Simple methods for comparing two predictive values with incomplete data

Statistical methods have been well developed for comparing the predictive values of two binary diagnostic tests under a paired design. However, existing methods do not make allowance for incomplete data. Although maximum likelihood based method can be used to deal with incomplete data, it requires iterative algorithm for implementation. A simple and easily implemented statistical method is therefore needed. Simple methods exist for comparing two sensitivities or specificities with incomplete data but such simple methods are not available for comparing two predictive values with incomplete data. In this paper, we propose two simple methods for comparing two predictive values with incomplete data. The test statistics derived by these two methods are simple to compute, only involving some minor modification of the existing weighted generalized score statistics with complete data. Simulation results demonstrate that the proposed methods are more efficient than the ad-hoc method that only uses the subjects wit complete data. As an illustration, the proposed methods are applied to an observational study comparing two non-invasive methods in detecting endometriosis.

Characterization of missing data patterns and mechanisms in longitudinal composite outcome trial in rheumatoid arthritis

Background

Composite measures, like the Disease Activity Score for 28 joints (DAS28), are key primary outcomes in rheumatoid arthritis (RA) trials. DAS28 combines four different components in a continuous measure. When one or more of these components are missing the overall composite score is also missing at intermediate or trial endpoint assessments.

Objectives

This study examined missing data patterns and mechanisms in a longitudinal RA trial to evaluate how best to handle missingness when analysing composite outcomes.

Design

The Tumour-Necrosis-Factor Inhibitors against Combination Intensive Therapy (TACIT) trial was an open label, pragmatic randomized multicentre two arm non-inferiority study. Patients were followed up for 12 months, with monthly measurement of the composite outcome and its components. Active RA patients were randomized to conventional disease modifying drugs (cDMARDs) or Tumour Necrosis Factor-α inhibitors (TNFis).

Methods

The TACIT trial was used to explore the extent of missing data in the composite outcome, DAS28. Patterns of missing data in components and the composite outcome were examined graphically. Longitudinal multivariable logistic regression analysis assessed missing data mechanisms during follow-up.

Results

Two hundred and five patients were randomized: at 12 months 59/205 (29%) had unobserved composite outcome and 146/205 (71%) had an observed DAS28 outcome; however, 34/146 had one or more intermediate assessments missing. We observed mixed missing data patterns, especially for the missing composite outcome due to one component missing rather than patient not attending thier visit. Age and gender predicted missingness components, providing strong evidence the missing observations were unlikely to be Missing Completely at Random (MCAR).

Conclusion

Researchers should undertake detailed evaluations of missing data patterns and mechanisms at the final and intermediate time points, whether or not the outcome variable is a composite outcome. In addition, the impact on treatment estimates in patients who only provide data at milestone assessments need to be assessed.

Trial Registration ISRCTN Number

37438295.

Using Multiple Imputation with GEE with Non-monotone Missing Longitudinal Binary Outcomes

This paper considers multiple imputation (MI) approaches for handling non-monotone missing longitudinal binary responses when estimating parameters of a marginal model using generalized estimating equations (GEE). GEE has been shown to yield consistent estimates of the regression parameters for a marginal model when data are missing completely at random (MCAR). However, when data are missing at random (MAR), the GEE estimates may not be consistent; the MI approaches proposed in this paper minimize bias under MAR. The first MI approach proposed is based on a multivariate normal distribution, but with the addition of pairwise products among the binary outcomes to the multivariate normal vector. Even though the multivariate normal does not impute 0 or 1 values for the missing binary responses, as discussed by Horton et al. (Am Stat 57:229-232, 2003), we suggest not rounding when filling in the missing binary data because it could increase bias. The second MI approach considered is the fully conditional specification (FCS) approach. In this approach, we specify a logistic regression model for each outcome given the outcomes at other time points and the covariates. Typically, one would only include main effects of the outcome at the other times as predictors in the FCS approach, but we explore if bias can be reduced by also including pairwise interactions of the outcomes at other time point in the FCS. In a study of asymptotic bias with non-monotone missing data, the proposed MI approaches are also compared to GEE without imputation. Finally, the proposed methods are illustrated using data from a longitudinal clinical trial comparing four psychosocial treatments from the National Institute on Drug Abuse Collaborative Cocaine Treatment Study, where patients' cocaine use is collected monthly for 6 months during treatment.

Exact Bayesian p-values for a test of independence in a 2 × 2 contingency table with missing data

Altham (Altham PME. Exact Bayesian analysis of a 2 × 2 contingency table, and Fisher's "exact" significance test. J R Stat Soc B 1969; 31: 261-269) showed that a one-sided p-value from Fisher's exact test of independence in a 2 × 2 contingency table is equal to the posterior probability of negative association in the 2 × 2 contingency table under a Bayesian analysis using an improper prior. We derive an extension of Fisher's exact test p-value in the presence of missing data, assuming the missing data mechanism is ignorable (i.e., missing at random or completely at random). Further, we propose Bayesian p-values for a test of independence in a 2 × 2 contingency table with missing data using alternative priors; we also present results from a simulation study exploring the Type I error rate and power of the proposed exact test p-values. An example, using data on the association between blood pressure and a cardiac enzyme, is presented to illustrate the methods.

