Local influence for incomplete data models

This paper proposes a method to assess the local influence in a minor perturbation of a statistical model with incomplete data. The idea is to utilize Cook's approach to the conditional expectation of the complete-data log-likelihood function in the EM algorithm. It is shown that the method proposed produces analytic results that are very similar to those obtained from a classical local influence approach based on the observed data likelihood function and has the potential to assess a variety of complicated models that cannot be handled by existing methods. An application to the generalized linear mixed model is investigated. Some illustrative artificial and real examples are presented.     

Application of sensitivity analysis to incomplete longitudinal CD4 count data

In this paper, we investigate the effect of tuberculosis pericarditis (TBP) treatment on CD4 count changes over time and draw inferences in the presence of missing data. We accounted for missing data and conducted sensitivity analyses to assess whether inferences under missing at random (MAR) assumption are sensitive to not missing at random (NMAR) assumptions using the selection model (SeM) framework. We conducted sensitivity analysis using the local influence approach and stress-testing analysis. Our analyses showed that the inferences from the MAR are robust to the NMAR assumption and influential subjects do not overturn the study conclusions about treatment effects and the dropout mechanism. Therefore, the missing CD4 count measurements are likely to be MAR. The results also revealed that TBP treatment does not interact with HIV/AIDS treatment and that TBP treatment has no significant effect on CD4 count changes over time. Although the methods considered were applied to data in the IMPI trial setting, the methods can also be applied to clinical trials with similar settings.     

Α Markov model for longitudinal studies with incomplete dichotomous outcomes

Missing outcome data constitute a serious threat to the validity and precision of inferences from randomized controlled trials. In this paper, we propose the use of a multistate Markov model for the analysis of incomplete individual patient data for a dichotomous outcome reported over a period of time. The model accounts for patients dropping out of the study and also for patients relapsing. The time of each observation is accounted for, and the model allows the estimation of time‐dependent relative treatment effects. We apply our methods to data from a study comparing the effectiveness of 2 pharmacological treatments for schizophrenia. The model jointly estimates the relative efficacy and the dropout rate and also allows for a wide range of clinically interesting inferences to be made. Assumptions about the missingness mechanism and the unobserved outcomes of patients dropping out can be incorporated into the analysis. The presented method constitutes a viable candidate for analyzing longitudinal, incomplete binary data.     

Hybrid pairwise‐likelihood estimation methods for incomplete longitudinal binary data

In longitudinal data, missing observations occur commonly with incomplete responses and covariates. Missing data can have a ‘missing not at random’ mechanism, a non‐monotone missing pattern, and moreover response and covariates can be missing not simultaneously. To avoid complexities in both modelling and computation, a two‐stage estimation method and a pairwise‐likelihood method are proposed. The two‐stage estimation method enjoys simplicities in computation, but incurs more severe efficiency loss. On the other hand, the pairwise approach leads to estimators with better efficiency, but can be cumbersome in computation. In this paper, we develop a compromise method using a hybrid pairwise‐likelihood framework. Our proposed approach has better efficiency than the two‐stage method, but its computational cost is still reasonable compared to the pairwise approach. The performance of the methods is evaluated empirically by means of simulation studies. Our methods are used to analyse longitudinal data obtained from the National Population Health Study.     

Marginal correlation from an extended random-effects model for repeated and overdispersed counts

