Columbia University Biostatistics Technical Report Series

Meta-Analysis of “Sparse” Data: Perspectives From the Avandia Cases

Michael O. Finkelstein et al. — Mon, 23 Apr 2012 17:31:32 PDT

Combining the results of multiple small trials to increase accuracy and statistical power, a technique called meta-analysis has become well established and increasingly important in medical studies, particularly in connection with new drugs. When the data are sparse, as they are in many such cases, certain accepted practices, applied reflexively by researchers, may be misleading because they are biased and for other reasons. We illustrate some of the problems by examining a meta-analysis of the connection between the diabetes drug Avandia (rosiglitazone) and myocardial infarction that was strongly criticized as misleading, but led to thousands of lawsuits being filed against the manufacturer and the FDA acting to restrict access to the drug. Our scrutiny of the Avandia meta-analysis is particularly appropriate because it plays an important role in ongoing litigation, has been sharply criticized, and has been subject to a more searching review in court than meta-analyses of other drugs.

Formulas for Correcting the Mean and Variance of the Full-Data Sample Mean of the Primary Endpoint Under the Dose Selected at Stage One in a Two-Stage Trial with Selection Between Two Active Doses

Bruce Levin — Wed, 15 Sep 2010 17:25:36 PDT

Formulas are provided in the normal distribution case to correct the mean and variance of the sample mean from a two-stage phase II study in which a preferred dose is selected in the first stage, the second stage collects additional data with the selected dose, and then a non-superiority hypothesis is tested with the data from both stages.

Losses To Follow-Up In WARSS Collaboration Datasets: A Detailed Statistical Presentation of the Imputation Procedures

John L.P. Thompson et al. — Sat, 05 Dec 2009 16:06:54 PST

This document prospectively records the procedures which will be used for handling losses to follow-up (LTF) in statistical analyses of WARSS data. They were developed by B Levin Ph.D. (WARSS senior statistical consultant) and JLP Thompson Ph.D. (WARSS statistician), and have been approved by the SOCC (Statistical Oversight and Coordinating Committee of the WARSS collaboration). They have been accepted by the WARSS Principal Investigator (JP Mohr M.D.), and also by the Principal Investigators of APASS, PICSS, HAS, and GENESIS for use in these collaborating studies.

Appendix to "Bush v. Gore: Two Neglected Lessons from a Statistical Perspective" by Michael O. Finkelstein and Bruce Levin

Michael O. Finkelstein et al. — Sun, 22 Feb 2009 15:34:13 PST

This is the appendix to the article entitled, "Bush v. Gore: Two Neglected Lessons from a Statistical Perspective" which was published in Jurimetrics 44(2):181-200 (2004).

On a Conjecture of Bechhofer, Kiefer, and Sobel for the Levin-Robbins-Leu Binomial Subset Selection Procedures

Cheng-Shiun Leu et al. — Sun, 27 Jan 2008 14:17:44 PST

We state a general formula which provides a lower bound for the probability of various types of acceptable subset selection with the Levin-Robbins-Leu binomial subset selection procedure without elimination or recruitment. We prove the truth of a conjecture of Bechhofer, Kiefer, and Sobel for this procedure by applying the general lower bound. We also introduce a simple modification that allows sequential elimination of inferior populations and recruitment of superior populations. Numerical evidence indicates that the new procedure also obeys the general lower bound while reducing the expected number of observations and failures compared with non-adaptive methods.

Handling Missing Data by Deleting Completely Observed Records

Cuiling Wang et al. — Thu, 19 Apr 2007 10:08:36 PDT

When data are missing, analyzing records that are completely observed may cause bias or ineffciency. Existing approaches in handling missing data include likelihood, imputation and inverse probability weighting. In this paper, we propose three estimators inspired by deleting some completely observed data in regression setting. First, we generate artificial observation indicators that are independent of outcome given observed data and draw inferences conditioning on the artificial observation indicators. Second, we propose a closely related weighting method. The proposed weighting method has more stable weights than those of the inverse probability weighting method (Zhao and Lipsitz, 1992). Third, we improve the e±ciency of the proposed weighting estimator by subtracting the projection of the estimating function onto the nuisance tangent space. When data are missing completely at random, we show that the proposed estimators have asymptotic variances smaller than or equal to the variance of the estimator obtained from using completely observed records only. Asymptotic relative effciency computation and simulation studies indicate that the proposed weighting estimators are more e±cient than the inverse probability weighting estimators under wide range of practical situations especially when the missingness propor- tion is large.

