A mixed effects model for analyzing area under the curve of longitudinally measured biomarkers with missing data

A simple approach for analyzing longitudinally measured biomarkers is to calculate summary measures such as the area under the curve (AUC) for each individual and then compare the mean AUC between treatment groups using methods such as t test. This two-step approach is difficult to implement when there are missing data since the AUC cannot be directly calculated for individuals with missing measurements. Simple methods for dealing with missing data include the complete case analysis and imputation. A recent study showed that the estimated mean AUC difference between treatment groups based on the linear mixed model (LMM), rather than on individually calculated AUCs by simple imputation, has negligible bias under random missing assumptions and only small bias when missing is not at random. However, this model assumes the outcome to be normally distributed, which is often violated in biomarker data. In this paper, we propose to use a LMM on log-transformed biomarkers, based on which statistical inference for the ratio, rather than difference, of AUC between treatment groups is provided. The proposed method can not only handle the potential baseline imbalance in a randomized trail but also circumvent the estimation of the nuisance variance parameters in the log-normal model. The proposed model is applied to a recently completed large randomized trial studying the effect of nicotine reduction on biomarker exposure of smokers.     

Non-parametric MANOVA approaches for non-normal multivariate outcomes with missing values

Between-group comparisons often entail many correlated response variables. The multivariate linear model, with its assumption of multivariate normality, is the accepted standard tool for these tests. When this assumption is violated, the non-parametric multivariate Kruskal–Wallis (MKW) test is frequently used. However, this test requires complete cases with no missing values in response variables. Deletion of cases with missing values likely leads to inefficient statistical inference. Here we extend the MKW test to retain information from partially observed cases. Results of simulated studies and analysis of real data show that the proposed method provides adequate coverage and superior power to complete case analyses.     

On inference for Kendall's τ within a longitudinal data setting

Kendall's τ is a non-parametric measure of correlation based on ranks and is used in a wide range of research disciplines. Although methods are available for making inference about Kendall's τ, none has been extended to modeling multiple Kendall's τs arising in longitudinal data analysis. Compounding this problem is the pervasive issue of missing data in such study designs. In this article, we develop a novel approach to provide inference about Kendall's τ within a longitudinal study setting under both complete and missing data. The proposed approach is illustrated with simulated data and applied to an HIV prevention study.     

Did the Military Interventions in the Mexican Drug War Increase Violence?

We analyze publicly available data to estimate the causal effects of military interventions on the homicide rates in certain problematic regions in Mexico. We use the Rubin causal model to compare the post-intervention homicide rate in each intervened region to the hypothetical homicide rate for that same year had the military intervention not taken place. Because the effect of a military intervention is not confined to the municipality subject to the intervention, a nonstandard definition of units is necessary to estimate the causal effect of the intervention under the standard no-interference assumption of stable-unit treatment value assumption (SUTVA). Donor pools are created for each missing potential outcome under no intervention, thereby allowing for the estimation of unit-level causal effects. A multiple imputation approach accounts for uncertainty about the missing potential outcomes.     

Extending the Mann–Whitney–Wilcoxon rank sum test to longitudinal regression analysis

Outliers are commonly observed in psychosocial research, generally resulting in biased estimates when comparing group differences using popular mean-based models such as the analysis of variance model. Rank-based methods such as the popular Mann–Whitney–Wilcoxon (MWW) rank sum test are more effective to address such outliers. However, available methods for inference are limited to cross-sectional data and cannot be applied to longitudinal studies under missing data. In this paper, we propose a generalized MWW test for comparing multiple groups with covariates within a longitudinal data setting, by utilizing the functional response models. Inference is based on a class of U-statistics-based weighted generalized estimating equations, providing consistent and asymptotically normal estimates not only under complete but missing data as well. The proposed approach is illustrated with both real and simulated study data.     

Robust logistic regression of family data in the presence of missing genotypes

Large cohort studies are commonly launched to study the risk effect of genetic variants or other risk factors on a chronic disorder. In these studies, family data are often collected to provide additional information for the purpose of improving the inference results. Statistical analysis of the family data can be very challenging due to the missing observations of genotypes, incomplete records of disease occurrences in family members, and the complicated dependence attributed to the shared genetic background and environmental factors. In this article, we investigate a class of logistic models with family-shared random effects to tackle these challenges, and develop a robust regression method based on the conditional logistic technique for statistical inference. An expectation–maximization (EM) algorithm with fast computation speed is developed to handle the missing genotypes. The proposed estimators are shown to be consistent and asymptotically normal. Additionally, a score test based on the proposed method is derived to test the genetic effect. Extensive simulation studies demonstrate that the proposed method performs well in finite samples in terms of estimate accuracy, robustness and computational speed. The proposed procedure is applied to an Alzheimer's disease study.     