Vangeneugden et al. [15 Vangeneugden, T., Molenberghs, G., Laenen, A., Geys, H., Beunckens, C. and Sotto, C. 2007. Marginal correlation in longitudinal binary data based on generalized linear mixed models, Tech. Rep., Hasselt University. submitted for publication [Google Scholar]] derived approximate correlation functions for longitudinal sequences of general data type, Gaussian and non-Gaussian, based on generalized linear mixed-effects models (GLMM). Their focus was on binary sequences, as well as on a combination of binary and Gaussian sequences. Here, we focus on the specific case of repeated count data, important in two respects. First, we employ the model proposed by Molenberghs et al. [13 Molenberghs, G., Verbeke, G. and Demétrio, C. G.B. 2007. An extended random-effects approach to modeling repeated, overdispersed count data. Lifetime Data Anal., 13: 513–531. [Crossref], [PubMed], [Web of Science ®] [Google Scholar]], which generalizes at the same time the Poisson-normal GLMM and the conventional overdispersion models, in particular the negative-binomial model. The model flexibly accommodates data hierarchies, intra-sequence correlation, and overdispersion. Second, means, variances, and joint probabilities can be expressed in closed form, allowing for exact intra-sequence correlation expressions. Next to the general situation, some important special cases such as exchangeable clustered outcomes are considered, producing insightful expressions. The closed-form expressions are contrasted with the generic approximate expressions of Vangeneugden et al. [15 Vangeneugden, T., Molenberghs, G., Laenen, A., Geys, H., Beunckens, C. and Sotto, C. 2007. Marginal correlation in longitudinal binary data based on generalized linear mixed models, Tech. Rep., Hasselt University. submitted for publication [Google Scholar]]. Data from an epileptic-seizures trial are analyzed and correlation functions derived. It is shown that the proposed extension strongly outperforms the classical GLMM.     

A Bayesian approach to analyse overdispersed longitudinal count data

In this paper, we consider a model for repeated count data, with within-subject correlation and/or overdispersion. It extends both the generalized linear mixed model and the negative-binomial model. This model, proposed in a likelihood context [17 G. Molenberghs, G. Verbeke, and C.G.B. Demétrio, An extended random-effects approach to modeling repeated, overdispersion count data, Lifetime Data Anal. 13 (2007), pp. 457–511.[Web of Science ®] , [Google Scholar],18 G. Molenberghs, G. Verbeke, C.G.B. Demétrio, and A. Vieira, A family of generalized linear models for repeated measures with normal and conjugate random effects, Statist. Sci. 25 (2010), pp. 325–347. doi: 10.1214/10-STS328[Crossref], [Web of Science ®] , [Google Scholar]] is placed in a Bayesian inferential framework. An important contribution takes the form of Bayesian model assessment based on pivotal quantities, rather than the often less adequate DIC. By means of a real biological data set, we also discuss some Bayesian model selection aspects, using a pivotal quantity proposed by Johnson [12 V.E. Johnson, Bayesian model assessment using pivotal quantities, Bayesian Anal. 2 (2007), pp. 719–734. doi: 10.1214/07-BA229[Crossref], [Web of Science ®] , [Google Scholar]].     

Bias from the use of generalized estimating equations to analyze incomplete longitudinal binary data

Patient dropout is a common problem in studies that collect repeated binary measurements. Generalized estimating equations (GEE) are often used to analyze such data. The dropout mechanism may be plausibly missing at random (MAR), i.e. unrelated to future measurements given covariates and past measurements. In this case, various authors have recommended weighted GEE with weights based on an assumed dropout model, or an imputation approach, or a doubly robust approach based on weighting and imputation. These approaches provide asymptotically unbiased inference, provided the dropout or imputation model (as appropriate) is correctly specified. Other authors have suggested that, provided the working correlation structure is correctly specified, GEE using an improved estimator of the correlation parameters (‘modified GEE’) show minimal bias. These modified GEE have not been thoroughly examined. In this paper, we study the asymptotic bias under MAR dropout of these modified GEE, the standard GEE, and also GEE using the true correlation. We demonstrate that all three methods are biased in general. The modified GEE may be preferred to the standard GEE and are subject to only minimal bias in many MAR scenarios but in others are substantially biased. Hence, we recommend the modified GEE be used with caution.     

Local influence for generalized linear mixed models

The authors describe a method for assessing model inadequacy in maximum likelihood estimation of a generalized linear mixed model. They treat the latent random effects in the model as missing data and develop the influence analysis on the basis of a Q‐function which is associated with the conditional expectation of the complete‐data log‐likelihood function in the EM algorithm. They propose a procedure to detect influential observations in six model perturbation schemes. They also illustrate their methodology in a hypothetical situation and in two real cases.     