Sample size determination for a matched-pairs study with incomplete data using exact approach

This research was motivated by a clinical trial design for a cognitive study. The pilot study was a matched-pairs design where some data are missing, specifically the missing data coming at the end of the study. Existing approaches to determine sample size are all based on asymptotic approaches (e.g., the generalized estimating equation (GEE) approach). When the sample size in a clinical trial is small to medium, these asymptotic approaches may not be appropriate for use due to the unsatisfactory Type I and II error rates. For this reason, we consider the exact unconditional approach to compute the sample size for a matched-pairs study with incomplete data. Recommendations are made for each possible missingness pattern by comparing the exact sample sizes based on three commonly used test statistics, with the existing sample size calculation based on the GEE approach. An example from a real surgeon-reviewers study is used to illustrate the application of the exact sample size calculation in study designs.

Drawing inferences from clinical studies with missing values using genetic algorithm

Missing data problem degrades the statistical power of any analysis made in clinical studies. To infer valid results from such studies, suitable method is required to replace the missing values. There is no method which can be universally applicable for handling missing values and the main objective of this paper is to introduce a common method applicable in all cases of missing data. In this paper, Bayesian Genetic Algorithm (BGA) is proposed to effectively impute both missing continuous and discrete values using heuristic search algorithm called genetic algorithm and Bayesian rule. BGA is applied to impute missing values in a real cancer dataset under Missing At Random (MAR) and Missing Completely At Random (MCAR) conditions. For both discrete and continuous attributes, the results show better classification accuracy and RMSE% than many existing methods.

Analysis of partially observed clustered data using generalized estimating equations and multiple imputation

Clustered data arise in many settings, particularly within the social and biomedical sciences. As an example, multiple-source reports are commonly collected in child and adolescent psychiatric epidemiologic studies where researchers use various informants (e.g. parent and adolescent) to provide a holistic view of a subject's symptomatology. Fitzmaurice et al. (1995) have described estimation of multiple source models using a standard generalized estimating equation (GEE) framework. However, these studies often have missing data due to additional stages of consent and assent required. The usual GEE is unbiased when missingness is Missing Completely at Random (MCAR) in the sense of Little and Rubin (2002). This is a strong assumption that may not be tenable. Other options such as weighted generalized estimating equations (WEEs) are computationally challenging when missingness is non-monotone. Multiple imputation is an attractive method to fit incomplete data models while only requiring the less restrictive Missing at Random (MAR) assumption. Previously estimation of partially observed clustered data was computationally challenging however recent developments in Stata have facilitated their use in practice. We demonstrate how to utilize multiple imputation in conjunction with a GEE to investigate the prevalence of disordered eating symptoms in adolescents reported by parents and adolescents as well as factors associated with concordance and prevalence. The methods are motivated by the Avon Longitudinal Study of Parents and their Children (ALSPAC), a cohort study that enrolled more than 14,000 pregnant mothers in 1991-92 and has followed the health and development of their children at regular intervals. While point estimates were fairly similar to the GEE under MCAR, the MAR model had smaller standard errors, while requiring less stringent assumptions regarding missingness.

Premorbid teacher-rated social functioning predicts adult schizophrenia-spectrum disorder: a high-risk prospective investigation

Social functioning deficits are a core component of schizophrenia spectrum disorders, and may emerge years prior to the onset of diagnosable illness. The current study prospectively examines the relation between teacher-rated childhood social dysfunction and later mental illness among participants who were at genetic high-risk for schizophrenia and controls (n=244). The teacher-rated social functioning scale significantly predicted psychiatric outcomes (schizophrenia-spectrum vs. other psychiatric disorder vs. no mental illness). Poor premorbid social functioning appears to constitute a marker of illness vulnerability and may also function as a chronic stressor potentially exacerbating risk for illness.

MCAR is not necessary for the complete cases to constitute a simple random subsample of the target sample

Missing data is the norm rather than the exception in complex epidemiological studies. Complete-case analyses, which discard all subjects with some data values missing, are known to be valid under the very restrictive assumption that the response mechanism is missing completely at random (MCAR). While conditions weaker than MCAR are known under which estimators of regression coefficients are unbiased, one often comes across the view in the literature that MCAR is necessary for the complete cases to form a simple random subsample of the target sample. In this paper, we explain why this is not the case, and we distill an assumption weaker than MCAR under which the simple random subsample condition holds, which we call available at random (AAR). Moreover, we show that, unlike MCAR, AAR response mechanisms can be missing not at random (MNAR). We also suggest how approximate AAR mechanisms might arise in practice through cancellation of selection and drop-out effects, and we conclude that before pooling partially complete and complete cases into an analysis, the investigator should consider how selection might impact on the representativeness of the cases included in the pooled analysis (compared to those comprising the complete cases only).