Likelihood‐based Inference with Missing Data Under Missing‐at‐Random

Likelihood‐based inference with missing data is challenging because the observed log likelihood is often an (intractable) integration over the missing data distribution, which also depends on the unknown parameter. Approximating the integral by Monte Carlo sampling does not necessarily lead to a valid likelihood over the entire parameter space because the Monte Carlo samples are generated from a distribution with a fixed parameter value. We consider approximating the observed log likelihood based on importance sampling. In the proposed method, the dependency of the integral on the parameter is properly reflected through fractional weights. We discuss constructing a confidence interval using the profile likelihood ratio test. A Newton–Raphson algorithm is employed to find the interval end points. Two limited simulation studies show the advantage of the Wilks inference over the Wald inference in terms of power, parameter space conformity and computational efficiency. A real data example on salamander mating shows that our method also works well with high‐dimensional missing data.     

Bayesian inference for stochastic multitype epidemics in structured populations via random graphs

Summary.  The paper is concerned with new methodology for statistical inference for final outcome infectious disease data using certain structured population stochastic epidemic models. A major obstacle to inference for such models is that the likelihood is both analytically and numerically intractable. The approach that is taken here is to impute missing information in the form of a random graph that describes the potential infectious contacts between individuals. This level of imputation overcomes various constraints of existing methodologies and yields more detailed information about the spread of disease. The methods are illustrated with both real and test data.     

An extension of the placebo‐based pattern‐mixture model

Pattern‐mixture models provide a general and flexible framework for sensitivity analyses of nonignorable missing data in longitudinal studies. The placebo‐based pattern‐mixture model handles missing data in a transparent and clinically interpretable manner. We extend this model to include a sensitivity parameter that characterizes the gradual departure of the missing data mechanism from being missing at random toward being missing not at random under the standard placebo‐based pattern‐mixture model. We derive the treatment effect implied by the extended model. We propose to utilize the primary analysis based on a mixed‐effects model for repeated measures to draw inference about the treatment effect under the extended placebo‐based pattern‐mixture model. We use simulation studies to confirm the validity of the proposed method. We apply the proposed method to a clinical study of major depressive disorders. Copyright © 2013 John Wiley & Sons, Ltd.     

Semiparametric inference for estimating equations with nonignorably missing covariates

We consider statistical inference of unknown parameters in estimating equations (EEs) when some covariates have nonignorably missing values, which is quite common in practice but has rarely been discussed in the literature. When an instrument, a fully observed covariate vector that helps identifying parameters under nonignorable missingness, is available, the conditional distribution of the missing covariates given other covariates can be estimated by the pseudolikelihood method of Zhao and Shao [(2015), ‘Semiparametric pseudo likelihoods in generalised linear models with nonignorable missing data’, Journal of the American Statistical Association, 110, 1577–1590)] and be used to construct unbiased EEs. These modified EEs then constitute a basis for valid inference by empirical likelihood. Our method is applicable to a wide range of EEs used in practice. It is semiparametric since no parametric model for the propensity of missing covariate data is assumed. Asymptotic properties of the proposed estimator and the empirical likelihood ratio test statistic are derived. Some simulation results and a real data analysis are presented for illustration.     

Data-driven sensitivity analysis to detect missing data mechanism with applications to structural equation modelling