A structured framework for assessing sensitivity to missing data assumptions in longitudinal clinical trials

The objective of this research was to demonstrate a framework for drawing inference from sensitivity analyses of incomplete longitudinal clinical trial data via a re‐analysis of data from a confirmatory clinical trial in depression. A likelihood‐based approach that assumed missing at random (MAR) was the primary analysis. Robustness to departure from MAR was assessed by comparing the primary result to those from a series of analyses that employed varying missing not at random (MNAR) assumptions (selection models, pattern mixture models and shared parameter models) and to MAR methods that used inclusive models. The key sensitivity analysis used multiple imputation assuming that after dropout the trajectory of drug‐treated patients was that of placebo treated patients with a similar outcome history (placebo multiple imputation). This result was used as the worst reasonable case to define the lower limit of plausible values for the treatment contrast. The endpoint contrast from the primary analysis was ? 2.79 (p = .013). In placebo multiple imputation, the result was ? 2.17. Results from the other sensitivity analyses ranged from ? 2.21 to ? 3.87 and were symmetrically distributed around the primary result. Hence, no clear evidence of bias from missing not at random data was found. In the worst reasonable case scenario, the treatment effect was 80% of the magnitude of the primary result. Therefore, it was concluded that a treatment effect existed. The structured sensitivity framework of using a worst reasonable case result based on a controlled imputation approach with transparent and debatable assumptions supplemented a series of plausible alternative models under varying assumptions was useful in this specific situation and holds promise as a generally useful framework. Copyright © 2012 John Wiley & Sons, Ltd.     

A robust pairwise likelihood method for incomplete longitudinal binary data arising in clusters

Clustered longitudinal data feature cross‐sectional associations within clusters, serial dependence within subjects, and associations between responses at different time points from different subjects within the same cluster. Generalized estimating equations are often used for inference with data of this sort since they do not require full specification of the response model. When data are incomplete, however, they require data to be missing completely at random unless inverse probability weights are introduced based on a model for the missing data process. The authors propose a robust approach for incomplete clustered longitudinal data using composite likelihood. Specifically, pairwise likelihood methods are described for conducting robust estimation with minimal model assumptions made. The authors also show that the resulting estimates remain valid for a wide variety of missing data problems including missing at random mechanisms and so in such cases there is no need to model the missing data process. In addition to describing the asymptotic properties of the resulting estimators, it is shown that the method performs well empirically through simulation studies for complete and incomplete data. Pairwise likelihood estimators are also compared with estimators obtained from inverse probability weighted alternating logistic regression. An application to data from the Waterloo Smoking Prevention Project is provided for illustration. The Canadian Journal of Statistics 39: 34–51; 2011 © 2010 Statistical Society of Canada     

Local influence diagnostics for generalized linear mixed models with overdispersion

Since the seminal paper by Cook and Weisberg [9 R.D. Cook and S. Weisberg, Residuals and Influence in Regression, Chapman &; Hall, London, 1982. [Google Scholar]], local influence, next to case deletion, has gained popularity as a tool to detect influential subjects and measurements for a variety of statistical models. For the linear mixed model the approach leads to easily interpretable and computationally convenient expressions, not only highlighting influential subjects, but also which aspect of their profile leads to undue influence on the model's fit [17 E. Lesaffre and G. Verbeke, Local influence in linear mixed models, Biometrics 54 (1998), pp. 570–582. doi: 10.2307/3109764[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]]. Ouwens et al. [24 M.J.N.M. Ouwens, F.E.S. Tan, and M.P.F. Berger, Local influence to detect influential data structures for generalized linear mixed models, Biometrics 57 (2001), pp. 1166–1172. doi: 10.1111/j.0006-341X.2001.01166.x[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]] applied the method to the Poisson-normal generalized linear mixed model (GLMM). Given the model's nonlinear structure, these authors did not derive interpretable components but rather focused on a graphical depiction of influence. In this paper, we consider GLMMs for binary, count, and time-to-event data, with the additional feature of accommodating overdispersion whenever necessary. For each situation, three approaches are considered, based on: (1) purely numerical derivations; (2) using a closed-form expression of the marginal likelihood function; and (3) using an integral representation of this likelihood. Unlike when case deletion is used, this leads to interpretable components, allowing not only to identify influential subjects, but also to study the cause thereof. The methodology is illustrated in case studies that range over the three data types mentioned.     

Robust estimation of mean and covariance for longitudinal data with dropouts

In this paper, we study estimation of linear models in the framework of longitudinal data with dropouts. Under the assumptions that random errors follow an elliptical distribution and all the subjects share the same within-subject covariance matrix which does not depend on covariates, we develop a robust method for simultaneous estimation of mean and covariance. The proposed method is robust against outliers, and does not require to model the covariance and missing data process. Theoretical properties of the proposed estimator are established and simulation studies show its good performance. In the end, the proposed method is applied to a real data analysis for illustration.     