Missing data methods in longitudinal studies: a review

Incomplete data are quite common in biomedical and other types of research, especially in longitudinal studies. During the last three decades, a vast amount of work has been done in the area. This has led, on the one hand, to a rich taxonomy of missing-data concepts, issues, and methods and, on the other hand, to a variety of data-analytic tools. Elements of taxonomy include: missing data patterns, mechanisms, and modeling frameworks; inferential paradigms; and sensitivity analysis frameworks. These are described in detail. A variety of concrete modeling devices is presented. To make matters concrete, two case studies are considered. The first one concerns quality of life among breast cancer patients, while the second one examines data from the Muscatine children's obesity study.

Joint generalized estimating equations for multivariate longitudinal binary outcomes with missing data: An application to AIDS data

In a large, prospective longitudinal study designed to monitor cardiac abnormalities in children born to HIV-infected women, instead of a single outcome variable, there are multiple binary outcomes (e.g., abnormal heart rate, abnormal blood pressure, abnormal heart wall thickness) considered as joint measures of heart function over time. In the presence of missing responses at some time points, longitudinal marginal models for these multiple outcomes can be estimated using generalized estimating equations (GEE) (Liang and Zeger, 1986), and consistent estimates can be obtained under the assumption of a missing completely at random (MCAR) mechanism. When the missing data mechanism is missing at random (MAR), that is the probability of missing a particular outcome at a time-point depends on observed values of that outcome and the remaining outcomes at other time points, we propose joint estimation of the marginal models using a single modified GEE based on an EM-type algorithm. The proposed method is motivated by the longitudinal study of cardiac abnormalities in children born to HIV-infected women and analyses of these data are presented to illustrate the application of the method. Further, in an asymptotic study of bias, we show that under an MAR mechanism in which missingness depends on all observed outcome variables, our joint estimation via the modified GEE produces almost unbiased estimates, provided the correlation model has been correctly specified, whereas estimates from standard GEE can lead to substantial bias.

Incorporating missingness for estimation of marginal regression models with multiple source predictors

Multiple informant data refers to information obtained from different individuals or sources used to measure the same construct; for example, researchers might collect information regarding child psychopathology from the child's teacher and the child's parent. Frequently, studies with multiple informants have incomplete observations; in some cases the missingness of informants is substantial. We introduce a Maximum Likelihood (ML) technique to fit models with multiple informants as predictors that permits missingness in the predictors as well as the response. We provide closed form solutions when possible and analytically compare the ML technique to the existing Generalized Estimating Equations (GEE) approach. We demonstrate that the ML approach can be used to compare the effect of the informants on response without standardizing the data. Simulations incorporating missingness show that ML is more efficient than the existing GEE method. In the presence of MCAR missing data, we find through a simulation study that the ML approach is robust to a relatively extreme departure from the normality assumption. We implement both methods in a study investigating the association between physical activity and obesity with activity measured using multiple informants (children and their mothers).

Bias Mechanisms in Intention-to-Treat Analysis With Data Subject to Treatment Noncompliance and Missing Outcomes

An analytical approach was employed to compare sensitivity of causal effect estimates with different assumptions on treatment noncompliance and non-response behaviors. The core of this approach is to fully clarify bias mechanisms of considered models and to connect these models based on common parameters. Focusing on intention-to-treat analysis, systematic model comparisons are performed on the basis of explicit bias mechanisms and connectivity between models. The method is applied to the Johns Hopkins school intervention trial, where assessment of the intention-to-treat effect on school children's mental health is likely to be affected by assumptions about intervention noncompliance and nonresponse at follow-up assessments. The example calls attention to the importance of focusing on each case in investigating relative sensitivity of causal effect estimates with different identifying assumptions, instead of pursuing a general conclusion that applies to every occasion.

Problems in dealing with missing data and informative censoring in clinical trials

Acommon problem in clinical trials is the missing data that occurs when patients do not complete the study and drop out without further measurements. Missing data cause the usual statistical analysis of complete or all available data to be subject to bias. There are no universally applicable methods for handling missing data. We recommend the following: (1) Report reasons for dropouts and proportions for each treatment group; (2) Conduct sensitivity analyses to encompass different scenarios of assumptions and discuss consistency or discrepancy among them; (3) Pay attention to minimize the chance of dropouts at the design stage and during trial monitoring; (4) Collect post-dropout data on the primary endpoints, if at all possible; and (5) Consider the dropout event itself an important endpoint in studies with many.