Tests If Dropouts Are Missed at Random

Dropouts are a common problem in longitudinal investigations where individuals are measured repeatedly over time. This holds also in a study on rheumatoid arthritis where an inception cohort was followed up over three years. The question arose whether or not these individuals caused a selection bias. Two tests taken from the literature could be used to answer this question. But since they do not use all available information, a new asymptotic test is proposed. In a small simulation study it is shown that the new test is more powerful than the others.     

Test of Random Subject Effects In Heteroskedastic Linear Models

A new test of random subject effects in linear regression models is presented. The test is robust against heteroskedasticity and its asymptotic distribution is derived under a sequence of local alternatives. The finite sample properties of the test are studied in a simulation experiment and an empirical example. The results presented show that the new test is to be preferred over earlier test proposed.     

Improved Doubly Robust Estimation When Data Are Monotonely Coarsened,with Application to Longitudinal Studies with Dropout

Summary A routine challenge is that of making inference on parameters in a statistical model of interest from longitudinal data subject to dropout, which are a special case of the more general setting of monotonely coarsened data. Considerable recent attention has focused on doubly robust (DR) estimators, which in this context involve positing models for both the missingness (more generally, coarsening) mechanism and aspects of the distribution of the full data, that have the appealing property of yielding consistent inferences if only one of these models is correctly specified. DR estimators have been criticized for potentially disastrous performance when both of these models are even only mildly misspecified. We propose a DR estimator applicable in general monotone coarsening problems that achieves comparable or improved performance relative to existing DR methods, which we demonstrate via simulation studies and by application to data from an AIDS clinical trial.     

A Model for the Censoring Process in Incomplete Repeated Measurement Experiments

Premature terminations or dropouts occur often in repeated measurement experiments. A number of methods have been proposed to analyze such data but most of them assume that the censoring mechanism is, within each group, unaffected by the mechanism generating the response variables. In this paper, we propose a model for the censoring mechanism that generates dropouts. We then show how this model can be used to check whether the censoring mechanism is affected by the response variables and other covariates. Finally, the methods of the paper are applied to the “Halothane” data set.     

Analyzing incomplete longitudinal clinical trial data

Using standard missing data taxonomy, due to Rubin and co-workers, and simple algebraic derivations, it is argued that some simple but commonly used methods to handle incomplete longitudinal clinical trial data, such as complete case analyses and methods based on last observation carried forward, require restrictive assumptions and stand on a weaker theoretical foundation than likelihood-based methods developed under the missing at random (MAR) framework. Given the availability of flexible software for analyzing longitudinal sequences of unequal length, implementation of likelihood-based MAR analyses is not limited by computational considerations. While such analyses are valid under the comparatively weak assumption of MAR, the possibility of data missing not at random (MNAR) is difficult to rule out. It is argued, however, that MNAR analyses are, themselves, surrounded with problems and therefore, rather than ignoring MNAR analyses altogether or blindly shifting to them, their optimal place is within sensitivity analysis. The concepts developed here are illustrated using data from three clinical trials, where it is shown that the analysis method may have an impact on the conclusions of the study.     

A Likelihood Ratio Test for the Nonparametric Behrens‐Fisher Problem

The nonparametric Behrens‐Fisher hypothesis is the most appropriate null hypothesis for the two‐sample comparison when one does not wish to make restrictive assumptions about possible distributions. In this paper, a numerical approach is described by which the likelihood ratio test can be calculated for the nonparametric Behrens‐Fisher problem. The approach taken here effectively reduces the number of parameters in the score equations to one by using a recursive formula for the remaining parameters. The resulting single dimensional problem can be solved numerically. The power of the likelihood ratio test is compared by simulation to that of a generalized Wilcoxon test of Brunner and Munzel. The tests have similar power for all alternatives considered when a simulated null distribution is used to generate cutoff values for the tests. The methods are illustrated on data on shoulder pain from a clinical trial.     

Doubly robust estimates for binary longitudinal data analysis with missing response and missing covariates

Longitudinal studies often feature incomplete response and covariate data. Likelihood-based methods such as the expectation-maximization algorithm give consistent estimators for model parameters when data are missing at random (MAR) provided that the response model and the missing covariate model are correctly specified; however, we do not need to specify the missing data mechanism. An alternative method is the weighted estimating equation, which gives consistent estimators if the missing data and response models are correctly specified; however, we do not need to specify the distribution of the covariates that have missing values. In this article, we develop a doubly robust estimation method for longitudinal data with missing response and missing covariate when data are MAR. This method is appealing in that it can provide consistent estimators if either the missing data model or the missing covariate model is correctly specified. Simulation studies demonstrate that this method performs well in a variety of situations.     