Underdispersion models: Models that are “under the radar”

The Poisson distribution is a benchmark for modeling count data. Its equidispersion constraint, however, does not accurately represent real data. Most real datasets express overdispersion; hence attention in the statistics community focuses on associated issues. More examples are surfacing, however, that display underdispersion, warranting the need to highlight this phenomenon and bring more attention to those models that can better describe such data structures. This work addresses various sources of data underdispersion and surveys several distributions that can model underdispersed data, comparing their performance on applied datasets.     

Using the EM-algorithm for survival data with incomplete categorical covariates

Incomplete covariate data is a common occurrence in many studies in which the outcome is survival time. With generalized linear models, when the missing covariates are categorical, a useful technique for obtaining parameter estimates is the EM by the method of weights proposed in Ibrahim (1990). In this article, we extend the EM by the method of weights to survival outcomes whose distributions may not fall in the class of generalized linear models. This method requires the estimation of the parameters of the distribution of the covariates. We present a clinical trials example with five covariates, four of which have some missing values.     

Sensitivity to imputation models and assumptions in receiver operating characteristic analysis with incomplete data

Modern statistical methods using incomplete data have been increasingly applied in a wide variety of substantive problems. Similarly, receiver operating characteristic (ROC) analysis, a method used in evaluating diagnostic tests or biomarkers in medical research, has also been increasingly popular problem in both its development and application. While missing-data methods have been applied in ROC analysis, the impact of model mis-specification and/or assumptions (e.g. missing at random) underlying the missing data has not been thoroughly studied. In this work, we study the performance of multiple imputation (MI) inference in ROC analysis. Particularly, we investigate parametric and non-parametric techniques for MI inference under common missingness mechanisms. Depending on the coherency of the imputation model with the underlying data generation mechanism, our results show that MI generally leads to well-calibrated inferences under ignorable missingness mechanisms.     

The Poisson Birnbaum–Saunders model with long-term survivors

In this paper, we propose a cure rate survival model by assuming that the number of competing causes of the event of interest follows the Poisson distribution and the time to event has the Birnbaum–Saunders (BS) distribution. We define the Poisson BS distribution and provide two useful representations for its density function which facilitate to obtain some mathematical properties. Two closed-form expressions for the moments of the new distribution are given. We estimate the parameters of the model with cure rate using maximum likelihood. For different parameter settings, sample sizes and censoring percentages, several simulations are performed. We derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and present some ways to perform a global influence study. We analyse a real data set from the medical area.     

A longitudinal study of children's aggressive behaviours based on multivariate mixed models with incomplete data

The authors consider children's behavioural and emotional problems and their relationships with possible predictors. They propose a multivariate transitional mixed‐effects model for a longitudinal study and simultaneously address non‐ignorable missing data in responses and covariates, measurement errors in covariates, and multivariate modelling of the responses and covariate processes. A real dataset is analysed in details using the proposed method with some interesting results. The Canadian Journal of Statistics 37: 435–452; 2009 © 2009 Statistical Society of Canada     

Progressive multi-state models for informatively incomplete longitudinal data

Progressive multi-state models provide a convenient framework for characterizing chronic disease processes where the states represent the degree of damage resulting from the disease. Incomplete data often arise in studies of such processes, and standard methods of analysis can lead to biased parameter estimates when observation of data is response-dependent. This paper describes a joint analysis useful for fitting progressive multi-state models to data arising in longitudinal studies in such settings. Likelihood based methods are described and parameters are shown to be identifiable. An EM algorithm is described for parameter estimation, and variance estimation is carried out using the Louis’ method. Simulation studies demonstrate that the proposed method works well in practice under a variety of settings. An application to data from a smoking prevention study illustrates the utility of the method.     

Local influence regression diagnostics for gamma data

The local influence diagnostics proposed in general terms irl Cook (1986), Thomas and Cook (1989) and Billor and Loynes (1993). arc adapted for gamma data. A data set prcviously analysed using different methods is then reexamined, and conclusions based on wing the new approach are made.