Right Censored Outcomes, Sunduz Keles, Mark J. Van Der Laan, Sandrine Dudoit

U.C. Berkeley Division of Biostatistics Working Paper Series

Over the last two decades, non-parametric and semi-parametric approaches that adapt well known techniques such as regression methods to the analysis of right censored data, e.g. right censored survival data, became popular in the statistics literature. However, the problem of choosing the best model (predictor) among a set of proposed models (predictors) in the right censored data setting have not gained much attention. In this paper, we develop a new cross-validation based model selection method to select among predictors of right censored outcomes such as survival times. The proposed method considers the risk of a given predictor based on the …

Tree-Based Multivariate Regression And Density Estimation With Right-Censored Data , Annette M. Molinaro, Sandrine Dudoit, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

We propose a unified strategy for estimator construction, selection, and performance assessment in the presence of censoring. This approach is entirely driven by the choice of a loss function for the full (uncensored) data structure and can be stated in terms of the following three main steps. (1) Define the parameter of interest as the minimizer of the expected loss, or risk, for a full data loss function chosen to represent the desired measure of performance. Map the full data loss function into an observed (censored) data loss function having the same expected value and leading to an efficient estimator …

Supervised Detection Of Regulatory Motifs In Dna Sequences, Sunduz Keles, Mark J. Van Der Laan, Sandrine Dudoit, Biao Xing, Michael B. Eisen

U.C. Berkeley Division of Biostatistics Working Paper Series

Identification of transcription factor binding sites (regulatory motifs) is a major interest in contemporary biology. We propose a new likelihood based method, COMODE, for identifying structural motifs in DNA sequences. Commonly used methods (e.g. MEME, Gibbs sampler) model binding sites as families of sequences described by a position weight matrix (PWM) and identify PWMs that maximize the likelihood of observed sequence data under a simple multinomial mixture model. This model assumes that the positions of the PWM correspond to independent multinomial distributions with four cell probabilities. We address supervising the search for DNA binding sites using the information derived from …

A Semiparametric Model Selection Criterion With Applications To The Marginal Structural Model, M. Alan Brookhart, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

Estimators for the parameter of interest in semiparametric models often depend on a guessed model for the nuisance parameter. The choice of the model for the nuisance parameter can affect both the finite sample bias and efficiency of the resulting estimator of the parameter of interest. In this paper we propose a finite sample criterion based on cross validation that can be used to select a nuisance parameter model from a list of candidate models. We show that expected value of this criterion is minimized by the nuisance parameter model that yields the estimator of the parameter of interest with …