Missing data are a common problem in almost all areas of empirical research. Ignoring the missing data mechanism, especially when data are missing not at random (MNAR), can result in biased and/or inefficient inference. Because MNAR mechanism is not verifiable based on the observed data, sensitivity analysis is often used to assess it. Current sensitivity analysis methods primarily assume a model for the response mechanism in conjunction with a measurement model and examine sensitivity to missing data mechanism via the parameters of the response model. Recently, Jamshidian and Mata (Post-modelling sensitivity analysis to detect the effect of missing data mechanism, Multivariate Behav. Res. 43 (2008), pp. 432–452) introduced a new method of sensitivity analysis that does not require the difficult task of modelling the missing data mechanism. In this method, a single measurement model is fitted to all of the data and to a sub-sample of the data. Discrepancy in the parameter estimates obtained from the the two data sets is used as a measure of sensitivity to missing data mechanism. Jamshidian and Mata describe their method mainly in the context of detecting data that are missing completely at random (MCAR). They used a bootstrap type method, that relies on heuristic input from the researcher, to test for the discrepancy of the parameter estimates. Instead of using bootstrap, the current article obtains confidence interval for parameter differences on two samples based on an asymptotic approximation. Because it does not use bootstrap, the developed procedure avoids likely convergence problems with the bootstrap methods. It does not require heuristic input from the researcher and can be readily implemented in statistical software. The article also discusses methods of obtaining sub-samples that may be used to test missing at random in addition to MCAR. An application of the developed procedure to a real data set, from the first wave of an ongoing longitudinal study on aging, is presented. Simulation studies are performed as well, using two methods of missing data generation, which show promise for the proposed sensitivity method. One method of missing data generation is also new and interesting in its own right.     

Empirical Likelihood Inference of the Partial Linear Isotonic Errors-in-variables Regression Models with Missing Data

This article is concerned with statistical inference of the partial linear isotonic regression model missing response and measurement errors in covariates. We proposed an empirical likelihood ratio test statistics and show that it has a limiting weighted chi-square distribution. An adjusted empirical likelihood ratio statistic, which is shown to have a limiting standard central chi-square distribution, is then proposed further. A maximum empirical likelihood estimator is also developed. A simulation study is conducted to examine the finite-sample property of proposed procedure.     

Using information on realized effects to determine prospective causal effects

The potential outcomes approach to causal inference postulates that each individual has a number of possibly latent outcomes, each of which would be observed under a different treatment. For any individual, some of these outcomes will be unobservable or counterfactual. Information about post-treatment characteristics sometimes allows statements about what would have happened if an individual or group with these characteristics had received a different treatment. These are statements about the realized effects of the treatment. Determining the likely effect of an intervention before making a decision involves inference about effects in populations defined only by characteristics observed before decisions about treatment are made. Information on realized effects can tighten bounds on these prospectively defined measures of the intervention effect. We derive formulae for the bounds and their sampling variances and illustrate these points with data from a hypothetical study of the efficacy of screening mammography.     

Analysing exponential random graph (p-star) models with missing data using Bayesian data augmentation

Missing data are often problematic in social network analysis since what is missing may potentially alter the conclusions about what we have observed as tie-variables need to be interpreted in relation to their local neighbourhood and the global structure. Some ad hoc methods for dealing with missing data in social networks have been proposed but here we consider a model-based approach. We discuss various aspects of fitting exponential family random graph (or p-star) models (ERGMs) to networks with missing data and present a Bayesian data augmentation algorithm for the purpose of estimation. This involves drawing from the full conditional posterior distribution of the parameters, something which is made possible by recently developed algorithms. With ERGMs already having complicated interdependencies, it is particularly important to provide inference that adequately describes the uncertainty, something that the Bayesian approach provides. To the extent that we wish to explore the missing parts of the network, the posterior predictive distributions, immediately available at the termination of the algorithm, are at our disposal, which allows us to explore the distribution of what is missing unconditionally on any particular parameter values. Some important features of treating missing data and of the implementation of the algorithm are illustrated using a well-known collaboration network and a variety of missing data scenarios.     

Estimating Causal Effects in Trials Involving Multi-Treatment Arms Subject to Non-compliance: A Bayesian framework

Data analysis for randomized trials including multi-treatment arms is often complicated by subjects who do not comply with their treatment assignment. We discuss here methods of estimating treatment efficacy for randomized trials involving multi-treatment arms subject to non-compliance. One treatment effect of interest in the presence of non-compliance is the complier average causal effect (CACE) (Angrist et al. 1996), which is defined as the treatment effect for subjects who would comply regardless of the assigned treatment. Following the idea of principal stratification (Frangakis & Rubin 2002), we define principal compliance (Little et al. 2009) in trials with three treatment arms, extend CACE and define causal estimands of interest in this setting. In addition, we discuss structural assumptions needed for estimation of causal effects and the identifiability problem inherent in this setting from both a Bayesian and a classical statistical perspective. We propose a likelihood-based framework that models potential outcomes in this setting and a Bayes procedure for statistical inference. We compare our method with a method of moments approach proposed by Cheng & Small (2006) using a hypothetical data set, and further illustrate our approach with an application to a behavioral intervention study (Janevic et al. 2003).     