An Improved Kernel Assisted Imputation Method in Missing Covariate Regression

Hui Zhang et al. — Thu, 19 Apr 2007 10:04:26 PDT

There are several methods of handling missing data in a non-likelihood framework, imputation and the inverse probability weighting method being the two main approaches. These methods require auxiliary mod- els, namely the probability of observation for the inverse probabil- ity and the conditional distribution of missing data and their correct specification. Wang and Wang (2001) proposed a kernel method for these auxiliary models and investigated the relationship among various kernel-assisted methods and showed some asymptotic equivalence. In this paper we delve into some questions arisen from Wang and Wang (2001). We first derive an improved imputation method which subtracts the projection of the original imputation score onto the nuisance tangent space. Further, we look into the asymptotic behavior of our method and do a comparison with those described in Wang and Wang (2001). Our results are in contrast with those of Wang and Wang (2001) in that the projection from the improved imputation method becomes negligible when the conditional expectation is estimated. It turns out that under the conditions where our projection becomes negligible, the projection from the inverse probability estimating function is of nonnegligible order.

Keywords: kernel estimator, Imputation methods, Improved imputation methods, Semiparametric methods

Importance Sampling Estimators for Cox Regression With Missing Covariates

Qiang Xu et al. — Wed, 18 Apr 2007 12:08:59 PDT

Missingness in covariates is a common problem in survival data. In this article, we propose an importance sampling method for estimating the regression parameters in the proportional hazards model with missing covariates. We also consider the augmented importance sampling method by subtracting the projection term onto the nuisance tangent space. The proposed methods provide consistent and asymptotically normally distributed estimators when the missing-data mechanism depends on outcome variables as well as the observed covariates. Simulation results indicate that the proposed importance sampling estimators are more efficient than the inverse probability weighting estimators for the regression coefficients of the missing covariates, and equally as efficient as or more efficient than the inverse probability weighting estimators for the regression coefficients of the completely observed covariates.

On Calculating Maximum Rank One Underapproximations for Positive Arrays

Bruce Levin — Fri, 30 Mar 2007 20:13:30 PDT

Given P, a rectangular array with positive elements, a rank one underapproximation for P is given by two positive vectors, say r and s, such that each component of rs' is no greater than the corresponding component of P, whence P can then be written as P=(pi)rs'+(1-pi)D for some constant pi, where the residual matrix D is non-negative. A maximal rank one underapproximation for P is such that pi is maximized over all possible rank one underapproximations for P.

We provide an algorithm for calculating the maximum rank one underapproximation and corresponding pi. We present an explicit expression in the special case of 2 x c tables, and show that the algorithm yields the correct solution in this case.

Note: this report was originally entitled "Technical Report No. B-48, January 1985" in the Columbia Biostatistics Tech Report Series. Due to early font styles, strict and non-strict inequalities may be difficult to distinguish in the scanned version. Magnifying the typeface will help.

Quadratic Distances On Probabilities: A Unified Foundation

Bruce G. Lindsay et al. — Thu, 29 Mar 2007 07:22:17 PDT

This work builds a unified framework for the study of quadratic form distance measures as they are used in assessing the goodness of fit of models. Many important procedures have this structure, but the theory for these methods is dispersed and incomplete. Central to the statistical analysis of these distances is the spectral decomposition of the kernel that generates the distance. We show how this determines the limiting distribution of natural goodness of fit tests. Additionally, we develop a new notion, the spectral degrees of freedom of the test, based on this decomposition. The degrees of freedom are easy to compute and estimate, and can be used as a guide in the construction of useful procedures in this class.

A Least Squares Estimation in Truncated Linear Regression

Wei Yann Tsai et al. — Thu, 22 Mar 2007 12:21:51 PDT

We investigate least squares estimation for regression coefficients of the covariates in the multiple linear regression model with truncated data and propose an alternative consistent least squares type estimator to the existing ones. The estimator is proved to have an asymptotic normal distribution with the same asymptotic variance matrix as the estimator proposed by Lai and Ying (1992b). However, the estimator is much simpler in computation than Lai and Ying's estimator. The estimation procedure does not require calculation of the nonparametric estimate of the error distribution. A simulation study shows that the estimator performs well even with a moderate sample size.