Simple Nonparametric Inference for Monotonic Processes

A scoring system for possibly censored monotonic processes is presented which allows to analyze such processes by means of standard parametric or nonparametric tests. In either case the analyses are invariant to monotonic transformations of time. Such analyses are of relevance whenever changes in degree of function are observed rather than the single and 100% loss of function of survival analysis. An example of gradually increasing function after particular reconstructive surgery is supplied.     

A note on MAR, identifying restrictions, model comparison, and sensitivity analysis in pattern mixture models with and without covariates for incomplete data

Summary Pattern mixture modeling is a popular approach for handling incomplete longitudinal data. Such models are not identifiable by construction. Identifying restrictions is one approach to mixture model identification ( Little, 1995 , Journal of the American Statistical Association 90 , 1112–1121; Little and Wang, 1996 , Biometrics 52 , 98–111; Thijs et al., 2002 , Biostatistics 3 , 245–265; Kenward, Molenberghs, and Thijs, 2003 , Biometrika 90 , 53–71; Daniels and Hogan, 2008 , in Missing Data in Longitudinal Studies: Strategies for Bayesian Modeling and Sensitivity Analysis) and is a natural starting point for missing not at random sensitivity analysis ( Thijs et al., 2002 , Biostatistics 3 , 245–265; Daniels and Hogan, 2008 , in Missing Data in Longitudinal Studies: Strategies for Bayesian Modeling and Sensitivity Analysis). However, when the pattern specific models are multivariate normal, identifying restrictions corresponding to missing at random (MAR) may not exist. Furthermore, identification strategies can be problematic in models with covariates (e.g., baseline covariates with time‐invariant coefficients). In this article, we explore conditions necessary for identifying restrictions that result in MAR to exist under a multivariate normality assumption and strategies for identifying sensitivity parameters for sensitivity analysis or for a fully Bayesian analysis with informative priors. In addition, we propose alternative modeling and sensitivity analysis strategies under a less restrictive assumption for the distribution of the observed response data. We adopt the deviance information criterion for model comparison and perform a simulation study to evaluate the performances of the different modeling approaches. We also apply the methods to a longitudinal clinical trial. Problems caused by baseline covariates with time‐invariant coefficients are investigated and an alternative identifying restriction based on residuals is proposed as a solution.     

Testing for bias in weighted estimating equations

It is very common in regression analysis to encounter incompletely observed covariate information. A recent approach to analyse such data is weighted estimating equations (Robins, J. M., Rotnitzky, A. and Zhao, L. P. (1994), JASA, 89, 846-866, and Zhao, L. P., Lipsitz, S. R. and Lew, D. (1996), Biometrics, 52, 1165-1182). With weighted estimating equations, the contribution to the estimating equation from a complete observation is weighted by the inverse of the probability of being observed. We propose a test statistic to assess if the weighted estimating equations produce biased estimates. Our test statistic is similar to the test statistic proposed by DuMouchel and Duncan (1983) for weighted least squares estimates for sample survey data. The method is illustrated using data from a randomized clinical trial on chemotherapy for multiple myeloma.     

The Nonparametric Behrens‐Fisher Problem: Asymptotic Theory and a Small‐Sample Approximation

A generalization of the Behrens‐Fisher problem for two samples is examined in a nonparametric model. It is not assumed that the underlying distribution functions are continuous so that data with arbitrary ties can be handled. A rank test is considered where the asymptotic variance is estimated consistently by using the ranks over all observations as well as the ranks within each sample. The consistency of the estimator is derived in the appendix. For small samples (n₁, n₂ ≥ 10), a simple approximation by a central t‐distribution is suggested where the degrees of freedom are taken from the Satterthwaite‐Smith‐Welch approximation in the parametric Behrens‐Fisher problem. It is demonstrated by means of a simulation study that the Wilcoxon‐Mann‐Whitney‐test may be conservative or liberal depending on the ratio of the sample sizes and the variances of the underlying distribution functions. For the suggested approximation, however, it turns out that the nominal level is maintained rather accurately. The suggested nonparametric procedure is applied to a data set from a clinical trial. Moreover, a confidence interval for the nonparametric treatment effect is given.     

A New Rank Test Against Ordered Alternatives in K-Sample Problems

The distribution-free test against ordered alternatives proposed by Jonckheere (1954) is based on the Kendall's rank correlation coefficient τ. A new rank test is proposed and illustrated with a numerical example. The proposed test is based on Spearman's σ and has similar functional structure as the Kruskal-Wallis test. A useful by-product is a test for departure from a trend.     

Test Based on Independent,but Non-identically Distributed Random Variables

The binomial test is applied for the problem of testing a hypothesis based on a sample of independent, but non-identically distributed random variables. The used basic idea is that each random variable indicates the presence of the hypothesis. Hence each random variable is transformed such that the binomial test can be used as a simple procedure.