Using a mixture model for multiple imputation in the presence of outliers: the 'Healthy for life' project

Summary.  We consider the problem of obtaining population-based inference in the presence of missing data and outliers in the context of estimating the prevalence of obesity and body mass index measures from the 'Healthy for life' study. Identifying multiple outliers in a multivariate setting is problematic because of problems such as masking, in which groups of outliers inflate the covariance matrix in a fashion that prevents their identification when included, and swamping, in which outliers skew covariances in a fashion that makes non-outlying observations appear to be outliers. We develop a latent class model that assumes that each observation belongs to one of K unobserved latent classes, with each latent class having a distinct covariance matrix. We consider the latent class covariance matrix with the largest determinant to form an 'outlier class'. By separating the covariance matrix for the outliers from the covariance matrices for the remainder of the data, we avoid the problems of masking and swamping. As did Ghosh-Dastidar and Schafer, we use a multiple-imputation approach, which allows us simultaneously to conduct inference after removing cases that appear to be outliers and to promulgate uncertainty in the outlier status through the model inference. We extend the work of Ghosh-Dastidar and Schafer by embedding the outlier class in a larger mixture model, consider penalized likelihood and posterior predictive distributions to assess model choice and model fit, and develop the model in a fashion to account for the complex sample design. We also consider the repeated sampling properties of the multiple imputation removal of outliers.     

A test of the missing data mechanism for repeated measures data

The occurrence of missing data is an often unavoidable consequence of repeated measures studies. Fortunately, multivariate general linear models such as growth curve models and linear mixed models with random effects have been well developed to analyze incomplete normally-distributed repeated measures data. Most statistical methods have assumed that the missing data occur at random. This assumption may include two types of missing data mechanism: missing completely at random (MCAR) and missing at random (MAR) in the sense of Rubin (1976). In this paper, we develop a test procedure for distinguishing these two types of missing data mechanism for incomplete normally-distributed repeated measures data. The proposed test is similar in spiril to the test of Park and Davis (1992). We derive the test for incomplete normally-distribrlted repeated measures data using linear mixed models. while Park and Davis (1992) cleirved thr test for incomplete repeatctl categorical data in the framework of Grizzle Starmer. and Koch (1969). Thr proposed procedure can be applied easily to any other multivariate general linear model which allow for missing data. The test is illustrated using the hip-replacernent patient.data from Crowder and Hand (1990).     

Inferences on missing information under multiple imputation and two-stage multiple imputation

In the presence of missing values, researchers may be interested in the rates of missing information. The rates of missing information are (a) important for assessing how the missing information contributes to inferential uncertainty about, Q, the population quantity of interest, (b) are an important component in the decision of the number of imputations, and (c) can be used to test model uncertainty and model fitting. In this article I will derive the asymptotic distribution of the rates of missing information in two scenarios: the conventional multiple imputation (MI), and the two-stage MI. Numerically I will show that the proposed asymptotic distribution agrees with the simulated one. I will also suggest the number of imputations needed to obtain reliable missing information rate estimates for each method, based on the asymptotic distribution.     

Parametric estimation of location and scale parameters in ranked set sampling

This paper develops parametric inference for the parameters of location-scale family of distributions based on a ranked set sample. Likelihood function incorporates within-set ranking errors into the model through a missing data mechanism. The maximum likelihood estimators of the location-scale and missing data model parameters are constructed and an EM-algorithm is provided. It is shown that the proposed estimator is robust against imperfect ranking error and provides higher efficiency over its competitors.     

Inference and Missing Data: Asymptotic Results

In Rubin (1976) the missing at random (MAR) and missing completely at random (MCAR) conditions are discussed. It is concluded that the MAR condition allows one to ignore the missing data mechanism when doing likelihood or Bayesian inference but also that the stronger MCAR condition is in some sense the weakest generally sufficient condition allowing (conditional) frequentist inference while ignoring the missing data mechanism. In this paper it is shown that (a slightly strengthened version of) the MAR condition is sufficient to yield ordinary large sample results for estimators and test statistics and thus may be used for (asymptotic) frequentist inference.