On Minimizing the Lower Bound for the Probability of Acceptable Subset Selection

Bruce Levin — Fri, 15 Dec 2006 14:29:00 PST

This report identifies the configuration that minimizes the (s,t)-lower bound for the probability of theta-acceptable subset selection, which is involved in the proof of a conjecture of Bechhofer, Kiefer, and Sobel for the Levin-Robbins-Leu binomial subset selection procedure with sequential elimination and recruitment.

Proof of the Lower Bound Formula for the Probability of Correct Binomial Subset Selection with the Levin-Robbins-Leu Sequential Elimination and Recruitment Procedure in the Case b=2, c=4.

Cheng-Shiun Leu et al. — Fri, 15 Dec 2006 14:17:55 PST

We prove the validity of a lower bound for the probability of correct subset selection for choosing the best b=2 binomial populations out of c=4. The procedure considered combines sequential elimination of inferior populations and simultaneous sequential recruitment of superior populations. This extends the results of Leu and Levin (2004) to this new procedure.

Decision trees for instantaneous hazard rates with adjustment for covariate effects

Moulinath Banerjee et al. — Tue, 27 Jun 2006 17:25:46 PDT

We investigate the problem of finding confidence sets for a threshold in the baseline hazard function of the Cox proportional hazards model. The aim is to find an effective way of condensing information in the baseline into a small number of estimable parameters. A binary decision tree is used as a working model, with the jump point (threshold) representing a time at which baseline risk changes abruptly between two levels. We find the asymptotic distribution of estimators of best-fitting parameters under an arbitrary misspecification of the working model. The estimators converge at cube-root rate to a non-normal limit distribution. Two alternate ways of constructing confidence intervals for the threshold are compared. Results from a simulation study and an example concerning a threshold for the age of onset of schizophrenia in a large cohort study are discussed.

Technical Report #B-92---Formulas for the Exact Probability of Correct Selection in the Binomial Levin-Robbins Sequential Selection Procedure in the Cases b=2, c=3 and b=2, c=4 for r=1

Bruce Levin et al. — Thu, 08 Jun 2006 19:01:22 PDT

The purpose of this report is to record some explicit expressions for the probability of correct selection in the Levin-Robbins procedure for selecting the best b out of c coins in the two cases in which such expressions are feasible.

Tech Report #B-91---Selecting the Best Subset of b Out of c Coins with the Levin-Robbins Sequential Elimination Procedure: Proof of the Lower Bound Formula for the Probability of Correct Selection in the Case b=2 and c=4.

Cheng-Shiun Leu et al. — Thu, 08 Jun 2006 18:54:19 PDT

The purpose of this report is to prove the validity of the lower bound formula for a sequential identification procedure with elimination of inferior coins in the special case of selecting the best b = 2 out of c = 4 coins for any r greater than or equal to 1. This is the first non-trivial case of a general conjecture not already proven in previous work and suggests a rigorous proof of the general conjecture may yet be possible.

A Generalization of the Levin-Robbins Procedure for Binomial Subset Selection and Recruitment Problems

Cheng-Shiun Leu et al. — Thu, 11 May 2006 14:17:46 PDT

We introduce a family of sequential selection and recruitment procedures for the subset identification problem in binomial populations. We demonstrate the general validity of a simple formula providing a lower bound for the probability of correct identification in a version of the family without sequential elimination or recruitment. A new application of the noncentral hypergeometric distribution is revealed. A similar theorem is conjectured to hold for the more efficient version which employs sequential elimination or recruitment.

Keywords: lower bound formula, probability of correct selection, recruitment, selection, sequential identification.

A Note on the Censoring Problem in Empirical Case-Outcome Studies

Michael O. Finkelstein et al. — Wed, 10 May 2006 13:38:18 PDT

In studies of the legal system investigators may collect information about cases within a study window and compile statistical information about their outcomes. Because there is frequently a long delay between the start time for cases and their resolution, a significant number of cases may be pending at the close of the study window. If there is a correlation between the outcome variable and being censored, exclusion of censored cases may bias the analysis in the sense that the reported outcomes will be systematically different from what would be reported if all the censored cases were followed to completion and included in the data. A prime example, which we will use to illustrate our approach, is the landmark study of reversals in death penalty cases in the state courts that was authored by a team led by Professor James S. Liebman of Columbia Law School. Two equivalent ways of estimating outcome rates accounting for the censored cases is the subject of this article.